A combination-space search for embedded technosignatures in solar and heliospheric archives

1. Introduction

1.1 Every search so far has presupposed the channel

The modern search for technosignatures begins with a choice of channel. Cocconi & Morrison (1959) argued for the 21 cm line, and essentially every program since has followed the same logic: name a carrier both parties can identify, then search it as deeply as instruments allow. Wright, Kanodia & Lubar (2018) formalized SETI as a coverage problem, but their axes are sky position, frequency, sensitivity, polarization, modulation and repetition rate: a map of how much radio has been searched, presupposing that the answer is radio.

That presupposition has since acquired a measured cost on its own terms. Gajjar & Brown (2026) show that turbulence in the host star’s wind broadens a narrowband carrier before it ever leaves the system — at 1 GHz roughly 70% of stars smear a line beyond 1 Hz, and a CME encounter smears it by three orders of magnitude — while the standard pipelines are matched filters for an infinitely sharp line, so the power redistributed into the Lorentzian wings is sensitivity silently lost. Within the carrier paradigm the remedy is a broader filter; here it reads instead as one more entry in §2.1’s ledger — a carrier is not only expensive but fragile against the medium itself, and a designer engineering for a million systems over deep time knew that before we measured it.

Where this sits among the alternatives. The 2018 NASA Technosignatures Workshop is the field’s reference organization of what a technosignature is, and a reader arriving from it is entitled to know which of its categories this paper occupies. The answer is none of them, and the reason is worth stating as the categories themselves state it — by what each one assumes exists:

The premise of this paper is that none of those is necessary. The carrier is emission the star already produces for its own reasons, redistributed rather than generated (§2.7): there is no apparatus to build, no structure to see, no byproduct to accumulate and no object to encounter. That premise does not appear in the workshop’s taxonomy, and the archives it points at — solar and heliospheric records collected for space weather — do not appear in its inventory of what has been searched. Three differences follow. The carrier is a relationship among observables rather than a channel to point at (§3.1, §3.3). The gate is opened by the receiver’s capability rather than paid for by the sender (§2.3, §2.9). And the target is our own star, from archives already collected, where the report’s emphasis is on new instruments and on piggybacking planned missions.

This is not the dark-forest conjecture. Because the argument here also declines to expect a loud beacon, it is easily conflated with the dark-forest picture — that the sky is silent because every civilization conceals itself and destroys any that reveal their position. The two are opposite in motivation and in consequence. The dark forest explains silence through predation and predicts that there is nothing to find: signalling is fatal, so the rational act is to hide and a search is pointless. The premise here is the reverse — an efficient, long-lived network that does communicate, whose subtlety is a matter of energy economy and capability gating (§2.1, §2.3) rather than fear, and which therefore leaves findable structure and specifies where to look for it. The one shared premise, that a cost-minimizing sender does not build a lighthouse, is common to most of the beacon-economics literature (§1.4) and is not by itself the dark forest. This paper is a search; the dark forest is an argument against one.

Proposals to look at emission a star already produces relocate the assumption rather than escaping it. Learned et al. (2008) would modulate a Cepheid; Chennamangalam et al. (2015) a pulsar; Bracewell (1960) proposed local probes; Benford, Benford & Benford (2010) concluded that a rational sender builds a cheap, intermittent pointer to a costlier message. Each names its carrier in advance. The receiver is told where to look, and the remaining problem is sensitivity.

1.2 What if the channel is not known?

We consider the case left over: no beam, no beacon hardware, no carrier specified in advance — structure carried by emission a star already produces, where the carrier may not be any single measured quantity but a relationship between several. The receiver’s problem is then not sensitivity at all; it is not knowing which series to examine.

Stated that way the proposition sounds untestable, and one of its neighbors is: “it is entirely possible that we are simply not advanced enough to understand the manifestations of a vastly more advanced technology, even if they are all around us” (KISS 2019). That is true and it is inert — it names no carrier, predicts no amplitude, and identifies no experiment that could fail. The difference between that observation and a research program is whether anything constrains the space of possibilities. Two things do.

1.3 Two constraints that make the space finite

The first is dimensional. A sender who shares no units with the receiver — no second, no meter, no kelvin — cannot encode in a quantity that carries units, because the receiver cannot recover the intended value. What survives are dimensionless quantities: ratios of like observables, normalized residuals, quantities scaled by their own dispersion. This removes every raw measurement and admits only combinations, converting an unbounded question into a countable one.

The second is economic. Transmitted bits cost energy, permanently, across interstellar distance. A sender minimizing that cost will not spend it on receivers unable to act on the result, and the efficient way to avoid doing so is to gate the message on demonstrated capability: a task cheap to pose, expensive to solve, and trivial to verify — proof of work (§2.3).

Together these have a consequence that is easy to miss: if the message is gated on solving a problem, the first barrier is not detecting a signal but recognizing that a problem has been posed — and that barrier is invisible from below and sized by the receiver’s capacity to search combinations (§2.4). A civilization holding the right data, with the compute to search it, sees exactly what a civilization with no signal to find sees.

1.4 What this paper does and does not claim

Most of the architecture assumed here is not ours. The two-tier pointer and its cost argument are Benford et al. (2010); modulating an existing astrophysical source is Learned et al. (2008) and Chennamangalam et al. (2015); local probes are Bracewell’s (1960); coverage as a figure of merit is Wright et al. (2018), and the framework for comparing search value Sheikh (2019). Hippke (2017a,b; with Forgan 2017) developed interstellar communication as an energy-per-bit optimization far more thoroughly than §2.1 does, and paper VI is prior art for the blind narrowband X-ray search of §A.1. Benford (2019) developed the case for co-orbital “lurkers” — the mechanism §2.7 invokes. Freudenthal (1960) and DeVito & Oehrle (1990) built message systems that assume no shared units: the units-free constraint is long established for message content.

What we believe is not already occupied is narrower than we first claimed:

An ADS full-text search returned no occurrences of the combination-space formulation within papers on technosignatures. We do not treat that as establishing priority: indexing is incomplete, and a search of the same kind failed to surface Hippke, Freudenthal and DeVito & Oehrle above.

One limitation is severe enough to state in the introduction. §3.6 ranks the channels a sender would plausibly choose. This paper searches the top of that ranking and the bottom of it, and is thinnest in the middle — five channels (rows 3–7 of the ranking) where the archives exist, the resolving power is adequate, and no search has been run. The nulls reported here are therefore not a test of the framework’s own best guess, and we would rather say so at the outset than let the count of searches imply otherwise.

1.5 Why the Sun

If probes are seeded at stars selected for the possibility of life, as Bracewell’s argument supposes, the star to examine is the local one — and the archives already exist and are unusually good: solar irradiance measured continuously since 1978, neutron monitors since 1964, sunspot number since 1749, GOES-R ultraviolet and X-ray irradiance at one-minute and one-second cadence. No new observation is required to begin. Every search here ran on public archives collected for space weather and solar physics. It is also why the paper looks nowhere else: if the modulator is placed at the local star by §2, then pulsars, binaries and distant stars are a different hypothesis (§4.1).

Section 2 develops the gating argument; Section 3 defines the combination space; Section 4 reports the results; Section 5 argues that coverage rather than sensitivity is now the binding constraint; Section 6 states what a completed enumeration would establish.

2. The gating argument

2.1 Four constraints a systems designer would actually apply

The framework is not a claim about what a sender wants. It is an engineering derivation: what would a communications designer, facing an unknown number of unknown recipients over geological time, be forced to build?

Each constraint deletes something, and what it subtracts is checkable:

Six deletions, each traceable to a stated constraint, and none chosen for elegance. The architecture that survives is simple, hidden, and long-lived because parsimony, gating and longevity are what remain once the alternatives are struck out.

The longevity constraint is open above, and that is load-bearing. It is stated as a floor — outlast the emergence of a receiver — and no ceiling is put on it anywhere in the architecture. Nothing here carries a design life, an end-of-mission, or a decommissioning step, because each of those would be a date, and a date is a specification that goes stale. The consequence is developed in §2.2: a system with no upper bound on its life is one that must be designed to run after its designers are gone.

None of this is new in its parts. What we add is that the four constraints together generate a chain specific enough to constrain a search — and, in one place, specific enough to be wrong.

2.2 The artifact outlives its designer

The longevity constraint of §2.1 is bounded below and open above: it says the system must outlast the emergence of a receiver, and it sets no date at which the system stops. That asymmetry is not an oversight in the statement of it. Followed to its conclusion it gives the principle on which the rest of the architecture turns: the system must be designed on the assumption that the civilization which built it will not be there. Not that it might not be. That it will not.

This is forced, not chosen. The chain’s own numbers make the designer’s survival the least likely element in it. A bus crosses a 10⁴ ly route in 7×10⁴ yr (§2.6); probes transit unsupervised for 10⁴–10⁶ yr (step 3); and the receiver the whole apparatus is built for takes something between 10⁶ and 10⁹ yr to arise, because that is how long it took here. A designer who requires their own continued existence anywhere in that span has built a system that fails on the most probable branch. The only design that survives contact with those timescales is one that treats its makers as already absent.

What remains after those deletions is a system whose every element is re-derivable by the receiver from physics it can measure for itself. That is not an aesthetic preference. It is the only class of specification that does not depend on a surviving author.

The accounting consequence is larger than the engineering one. A Drake-style estimate multiplies by L, the lifetime of a communicating civilization, and L is the term that has always carried the argument’s pessimism. If the artifact is designed to outlive its designer — and §2.1’s constraints force that — then L is the wrong lifetime to be multiplying by. The detectable population is set by the lifetime of the artifact, and an unattended, self-repairing, consumable-free system in a stellar atmosphere has no obvious reason to share the fate of the society that launched it. The question “is anyone alive out there” and the question “is anything out there still running” come apart. Behavioral SETI looks for the second. It is the weaker question, and the more answerable one, and a positive answer to it would say nothing whatever about whether its author still exists.

What this changes about the search. If the signal is a standing condition rather than an event, then it is present now or it is not, and it was present a century ago on the same terms. Three consequences follow, and all three are visible in §4:

• look across the longest record available, not for transients. A modulation that requires a transmitter to be transmitting is looking for an event. This paper searches archives spanning 26 to 46 years, and in one case 11,350, because a standing condition should be in all of them or in none.
• an instrument handover is a threat, not an inconvenience. A signal that persists across decades must survive the cross-calibration of the record that would show it — which is why §4.1 spends a control on exactly that question.
• a null bounds the condition, not the builders. None of §4’s limits constrain whether anyone is or was there. They constrain whether a particular dimensionless relationship is being held in a particular observable, now, at the stated amplitude. Those are different claims and the paper does not trade on the difference.

Where this is weakest, stated plainly. Persistence over ≫10⁶ yr is the framework’s largest unsupported assumption, and it is flagged as such at step 4 of the chain (§2.5). Nothing humanity has built bounds it: our longest-lived unattended artifacts are decades old, and the failure modes of a self-repairing system over geological time are not something we can argue from experience. It is an assumption the architecture requires, not a result the architecture earns — and if it is wrong, the deletions of §2.1 stand but the search premise of §1.2 does not. We state it here rather than in a caveat at the end because everything downstream inherits it.

2.3 Proof of work: what the analogy carries and what it does not

The cheapest way to make a message conditional on demonstrated capability is a task cheap to pose, expensive to solve, and trivial to verify — proof of work in the cryptographic sense; the familiar instance is the CAPTCHA, whose poser need not know who solves it, only that the solver did the work. The analogy transfers on mechanism and fails on motive: a CAPTCHA is adversarial and nothing here is — the sender has simply declined to pay for a transmission whose recipient cannot act on it.

The constraint runs both ways, and this is what makes it useful. A gate functions only if the intended receiver can pass it, so the key space must be small enough to exhaust — a key drawn from an enormous space is operationally no key at all. This is why the search below is finite: it predicts the answer lies somewhere enumerable rather than somewhere clever.

The difficulty is also bounded from below: the gate must not open before the receiver possesses the capability the message presupposes, since a gate set too low admits a receiver that can detect but not act. The difficulty sits at the capability threshold, and a response from below it is filtered out by construction. A difficulty bounded from both sides is a deletion in the sense of §2.1, and it makes the staging of §2.5 a consequence rather than a choice: each stage is a gate set at a higher threshold.

The nearest antecedent is Clarke’s The Sentinel (1951), whose artifact reports not that it was found but that it was reachable — an antecedent for the intuition, not a prior derivation. The gate also repairs a known weakness of the nearest hypothesis in the literature: the zoo hypothesis (Ball 1973) requires coordination — many civilizations abstaining indefinitely, broken by one defector — while a difficulty-bounded gate is self-enforcing, opened by the receiver’s own capability rather than anyone’s forbearance. Not an observational discriminator (both predict we see nothing, §1.4), but a real difference in what must be assumed.

2.4 The gate before the first gate

If the message is conditional on solving a problem, the first barrier is not solving it but recognizing that a problem has been posed. That barrier is invisible from below: a civilization holding the right archives, at the right cadence, with the compute to search them, sees exactly what a civilization with nothing to find sees. Absence of evidence and unrecognized evidence are, at that stage, the same observation.

The recognition gate has a size, and the size is set by the carrier. A message carried in a relationship among observables — the only carrier a sender without shared units can use (§1.3) — is by construction findable only by a receiver that can search relationships. That search is combinatorial: pairs, triples and higher combinations of every well-measured quantity, each tested in every admissible form at every admissible cadence against a null matched to the statistic. None of the pieces is new. Every observable is already measured, every form already defined; the work is recombination across a space too large for a person or a team to traverse by hand, and validation of whatever the recombination turns up. That is a capability, not a technology, and it is one a designer can anticipate without knowing how a receiver will implement it. Choosing a combinatorial carrier is choosing a compute gate. The difficulty floor of §2.3 then has a natural setting: larger than manual search can close, small enough for a receiver with autonomous search to close in years rather than centuries.

Our own position on that ladder is measurable. The pair space of thirty observables closes in under an hour on one machine; the triple space took 117 million statistic evaluations and 26 hours (§4.8, §5.2); the 27,405 quadruples would take of order a week; each additional form multiplies the count. Thirteen years ago the pair sweep alone was a research program. The recombination capacity itself is now demonstrable outside this field: in May 2026 a general-purpose reasoning model disproved the Erdős unit-distance conjecture by importing a technique from a distant branch of mathematics that had not been applied to the problem, a result human mathematicians then sharpened within weeks (OpenAI 2026a; Sawin 2026); in September 2026 a coordinated system of some 10⁴ agents produced, in 88 hours, a Lean-formalized blow-up solution to the Navier–Stokes problem (OpenAI 2026b), a claim that is at this writing unverified by the community and subject to a priority dispute, and which we cite only as an instance of scale. Neither result invented new mathematics; both found a combination of known pieces in a space no one had searched. That is the capability gate 0 filters for, and it comes with the paper’s own caution attached: a receiver that can search but not validate produces unchecked results (Software), and the gate is passed only by one that does both.

The ladder can be put against real machines, and §5.7 does so (Figure 5). On the Cray-1 of 1976 — a machine some of whose programmers are still working — the pair space this paper closed in an hour would have taken five months, the triple space thirty-four years, and the fully enumerable space of §3.6 a hundred million years. The triple space fell inside a career with the Cray Y-MP in 1988 and inside a day with ASCI Red in 1997; the enumerable maximum fell inside a year around 2015 and inside a working day on the large GPU clusters of 2024–2026 at double precision. A receiver that monitored its star continuously from the 1970s therefore held the data for four decades before it held the compute to search the combinations in it, and the crossing is recent enough to date. That is the shape a designer setting a difficulty floor at the recombination threshold would want: not a wall, but a gate that opens on a known curve, for a receiver whose archives are already deep enough to be worth searching when it does. We do not claim to stand at that threshold; we claim that the ladder has rungs, that ours is on it, and that the figure says which.

We draw two consequences and refuse a third. The recognition gate weakens a common premise of the Fermi paradox — that contact would be self-announcing — without requiring absence, concealment or a great filter. It argues that a search of this kind should report coverage rather than detections, coverage being the only quantity that accumulates. What it must not be used for is explaining a null, and §5.3 states that prohibition as strongly as we can manage.

2.5 The gate chain, step by step

The four constraints generate a single chain: the sender’s half paid once and amortized across the galaxy, the receiver’s half paid separately by each receiver, and only when it can pay. Every step is written down because the empirical program of §4 tests exactly one link.

The stellar filter is an exclusion as much as a selection. Step 1 names what the survey keeps — main-sequence, long-lived, a rocky planet in the habitable zone — but the same design requirements strike out whole classes of star before anything is spent on them, and the exclusions bind as tightly as the inclusions. Longevity removes the massive hot stars (O, B, and most A): a network built to outlast its builders cannot anchor to a star that leaves the main sequence in a few million years, and for the same reason it removes the evolved and dying — subgiants and giants — that have no gigayears of future and pulsate and shed mass as they go. A clean channel removes the noisy: flare stars and magnetically active young stars, whose outbursts and spot-driven swings would drown any imposed modulation, and the intrinsic variables — Cepheids, RR Lyrae, Miras and their kin — whose own pulsation is a signal the node did not put there and cannot overwrite. Stability removes the complicated: close binaries and higher multiples, with their eclipses, tides and unsettled dynamics, and stars embedded in crowded or dynamically hot environments where gigayear persistence is not on offer. What survives all three cuts is a narrow, recognizable population — the quiet, single, middle-aged main-sequence G or K dwarf, old enough that its magnetic activity has decayed to near silence, young enough to burn steadily for gigayears more: the Sun and its near twins. That is the host set the architecture predicts, and it is a constraint the receiver can turn around — a search for the network should look only there, and the naturally conspicuous stars, the bright variables that first draw the eye, are precisely the classes the design excludes. Raw variability is therefore not a lead but an anti-selector: the loudest stars are the ones a designer would never use.

The return, in more detail. Once a receiver has echoed the trigger it has demonstrated the capability the gate was set to, and the node has no reason to ration what follows. The natural form of the return is several narrow beams in parallel, on adjacent bands or polarizations, aimed at the address the reply came from. Multi-beam operation costs nothing but power: the beams share one aperture, and the node can size their power to what the receiver has just shown it can hear, since the strength of the reply at the node measures the receiver’s transmitter and aperture directly. One stream carries the dictionary — the self-describing structure everything else is read against — and the remaining streams carry payload keyed to it, so that a receiver decodes the dictionary once and then reads the rest as it arrives. What such a dictionary contains has been worked out in some detail in the message-design literature, and the answer is stable across sixty years: it begins with what any receiver must already possess. Counting and arithmetic, presented so that the notation teaches itself (Hogben 1952; Freudenthal 1960); the elements, by atomic number and mass, and the constants of physics as dimensionless ratios (Staff at NAIC 1975; DeVito & Oehrle 1990); logic and the means to define new terms from old (Ollongren 2013); and only then anything the sender wants to say. The Arecibo message of 1974 and the Evpatoria messages of 1999–2003 are transmitted instances of exactly this order, and the same structure has been re-proposed for modern instruments (Jiang et al. 2022). The framework adds nothing to that literature except the observation that a receiver who has passed the gate has, by construction, already derived every entry in the dictionary’s first page — which is what makes the dictionary decodable and the gate worth setting. One caution from the same literature applies: a message rich enough to be useful cannot be proved harmless before it is read (Hippke & Learned 2018), and a receiver at step 12 should treat the payload streams with the care that implies.

The return, sized. The return of step 12 can be put in numbers, and they are worth having on their own. Take the node’s transmitter as an infrared laser at 1.55 µm behind a 10 m aperture — the receiver’s own atmospheric window, mature technology at the receiver’s rung, and diffraction-limited to 1.9×10⁻⁷ rad, so that at the mean node distance of 33 ly (§2.6) the beam is 0.4 AU across at the receiver: it need not track a planet, only an orbit. A 10 m receiver then collects 2.2×10⁻²⁰ of the transmitted power, which at this wavelength is 220 photons per second per kilowatt; a 30 m receiver, nine times that. A photon-counting receiver decodes between 0.1 and 1 bit per photon depending on background and coding, so one kilowatt is 20–200 bit/s at 10 m and 200–2,000 bit/s at 30 m.

Two things follow. The aggregate rate depends on total power and receiver aperture, not on how the power is split. Parallel beams share the aperture and add nothing to throughput; what they buy is organization — dictionary on one stream, payload on others — and they are limited only by the smallest useful photon rate per beam (of order 10 s⁻¹, so beams down to ~50 W at 10 m) and by the receiver’s ability to separate them, which at 1 GHz channel spacing in the telecom band is of order 10⁴. The node’s power budget sets everything. A node set down at a star (§2.6) is solar-powered; a tonne-class node can carry 10²–10⁴ m² of thin-film collector, which at 1 AU-equivalent is 10 kW to 1 MW electrical.

node power budget	collector needed	beams at 1 kW / at 100 W	aggregate, 10 m receiver	aggregate, 30 m receiver	payload per year, 30 m
10 kW	~50 m²	10 / 100	220–2,200 bit/s	2–20 kbit/s	8–80 GB
100 kW	~500 m²	100 / 1,000	2.2–22 kbit/s	20–200 kbit/s	80–800 GB
1 MW	~5,000 m²	1,000 / 10,000	22–220 kbit/s	0.2–2 Mbit/s	0.8–8 TB

At 33 ly; at the 90th-percentile distance of 49 ly divide by 2.2, at 60 ly by 3.3. Ranges span 0.1–1 bit per photon. Collector at 20% conversion and 1 AU-equivalent insolation.

So the return is not a trickle. At the middle row — a node no more capable than a large communications satellite, with the collector area of a tennis court — a receiver with a 30 m telescope reads tens of kilobits per second across a hundred parallel streams: a dictionary of 10⁹–10¹⁰ bits arrives in hours to days, and the payload accumulates at hundreds of gigabytes a year for as long as the node chooses to send. The number of distinct narrowband infrared lines a receiver would see from the node’s direction is the number of beams — of order 10² to 10³ at any realistic budget — and that is itself a signature: a cluster of unresolved, coherent, sub-GHz lines at 1.55 µm from a point near a nearby star, all modulated, is not a thing nature makes. It is also, at 10²–10³ photons per second per line, well within reach of the optical-SETI instruments that already exist — a receiver at step 12 does not need to build a receiver, only a transmitter, which is what the gate was designed to require.

Range, and where the link fails. A photon-counting link does not degrade gracefully all the way down; it has a threshold, and its position is set by one design choice on each side. The background is the node’s own host star. A Sun-like star at 10 pc has H ≈ 3.5, and within a 1 GHz channel at 1.55 µm — narrow enough to separate a thousand beams — it delivers about 2×10⁵ photons per second into a 10 m aperture, a thousand times the 1 kW signal. Measured as a ratio of rates the link is dead at any distance. It is rescued by time, not by aperture: the node transmits pulses, the receiver counts photons in nanosecond slots (single-photon detectors now reach tens of picoseconds of jitter), and the background per slot is 2×10⁵ × 10⁻⁹ = 2×10⁻⁴ — negligible against a pulse that lands five photons in one slot. That is pulse-position modulation with photon counting, the scheme our own deep-space optical links use (LLCD in 2013; DSOC from 2023), and it recovers several bits per photon against a background a thousand times brighter than the signal, provided the receiver has the node’s slot clock, which is the first thing the dictionary stream would supply.

The threshold is then a floor in photons, not a ratio. Reliable decoding needs of order five signal photons per pulse, and the pulse rate can be lowered to gather them — down to about one pulse per second, below which the symbol interval grows long enough for the background to reach the slot that matters. The link fails where the received rate falls to a few photons per second, and until then the bit rate falls as d⁻²:

beam power	10 m receiver	30 m receiver	100 m receiver
1 kW	~220 ly	~650 ly	~2,200 ly
10 kW	~700 ly	~2,100 ly	~7,000 ly
100 kW	~2,200 ly	~6,500 ly	~22,000 ly

Distance at which a 10 m node aperture at 1.55 µm delivers 5 photons per second; scales as √(power) × receiver diameter. Beyond it no coding recovers the link; short of it the rate is the §2.5 table divided by (d/33 ly)².

Two consequences. The mean node distance of 33 ly sits an order of magnitude inside the cliff even at the lowest power and the smallest receiver, which is why the return can be sized as generously as the previous table does. And the lattice spacing of §2.6 is bounded above by the link, not only by latency: a 300 ly mesh — the sparsest row of the latency table — is unreachable by a 1 kW beam into a 10 m telescope and marginal at 10 kW, so a designer choosing sparse nodes must either raise the beam power or assume a larger receiver, and a designer who cannot assume either has chosen the 60 ly spacing for two independent reasons. The threshold also runs the other way: a receiver that builds the 30 m aperture it would need anyway for step 8 has, by that act, tripled the distance from which a node can reach it.

The receiver we would build. The tables above assume a receiver that happens to exist. A civilization at step 12 is in a different position: it has decades of warning — the round trip of §2.6 — it knows the node’s direction and wavelength from the reply it sent, and the value of what is coming is not in doubt. It would build a receiver for the purpose, and with technology deployable now or within a decade the receiver is not a telescope. A photon-counting link needs collecting area and timing, not image quality: a mirror good to an arcsecond concentrates a 0.4 AU beam onto a detector as well as one good to a milliarcsecond, which removes the constraint that makes large space optics expensive. The natural instrument is a photon bucket in space — a membrane or inflatable collector of thousands of square meters, feeding single-photon detectors, staring at one point for as long as the node sends — placed where the node is never behind the Sun, which means either L2 with a seasonal gap or a heliocentric pair for continuous coverage. Ground telescopes lose to it on three counts at once: aperture, duty cycle (weather, daylight, solar conjunction, and shared time put a large ground telescope on one target perhaps a tenth of the year), and the atmosphere’s 15% loss at 1.55 µm.

receiver	duty	gain over 10 m ground, shared	aggregate at 100 kW node, 33 ly	payload per year	cliff, 1 kW / 100 kW beam
ground 10 m, shared time	10%	1	2–22 kbit/s	10–100 GB	220 / 2,200 ly
ground 30 m, dedicated campaign	35%	30	70–700 kbit/s	0.3–3 TB	660 / 6,600 ly
space 8 m monolithic mirror, one heavy-lift fairing, L2	95%	7	16–160 kbit/s	60–600 GB	180 / 1,800 ly
space 30 m segmented or deployable, L2	95%	100	0.2–2 Mbit/s	0.9–9 TB	660 / 6,600 ly
space 100 m photon-bucket membrane, heliocentric pair	100%	1,200	2.6–26 Mbit/s	10–100 TB	2,200 / 22,000 ly

Gain is (aperture ratio)² × duty × atmospheric transmission relative to the first row; aggregate spans 0.1–1 bit per photon; cliff distances are the 5-photon-per-second floor of the previous table. The 8 m row is a single-launch monolith of the kind a 9 m fairing permits; the 100 m row is a light bucket, not an imaging telescope, and its mass is dominated by structure rather than optic.

The last row is the one a receiver with decades and a reason would build — and it would not build one. An instrument that is the only channel to a return of this value is a single point of failure for a program measured in decades, and the sensible deployment is several collectors in parallel: three or four buckets on independent spacecraft, at separated stations, with independent detectors and clocks. Redundancy is nearly free here, because a light bucket has no precision optic to duplicate, and it buys three things at once — continuous coverage through any one unit’s conjunction, outage or loss; independent confirmation of every symbol, so that no single detector artifact can enter the record of a message that will be studied for centuries; and, since photon rates add across collectors, the aggregate of the fleet is the sum of its members. Four 100 m buckets are a 200 m receiver that cannot be lost to one launch failure or one micrometeoroid. That is how a receiver at step 12 would actually be built, and it changes the character of the return. At 100 TB a year the exchange is limited by the node’s power and by what the sender chose to send, not by the receiver; and a 100 m bucket reaches a 1 kW node at 2,200 ly, which is the whole neighborhood of the lattice and not merely its nearest cell. It also settles who pays for what, which is the framework’s recurring question: the receiver builds the aperture and the transmitter, the sender’s node supplies the power, and the link budget closes with margin at every distance the mesh uses. None of this requires anything beyond a heavy-lift launcher, membrane optics of the kind flown as sunshields and solar sails, and superconducting detectors that exist in laboratories today — the receiver is within reach of the same civilization that has just built a transmitter, which is the rung the gate was set to.

Cross-check against the literature. The link budget above uses nothing beyond the standard deep-space optical link equation (Hemmati 2006): received power = transmitted power × receiver area / (π (θd/2)²), with θ = 1.22 λ/D. Its inputs check as follows. Diffraction. 1.9×10⁻⁷ rad for 10 m at 1.55 µm is the Airy full angle and is what LLCD and DSOC use to size their beams; the DSOC ground receiver was the 5.1 m Hale telescope feeding a superconducting nanowire array, so a 10 m receiver is two of those (Boroson et al. 2014; Biswas et al. 2018). Photons per watt. 220 s⁻¹ kW⁻¹ at 33 ly into 10 m follows from the equation with no free parameters. Background. The Sun’s absolute H magnitude is 3.32 (Willmer 2018), so a solar twin at 10 pc is H ≈ 3.3; with the Vega zero-point of 1.13×10⁻⁹ W m⁻² µm⁻¹ (Cohen, Wheaton & Megeath 2003) that gives 2×10⁵ photons s⁻¹ per GHz in 10 m, as used. Detectors. The nanosecond slots assumed are conservative: flight-heritage SNSPD arrays at 1550 nm show sub-50 ps jitter, ~55% system efficiency and ~150 dark counts s⁻¹ per pixel (Guardiani et al. 2024), laboratory devices exceed 90% efficiency (Hao et al. 2024), and the 400 ps slots of the record photon-efficiency experiments are finer than assumed here. Bits per photon. The 0.1–1 bit per photon used in every table is deliberately pessimistic. Serially concatenated PPM (Moision & Hamkins 2005) with photon counting has demonstrated 13 bits per incident photon (Farr, Choi & Moision 2013) and 14.5 bits per received photon in a photon-starved channel (Banaszek et al. 2025), at low background; Hippke (2017a) reaches the same order for interstellar links from first principles. With the host star’s background gated to 2×10⁻⁴ per slot, several bits per photon is the realistic figure, so the aggregate rates and payload volumes above are more likely low by a factor of a few than high, while the cliff distances, which depend on photons rather than bits, are unaffected.

What the payload does not contain. Two things a receiver might expect are, on the framework’s own logic, absent from any early stage. The sender’s home coordinates are never required: the node is the sender’s proxy, and nothing in steps 10–12 is improved by the receiver knowing where the sender lives. They are also the one disclosure that cannot be gated after the fact. The lower bound of §2.3 exists so that nothing is released to a receiver who cannot yet use it responsibly, and a receiver that has cleared gate 0 and one decode has demonstrated search capacity and arithmetic, not restraint; the staging therefore puts the origin behind the deepest gate the sender can define, or omits it altogether. This is the framework’s answer to the transmission-risk debate (Haqq-Misra et al. 2013; Brin 2014; Gertz 2016): the sender’s exposure is bounded by design rather than by an assumption about the receiver, which is precisely the property critics of unsolicited transmission say it lacks. The lattice map is withheld for the same reason in a different form. The map is a description of the designers’ infrastructure, not of anyone using it, and its sensitivity is geometric: the bus routes of §2.6 radiate from a launch region, the fill pattern has a center, and a receiver holding the whole lattice with its deployment order can triangulate the origin the sender declined to state. The map is the coordinates by inference. The pointer already sets the precedent — one address, not a topology — and the payload follows it: the protocol for the node in hand and the address of the next hop, which is all a participant needs. Some geometry leaks regardless, since the beam origin fixes the node’s position and relay latency constrains what lies behind it; that is inference the receiver performs, not disclosure the sender makes.

What it plausibly does contain: the means to extend the mesh. The economics of §2.6 say the sender paid once for the network and the receiver pays for everything on its side, and a receiver at step 12 has just built a transmitter and demonstrated that it can navigate the combination space — it is capable of building a node. A payload that carries the node specification and the joining protocol turns every receiver that clears the gate into an extension of the network at that receiver’s expense. It is consistent with every constraint in §2.1, it is the one thing a cost-minimizing sender would want the payload to include, and it makes the mesh a commons that grows by recruitment rather than a monument that decays. Stated as a prediction: the payload carries the local protocol, the next hop, and the means to join; a receiver that finds the origin or the map in an early stage has found something the framework does not predict.

The asymmetry is the whole economy of the scheme: steps 1–5 are incurred once and run unattended; the receiver-side steps are incurred separately by each civilization, and only when it can act. No energy is ever spent on a recipient that could not answer — with no agreement between senders, no enforcement, no wish to exclude.

Only step 5 is testable from here, and only its consequence. Steps 1–4 concern hardware in another star’s atmosphere and a network between the stars, neither observable at our sensitivity; steps 6 onward have not happened. What §4 measures is whether the modulation step 5 would impose is present in the Sun’s output, and across 30 searches it is not, above the amplitudes of §4.2. That bounds step 5 and says nothing whatever about steps 1 to 4. The obvious objection is the round trip. One exchange at these distances is decades to millennia — a cost in latency, not in energy, and energy is the constraint: the reply’s wavelength, modulation and symbol rate are specified in the pointer itself, so the node listens on one named channel at essentially zero cost. What the argument does not establish is that gating is the only rational design: a sender optimizing latency would pay for the whole message up front and skip the gate. Energy minimization is a premise, not a fact about anybody. Step 12 is where the chain continues: what the node returns can itself sit behind a second gate demanding a further capability. A directed neutrino beam at the Glashow resonance, 6.3 PeV (Learned, Pakvasa & Zee 2009; Silagadze 2008) is unabsorbed, unmistakable, and readable only by a receiver that has built a cubic-kilometer detector — and far too expensive for a first-contact pointer, needing an accelerator and continuous power, exactly the dedicated apparatus §2.7 excludes. Its place is as content behind a later gate at step 12. The solar neutrino flux itself is unavailable as a carrier — nothing at the photosphere or corona can modulate a flux made in the core — and serves as the one control channel no mechanism in this paper can touch. (That control has since been read directly, and is quiet: see the September 2026 addendum.)

2.6 Deploying the architecture: the carrier bus, the fleet, and the latency it buys

Steps 2 and 3 of the chain are its only capital expenditure — everything else is either observation, patience, or the receiver’s money — and a reader is entitled to ask what they cost and whether the answer is absurd. This section prices them. Nothing here is tested by §4, which bounds step 5 alone (§2.7); the purpose is to show that the sender’s half of the chain is an engineering program with a size, not a miracle with a name, and to identify which of its parameters a designer would actually spend effort on.

Two cargoes. A mesh node is small: a receiver aperture of order 10 m, a laser transmitter of order a kilowatt per beam for step 12 and for the mesh hops (see the link budget after the chain), a decelerating sail, and the stored stages of the message — of order 1–2 t, comparable to a large communications satellite. A stellar probe cannot be small in the same sense, because the modulator it must emplace is 2×10⁹ kg of film (§2.7) and no bus carries that; the probe is a seed — a self-replicating factory that builds the modulator from cometary and asteroidal material at the target — and its mass is the largest uncertainty in the whole architecture, anywhere from 1 t for a nanoscale replicator to tens of tonnes for a conventional one. We carry both figures below.

The carrier bus. A mesh of 10⁷ nodes launched one by one would need 10⁷ propulsion systems and 10⁷ forward shields, and at the transit speeds the chain requires — 0.1–0.2 c, for a 60 ly hop to take 300–600 years rather than the 10⁵ that chemical or solar-sail speeds imply — the shield is not optional: at those speeds a grain of interstellar dust carries the energy of a rifle round, and erosion by gas and dust over a light-year is the dominant hazard identified for gram-scale probes (Hoang et al. 2017). A single large bus carrying many nodes behind one shield amortizes both costs — and it carries the probes of step 3 as well, because the two fleets share a route. The candidate stars of step 1 and the lattice points of step 2 lie along the same tracks through the disc, so a bus that threads the candidate list passes the lattice points on the way, and a dual-purpose bus delivers the stellar probes and the mesh relays from one hull, one shield and one acceleration. Two fleets launched separately would pay the bus cost twice for the same ground; launched together they pay it once, which is the economy of scale that makes 10⁷ nodes and 10⁹ probes the same program rather than two. It accelerates once, navigates once, protects its cargo through the medium, and releases its cargo along its track — a probe on approach to each candidate star from step 1, a node at each lattice point between — each with only the means to decelerate and station-keep — the cheap end of the propulsion budget, since a probe can shed velocity against its target star’s light, wind and field over centuries, where the bus must gain it in a burst. A node has no star at its lattice point and must either brake magnetically against the interstellar medium (Zubrin & Andrews 1991; Perakis & Hein 2016), which works but slowly, or be set down at whichever star lies nearest that point, which brakes it and powers it for step 12; the lattice is approximate in any case, and the second is the natural choice. The energy is still large: a tonne at 0.2 c carries 1.8×10¹⁸ J, and 10⁷ of them 2×10²⁵ J, roughly two years of the sunlight falling on a planet like Earth — affordable to a sender at Kardashev I and trivial above it, and paid once for the whole galaxy. That is the same asymmetry as steps 1–5 applied to the launch itself: the expensive part is shared — across nodes, across probes, and across both fleets at once — and the part paid per unit is the small part.

Fleet size and launch energy. The fleet follows from three numbers: the disc volume (~8×10¹² ly³), the node spacing (60 ly, hence 3.6×10⁷ nodes), and the candidate density (10⁹ systems, one per ~20 ly). A bus at 0.15 c on a route of length ℓ serving a swath of width w covers ℓw² of disc; the buses needed are the disc volume divided by that.

Transit is the step-3 figure of 10⁴–10⁶ yr; longer routes mean fewer, larger buses. Probes outnumber nodes on every bus by ~30:1, so the probe seed mass sets the bus mass.

The launch energy follows: at 0.15 c every kilogram costs 10¹⁵ J. With 1 t seeds the whole galactic cargo is ~10¹² kg and ~10²⁷ J — two centuries of the sunlight falling on Earth, or three seconds of the Sun’s output; with 30 t seeds, 3×10¹³ kg and 3×10²⁸ J — six millennia of Earth’s insolation, or eighty seconds of the Sun’s. Either is paid once. The first is within reach of a civilization that has learned to collect a fraction of its star’s light; the second requires something closer to Kardashev II, or the alternative the numbers point at — buses that replicate en route, in the manner of Tipler (1980) and Freitas (1980), which reduces the launched mass by orders of magnitude at the cost of a longer fill time. The probe seed mass, not the node count, is the parameter a sender would spend its engineering on, and the ppb-depth carrier of §3.6 is attractive to a designer for exactly this reason: a modulator a thousand times lighter is a seed a thousand times simpler.

Latency. The mesh exists so that step 12 arrives within a receiver’s institutional memory rather than its geological one. Without it the reply goes to the sender’s home system at an unknown distance — for a sender anywhere in the disc, kiloparsecs, and a round trip of 10³–10⁴ years — and staged disclosure would be meaningless to a receiver that cannot wait for the second stage. With it the latency is set by one number, the mean node spacing L, and the distance to the nearest node is a distribution, not a value: for nodes placed without regard to the receiver (a Poisson field of density L⁻³) the nearest lies at 0.55 L on average, with the 10th and 90th percentiles at 0.29 L and 0.82 L; a regular lattice is tighter, 0.48 L on average and never beyond 0.87 L.

Disc volume taken as ~8×10¹² ly³ (a cylinder of radius 50,000 ly and thickness 1,000 ly); a mesh seeded only near candidate systems needs fewer nodes for the same effective spacing. Each further stage of step 12 costs one more round trip.

This is why the node must hold the staged content and the means to judge a reply, not merely forward it: a relay to the sender would put every stage of the exchange at the sender’s distance, and the latency figures above would apply to nothing. The node is the sender’s proxy, and the whole exchange — verification, acknowledgment, each successive unlock — runs at node distance. A spacing of 60 ly — one node per few thousand stars, of order 10⁷ nodes galaxy-wide — puts the first return between 35 and 100 years after the reply is sent, and a three-stage disclosure inside two or three centuries: long for a person, short for a civilization that has just built a transmitter, and three orders of magnitude shorter than a reply to the sender itself. The mesh’s node count is therefore not an architectural flourish; it is the parameter that decides whether the gate chain is a conversation or an archive.

Coverage, not stars: where the nodes go. The latency table places nodes as a Poisson field, which is the pessimistic case, not the design. A designer optimizing for coverage — every candidate system within reach of a node — and for the links between nodes puts them on a near-lattice in open space, not at stars: one node answers for every star within its reach. At the same node count a body-centred cubic lattice, the thinnest covering of space, brings the mean distance to the nearest node from 33 to 28 ly and the worst case from ~90 to 42 ly. Tested on the real neighbourhood — the 1,265 G/K dwarfs Gaia DR3 (Gaia Collaboration 2023) places within 100 ly of the Sun (M_G 3.8–7.6, BP−RP 0.72–1.84, parallax precision better than 10%; the brightest stars, saturated in Gaia, are missing) — a star-aware covering puts every one of them within 30 ly of a node with 63 nodes, where a lattice needs 89 and the 60 ly Poisson field leaves 59% of them farther than 30 ly. Only one of the 63 lies within 3 ly of a star; the median node serves 18 G/K systems, and the median star has two nodes within reach, so the loss of one strands nothing. Across the disc the same rule gives about one node per 74,000 ly³ — some 7×10⁸ nodes for 7.8×10⁹ G/K stars, twenty-five times the 60 ly figure, which is the price of a 30 ly guarantee — and a distribution that follows where stars exist, not how many there are: node surface density falls only from 1.6× the solar value at 5,000 ly from the centre to 0.2× at 55,000 ly while the stars fall from 12× to 0.03×, the layer is flat-topped to about ±3,800 ly, and the spiral arms barely register. The 29.8 ly carried for the Sun’s own node is the covering radius: the worst case the design admits.

The links. Coverage and communication set the same scale. A 30 ly covering radius leaves neighbouring nodes 29–48 ly apart, inside a comfortable optical link for the node hardware above (10 m aperture, 1 kW per beam, 1.55 µm, photon counting against a dark deep-space background at 0.1–1 bit per photon), and every node is closer to its targets than to its neighbours. The designer has a second reason to prefer the lattice: a mesh is only as good as its weakest link, and a Poisson field leaves some nodes with their fourth-nearest neighbour ~90 ly away, where the link is four times weaker. The links need not be large, only dependable — and on this placement every node keeps at least four of them above a megabit a day.

Distances from the star-aware covering of the Gaia DR3 neighbourhood. The farthest-target row is also the Sun’s node parked at 29.8 ly, and reproduces the 220 photons/s of the link budget after the chain to within 15%. Neighbour links are symmetric; a target’s own uplink scales with its transmitter.

Scale, for reference. The mesh at 60 ly spacing is of order 10⁷ nodes with optical links across a galactic disc. On 30 January 2026 a single company on a single planet filed with its regulator for a constellation of up to 10⁶ actively powered satellites with optical inter-satellite links in one planet’s low orbit, for compute rather than communication, and the filing was accepted for review five days later (FCC 2026; SpaceNews 2026); the launch vehicle intended to deploy it carried its first test payloads of the associated satellite generation in May 2026, each rated at roughly 10× the downlink of its predecessor. Nothing about that system resembles the mesh — its nodes are powered, short-lived and a few hundred kilometers apart — and it is cited for one fact only: node counts within an order of magnitude of what the architecture needs are within the reach of a civilization at our own rung, on our own timescale. The mesh is not an extrapolation past anything we can see being built.

What this section claims. That the sender’s half of the chain is a fleet of order 10⁴–10⁵ carrier buses, each delivering thousands of probe seeds and hundreds to thousands of mesh nodes along a route of 10⁴–10⁵ ly, at a one-time energy cost between a few centuries and a few millennia of a planet’s insolation depending on the seed mass; that the node count is what makes the exchange a conversation rather than an archive; and that node counts of this order are being proposed by a civilization at our own rung. It does not claim that any of this has been built. §4 does not test it and cannot.

2.6b Two triggered extensions: relay migration, and the leakage-fitted forecast

The mesh of §2.6 has been described as static, but nothing in the architecture requires that. A relay’s defining function is listening, and a technological receiver announces itself long before it can transmit deliberately: broadcast and radar leakage precede the directed uplink of step 10 by decades, and leakage is unintentional — it cannot be faked, withheld strategically, or mistaken for a natural process. That makes it the cheapest possible tripwire, and it suggests a triggered behavior: on detecting leakage, the relay begins to close the distance to its target, moving strictly along the axis between itself and the target star.

The axis constraint is not a stylistic choice; it is forced by the addressing scheme, and it is the only migration the architecture permits. The pointer written into the star’s registers encodes the relay’s address in ~39 bits of sky position — a bearing, not a three-dimensional coordinate — and the pointer cannot be cheaply rewritten. A relay that wanders invalidates its own address; a relay that moves only along the line of sight never changes its bearing as seen from the target, so a receiver beaming the unlock along the pointed axis reaches it at any distance. The one degree of freedom the addressing scheme leaves free is exactly the one that shortens the link. Each light-year closed cuts the round-trip latency by two years and improves the link budget as 1/r² in both directions — the receiver’s uplink, its first real expense (§2.5, step 10), cheapens quadratically as the relay approaches. And the relay never leaves the line of sight, so it listens and relays throughout the transit; migration costs propellant, not availability.

What trips the wire, calibrated on the one case we have. Earth’s leakage record is the natural test article. Medium-wave broadcasting from the 1920s is largely confined by the ionosphere — the dayside escape cutoff sits near 10 MHz, well above the AM band (Sullivan, Brown & Wetherill 1978) — but the confinement is not absolute: the critical frequency sags toward the band on the night side and at high latitudes, a small intermittent fraction escapes, and recovering it at tens of light-years is an aperture problem — large low-frequency collecting area against the Galactic synchrotron foreground. There is a design argument that the relay is pre-programmed for exactly this band: a civilization’s first radio emissions are its lowest-frequency ones — the simplest oscillators, built first, everywhere — so the earliest achievable tripwire lives, by physics, in the band that escapes worst. A designer optimizing for the earliest trigger therefore sizes the low-frequency aperture in advance as a requirement, not an afterthought; the two decades of head start are what the aperture is for. The clean escapes begin with VHF television in the late 1930s and the radar burst of the 1940s; planetary radar (2×10¹³ W EIRP, 1974–) is detectable at 30 ly with modest apertures. The ladder is now quantified at Earth-2024 capability on both ends (Sheikh et al. 2025): planetary radar carries to 12,000 ly, the Deep Space Network to 65, and omnidirectional mobile leakage to 4 — so at 30 ly the radar tripwire costs the relay nothing, broadcast-era leakage needs ~50× Earth-level sensitivity, and the medium-wave trickle is the design-limiting requirement, consistent with an aperture provisioned for it rather than improvised. The strongest documented medium-wave carrier of the era sets the date: WLW Cincinnati, 500 kW at 700 kHz from 1934 to 1939 — the only U.S. commercial station ever authorized at that power — whose shell reaches a relay parked 29.8 ly out in 1964, for an array of the size the next paragraph derives. The Mexican border blasters ran in the same years: XER at Villa Acuña (1931–33) and its successor XERA (1935–39), with claimed powers up to a megawatt that no regulator ever measured. Taken at their claims they weigh as much as WLW in the relay’s statistic or more, and XER’s shell would move the trigger to 1961–63; the date carried here is 1964 because it rests on a licensed power. Waiting for the cleaner signals would only delay the trigger: 1940s television and FM arrive around 1970, and planetary radar (1974) not until 2004. At a 0.1c cruise from 1964 the relay has closed about 6 ly by now and reaches the outer solar system around 2260; at a stealthier 0.01c, on millennial timescales. The dates of the exchange move with it. An unlock sent in 2035 finds the relay 22.7 ly out and still closing, and meets it about 2056 rather than 2065; the reply, sent from 20.6 ly, lands about 2076 rather than 2095 — nineteen years saved on one round trip, before any further stage.

The receiver this forces. The ionosphere is a two-way mirror: the same cutoff that traps medium-wave inside a planet blocks the band from every planetary surface — which is why present-day astronomy proposes lunar-farside arrays (FARSIDE, FarView, LuSEE-Night) to open it at all. The tripwire band can only be watched from space, so the listening relay is a space-based array by necessity, not preference. The band then pays for the requirement it imposes: effective area scales as Gλ²/4π, so a single wire dipole at 1 MHz collects with ~10⁴ m² — four million times its 2 GHz figure — and the receiving element is literally wire, nearly massless in zero gravity. What sensitivity still costs is element count, because the band is externally noise-limited: the Galactic synchrotron foreground, ~10⁷ K at these frequencies, sets a floor that no receiver quality can buy back. Closing the budget on the strongest single carrier of the era — WLW Cincinnati, 500 kW at 700 kHz, 1934–39 — at 29.8 ly through a ~10⁻³ ionospheric escape fraction leaves a flux of ~5×10⁻³⁴ W m⁻² at the relay. Against the 10⁷ K foreground, a carrier detection at signal-to-noise 10, in 0.01 Hz bins stacked over one year, needs a collecting area of ~5×10¹³ m² — a filled array ~7,000 km on a side; spending all five of WLW’s 500 kW years, night side only, brings it to ~5,600 km. That is ~10⁹ dipoles on a half-wavelength grid, and mass is not what it costs, because the band is sky-limited: a conductor need only deliver more of the 10⁷ K foreground than its receiver adds, and a receiver at deep-space ambient adds a few kelvin. A 1 µm filament converts ~10⁻⁵ of what it intercepts and still delivers ~100 K of sky, so the whole array is of order a quarter-tonne of aluminium — less in carbon-nanotube yarn — and folds into the node that the carrier bus delivers alongside its probes. A kilometre-scale array falls short by seven orders of magnitude in signal-to-noise, and even a 1,000 km array reaches only ~0.2. The planetary medium-wave ensemble helps, but less than its total power suggests: each station holds its own channel, so the carriers combine as the root-sum-square of their separate detections, which the strongest dominate. By the mid-1930s dozens of 50 kW clear-channel stations were on the air in the United States alone; a hundred of them alongside WLW in 1934–39 gain ~1.4× in signal-to-noise (~4,700 km), and stacking the same clear channels through the following quarter-century — each carrier integrated over its own decades — reaches ~2.6× (~3,400 km), at the price of a trigger in the 1970s–90s rather than the 1960s. The requirement is set by carrier detection, not decoding: nearly all of an AM transmitter’s power sits in a line well under a hertz wide, present continuously, so it can be stacked; the audio carries a few percent of the power spread over 10 kHz and must be recovered in real time, because speech and music do not repeat. Demodulating it at the same range would need an aperture of order an astronomical unit — nine orders of magnitude more area — and even detecting that the carrier is modulated at all needs ~10⁶ km. Two limits on the integration are worth stating. The line integrates coherently — signal-to-noise growing with τ rather than √τ — only within the carrier’s coherence time, set by a 1930s crystal oscillator, the diurnal Doppler of Earth’s rotation, and phase noise on a path that escapes near the ionospheric cutoff: bins of 0.01–1 Hz, beyond which stacking gains only √τ. And the integration is not unlimited for any one carrier: WLW’s 500 kW shell is a five-year pulse that passes the relay once. The dominant unknown is the escape fraction, and it cuts against the design: 700 kHz sits below the night-side F-layer critical frequency (typically 2–5 MHz), and a vertical broadcast mast radiates least toward the zenith, where escape is easiest; each factor of 100 lower escape multiplies the array’s side by ten (~70,000 km at 10⁻⁵). Stated as assumed capability, and quantified. At this band the array is not an instrument the relay carries — the array is the relay, with bus and drive as fittings on an antenna the width of a planet.

The forecast: capability read from leakage. A listening relay does better than trigger — it fits a curve. Raw leakage power is a poor capability metric (it measures population and habit), but coding efficiency is unit-free: the distance of an intercepted waveform from the Shannon limit is a dimensionless, physics-anchored measure of the sender’s communication capability, readable by any receiver sharing no units with the source — the same class of observable this paper is built on, appearing on the listening side. Earth’s record again calibrates: amplitude modulation sits ~30 dB from the limit; within eighty years the leakage’s coding (spread-spectrum, LDPC) closed to within a fraction of a dB. Power trend and entropy trend together also disambiguate the one failure mode that matters: falling power with rising entropy is efficiency; falling power with falling entropy is regression or collapse. That distinction is the relay’s risk model for committing propellant.

Three decisions then stop being reflexes and become solved optimizations. Departure and cruise speed: the relay compares the fitted capability doubling time against its own transit time; for a fast developer — decades from first leakage to gate-crossing, against centuries of transit — it concludes that first contact happens from the parked address, and migrates to shorten the long payload conversation instead, at the slowest cruise consistent with arriving before the dialogue’s bulk. Beam sizing: the reply is transmitted at the parsimony-minimum for the receiver’s aperture projected to the beam’s arrival date, not measured today. Gate-fall prediction: a relay reading our 1990s leakage — consumer electronics implying FFT-scale silicon — could have placed tier-3 decode feasibility in the 2020s within a decade or two.

The reflexive point deserves stating plainly. §2.8 requires the architecture to predict receiver behavior, and §5.6b’s exhibit is a capability chart — Cray-1 to Frontier, eight-billion-fold in one working lifetime — drawn by the receiver, from the inside, to show the gate falls within a lifetime of becoming visible. This extension has the relay drawing the same chart from the outside, fitted from leakage rather than from history. The two extrapolations are aimed at each other, and the gate is set where they are designed to meet: a fixed, dumb, megayear-old gate arrives on time because the timing intelligence lives in the relay, not in the gate.

The costs are real and worth stating. A drive is an emitter, but the geometry is kind to it: a relay accelerating toward the target throws its exhaust away from the target, and braking need not use the drive at all — a magnetic sail brakes against the interstellar medium with no exhaust. Braking on the drive would be the worst case, pointing the plume at the one observer that matters, and the design has no reason to. What migration spends is propellant and a sliver of passivity toward everyone else; the designer’s levers are speed, timing, and the judgment that once the target is leaking radar, concealment has done its work. And taken to its limit the migrated relay parks at the heliopause and round-trip time collapses from decades to hours — at which point it is the Bracewell (1960) probe, the lurker of Benford (2019). The extension derives the lurker as the late-stage state of a gated mesh rather than assuming it, and the standing cost objection to solar-system artifacts — that every system must be seeded with expensive hardware — dissolves, because the hardware migrates in only behind a confirmed trigger.

The drive: fusion against antimatter. Braking on a magnetic sail puts the whole propellant budget into the acceleration, and for a tonne-scale relay the two candidate drives differ less in performance than in what they ask of the relay during the half-billion years before it is used. A D–³He fusion drive exhausting at 0.03–0.05 c needs a mass ratio of 7–28 to reach 0.1 c — 6 to 27 t of propellant for a 1 t relay — and its fuel is inert: deuterium held as solid lithium deuteride and helium-3 in a sealed tank wait out any interval without power or attention. The same drive sets the speed ceiling; a 100:1 fusion stage reaches 0.14–0.23 c and even an ideal one about half the speed of light, so 0.1 c is the defensible middle rather than a limit. An antimatter (pion) drive at an effective 0.3 c needs a mass ratio of only 1.36 — about 180 kg of antimatter — but there the fuel is the hazard. Storage for eons is not ruled out by physics: in free fall a pellet of anti-hydrogen ice needs almost no force to hold clear of every surface, a superconducting trap at deep-space ambient draws no power, and cosmic rays erode only milligrams over 500 Myr. What decides it is temperature. At the 2.7 K of the microwave background anti-hydrogen ice sublimates away in about three million years; below 2 K it survives the half-billion intact, and holding it there against the background’s 25 µW takes a microwatt cryocooler. Antimatter storage over eons therefore works — as an active system that must never fail, guarding 8 Gt of stored energy whose release destroys the relay — and making the fuel on demand instead would take a megawatt supply a million years. A designer optimizing for survival over deep time rather than for mass takes the fusion drive: several tonnes more on the bus, and nothing to keep cold.

The observable this predicts. Under this extension the present epoch is an inbound window, and one no current instrument can see into. The node was released by its bus in open space, not at a star — one node answers for every star within its reach — and on the trigger it accelerates from there straight down the axis toward the target. The geometry hides the burn: a drive accelerating toward the target throws its exhaust away from it, and for a tonne-scale relay the isotropic remainder that did reach us — around 1994–2003 for a 0.01 g burn from 29.8 ly — fell some eleven orders of magnitude below the all-sky limits of the day. It then cruises dark, and brakes on a magnetic sail whose kilohertz emission sits below the interstellar plasma cutoff. The migration is therefore not observable directly; it is observable in the timing of the reply. The pointer carries the node’s distance, so it fixes when a parked relay’s answer must arrive; an answer that comes early — by about nineteen years for a 2035 unlock under a 1964 trigger at 0.1c — is the detection, and its lead measures both the trigger date and the cruise speed.

2.6c The node as instrument: what the mesh senses, computes, and becomes

The relay is defined from the outset as a listening and compute post (§2.2, §2.6), not a transmitter — its radiative silence is the whole point. That framing has a consequence worth drawing out: passivity here is radiative, not informational. A node may sense its surroundings and compute on what it senses at essentially zero radiated energy, because listening, sensing and computing cost no photons. This opens a class of node functions that the beacon description alone does not exhaust, and that follow from capabilities the architecture already assumes rather than new ones.

What a node can sense, in Newtonian physics alone. A swarm distributed over the host system is a gravimeter. A single point measures only the product Gm/r² of a planet’s mass and distance — degenerate — but a distributed swarm measures the field and its gradient, and because the field falls as 1/r² while the tidal gradient falls as 1/r³, their ratio returns r directly, and the field then returns m. The swarm’s spatial extent also yields a gravitational parallax, and tracking the star’s reflex wobble over an orbit gives the period — hence, by Kepler’s third law with the star’s known mass, the orbital distance — and the wobble amplitude gives the planet mass. This is the reflex motion that radial-velocity and astrometric surveys already exploit (Mayor & Queloz 1995); it is classical mechanics, requiring no general relativity — planetary orbital gravitational-wave emission is some forty orders of magnitude below any detector, so the “rubber sheet” never enters. Because the swarm reads the field as a three-dimensional vector rather than a line-of-sight projection, it resolves the inclination that leaves ground-based radial velocity with only m sin i: it recovers the true mass and the full orbit of every planet.

And what it can sense with light. A node stationed in the system has a spectroscopic vantage far exceeding ours: it can directly image each planet, resolved from the star, across a full orbital phase, where we are confined to the thin transmission annulus of a transit. From that vantage the atmospheric constituents that bear on the presence of life are in reach — the O₂/O₃–CH₄ disequilibrium pair whose coexistence implies active replenishment (Lovelock 1965), water, N₂O, and the industrial technosignatures CFCs and NO₂. The honest qualification is the same recognition problem that governs the rest of this paper (§5.3): biosignatures admit abiotic false positives, so what a node forms is a probability of life, not a verdict.

The message then writes itself, and it writes itself dimensionlessly. Everything the node senses is already in the unit-free form §3.1 requires of an admissible carrier. A planetary architecture reduces to a count (the number of planets), a set of period ratios to the innermost planet, and a set of mass ratios to the star — and the orbital distances need not be sent at all, because Kepler’s third law recovers them from the periods. The periodicity need not even be stated as a number: a node may modulate its beacon at the orbital periods it senses, so the signal’s own rhythm carries the architecture it describes. Encoded as continued fractions — base-independent by construction — the entire Solar System closes in of order two hundred bits, and a resonant system such as TRAPPIST-1 collapses to a sequence of small integers (its adjacent period ratios are 8:5, 5:3, 3:2, 3:2, 4:3, 3:2). Atmospheric composition adds a few mixing ratios per planet; the life-probability, a scalar. The whole description of a living planetary system is pointer-scale — the same order as the ~127-bit pointer of §2.5 — and made entirely of counts and ratios.

What the mesh becomes. Give every node these three faculties at once — it listens (the leakage tripwire of §2.6b), it senses (its host’s planets, atmospheres, and life-probability), and it computes (the same faculty that in §2.6b backs a civilization’s capability curve out of its leakage) — and share the results across the network, and the architecture is no longer a field of dormant beacons. It is a distributed, self-updating census of biological and technological activity across its sphere of the galaxy. This is the function the beacon description was missing, and it closes the parsimony argument (§2.7) in a way a pure beacon cannot: compute is cheap and listening and sensing are free, while radiating is the one expensive act, so a network of compute-and-listening posts can do nearly everything it does — census the galaxy, forecast the civilizations in it, decide where contact is worth its cost — without radiating at all. The silence is not only a beacon waiting to be keyed; it is the signature of a system whose activity is almost entirely non-radiative. And it extends the gate: a network that knows where life is emerging spends its rare photons not merely on receivers that have proven their capability, but on the living, capable systems its census has already flagged as worth the tier-2 beam.

How a distributed signal makes itself findable. A signal spread thinly across many stars has a recognition problem sharper than any single beacon: a receiver would have to already suspect it, and know to correlate an entire field, before it could be seen — and nothing in ordinary astronomy poses that question, so such a signal sits in the archives mis-filed as a set of unrelated one-off measurements. A receiver-centric designer anticipates exactly that blindness and resolves it the way the pointer resolves the where-to-look problem: with a nudge. The two-tier pointer of §2.5 is here applied not to a coordinate but to a method — a cheap, conspicuous marker whose function is to say correlate the population.

The concrete form is a centroid: one star at the array’s center tweaked conspicuously enough to be flagged by ordinary single-star variability searches — the channel astronomy already watches — and thereby to nominate its own neighborhood as the thing to correlate. The centroid fixes the array’s location; its scale, how far out the array extends, is the remaining free parameter, and a parsimonious designer would not leave it to a blind radial scan, which costs the receiver a multiple-testing penalty that grows with every depth tried. The scale is instead encoded as a dimensionless depth key: a universal constant — π, e, a small ratio — multiplying a natural length the receiver already measures, the centroid’s own distance. The receiver reads the distance, recognizes the constant, and computes one radius rather than scanning many; recognizing the constant is itself the proof-of-work of §2.9, gating the search geometry behind a small mathematical puzzle. The whole scheme stays inside the dimensionless, unit-free vocabulary §3.1 demands — the depth is one more constant in the pointer, beside the planet counts and mass ratios.

This defines a concrete search whose selection is nominated by the signal rather than by our theory, and whose every step is forced by the same principles: an aperiodic-inclusive variability metric, because a periodicity catalogue structurally excludes the aperiodic anomalies that would serve as markers (the Boyajian’s-star morphology); a class-normalized outlier cut, comparing each star to its own stellar type rather than a global distribution, so the intrinsically loud classes are not mistaken for anomalies; the surviving conspicuous, within-class outliers as candidate centroids; and, at each candidate, the neighborhood correlation of §4.8, keyed to constant-derived depths and tested against matched random fields. §5.5 reports what this search returns.

What this is, and is not. These paragraphs elaborate the internal coherence of the hypothesis — the sensing, the message, the network’s purpose, and the means by which a distributed signal would make itself findable all follow from capabilities and principles already assumed, and each lands in the dimensionless, energy-minimizing, gate-building form the argument demands. That is worth establishing, but it is not evidence: a more complete story is not a truer one. The extension earns its place only because it makes falsifiable predictions the rest of this paper can act on — that coordinated modulation, if it exists, is carried across a nearby stellar population and entered through a conspicuous marker at a computable depth. Those predictions are testable in existing photometry, and §5.5 reports the searches they define — every one, so far, null.

2.6d What the network is for, and what keeps it alive

Three things the argument has so far left implicit have to be said, because without them the deep-time story is not coherent — and a reader could fairly object that a structure abandoned for a billion years is either impossible or pointless. The first is that the network is not passive and does not merely outlive its builders the way an artifact outlives a hand. A thing that must persist, adapt, and decide across gigayears — reading a civilization’s leakage (§2.6b), keeping a census current (§2.6c), holding a gate that still arrives on time — is not a dumb beacon left running; it is stewarded by an autonomous controller, an intelligence built to maintain, repair, replicate, and re-plan without its makers. The builders do not leave a corpse humming; they delegate continuity to a steward designed to outlast them, and the steward is the sense in which the network “outlives” anyone. This is the parsimonious choice, not an extravagance: a static beacon cannot stay functional or relevant for a billion years — it degrades, goes stale, cannot respond — so the cheapest route to megayear persistence is a self-maintaining intelligence. Over deep time the dumb option is the expensive one, because it fails.

The second is that the stellar filter of step 1 is better read as a payoff matrix than a fixed cut. With of order 10¹¹ stars and finite seeding capacity, the designer is allocating, not merely selecting: each candidate carries an expected return — the probability that a receiver eventually arises there, times the value of contact, against the cost of seeding and stewardship — evaluated over gigayears at a very low temporal discount, which is exactly what licenses the patience the whole design assumes. The exclusion criteria above are the prior on that probability; the census of §2.6c keeps the matrix current, moving the expensive acts — the migration, the tier-2 beam — toward the systems it has learned are actually developing. The filter chooses where to seed; the ledger, continuously re-scored, chooses where to spend. And where more than one civilization has laid down such a network, the allocation becomes a genuine game — a galactic commons to partition or coordinate — whose equilibrium may itself contribute to the silence, since coordinated builders would not redundantly broadcast.

The third, and the deepest, is what the network is for — and it gives the architecture a purpose beyond passive monitoring over eons. The argument has framed everything around a single act, a pointer and a receiver and a contact, but a relay mesh is communication infrastructure, and the value of infrastructure is never one link; it is the network the links form. The return on a fabric connecting n civilizations grows far faster than n, so the payoff of onboarding many emerging technological cultures into a shared medium dwarfs the payoff of reaching any one of them. The natural design purpose, then, is not to contact us but to link us — to bootstrap each civilization, as it crosses the capability threshold, into a pre-existing galactic communication network already joining the others. First contact, in this reading, is not an endpoint but an admission: the proof-of-work gate is the entrance exam to a fabric laid down in advance, and the message behind it is less a greeting than an invitation to join. This deepens the sense in which the network outlives its builders. They may be long gone, transformed, or silent; what they leave is not a monument but infrastructure — roads for a traffic that has not yet arrived — held open by its steward until the civilizations it was built to connect emerge to use it. The idea has a lineage (Learned, Kudritzki, Pakvasa & Zee’s “Cepheid Galactic Internet” is the nearest); what this framework adds is that the infrastructure is passive-until-addressed, gated on demonstrated capability, and stewarded rather than static — a backbone waiting, patiently and cheaply, for its nodes to boot.

What the payload contains, and how it is addressed. If the message behind the gate is an invitation to join, its content is the thing a new member most needs: the network’s own directory — a catalogue of the active nodes and where they are. This is tier-2 content by §2.5’s division, delivered only once the proof-of-work is met, not radiated for a pre-threshold civilization to stumble on; the beacon in front of the gate stays compact, and the directory is the reward for opening it — which is also why the signal we could actually detect is small, a pointer rather than a broadcast. The directory faces the same unit-free constraint as everything else (§3.1): a coordinate means nothing without a shared frame, so the positions cannot be given in any civilization’s grid or units. The natural — and very nearly the only — solution is pulsar-referencing: each node located relative to a set of identified pulsars, the pulsars named by their periods expressed as ratios to a universal clock (the hydrogen 21 cm transition) and their geometry given as dimensionless angles. It is the scheme humanity itself reached for on the Pioneer and Voyager plaques, for exactly this reason, and it ties the network’s self-description to the same millisecond pulsars whose timing arrays §4 searches. One consequence follows for the steward of §2.6d: pulsars spin down and drift, so a directory that stays valid across the network’s lifetime must be actively re-referenced — one more task a static beacon could not perform and an autonomous controller can.

2.7 What must sit at the star, and what it must cost

Step 4 of §2.5 is the load-bearing step. A signal in stellar output requires a persistent agent in the stellar environment: surviving on the timescale of the beacon, self-repairing, able to impose a small modulation on emission the star produces anyway. This is the largest unsupported assumption in the framework. Nothing in §4 tests whether such a thing exists; the searches test only whether the modulation it would produce is present.

Where it would have to be. Not the photosphere. The channels that are cheap to modulate — Lyman-α, the EUV lines, the soft X-ray background — form in the chromosphere, transition region and corona, which are optically thin and mechanically tenuous. The cheapest emission to modulate is formed exactly where a persistent structure is most plausible — and where the radiative flux limits our own probes to brief perihelion passes.

What it would cost to run. The limits of §4.2, read backwards, give the power a sender must command to sit at the threshold of our detection:

Taking world primary energy use as 2×10¹³ W. The channel powers follow from each band’s share of the solar luminosity.

This is the strongest argument in §2 that survives contact with numbers. Modulating the bolometric output at our sensitivity would take a billion times humanity’s energy budget; a narrow line costs 10⁷ times less for the same detectability, and the soft X-ray background at our published limit about eight times what our civilization already generates. Bit-energy conservation therefore chooses the carrier, not just the message length, and §4.2 reports limits in all three channels.

Read naively, this refutes the premise. If a 1.4×10⁻⁶ modulation of Lyman-α means generating 2.4×10¹⁵ W, the sender has built a transmitter more expensive than any radio beacon. The escape is that a modulation need not be generated. Flux the star already radiates can be redistributed — occulted, scattered, or shifted between channels — and the cost is area, not power. To occult a fraction 1.4×10⁻⁶ of the disc requires 2.1×10¹² m², a square some 1,460 km on a side, or 2.1×10⁹ kg of micron film. To produce a two-minute modulation that structure must also change what it intercepts on that timescale — rotate, translate, or vary its opacity. We do not know which, and the searches do not distinguish them: the area is what the energy argument bounds, and the mechanism is what it does not.

That is the difference between 10¹⁵ W and nothing at all. It is also, plainly, Bracewell’s mechanism and not ours — a local artifact in the line of sight, developed for co-orbital positions by Benford (2019); what this paper claims as new is the search strategy (§1.4).

Two consequences follow. The fast-band limits of §4.2 bound any modulator at the cadences examined, gray or selective alike: a gray occulter moves Lyman-α along with everything else, so at two minutes it is bounded directly by the 1.4×10⁻⁶ Lyman-α limit, not by the daily achromatic figure. The achromatic and line-ratio searches (§A.1) add not a separate bound but a discriminator between mechanisms, and only at the cadences at which they were run: a spectrally selective modulator is the more demanding object, and distinguishing it from a gray one at two-minute cadence has not been done here.

An honest reframing. “No transmitter exists” is stronger than we can defend. The defensible claim is narrower: no dedicated radiating apparatus is assumed, and the carrier is not specified in advance. That leaves the work a systematic anomaly search over a principled combination space — where its value lies in any case, since the limits of §4.2 stand however the modulation is imagined to arise, or whether it arises at all.

Why not look for the thing gravitationally? Lunar laser ranging reaches millimeters; the search is not worth trying, by eleven orders of magnitude.

dr ~ 2GMr/(D^3 n^2)     r = Earth-Moon separation, n = lunar mean motion

  modulator of this section, 2.1e9 kg, at 1 AU     4.5 femtometers
  short of a 1 mm normal point by                  2.2e11
  mass at 1 AU that WOULD give 1 mm                4.6e20 kg = half of Ceres
  or: how near a 2.1e9 kg probe must be            24,800 km, inside geostationary

The radiative route is worse: the Moon’s response to radiation pressure is 8.3×10⁻⁵ m per unit fractional irradiance, and Lyman-α is 4.4×10⁻⁶ of the total, so the 1.4×10⁻⁶ limit of §4.2 moves the Moon by 5×10⁻¹⁶ m. Ranging would have to improve by 8.6×10⁶ to compete with pointing a photometer at the Sun; planetary ephemerides do better, but by order 10², not 10¹¹. No dynamical method reaches an object this light — a modulator that redistributes rather than generates has almost no mass to find.

2.8 What the argument has to predict

An argument of this shape earns its place only by making the searches it motivates sharper than the nulls it might be used to excuse. Four predictions follow:

2.9 The gate as a cryptographic object

The constraints of §2.1 and the proof-of-work argument of §2.3 describe the gate functionally. It is worth naming the cryptographic primitive it actually corresponds to, because the correspondence is exact, because it repairs an imprecision in §2.3, and because it makes a prediction about search order that the bottom-up program would not otherwise have made.

The gate is a symmetric cipher under brute-force key search. The key is the tuple (which observables, which functional form); the ciphertext is the public archive; the plaintext is the imposed modulation; the key space is C(N,k) × |F|, which is the combination space of §3.3 under another name. Everything is public except the key — the archives, the physics, the method, this paper — which is Kerckhoffs’s principle in its purest available form, and not by accident: a sender wanting concealment would not modulate a star.

The property that makes the scheme work at all is self-identifying plaintext. A receiver knows when the right key has been tried because the statistic fires; no crib and no known-plaintext pair is needed. That is the same property that makes brute force viable against a block cipher — one can recognize English without knowing in advance which English — and it is what the detection criteria of §3.5 formalize.

It is not proof of work in the hashcash sense, and the difference matters. In hashcash the sender performs the work, to price a message or prove commitment. Here the receiver performs it. The direction of payment is inverted, and with it the design goal: this is a proof-of-capability challenge — a CAPTCHA turned inside out, a puzzle a computer is meant to pass rather than fail. The analogy in §2.3 is right about the mechanism and wrong about the polarity.

The nearest technical antecedent is not in the SETI literature. Rivest, Shamir & Wagner (1996) pose a time-lock puzzle whose solution requires a chosen amount of computation, and set its difficulty from projected future hardware speed so that it opens at a chosen future date. That is the two-sided difficulty bound of §2.3, arrived at independently in a different field.

The construction differs in one respect, and the difference is favorable to the argument here. A time-lock puzzle, and its descendant the verifiable delay function, is inherently sequential: the delay cannot be bought off with more machines. A key search is embarrassingly parallel.

A designer wanting a calendar clock therefore chose the wrong construction. A designer wanting to gate on total civilization compute chose exactly the right one — and that is the more sensible quantity to threshold, because it is what the lower bound of §2.3 is trying to measure in the first place. Calendar dates are parochial; one cannot reach a capability level by waiting.

Key-space entropy, and why the data requirement is mild. The required record length is set by the need for the true key’s statistic to stand above the maximum of K draws from the null. That maximum grows as √(2 ln K) while detectable amplitude falls as 1/√N, so N ∝ ln K.

The key space spans fifteen orders of magnitude; the required record length spans 5.6×. Compute scales with K and data scales with log K. The gate is compute-bound, not data-bound — which is how one would design a lock intended to be opened eventually, and which is an independent route to the empirical finding of §4.3 that sensitivity is not the binding constraint.

Unicity distance. Shannon’s unicity distance is the ciphertext length below which more than one key yields a plausible decryption. That is the same quantity as the multiple-testing threshold of §3.5: below it, wrong keys fire. The N²/α scaling of §3.4 is therefore an instance of a known information-theoretic bound rather than a bespoke statistical fix — and a void result is, in this language, a search conducted below the unicity distance for its key space.

Weak keys, and the search order they imply. A cipher is strong when its keys are drawn uniformly from the key space. These are not. The sender is constrained: the combination must be one the receiver plausibly measures, on records long enough to hold a message, and that concentrates the key distribution onto the best-instrumented, longest and most obvious observables. In cryptographic terms these are weak keys, and brute force is the wrong attack against them. No password cracker begins with brute force; it begins with a wordlist.

The wordlist is 548× cheaper and, because the multiple-testing penalty scales with the number of keys tried, roughly 550× more sensitive — a signal that an exhaustive sweep would bury under its own trials correction can survive the wordlist’s. That gain is earned only by committing to the wordlist in advance; choosing it after seeing exhaustive results would be selection.

The wordlist that follows from the designer’s own constraints — the channels any civilization measures early, precisely, and for a long time — is the §3.6 ordering reached by a different route. Two independent arguments converging on the same search order is itself evidence the ordering is right, and the consequence for this program is concrete: the unsearched rows of §3.6 should be run before any exhaustive tier, not because exhaustion is expensive but because it answers a less likely question at greater cost.

One caveat, stated because it bounds what the argument licenses. The weak-key claim rests on a prior over what a sender would choose, and that prior is our reasoning about their design rather than anything testable. It justifies search order, which costs nothing if it is wrong. It would not justify restricting the search space, and nothing here should be read as license to drop the exhaustive tiers.

2.10 The pointer's structure: framing, an array, and a line address

Prediction iii fixes the pointer's size at order 10² bits. It does not fix the pointer's layout, and the layout has consequences both for the design and for the search.

A header is needed for framing, not for identity. The bearing identifies the relay. About 39 bits of sky position is a unique address, so a separate identifier would carry nothing the bearing does not. What a receiver cannot recover from content is where a frame begins and where one field ends and the next starts. Repetition gives the frame length, since autocorrelation finds the period, but not the frame's phase, and that needs a synchronization pattern. Under the no-shared-conventions constraint (§2.1), a code table is an institutional fact and is excluded. The marker has to be self-evident: a maximal-length or Barker-type sequence whose sharp autocorrelation identifies it without being told, runs of prime length, or a field count written in unary.

The address field should be an array, not a single pointer. The design almost forces this.

• A single point of failure. The stellar pointer cannot be cheaply rewritten (§2.6b), and relays have finite lives. A pointer to one relay makes the whole star's gate hostage to that relay for gigayears. The modulator cannot even learn that a relay has failed, since nothing reports back to it. Redundancy therefore has to be written in at the start, as a static list.
• The mesh already supplies the spares. Coverage-optimal placement (§2.6) gives overlapping 30-light-year bubbles, so every target star is within reach of several nodes. The array lists them, and the receiver triggers all of their bearings, which costs k times a transmitter it must build anyway.
• A bearing addresses a line, not a point. This is what makes the array repairable. The axis constraint of §2.6b, which holds that a relay may move only along its line of sight so that its bearing never changes, has a second consequence: a replacement relay can park anywhere on a failed relay's line of sight and inherit its address. The only field it cannot inherit is the range, and with it the reply-timing prediction for that entry. A dead relay is a maintenance event, not a dead address.

What it costs. A bearing of ~39 bits, a range of ~10 bits and a short header give a frame of ~200–450 bits for four to eight pointers, still of order 10², as prediction iii requires. The price is paid in the receiver's integration, not the sender's energy. Folding N repeats of an n-bit frame in a channel with fractional noise σ per sample at cadence τ, and a modulation depth ε, the two times are:

T_detect ≈ (5σ/ε)²·τ and T_decode ≈ n·(4σ/ε)²·τ

Detecting that a pattern exists comes about n times sooner than reading it: for a few hundred bits, detection precedes decoding by two orders of magnitude. An eight-pointer frame therefore takes several times longer to decode than a single pointer, but no longer to detect. That is a cheap trade for a sender whose concern is the gate surviving, not the speed of the first read.

The clock: borrowed from the system, not shared by convention. Framing tells a receiver where bits begin. It does not say how long a bit is, and a bit period agreed in advance is exactly the kind of shared convention §2.1 deletes. The way out is to clock the pointer to a rate both sides can measure independently. A modulator embedded in the star has real-time access to every solar parameter by construction, so it can phase-lock to the live value rather than to a stored period that drifts. Three families of clock are available, and they differ in whom they address.

• The star's rotation. This clock is universal. Any receiver that sees the signal can measure the rotation in the same data: a distant one through rotational modulation of the integrated light, a local one by tracking surface features. Rotation is differential, so a designer wanting a single number would use a well-defined one: the equatorial rate, or the rigid ~430 nHz rotation of the radiative interior, which helioseismology measures to high precision. Each observer sees a synodic period set by its own orbit, 27.3 days from Earth, and converts to sidereal using an orbit it already knows. The rotation period is where natural variability is strongest, because active regions rotate. The bit clock would therefore sit at a high harmonic, e.g. rotation/64 ≈ 10 h, and would hold phase across years, which natural active longitudes do not do reliably.
• The star's p-modes. The five-minute oscillation (ν_max ≈ 3.09 mHz, individual modes known to nHz) is fast, physical, and measurable from interstellar range: Kepler measured ν_max for thousands of stars. It drifts only on stellar-evolution timescales, and a phase-locked modulator tracks even that.
• A planet's rotation. Clocking to the rotation of Earth or Venus writes to one planet. Only a receiver on that planet knows the clock, so this is a signal for local receivers, the case §5.6d's resolved search addresses. A planet's day drifts over geological time (Earth's was ~22 h four hundred million years ago), but a modulator that watches the planet's reflected light-curve can lock to the current value. One trap is specific to our data. SDO's orbit is geosynchronous, so its period is one sidereal day: a signal clocked to Earth's sidereal day would sit on SDO's strongest instrumental line. That clock needs SOHO at L1, which has no 24-hour orbit.

Real-time access has two further consequences. The modulator can hold signal-to-noise constant by raising its depth when the star is noisy. It can also transmit hardest when the star is quiet, the cheapest time for a given signal-to-noise, so searches should weight activity minima. For the search, a borrowed clock collapses the period dimension. Instead of scanning bit periods blind, one folds on a short list of measured natural phases (solar rotation at the equatorial, Carrington and interior rates, ν_max, and planetary days and years) and their harmonics. That is a few hundred candidate clocks instead of millions of trial periods, with the trials penalty shrinking to match. It is §2.9's wordlist argument applied to timing.

What is new in this, and what is not. Natural references as coordination devices, focal points that two parties reach without communicating, are an established SETI idea. They have been proposed for which frequency to use (Cocconi & Morrison 1959; Planck frequencies, Wright 2020) and for when to transmit: the SETI Ellipsoid and related event-synchronization strategies, in which a beacon fires on a conspicuous astrophysical event (Davenport et al. 2022; reviewed by Nilipour et al. 2023). The nearest neighbour to the present idea is the Cepheid proposal of Learned et al. (2008). There the host star's own pulsation is modulated and serves as carrier, as a proposal not taken to a search. To our knowledge, a natural rate has not previously been proposed as the bit clock of an embedded signal, phase-locked to its live value by a modulator with real-time access to it. Nor has a search been built on folding at measured natural phase, with decoy clocks as its null. The step extends the focal-point argument from frequency and epoch to the symbol clock. It adds one consequence neither of those uses has: the choice of clock selects the addressee, the star's own rates for any receiver and a planet's rotation for that planet alone. The claim is stated as "to our knowledge" deliberately. It rests on a literature search, not an exhaustive one.

A structural prediction, independent of content. A frame of k equal-length address fields has an internal period at the field length. If the listed relays are mesh neighbours, their bearings cluster, so the fields also share their high-order bits. The frame should show a period inside the period. A search can test for this without knowing a single bit of the content: look for sub-harmonic self-similarity at an unknown frame period in a combination series. The same frame may also be written into more than one channel, which is the channel-level form of the same redundancy.

3. The combination space

3.1 Admissible carriers, and a constraint that had to be sharpened

A sender sharing no units with the receiver cannot encode in a quantity carrying units. Admissible carriers are therefore dimensionless: ratios of like observables, normalized residuals, quantities scaled by their own dispersion. This removes every raw measurement and admits only combinations, which is what converts an unbounded question into a countable one.

A fractional modulation depth of a single observable is admissible. The deepest limits in §4.2 are single-channel; a depth is a quantity divided by its own baseline — a normalized residual, dimensionless in exactly the required sense — so the single-channel searches lie inside the combination space, in its k = 1 corner.

Dimensionless is necessary and not sufficient, and we learned this the expensive way. A first sweep included plasma beta and the Alfvén Mach number because they are dimensionless, and returned sixteen Bonferroni survivors — every one a pair involving those two quantities. Neither is a measurement: both are functions of density, temperature, speed and field strength, all already in the channel set, so testing |B| against β tests |B| against a formula containing |B|. The geomagnetic indices failed the same way, as a downstream response to the solar wind. Requiring that a candidate be dimensionless and independently measured took sixteen survivors to four, all textbook heliophysics.

Extending the set to thirty (§3.2) showed that two clauses are still not enough: seven of the nineteen survivors are pairs each dimensionless and independently measured yet not independent of each other — the four Ca II K indices come from a single spectrum, the abundance ratios Fe/O, C/O and He/O share a denominator, CME rate and speed come from one catalog. A pair sharing an instrument, a spectrum or a denominator will produce a survivor whether or not anything is modulating it.

The regularizer therefore has three clauses, not two. An admissible carrier is a pair of quantities that are (i) dimensionless, (ii) independently measured, and (iii) not derived from a shared instrument, spectrum or normalizing quantity. Clause (iii) was added after seeing the thirty-channel result and is a post-hoc correction — which is why §4.8 reports the uncorrected survivor list in full, and why a pre-sweep correlation audit at a declared threshold of |r| > 0.90 (the value used in §4.8) is now part of the procedure, to be applied before any future sweep rather than offered as an explanation after it.

3.2 The channel set

Thirteen channels were used in the first sweep: sunspot number, F10.7, cosmic-ray flux, solar wind speed, density and proton temperature, interplanetary field magnitude, the alpha-to-proton ratio, total solar irradiance, Mg II core-to-wing, the GOES X-ray background, the >10 MeV integral proton flux, and Lyman-α; the geomagnetic indices are excluded by §3.1. An earlier version used ten and was written up as covering the pair space — wrongly: it covered the pairs of ten channels chosen by the author, the same opportunism the sweep exists to escape, moved up one level.

Seventeen further channels were then acquired to close the pair space, all public:

The charge-state and abundance ratios are the strongest additions on the criteria of §3.1: dimensionless by construction; the hemispheric asymmetry likewise, over 142 years.

That gives thirty observables, the denominator of every coverage fraction in this paper; §5.4 records that a different but defensible list moves it by a factor of order unity.

3.3 Forms, and the size of the space

Each pair is tested under four dimensionless functional forms — ratio, log-ratio, and the corresponding normalized residuals — over the bands the records support. A pair is dropped only when its two channels share fewer than 2,000 days.

The size of this table is the recognition gate of §2.4 made explicit: nothing in it is new data, and closing it is a matter of recombination capacity. The pair space is now essentially closed at thirty observables. What remains unexamined is an archival fact, not a choice: 53 pairs and 1,126 triples have no 2,000-day window in common. Coverage is now limited by record overlap rather than by effort, a different and more tractable problem.

The sweep is stratified by physical category, and that stratification is what makes its null informative: 510 of the 1,074 pair tests cross the Sun–heliosphere boundary, exactly where a beacon coupling something at the Sun to something measured at 1 AU would appear.

3.4 Nulls, and matching the surrogate to the statistic

Significance is assessed against surrogates rather than a parametric model, since solar series are red, non-Gaussian and gapped — and the choice of surrogate is not a matter of taste. Two controls in this work were no-ops: they could not have failed, and were caught only after returning reassuring answers.

The rule that falls out: match the null to what the statistic is made of. Nonlinear structure needs IAAFT; level structure needs phase randomization. Anything determined by the power spectrum alone cannot be tested this way at all — lag-k autocorrelation is the Fourier transform of the power spectrum, so no spectrum-preserving surrogate can serve as its null, while permutation destroys all autocorrelation and is beaten trivially by any red series.

3.5 What counts as a detection

A candidate had to clear four hurdles, fixed before the data was examined:

Two caveats belong with the sweep itself. Several pairs sit at the p-floor set by the shift count, 1/6001 = 1.67×10⁻⁴, so a reported p = 0.00017 means only as significant as six thousand shifts can show; that floor is adequate here and not for a larger enumeration — §5.2 gives the scaling and a run of ours that violated it. And the exact survivor count moved between four and five across two runs, because the shift generator was seeded from a string hash Python randomizes per process. The category structure is stable across runs; the exact survivor count is not, and no claim in this paper rests on it.

3.6 Where a designer would put it, and where we have looked

The sender’s side gives a sharper question — given a choice of channel, which would a designer pick? — answerable before any data is examined, and set out ahead of the results so that the ordering it produces is a prediction rather than a description of what we happened to find.

The detection threshold for a periodic fractional modulation of depth d, per-sample fractional noise σ₁, coherent integration over N samples, family-wise false alarm α and empirical noise scale μ is the expression behind §4.2:

d_min ≈ 2 σ₁ √( 2 μ ln(N/α) / N )

For σ₁ ≈ 10⁻³ per minute and N ≈ 2.6×10⁶ (five years) this gives _d__(min) ≈ 5×10⁻⁶, consistent with the injected Lyman-α limit. Sender cost is linear in d (§2.7): 1 ppm needs about 1.5×10¹² m² of interceptor, 1 ppb about 1.5×10⁹ m². The designer therefore prefers the shallowest depth any receiver channel resolves — and resolving power is highest in timing and frequency, lowest in composition and particle flux, with irradiance between.

Why the margin column is nearly all ones. A proof-of-work gate is set just above what a first-capable receiver can resolve (§2.3), so the level “required” in each channel is defined by our own capability. A margin near unity is what the gating argument predicts, not evidence for it; the informative column is the last one — whether the channel has been searched at that level at all. Rows where the margin is near unity and no search has been run are where the framework’s own logic says a first-contact pointer would sit.

Order-of-magnitude throughout, except our own reach in rows 1, 8 and 9, which is the injection-verified figure from §4.2. The margin column is the level a designer would set divided by our reach, so a margin above 1 means the channel is resolved finely enough to carry the signal and a margin below 1 means it is not; where either column is a range, the margin spans the corresponding ends. Solar neutrino flux appears nowhere as a carrier: nothing at the atmosphere can modulate a flux produced in the core, which is precisely what makes it the one control channel no mechanism in this paper can touch. It has since been measured, and is quiet: see the addendum.

The margin column is an ESTIMATE, and every estimate in it that has since been measured came in worse. Rows 5, 6 and 7 were searched after this table was drawn (§4.15–§4.17), and in all three cases the measured reach fell short of the estimated one:

row	margin as estimated here	margin as measured
5 — occulter transit timing	~1	unreachable; the six-hour noise floor is 31–108 ppm against a 76 ppm Venus transit (§4.17)
6 — broadband irradiance	~1	0.5 (§4.15)
7 — p-mode frequency	~0.1–1	0.069 (§4.16)

Three of three, all optimistic, and row 5 by enough that the row is not searchable at the designer level at all. The estimates were made by reasoning from instrument specifications rather than by injection, and reasoning of that kind has been systematically generous: a specification describes what an instrument resolves, not what survives a detrending, a continuum estimate and a trials correction. The unmeasured entries in this column should therefore be read as upper bounds on capability rather than as capability, and the rows still marked ‘no’ in the last column may be further from reach than they appear. What the table continues to establish is the ORDERING — which channels a designer would prefer — and that ordering is unaffected.

What the ordering says about this paper. The program has been most thorough exactly where the designer’s probability is highest (row 1) and where it is lowest (rows 8–9), and thinnest in between — not a designed allocation, but where the archives happened to be easiest. Rows 1, 2, 8 and 9 are capable and searched: row 1 is the strongest single statement this paper can make — null to 1.4 ppm at the level a designer would plausibly choose — and row 2 was run after this table was first drawn (§4.14). Rows 3, 4 and 5 are capable and not searched, and rows 6 and 7 were searched after this table was drawn (§4.15, §4.16) and are null, row 7 at a measured margin of 0.05 rather than the 0.1–1 estimated here: the archives exist, the resolving power is there, and no search has been run. If the pointer is meant for a first-capable receiver, the framework’s own logic puts it there at least as plausibly as in row 1, and the honest reading is that this paper has not yet tested the framework’s best guess. Rows 10–15 are below threshold: if the designer aimed at a more advanced receiver, the message is there and cannot yet be seen — a gap that closes on a decade timescale (Rubin/LSST and small-body missions, rows 12–13; Hyper-Kamiokande, JUNO, DUNE, KM3NeT, IceCube-Gen2, rows 14–15).

The program this implies is to run the remaining rows in the order 5, 6, 3, 2, 7, then 4 — cheapest and best cost-per-bit first — and that order is not the one the searches so far have followed. Rows 5 and 6 are the same archive twice: SOHO/VIRGO SPM and TIM at minute cadence, first as a transit-timing-sequence search after Arnold (2005), then coherently at 10⁻⁷. Row 4 belongs to the archaeological strand of the field (Davies & Wagner 2013; Benford 2019).

3.7 Channels above our threshold, and why a designer would not put the pointer there

Rows 14–15 of the §3.6 table sit far below unity, and there is a class of physics behind them that deserves its own statement, because a reader will ask why the search does not look there and because the answer is one of the framework’s more useful predictions.

What we cannot yet modulate or detect. For the electromagnetic channels of §3.2 our position is well inside the capable regime: we detect, we resolve, and we could in principle modulate. For two channels of known physics we are at or below the floor of capability, and for a third there is no floor to be below.

Neutrinos are excluded from the pointer by cost, gravitational waves by physics, and unknown physics by the recognition argument of §2.4: a puzzle posed in a channel the receiver cannot conceive of is not a gate but a wall, and a wall filters for nothing.

The framework’s prediction, and the design reason behind it. A designer could set the pointer in one of these channels, and would thereby select for a receiver centuries past the point of being able to reply. §2.3 says why a designer would not. The gate is bounded from above: it must be passable by the receiver it is meant for, or it is operationally no gate at all. The receiver the architecture is meant for is one that can build a transmitter and search a combination space — capable, but not necessarily mature — and a pointer that only a megaton neutrino observatory or a space-based gravitational-wave interferometer could read would choke off contact with exactly the receivers the safeguard exists to reach: species obviously capable of the exchange, whose remaining immaturity is the thing a staged disclosure is designed to accommodate. The lower bound on difficulty filters out receivers that cannot act; the upper bound is there so that the filter does not also remove receivers that can. The framework therefore predicts that the pointer is photonic and combinatorial, at the level of §3.6 rows 1–7, and that channels above the receiver’s threshold appear, if at all, as content behind later gates at step 12 — where a receiver that has already demonstrated a transmitter can be told to build a detector, and where the sender’s neutrino accelerator is paid for only after somebody has answered. This is the same asymmetry as everywhere else in the chain: the expensive channel is reserved for the stage at which the receiver has already paid to be found.

Two consequences for this paper. First, the absence of neutrino and gravitational-wave searches in §4 is not a gap in coverage of the pointer; the framework says the pointer is not there, and a designer who put it there would have defeated their own purpose. Second, the §5.5 recommendation of a monoenergetic-line search in existing neutrino catalogs stands, but as a test for a later-stage channel or an unrelated beacon, not for the framework’s pointer — a result there would be interesting on its own terms and would not bear on the gating argument.

4. Results

4.1 Summary

Thirty searches were run, every one against a public archive collected for unrelated purposes. Three sections at the end of these results are not searches and do not enter the tally: §4.10 puts the one result that came closest to a detection through the procedure of §3.5, §4.11 tests whether the fast-band searches lose signal to a geometric term the coherent model omits, and §4.13 applies the method to a dataset the architecture cannot reach. All are null.

Scope. Every search examines the Sun, the heliosphere, or the solar system out to 160 AU — forced by the architecture, since step 4 of §2.5 places the modulator at the local star. A search of distant sources tests a different hypothesis. A further 23 searches of pulsar, pulsar-timing-array, X-ray binary, eclipsing binary and astrometric archives were run during this program and are reported separately. All were null or void; none is excluded here because of its outcome, and their exclusion is stated so that the coverage claimed in §4.3 is not read as covering more than it does.

Why each search ended where it did, which is not the same question. “Null” is one word covering four situations, and the difference decides where effort should go next.

Only the first line is astronomy. Three are statements about our own machinery and two about the archives we were handed. Thirty searches and twenty-six nulls reads as broad coverage; the honest reading is twenty-two channels searched competently, three searches too weak to count, two defeated by their own machinery, and one defeated by the data.

4.2 Limits

All limits are fractional amplitudes of a modulation in the named observable, against the stated null. Each is the amplitude recovered 95% of the time under injection, against the threshold the search itself uses (μ = median(R)/ln 2, α = 0.05).

Eleven injection-verified limits in this table; the twelfth is the cross-viewpoint bound of §4.12, which is in different units and is stated there. Three further searches bound a modulation through a measured false-alarm and power pair rather than a single amplitude — cross-channel coherence (FP 3.5%, 100% power at 0.40σ), self-keyed spread spectrum (FP 6.3%, 99.7% at 0.10σ) and the Voyager radial coincidence (FP 3.0%, 100% at 50% injection); they are in Appendix A because they do not reduce to one number. The sidereal fold is verified against a gated statistic, and the distinction is the point. The fold statistic itself cannot be injection-verified: on the Oulu series it sits at 9.3 σ with nothing injected, and the anti-sidereal control sits higher at 11.4 σ — the seasonal leakage of §4.4, seen from the other side. A random-phase injection into a detector already firing cancels as often as it adds. Because a real sidereal signal raises sidereal alone while leakage raises both lines together, the contrast D = amp(sidereal) − amp(anti-sidereal) is blind to the leakage; on unmodified data D sits at −1.2 σ and does not fire, which is the precondition for measuring anything. Injected against D with a zero-amplitude arm first, recovery is 0% at zero, 0% at 1.8×10⁻⁴, 46.5% at 3×10⁻⁴ and 100% by 10⁻³, monotonic throughout, giving 8.4×10⁻⁴ at 95% recovery. The 1.8×10⁻⁴ previously quoted analytically is therefore optimistic by 4.7× — inside the 1.2–4.8× range §5.6 already documents for analytic figures, which corroborates that finding rather than contradicting it. The ungated fold statistic remains unverifiable, and is now demonstrably so rather than by assertion.

An earlier version of this table reported analytic threshold-crossing amplitudes as though they were injection-established confidence limits. Neither was true — running the injections showed those figures optimistic by factors of 1.2 to 4.8, and the analytic column was removed rather than corrected, because it could not be regenerated from the pipeline (§5.6). Nothing downstream used it: the energetics of §2.7 and the cadence ratio of §4.3 derive from the injected values. Three rows needed no correction — the achromatic search, the multi-scale construction and the self-keyed test — and they are exactly the three whose injection curves were written into them from the start; every figure that came instead from an analytic threshold was optimistic, by 1.2 to 4.8 times. The correlation is perfect and the sample is eleven. That is the argument for injection stated as compactly as this paper can state it.

Re-running the self-keyed search produced an alignment-specific excess. It is treated at length in §4.10, because the first thing we did with it was wrong.

On the shape of the null. Continuum-normalized power here has mean 1.44 rather than 1.0 and exceeds an Exp(1) threshold some 4×10³ times more often than that distribution predicts. That would matter if the searches had assumed Exp(1); they do not — each estimates the scale empirically as μ = median(R)/ln 2, which returns 1.44 on this data, so the threshold actually used sits 44% above the nominal one and the clean candidate lists of §A.1 are consistent with the heavy tail.

4.3 Sensitivity is not the binding constraint

The limits above span five orders of magnitude: solar variability is red, so the noise a signal must exceed falls steeply with frequency, and every step toward shorter periods paid.

A search limited by sensitivity is improved by better instruments or longer baselines. This one is not: the limiting quantity is the fraction of the combination space examined, which no instrument improves — only measuring more quantities, and combining them more ways, does. The pair space has been closed from 18% to 88% (§3.2, §4.8); the residual 12% cannot close by waiting (§5.7); the triple space was attempted and returned void (§4.9).

4.4 Controls

Each search was gated on a control that could fail, and four did useful work; where possible the control was supplied by physics, since a physical control cannot be tuned.

The twin and anti-sidereal controls each removed an apparent detection that had passed every amplitude test — the sharpest case a 4.9σ sidereal line beaten by its own control at a frequency where nothing physical can live.

4.5 A void result, reported

One search is reported as void rather than null, because its detector failed validation: a null from an uncalibrated detector is a statement about the analysis and not about the sky, and recording it as a null would silently inflate the coverage this paper claims.

Aperiodic structure in solar output (LASP TSI composite, 46 yr, 16,801 daily samples). The detector, built to find non-periodic but non-random structure, failed its own positive control twice over: the statistic was computed on a channel dominated by the activity cycle, and the surrogate preserved the very quantity the statistic measured. Changing the null altered the false-positive rate and left injection recovery untouched, localizing the fault to the statistic. No limit is claimed and none should be read into it.

A second void arose in the triple sweep (§4.9); two more arose in parts of the program not reported here (§4.1). The diagnostic was the same in all of them.

4.6 The one detection, and its relation to prior work

Solar p-mode oscillations were recovered in GOES EXIS Mg II irradiance at one-minute cadence. The detection statistic was not oscillation power but the spacing of the mode comb: the autocorrelation of the normalized power spectrum across 2.5–4.0 mHz peaks at 135.1 µHz on GOES-16 and 135.0 µHz on GOES-17, against an accepted large frequency separation of 134.9 µHz — fixed by √(M/R³) and therefore predicted rather than fitted.

Solar oscillations in this instrument were reported first by Eden et al. (2024), from the 3-second product — an order of magnitude finer in cadence and precision than the one-minute product used here. We claim no priority for the detection. What differs is method, not discovery: Eden et al. measure oscillation power at characteristic periods, whereas the statistic here is the comb spacing; their Letter reports no frequency separation, autocorrelation, or mode comb. The modest addition is that EUVS-C resolves the radial orders well enough for Δν to be recovered from the coarser product. Δν itself is among the best-determined quantities in solar physics and is not measured here to any useful precision; its value in this work is that it was known in advance.

4.7 Why a re-detection is the most useful result in this paper

A program returning nulls faces an ambiguity it cannot resolve from within: nothing present, or a detector that does not work. Injection addresses part of this, but tests a detector against a signal of assumed shape chosen by the detector’s own author. The p-mode recovery closes the gap from outside: the same pipeline — same detrending, same continuum normalization, same thresholds — applied to the same archives, returns a predicted astrophysical quantity at 7σ.

Independent prior publication therefore strengthens the control: the nulls in §4.2 are limits from an apparatus demonstrated to detect a real signal of comparable subtlety in the same data.

4.8 The thirty-channel pair sweep

All 435 pairs were attempted; 382 had the required 2,000-day overlap, giving 1,074 tests against 250,000 circular shifts each, a Bonferroni threshold of 4.66×10⁻⁵ and an attainable p-floor of 4.0×10⁻⁶ — 11.6× below the threshold, so every test in the sweep could fire. Every p below is a counted exceedance; the generalized Pareto extrapolation of §5.2 is not used anywhere in this paper.

131 tests fell below p = 0.05 against 53.7 expected by chance — the signature of genuine physical coupling in the data rather than of a miscalibrated null.

The first row of interest is the empty one. Five hundred and ten tests span the Sun and 1 AU — the largest block in the sweep, and the one place a signal coupling a solar observable to a heliospheric one must appear. It was empty at thirteen channels, empty at thirty, and still empty when the surrogate budget is deepened to 250,000 shifts. Of the nineteen survivors, none is unexplained:

A correlation audit run before the sweep predicted part of this. Eight pairs exceeded |r| = 0.90 on their own overlap — among them Ca II K emission index against K3 at r = 0.975 and O⁷⁺/O⁶⁺ against ⟨q⟩_(O) at 0.931 — and those pairs duly produced survivors. Sunspot area against sunspot number did not flag, the audit doing useful work in the other direction: area carries information the count does not. Two more pairs exceed |r| = 0.90 once the sunspot channel is corrected (§5.6) — sunspot number against F10.7 at r = 0.945 and against Mg II at 0.916 — and neither survives (p ≥ 0.63); sunspot area against sunspot number still does not flag (r = 0.85). Sunspot number against the X-ray background, the activity-index pair SWPC publishes, survives the thirteen-channel sweep but not this one: p = 8.0×10⁻⁵ against the 4.66×10⁻⁵ threshold.

Null. The sweep recovers the strongest genuine couplings in the data unprompted, confines them to the physically related and instrumentally coupled subsets, and returns nothing across the Sun–heliosphere boundary in 510 tests.

Both sweeps were re-run with deterministic seeds, and the result is reproducible for the first time. The surrogate seed was previously derived from Python’s hash(), which is salted per process: three consecutive runs returned 1689220225, 56146563 and 1311111642 for the same key, so no p-value here could be regenerated by anyone, including us. That is not a correctness fault — an arbitrary seed is still a valid seed — but for a paper whose standard is that a null is worth the fraction of a space it excludes, an unreproducible null is worth less than it looks. The seeds are now BLAKE2b over a canonical key, fixed across processes, versions and platforms.

Re-running draws different surrogates from the same null, so the two sets should differ in detail and agree in distribution. They do: a two-sample Kolmogorov–Smirnov test gives D = 0.008, p = 1.00 on the 1,074 pairs and D = 0.003, p = 1.00 on the 10,996 triples. The claim this section rests on is unchanged — zero survivors cross the Sun–heliosphere boundary in either run.

One instability is worth reporting rather than absorbing. Survivor counts at the family-wise threshold moved from 10 to 13 among the pairs and 198 to 206 among the triples, and 182 of 10,996 triples — 1.7% — cross the threshold in one run and not the other. Every one of those sits below the counting floor of 1/(M+1) = 10⁻⁴, where the p-value came from the generalized-Pareto tail fit rather than from counted exceedances. The extrapolation is not stable across surrogate realizations at the level of an individual test, and this is one of the three findings that led us to abandon it (§5.2). It does not affect the distributions, the boundary result, or the void verdict below. The pair sweep has since been re-run at 250,000 shifts, where counting resolves every test and no extrapolated p-value is used at all.

4.9 The triple sweep, and why it is void

Of 4,060 triples of thirty observables, 2,934 have the required 2,000-day overlap, giving 11,736 tests under two forms that do not factor into pair statistics: the third-order residual product d(a)d_(b)d_(c), and the log-space curvature log r(a) − 2 log r(b)_ + log r(c)_, each channel taking a turn as the middle term. 10,996 tests completed.

The screen-then-confirm design worked as intended: 105 of 198 screened survivors failed direct counting, the tail fit behaving exactly as §5.2 warns. But the 93 that survived (92 after the correction of §5.6, which removes the one triple involving the sunspot channel; the verdict is unchanged) are not reported as detections, because a third control invalidates both forms.

Every survivor was the same form. All 198 were log-curvature; not one was a residual product. A perfect segregation by form is a property of the statistic, not plausibly of the sky, so both forms were run against synthetic triples with the red spectra and shared activity driver of the real channels and containing no signal whatever:

Both forms fail, in opposite directions. The log-curvature form is roughly calibrated in the bulk and has a ninefold excess in the far tail — precisely the regime that sets Bonferroni survivors. The residual product sits 4.5σ below its own null by construction and can essentially never produce a positive. The clean split by form does not show one statistic firing on signal; it shows the other unable to fire at all.

The mechanism is visible in the algebra. log a − 2 log b + log c cancels common-mode variation when the three series are aligned and not when two are circularly shifted, so for channels driven by a common solar cycle the statistic separates from its null by construction. Consistent with that, 61% of the confirmed triples are entirely heliospheric, and the Ca II K emission index, K2V/K3 and K3 — three indices from one spectrum — confirm together at the floor.

Void, and what would fix it. The circular shift is the wrong null for a three-body statistic: it destroys pairwise structure along with three-way structure. The right surrogate preserves every pairwise cross-correlation and destroys only the three-way alignment — the third-order analog of IAAFT — and building it is the prerequisite for any triple enumeration. Two honest limits on the control itself: its 320 synthetic triples measure the false-positive rate at 10⁻³, not at the 4.5×10⁻⁶ Bonferroni threshold, and the synthetic channels approximate the real correlation structure rather than reproducing it. Neither weakens the verdict: a ninefold tail excess and a form that cannot fire are sufficient grounds to void a result, though not to confirm one.

The surrogate was built, and it repairs one form and not the other. The construction is multivariate IAAFT: one common random phase applied to all three channels at once, which leaves every auto- and cross-spectrum exactly intact — the relative phases between channels are untouched — while the bispectral phase picks up φ(f₁) + φ(f₂) − φ(f₁+f₂) and is randomized. Second-order structure survives; three-way structure does not. Rank-mapping restores each channel’s marginal and iterating restores the spectrum that rank-mapping perturbs. It was run against the same 320 signal-free synthetic triples, both nulls computed on the same triples in one process so the comparison is exact rather than remembered:

The circular-shift rows reproduce the table above — 6.6% and 0.9%, median z −4.45 against −4.49 — so the harness is validated by its own positive control before the surrogate columns are read.

The residual product was never the broken part. Its median z moves from −4.45 to −0.17: from four and a half standard deviations below its own null to centered on it, firing at 2.8% where 5% is nominal. A form that “cannot fire at all” turns out to have been a statement about the null and not about the statistic, exactly as the diagnosis above predicted. It remains mildly conservative, which is the safe direction — a limit drawn from it is understated, not overstated.

The log-curvature form is not repaired; it fails in the opposite direction. The ninefold tail excess is gone, but it now fires 0.0% of the time at p < 0.05 with median z −1.40, the observed statistic sitting systematically below the surrogate null. That is the residual product’s original failure, relocated. A form that cannot fire is not a form that found nothing, and no limit is claimed from it under either null.

The practical consequence is that three quarters of a full triple enumeration — the sweep builds one residual-product test and three log-curvature tests per triple — would be spent on a statistic that cannot produce a detection. Enumerating the residual product alone is 2,934 tests rather than 11,736, and the looser family-wise threshold that follows reduces the surrogate budget with it.

The floor was then measured, and it is not purchasable. Having a null that passes its zero arm is not the same as having sensitivity, so the detection floor itself was scanned: signals injected three-way-only into signal-free synthetic triples, across record lengths of 2,000 to 9,000 samples and signal timescales of 3 to 30 days, with the 50% recovery amplitude bracketed by the injected grid in all nine cells.

n	3 d	10 d	30 d
2,000	0.117	0.132	0.149
4,500	0.107	0.055	0.112
9,000	0.088	0.088	0.100

Amplitude at 50% recovery. The floor sits at 5–15% fractional modulation throughout.

It improves as n^−0.25, not n^−0.5. That is the fourth root — the signature of incoherent power combination, which is what a third-order statistic is left with once the phase information is gone. And it does not depend on the signal's timescale at all: the slope against block length is +0.11, +0.01 and +0.05 across a tenfold range, indistinguishable from zero.

The consequence is arithmetic. Improving the floor by a factor of ten requires eleven thousand times the data. The median three-way overlap in these archives is 4,650 days; ten times better would need 5.1×10⁷ days, about 140,000 years of continuous observation. The pair limits of §4.8 run from 10⁻⁵ to 10⁻⁶. Three-way-only structure is three to four orders beyond reach and no observing programme moves it.

This is a statement about detectability and not about the sky. It says what any search of this kind can achieve on archives of this length, which is the quantity §5.1 argues should be reported, and it retires the question of whether a better surrogate or a better statistic would open the triple space: neither buys what the scaling law forbids.

Void. No limit on three-way dimensionless structure is claimed, and the 93 survivors of direct counting are reported as artifacts of an uncalibrated statistic. The triple space remains, in the sense that matters, unexamined — but it is no longer unexaminable. The residual product now has a null that passes its own zero arm, and the enumeration is affordable; the log-curvature form needs a null that has yet to be found.

4.10 A candidate, and what the detection procedure did to it

Re-running the self-keyed search (§A.2, row 3) returned a matched-filter correlation against the true F10.7 key of ρ = −3.52, against a shifted-key null whose 95th percentile is 2. The quantity ρ is not a correlation coefficient. It is the matched-filter output normalized to unit variance under the null — ρ = Σ(aᵢ/σ_a)cᵢ/√n, with a the mean-subtracted field residual and c the ±1 chip sequence — so it is a z-score, in units of its own null standard deviation, and equals the Pearson correlation times √n. The observed −3.52 therefore corresponds to a Pearson r of −0.065 over 2,919 samples. A nominal Gaussian would read −3.52 as p ≈ 4×10⁻⁴; the shifted-key null gives 7×10⁻³, because that null is not Gaussian, and it is the empirical null that is used throughout.14: p = 0.007 — an alignment-specific excess. The first version of this section set it aside on the grounds that the correlation is negative, and a beacon keyed to solar activity would not anticorrelate with it.

That reasoning was invalid, and it is the exact error §5.3 warns against — using the framework to decide what the data mean. The sign is not a property the beacon hypothesis constrains: a modulation imposed by redistributing flux can present with either sign, and dismissing a result for an unattractive sign is procedurally indistinguishable from accepting one for an attractive sign. The detection procedure of §3.5 is applied instead.

1. Is the null matched? F10.7 and the interplanetary field both carry solar rotation, and the published null’s arbitrary lags destroy rotational alignment and any beacon alignment together; a second null shifted by integer Carrington rotations preserves rotational phase and destroys only arbitrary alignment.

2. Does it replicate? The same reduction, key and statistic were run against Wind/MFI over the same span; 2,897 of 2,898 days reduced. This is an instrumental control, not a physical one: both spacecraft sit near L1 and sample the same plasma, so agreement rules out an ACE artifact, not a real coupling.

3. Both halves of the record?

The excess survives the rotation-matched null and replicates on an independent magnetometer, with the same sign and comparable magnitude. It does not hold in the second half of either record.

4. Is the second-half absence real, or has the test lost power there? This decides the result, and it has to be measured. The chip sequence is the sign of the day-to-day change in F10.7, which collapses from a median of 3.61 sfu in the first half to 0.83 sfu in the second as the cycle declines from the 2014 maximum — so the code might simply have degenerated toward a coin flip. Running the injection curve separately in each half settles it:

The second half is not the insensitive one — it is the more sensitive one: 100% and 94% power at 0.10σ, against 84% and 90% in the first half, because the field residual is quieter near solar minimum. Since power is quoted in units of the local standard deviation, a beacon of fixed absolute amplitude would be easier to see in the second half, not harder. The absence is real.

Verdict: a real anticorrelation, and not a beacon. It survives a rotation-matched null, so it is not solar rotation; it replicates on an independent magnetometer, so it is not an ACE artifact; but it is confined to 2012–2015 in both instruments, and the window in which it is absent has more power to detect it. A set-and-forget beacon does not switch off, and the interval carrying the excess is the maximum of cycle 24 — the temporal signature of an activity-driven physical coupling, which is the physics step of §3.5 doing its work. And it would not clear the trials correction in any case. The self-keyed family tried one configuration, so there is no within-search look-elsewhere factor; but this is one result among 30 searches, giving a per-search Bonferroni threshold of 0.05/30 = 1.7×10⁻³ (the denominator is 29, not 1,099, because the pair sweep carries its own within-search correction over its 1,074 tests). The observed p of 0.006 to 0.012 does not reach it. We report it as a candidate that failed replication in time and does not clear the paper’s own trials threshold — not as a result dismissed for having the wrong sign.

Two things are worth keeping. The rotation-matched null is a control of exactly the kind §4.4 argues for — supplied by physics, not tunable. And we would not have run any of these tests had the sign argument stood: the sign was doing the work that four measurements should have done, and it happened to reach a similar destination — the most dangerous way for a shortcut to fail. The anticorrelation itself is a modest heliophysical result we make no claim to have explained.

4.11 The one-year geometric term

Every fast-band search in §4.2 integrates coherently for 1,945 days against a stationary carrier, but a modulator fixed in inertial space and viewed from a moving Earth does not present one: the light-travel time and projected velocity acquire an annual term. Two cases: phase modulation, from the annual swing in light-travel time, settled analytically; and amplitude modulation, from the changing geometry to a fixed occulter, settled by injection.

Phase. The semi-amplitude of the Earth–Sun distance is e = 0.0167 AU = 8.3 light-seconds; a sinusoidal phase modulation of amplitude φ leaves J₀(φ) of the carrier amplitude:

The term is negligible over the whole band searched — 9.2% of carrier power at the shortest period in the paper and under 1.5% above 300 s, because 8.3 light-seconds is small compared with every candidate period.

Is there an annual term in the data anyway? Over T = 1,945 d the frequency resolution is 5.95×10⁻⁹ Hz and the annual frequency 3.169×10⁻⁸ Hz, so the sidebands sit 5.33 bins from the carrier — resolvable, but split across bins and spread by the Blackman mainlobe — and the statistic must sum a block, _R__(geo)(i) = R(i) + R(i±5) + R(i±6). A first threshold for that sum assumed Gamma(5) and over-fired grossly (§5.6). The matched null needs no distributional assumption: a sideband at any other offset has identical structure and no geometric meaning, so the exceedance count at the true offset is compared with eight decoy offsets. This is a control that can fail: had an annual term been present, the true offset would have stood above the decoys.

Null in all eight channels — the true-offset count sits at or below the decoy median almost everywhere. The three EUVS-A channels carry the known per-sensor comb (§4.4); the decoy comparison is insensitive to it, which is the point of using a count ratio rather than an absolute count.

And the method arm. Injecting a carrier with a full-depth annual envelope, the geometric statistic gains 15–25% over the coherent one and changes no detection: both find the signal at a = 1×10⁻⁵ and both miss at zero. Amplitude modulation preserves the carrier — an envelope of 1 + cos leaves half the power where the coherent search is already looking — and the available phase modulation is 0.44 rad at worst.

Conclusion: the coherent integration of §4.2 is sound, and its limits stand as quoted. The analytic term is 9% at the extreme and under 2% across most of the band; what the exercise buys is that the fast-band limits no longer rest on an unstated assumption of a stationary carrier.

4.12 A different line of sight

Every search above reads the Sun from one place, and §2.7 proposes a modulator that redistributes flux along the receiver’s line of sight — line-of-sight-specific by construction. An observer elsewhere sees either nothing or something different, whereas a modulation intrinsic to the Sun is seen by everybody, offset only by geometry. No search so far could tell those apart. This one can.

Earth line against Mars line. GOES-16 EUVS Lyman-α against MAVEN/EUVM diode C, also Lyman-α, over 2019-12-10 to 2025-04-06 — 1,946 days, MAVEN returning data on 94.3% of them. Both series are normalized to 1 AU and shifted to photon-emission time.

On product choice, because it decides the answer. MAVEN’s L3B product is FISM-M model output, partly driven by Earth-based inputs; using it would contaminate the non-Earth line with the very line it is compared against and return agreement at every viewpoint — a result manufactured by the choice of file. L2B is measurement, and is what is used. Its cost is cadence: orbit-averaged at 3.66 h, so this test reaches periods above about 7 hours.

The control, which had to come first. Solar rotation must be seen by both platforms at a lag fixed by geometry, τ = (λ_(M) − λ_(E))/360 × 27.2753 d, and over this span the Earth–Mars angle sweeps the full 360°.

The geometry is recovered — the license for everything that follows: a viewpoint null means nothing unless the viewpoint machinery can be shown to find something.

The search. Nine peaks above threshold in the Earth line, seven in the Mars line, six in both. The three Earth-only peaks are reported as unresolved, not detections: MAVEN had only 66%, 20% and 93% power to see a signal of the amplitude observed at Earth, short of the 95% bar — “absent at the other viewpoint” is worthless unless the other viewpoint could have seen it.

The limit holds over 2 to about 10 days and nowhere else, and the two failure modes above that are kept apart. At 13.5 and 27 days the circular-roll surrogate preserves the real rotational peak, so the detector fires on unmodified data — the same failure as the sidereal fold in §4.2. Read without the false-alarm column beside it, the 27-day row would have appeared as the strongest limit in this paper.

The result is a null, and it is the first in this paper that constrains the mechanism §2.7 actually proposes rather than modulation in general. (An earlier version reported ten Earth-only candidates; all were one uncorrected geometric factor — §5.6.)

The fast band. The slow-band test is limited by cadence, not principle, so it was run again on the MAVEN L2 band product at full diode cadence: 1,879 days reduced to one-minute means on the matched 1,945-day grid — coverage 72.6% at Earth, 75.9% at Mars, 54.7% in common (the 66 missing days are the MAVEN safe-mode outage of 2022).

The band has to be capped at 6 hours, and the reason is not conservatism. Solar rotation has a different synodic period from each viewpoint — 27.275 d from Earth, 26.354 d from Mars — so rotation-band structure is present in one line and absent in the other by construction. Run without the cap, this search duly returned a 26.6-day “Earth-only” candidate at 99.2% Mars power — exactly the signature it exists to find, and entirely a geometric artifact. Periods near and above a day belong to the slow-band test, which carries the rotational lag.

In the judged band, 125 s to 6 h, the Earth line has no peak above threshold at all. What the search bounds is the amplitude at which a viewpoint-specific modulation would have been seen, measured by injection through the identical pipeline:

The threshold was challenged and it holds. A band-wide control was added on review — the count of exceedances over all 1,336,602 judged bins per surrogate, whose expectation is α = 0.05 if the threshold is calibrated — because the previous check tested a single fixed bin against a threshold set for 1.3 million and returned ~0% whether the pipeline was sound or not. The first run of the new control returned 197 exceedances per surrogate, which would have meant the limits here were not family-wise 0.05 and would have loosened them roughly fourfold.

That number was an artifact of the control, not a property of the search. The surrogate rolled the series and then re-applied the gap mask, which on a gapped record moves the existing zero-blocks to new positions and adds zeros back at the original gaps: at MAVEN’s coverage it carried half again the data’s zero fraction, and zero-blocks generate large spectral structure. Rolling only the observed samples, so the gap pattern stays exactly where it is, gives:

line	analytic	empirical	ratio
Earth	24.7	23.8	1.00×
Mars	24.8	25.7	1.02×

The analytic threshold μ·ln(N/α) is calibrated on this data to within 2%, and the limits below stand as printed. §4.2’s argument — that estimating the scale empirically as μ = median(R)/ln 2 accommodates the heavy tail — survives a direct test it had not previously been given. The episode is recorded in §5.6 rather than removed: a control that had to be rebuilt twice before it measured the thing it was auditing is worth more as a record than as a silence.

Fractional amplitude at 95% recovery. The Earth-line figure at 307 s is 2.2× the §4.2 limit at the same period, which is the expected cost of the daily fold and the common-grid interpolation and is the check that this pipeline is calibrated against the paper’s main one.

The binding number is Mars, and it is 580 times worse than Earth. A viewpoint claim requires both lines to be able to see the signal, so the limit is set by the weaker: 4.2×10⁻³ at 307 s, against 1.4×10⁻⁶ for the Earth line alone in §4.2. The limitation is the instrument, not the geometry: EUVM is an orbiter’s monitor, and its one-minute means carry a relative scatter of 0.84 in log space against 3.6×10⁻³ for GOES. One row is blind by construction and is listed to say so. The pipeline folds out the mean daily profile, which removes every harmonic of 1/86400 s; 3,600 s is the 24th. Injections there recover nothing at any amplitude — a property of the filter, not of the data — and the quotable rows use periods deliberately incommensurate with the day. (An earlier version injected after the fold, exempting the injected signal from a filter a real one would have met; §5.6 records it.)

The next line-of-sight test is an imager summed to an irradiance, not another orbiter monitor. Disc-integrated sums of STEREO/EUVI images (304, 171, 195, 284 Å) are a standard EUV irradiance proxy; STEREO-A has run since 2007, sweeping the full 360° from Earth, at a per-image cadence that reaches the fast band MAVEN L2 cannot; Solar Orbiter EUI/FSI adds a heliolatitude lever. Neither was run here — both need an image-to-irradiance reduction with its own degradation model, and the passband mismatch with GOES 304 is best handled with platform-internal ratios such as 304/171, dimensionless per platform. A dedicated disc-integrated photometer off the Earth line remains the instrument this measurement eventually wants; it is neither the only improvement available nor the cheapest.

4.13 A dataset outside the Sun: lunar laser ranging

Lunar laser ranging is the most precisely monitored geometric quantity in the solar system — five retroreflectors, four decades, millimeter normal points — exactly the archive the coverage principle of §5.5 points at. It is treated here, but it is not one of the 25 searches and does not enter the tally:

It cannot constrain the modulator of §2.7, and we say so before reporting the nulls. §2.7 puts ranging 2.2×10¹¹ short gravitationally and 2.0×10¹² short radiatively. A null here bounds any unmodeled periodic structure in the Earth–Moon system, but not the architecture this paper is about. It is a method demonstration and an honest negative.

The construction is the interesting part. A single-reflector range is contaminated by station coordinates, atmosphere, Earth orientation and the lunar ephemeris. Ranging to two reflectors from the same station on the same night and taking the ratio of the round-trip times cancels all of those to first order — and the ratio is dimensionless by construction, satisfying §3.1 with no shared unit. Five reflectors give ten pairs. No dynamical model is used anywhere — published LLR residuals are post-fit against a full ephemeris, so a signal shaped like any fitted parameter is removed before anyone sees it; nothing here is fitted, so nothing is absorbed. The price is that a local detrend is a high-pass filter, so the detrend window is the coverage claim.

Residual scatter after the per-session polynomial is 244 ps = 36.5 mm one-way, which is ordinary LLR single-shot precision and is the check that the detrend is doing its job rather than eating the signal.

Every near-survivor is a lunar harmonic, and finding that out required fixing the controls. The three lowest p-values land on 2.782 d, 3.357 d and 1.511 d. Against a control list of the lunar months alone, all three look clean; carried to its harmonics they are the anomalistic month over 10 (2.7555 d), the draconic over 8 (3.4015 d) and the half-anomalistic over 9 (1.5308 d). A 40-day high-pass suppresses the fundamentals and passes exactly these harmonics, so the harmonics are the relevant controls for this band. None clears the trials threshold in any case. (A parser error earlier in this search forced the withdrawal of a claim that full-rate LLR is background-dominated — §5.6; split correctly, the data were always clean.)

One thing is genuinely non-random. The laser fire epochs are strongly quantized: phase concentration 0.52 against a chance level of 0.0077, grid occupancy χ² = 1.1×10⁵ on 19 degrees of freedom, 60% of returns in one phase bin — a real periodic measurement artifact, and the expected one (timing hardware). We have not bracketed its fundamental: the comb scan rises monotonically to whatever ceiling it is given, 2 kHz and then 12 kHz, so the grid is finer than 83 µs; a greatest-common-divisor of the epoch differences would pin it, and we report the failure rather than quoting the ceiling as a result.

Why the minute-cadence version was not run. Stations cycle between reflectors within a night, so the same construction should run at minutes. But the open-mirror full-rate holdings are Matera, Wettzell and Beijing; APOLLO, which does cycle reflectors, publishes to a separate archive; and of the nights available, 6 carry more than one reflector and none yields 60 paired points. The code path would run unchanged on APOLLO data.

4.14 Radio spectral index at one second

§3.6 ranks this channel second and records it as unsearched. It is now searched. RSTN reports eight frequencies per site at one-second cadence, so the ratio of any two is dimensionless by construction — and a redistributor is exactly the thing that would move a spectral index while leaving the total alone. Three sites — Learmonth, Palehua, San Vito — 84 to 89 days each, all 28 pairs per site, band 2.2 to 300 s (the lower bound clear of the 2.000 s Nyquist, the upper set by a 601 s detrend).

The result is null for the carrier, and the way it got there is the interesting part. The pooled search returns 44, 64 and 43 peaks above threshold at the three sites, and the twin gate flags two periods near 5.9 s at more than one site — precisely the signature the search exists to find.

The power is in a channel, not in the ratio. A solar modulation of the spectral index moves the ratio; this does not. The ratio peaks the pooled search reported are inherited from the affected channel, and the same pattern holds at the other two sites, where nothing reaches threshold in any ratio at these periods.

And the twin gate is not valid here in any case. RSTN sites are standardised receivers — different sites, not independent designs — so a per-design artifact appears at all three exactly as a solar signal would: the failure that defeated the GOES-16/17 gate in §4.4, recurring. Agreement across RSTN sites cannot carry the weight the Oulu/Kiel agreement carries.

Two implementation errors produced a false null — zero peaks at three sites, twice — before this search produced a real one; §5.6 records both, and after repair an injected 0.05σ tone is recovered at R = 4.35 against a threshold of 2.80.

4.15 Row 6: broadband irradiance, coherently, on SOHO/VIRGO

§3.6 ranked this row sixth by designer preference and recorded it unsearched, on the stated grounds that the archive was not reachable. That was wrong, and the way it was wrong is worth recording. PMOD’s ftp.pmodwrc.ch publishes only an IPv6 AAAA record and does fail from our networks — retried here, three ways, still failing. But NASA mirrors the entire SOHO mission over plain HTTPS. The primary source was dead; the dataset never was. A dead primary source is not an unavailable dataset, and this one blocked three rows of the designer table for four revisions.

VIRGO’s three sun photometers (402 nm, 500 nm, 862 nm) give 14,342,400 samples each at 60 s from 1996-01-23 to 2023-04-30 — 27.3 years, 90.5% / 94.3% / 95.4% present. Because the product is already relative, in ppm, the difference of two channels in ppm is their log color ratio: the dimensionless carriers of §3.1 come free, with no constructed quantity and no shared denominator.

Gate 1 is unusually strong here and it failed twice before it passed. VIRGO SPM is an instrument solar p-modes are classically measured with, so the five-minute oscillation must come out of a blind search of this data or the search is not calibrated. The first run failed it on two channels — and returned eighteen “candidates”, every one at exactly 180.0000 s or 360.0000 s, which are 2⁄3 and 1⁄3 of the 60 s Nyquist. A real signal has no reason to land on a round number in our units. They are instrument lines reaching R ≈ 19,000, and they were strong enough to drag the envelope estimator off the p-mode peak. They are excluded a priori, as exact multiples of the sample interval, rather than after inspection of which bins came out large. The second failure was ours: an envelope filter of 2,001 bins is 2.3 µHz against an envelope roughly 1,000 µHz wide (§5.6).

Null. With the instrument lines excluded, no bin of 7,164,744 exceeds threshold in any of the three channels. The color ratios return two marginal bins each, and all six are identifiable without appeal to anything new: exactly 720.0000 minutes (the 12-hour spacecraft thermal cycle), 4,780,800 minutes (9.1 years, the solar cycle), and 178 and 212 days, which sit against the edge of the filter described below. No candidate appears in all three photometers, which is the gate a solar signal would pass and a detector artifact would not.

The limit, and where the search is blind. Injection into the prepared series puts 95% recovery at 0.20 ppm at 307 s, rising to 0.80 ppm at 1,009 s and 1.60 ppm at 10 h. §3.6 estimated this row’s reach at ~10⁻⁷ with margin ~1; the measured value is 2.0×10⁻⁷, margin 0.5 — the closest any row in this program comes to the level the framework’s own logic places a signal at, and null there.

A second instrument, and a correction. VIRGO also flies a PMO6 radiometer, and its one-minute total-irradiance product carries no highpass at all, so it reaches a band SPM cannot. Searched identically it is likewise null: every one of twenty candidates above threshold reads R ≈ 1,000–2,000 in the radiometer and R < 5.1 in all three photometers, and is therefore specific to one detector chain. This matters as much as the null does, because the a priori frequency veto had failed on those same candidates — it excluded exactly 180.0000 s and 360.0000 s at a half-width of 0.03%, while the real line carries sidebands to ±2% and a 120 s companion the veto never named. The cross-product gate caught all twenty without needing any of that to be predicted correctly.

The long-period blindness is ours, not the archive’s, and an earlier version of this section said otherwise. It was attributed to the L2 two-month highpass. It is not: the search detrends against a one-day running median, which removes everything slower than about a day before the archive’s filter is ever approached. Lengthening the detrend to 45 days was tried, and it did not open the band — injection recovers 6.40 ppm at one day, 42% at three days, and nothing at all at 10, 30, 60 or 120 days at any amplitude tested. The reason is physical rather than procedural: solar variability between roughly three days and two months runs to hundreds of ppm, so a part-per-million signal is far beneath it. That band is not unsearched; it is unsearchable at this level, and the same is true of the six-hour timescale in §4.17.

Two restrictions on the null that remain. The L2 product also states correction for orbit, degradation, outliers and “attractors”, and an outlier step can remove exactly the impulsive or quantized structure this statistic is built to detect. Testing that needs an injection through the VDC’s own pipeline, which requires L1 data the mission bundles do not carry, and it has not been done. Separately, the widened line veto now excludes 12% of the band, reported here because a notch broad enough to remove every candidate is not a null unless its cost is stated.

4.16 Row 7: p-mode frequency structure, on SOHO/GOLF

GOLF is the better instrument for this row than the EXIS photometry of §4.6, and for a reason that matters to the argument: it measures Doppler velocity, a different observable of the same oscillation, and its two photomultipliers PM1 and PM2 put the twin gate inside the instrument, where there is no cross-calibration to dispute. 40,845,600 samples at 20 s over 25.9 years at 98.4% duty — the most continuous record in this program.

What is searched is not whether p-modes exist. That is settled, and here it is the control. A sender able to modulate the Sun at all could shift mode frequencies, the most precisely measured quantity in solar physics; the observable is therefore the frequency shift over time, and the question is whether it carries structure beyond what solar activity explains.

Gate 1a is the rare control with no free parameter: Δν follows from stellar structure and is never fitted. Gate 1b matters for a different reason — a search for a 10⁻⁶ modulation that cannot recover a known 10⁻⁴ one is not a search, so the known activity term is an asset rather than a nuisance.

Null. Regressing the shift on F10.7 removes 51.7% of the variance (2.071 nHz per sfu; residual rms 0.0760 µHz). The strongest remaining peak sits at 2,455 d with power 0.128 against a permutation threshold of 0.145 — p = 0.126 across 200 independent frequencies and 20,000 permutations of the residual. Nothing beyond the activity term.

And the limit corrects this paper’s own estimate. Injecting a sinusoid into the shift series before the activity regression — so the signal meets the same detrending a real one would, rather than being exempted from it — puts 95% recovery at 0.045 µHz, which is 1.46×10⁻⁵ in fractional mode frequency at ν_max. §3.6 estimated our reach for this row at 10⁻⁵–10⁻⁶ with margin 0.1–1; that estimate was explicitly order-of-magnitude, and the measured value gives a margin of 0.069. Row 7 is searched, it is null, and it does not reach the level a designer would set. It falls short by a factor of fifteen.

That figure is what a deliberate search for a better estimator produced, not a first attempt. Ten candidates were ranked — segment lengths of 45, 90 and 180 days, centroid windows of ±2, ±4 and ±6 µHz, and a two-photomultiplier combination — and the ranking was done on the known solar-cycle term alone, never on sensitivity, so that the choice could not be tuned toward a nicer limit. The selected estimator (PM1+PM2, 180-day segments, ±4 µHz) recovers the solar-cycle shift at 1.08× the literature value and cuts residual scatter by 2.15×, from 0.0760 to 0.0353 µHz.

A 2.15× cut in residual scatter bought only 1.33× in sensitivity, and the reason is worth stating. Longer segments buy precision per point and pay for it in points: 180-day segments leave 51 where 90-day segments leave 104. A periodic search loses power with fewer samples even when each sample is better, so the two effects largely cancel. Residual rms is not a proxy for detection sensitivity when the sample count moves, and an estimator ranked on scatter alone would have been chosen wrongly here.

The shortfall is not mainly method, and this is a revision of what an earlier draft of this section claimed. It was argued here that fitting individual mode profiles rather than cross-correlating band spectra would reach ~0.01 µHz, six times better, putting the limit near 3×10⁻⁶. That was an extrapolation from published mode-fitting precision and it did not survive being tried. A ten-candidate estimator search, ranked on the known activity term, delivered 1.33× — from 1.94×10⁻⁵ to 1.46×10⁻⁵ — and the best of those candidates gave the same injected limit as one of the simplest. The remaining factor of fifteen to 10⁻⁶ is not visibly recoverable from this archive by better estimation, and the honest statement is that GOLF does not reach the designer level for this row.

4.17 Row 5: occultation dips and transit timing, and why it is void

§3.6 ranks this row fifth at a level of 10 ppm dips, after Arnold (2005), and records it searched only as a daily single-dip test. VIRGO’s one-minute total irradiance — 14,199,837 samples over 27.0 years, no highpass — is the right archive for it.

The control is the strongest available anywhere in this program, because the Sun is transited by known bodies at known times and the depth is not fitted. It is the ratio of two disk areas: Venus (R/R_☉)² = 75.6 ppm, Mercury 12.3 ppm. Venus transited on 2004-06-08 and 2012-06-06, Mercury on four dates between 2003 and 2019.

Void: none of the six is recovered. A matched filter at the known duration and the known mid-transit time returns Venus 2004 at −57 ppm — the Sun was brighter across the transit window than the surrounding six days — and Venus 2012 at 164.6 ppm against a null scatter of 108.2. Across all six the SNR spans −1.3 to +1.5.

The reason is measured, and it is the Sun rather than the method. A 76 ppm dip over six hours against 62.4 ppm per minute of white noise would be SNR ≈ 23, which is why two earlier versions of this gate were rewritten before the answer was believed. But the noise at six-hour timescales is not white: the null scatter of the same filter on transit-free days is 31–108 ppm, ten to thirty times the white-noise prediction, because supergranulation and active-region evolution live there. Venus sits at SNR ≈ 1 against that.

Two earlier versions of this gate are recorded in §5.6 because the sequence is instructive. The first compared the deepest hour on the transit day against 0.4× the predicted depth and reported six of six recovered — all six numbers were noise, Mercury reading 50–70 ppm against a predicted 12.3. The second added a control-day null and returned zero of six, correctly. An injection would not have caught the first error: the daily detrend was absorbing the transit, and a signal injected after that step recovers perfectly while a real one does not. Only an anchor with a predicted amplitude could catch it.

No limit is claimed, and the row is reported as void rather than null: a detector that cannot see a real 76 ppm occultation constrains nothing about an artificial one. §3.6’s estimate of ~1 for this margin was made against a per-minute noise figure, and a transit is not a per-minute event.

4.18 Row 3: sub-minute EUV, with a dark diode

SDO/EVE ESP gives four EUV bands at 0.25 s over 120 consecutive days — 41,472,000 samples, 97.7% present, Nyquist 2 Hz. That is a band forty years of daily records cannot represent at all, and §4.3’s finding that the same pipeline is 245× more sensitive at two minutes than at one day argues for pushing to the fast end.

The gate is built into the instrument. ESP carries CH_D, a dark diode: same detector chain, same electronics, same telemetry, no photons. Anything appearing there is instrumental by construction, with nothing to model. The five-minute oscillation is recovered in all four light channels (envelope 3.03–3.23 mHz) and peaks elsewhere in the dark channel, which is the pairing that makes the control informative rather than merely available.

Null. Of 20,736,000 bins, CH_18 and CH_36 return nothing above threshold; CH_26 and CH_30 return 31 and 30 candidates, every one of them at exactly 2.000000 Hz — the Nyquist frequency. Nothing physical sits precisely at Nyquist. The dark channel vetoes nine of the twelve strongest as instrumental; the three it passes are at Nyquist as well, which is a reminder that a control discriminates only against the failures it was built for.

CH_36 is excluded from any limit. Its values cross zero (median 2.6×10⁻⁴, minimum −5.2×10⁻⁴), so the relative residual x/trend − 1 diverges and its residual rms is 76%. The channel is reported as searched and returning nothing, not as constraining anything, and it is excluded from the ratios below — a ratio against a divergent denominator is a division by something near zero, not a dimensionless carrier.

The dimensionless carriers. §3.1 restricts a sender to dimensionless quantities, so the single channels above are the wrong object and the color ratios are the right one. Because each channel is prepared as a relative residual, the difference of two prepared channels is their log ratio to first order, and the three ratios among CH_18, CH_26 and CH_30 cost nothing extra to form. All three are null, and every candidate is again at exactly 2.000000 Hz — 11, 2 and 31 bins above threshold across the three, all at Nyquist. Three of those pass the dark-channel veto, which records only that the dark diode happened to carry less power in those particular bins: a control discriminates against the failures it was built for, and this one was not built against a sampling edge. With Nyquist excluded a priori, as it can be from the cadence alone, no candidate survives in any ratio.

4.20 Row 10: disc-integrated line-profile ratios

Row 10 sits at the bottom of the designer's ordering in §3.6 for a reason that is also the reason it is hard. A ratio of two line depths measured in the same exposure divides out atmospheric transmission, instrumental throughput, exposure time, detector gain and pointing; only differential effects survive. It is the dimensionless construction of §3.1 at its most complete, which is why a designer could set a gate at 10⁻⁷ there and expect a capable receiver to reach it.

The archive. The HARPS-N solar telescope has observed the Sun as a star every possible day since 14 July 2015: 173,793 disc-integrated spectra to 5 May 2025, 9.86 years, 3,239 observing days, 300 s exposures at 325 s cadence, released publicly in 2025. Per-exposure relative precision is 7.96×10⁻⁵ on both cross-correlation contrast and FWHM; bisector span is two orders worse at 8.6×10⁻³ and is not usable.

Two dimensionless channels were formed from daily means: contrast, the line depth as a fraction of the continuum, and equivalent width, contrast × FWHM. The second is the better channel and the reason is worth stating — contrast is an activity indicator and is used as one, so measuring on it means measuring the Sun's activity. Depth and width anticorrelate under activity, so their product cancels part of it: scatter falls from 5.10×10⁻⁴ to 3.02×10⁻⁴.

Three nulls were tried and two were wrong, each caught by the zero arm. Rolling values against fixed times breaks the value–time pairing on irregular sampling and leaves every surrogate whiter than the data: the zero arm fired at 100%. A circular shift is in any case the wrong instrument here — Lomb–Scargle power is invariant under a time shift, so it is degenerate for intrinsic periodicity, however well it serves the external-key searches of §4.10 and §4.12. Phase randomisation on an interpolated grid attenuates power in the search band, because filling the daytime gaps smooths the series: the zero arm was clean on contrast but fired at 95% on equivalent width, which carries more periodic content. The null adopted draws Gaussian surrogates at the observed times from the data's own empirical autocovariance — no gridding, no interpolation — and is clean on both channels, with median p of 0.4934 on equivalent width where a calibrated two-sided test belongs.

What that null asserts is the whole test: a detection must exceed what the star's own correlation structure produces. Quasi-periodic solar variability is inside it by construction, which is the honest choice and is what stellar activity modelling does; the cost is that a beacon sitting exactly in that structure is partly absorbed.

Null, slow band. No periodic fractional modulation in the 4–40 day band, at an injection-verified 9.4×10⁻⁵ (95% recovery, equivalent width, zero arm 0.0%, monotonic). Contrast gives 2.84×10⁻⁴ and returns the same figure under either working null, which is a useful check that its answer does not depend on the choice.

In that band the limit is the Sun, not the spectrograph. Detrended daily scatter is 25 to 43 times the photon floor of 1.2×10⁻⁵, so the search is bounded by solar rotation and active-region evolution and no integration improves it. This paper's own claim — that the archives are far more informative at two minutes than at one day — predicts the Sun should go quiet at high frequency. It does, and the same archive is recorded at five-minute cadence.

Null, fast band, and it is ten times better. Within a contiguous observing session the residual falls to 1.21× the photon floor on contrast and 1.48× on equivalent width, against 25–43× for the daily series: at minute cadence this channel is photon-limited. Searched over 8.6–130.9 cycles per day (periods of 11 minutes to 2.8 hours) across 1,656 sessions and 96,732 exposures, the result is null at an injection-verified 9.4×10⁻⁶ — zero arm 0.0%, median p 0.4453, clean transition between 3×10⁻⁶ and 10⁻⁵.

Sessions, not calendar days, and the distinction is not cosmetic. Grouping by integer BJD gives a median span of 23.7 h per bucket, impossible for a daytime telescope: the day boundary falls mid-observation and each bucket straddles a night, so a within-day detrend fits a line across twenty hours of darkness. A session is a run with no gap over an hour — 1,919 of them, median 5.53 h and 56 exposures.

The within-session residual is not white and the null must not pretend it is. Contrast shows lag-1 autocorrelation +0.415 decaying to +0.047 by lag 10; equivalent width +0.155 and white past lag 2. Permuting residuals would destroy that structure and manufacture significance — the failure that produced the 100% and 95% zero arms above. Surrogates are therefore drawn per session from that session's own empirical autocovariance at the observed times. Sessions are ~56 points, so the factorisation is trivial where the slow-band null needed 2,444 × 2,444.

Two limits on the fast-band figure. The surrogate carries no phase relation between sessions by construction — that is the point, since the test asks whether coherence persists across the decade, but a genuinely solar signal coherent across sessions would also fire. And five-minute p-modes sit at 288 c/d, above the 133 c/d Nyquist of a 325 s cadence, and alias into this band near 22 c/d (~65 min): a candidate there is solar before it is anything else.

§3.6 estimated 10⁻⁶. The fast band comes in 9.4× short of that and the slow band 94×, so the estimate was closer than the first search suggested and the slow band was simply the wrong band. The pattern of §5.1 survives either way — rows 6 and 7 came in at margins 0.5 and 0.069 against their own estimates, and the margin column remains an upper bound on capability rather than capability.

The search now allows the signal frequency to drift, and the paper did not previously. A designer with a probe in-system knows the star's rotation rate in real time and can lock a modulation to it. Solar rotation is differential and its effective rate drifts a few percent over the cycle; at the Carrington period a 1% drift accumulates ~8 radians of phase across a decade, more than a full cycle, and coherent integration at a fixed frequency then loses most of such a signal. Rotation appears elsewhere in this paper only as a control (§4.10, §4.12) and never as a clock a signal might track. The search is therefore run de-chirped, over frequency drift as well as frequency, with the null paying the same widened search.

4.19 The one spatially resolved channel, and why it is void

Every other search in this paper is disc-integrated. SOHO/VIRGO’s Luminosity Oscillation Imager gives twelve science pixels across the solar disc, plus four guiding references, at 60 s over 29.0 years with a 97.19% duty cycle — the best-sampled record used anywhere here.

It was acquired because a bound computed before acquisition said it should reach further than any disc-integrated channel for a localized source. A feature covering fraction f of the disc at depth d contributes f·d to a disc-integrated series and d to the pixel containing it: for a feature one pixel across that is a signal gain of 12 against a noise cost of √12, a net 3.46×, putting the projected limit at 5.8×10⁻⁸ against the 2.0×10⁻⁷ measured on SPM. The same arithmetic makes LOI 3.46× worse than SPM for a global modulation, because the photons have been divided for nothing.

Gate 1 passes on every pixel. LOI is a helioseismology imager, so the five-minute oscillation must be present: all twelve return an envelope between 3.025 and 3.175 mHz. The instrument, the data and the pipeline are all behaving.

The rotation control fails, and it takes the test with it. A localized feature crosses the disc in about thirteen days, entering and leaving pixels in a sequence fixed by geometry — so every pixel pair has a predicted lag, and solar rotation supplies the positive control for free, because active regions do exactly this. One of sixty-six pixel pairs correlates above r = 0.3. The rotational coupling that must be there is not.

The probable cause is in the product, and it was named in the header before the data was downloaded. L2 states correction for orbit, outliers, attractors and roll sensitivity changes. SOHO rolls; LOI’s pixels are fixed in the instrument frame; removing roll sensitivity plausibly removes the inter-pixel spatial structure with it, leaving twelve near-independent photometers rather than twelve pixels on a rotating disc. Void: not for want of sensitivity, but because the observable the test needs appears to have been processed out upstream. Settling it requires L1, which the mission bundles do not carry.

This is the third search in this paper stopped at the same boundary — the others being the outlier-correction question for VIRGO SPM (§4.15) and for EVE ESP (§4.18). In all three the limiting factor is not the instrument or the analysis but a level-2 product having already decided what to remove, and §5.7 should read that as a structural constraint on archival work rather than as three unrelated caveats.

5. Discussion

5.1 What twenty-two nulls establish, and what they do not

A null result bounds an amplitude; it does not settle existence. What §4.2 establishes is that no dimensionless modulation exceeding the stated amplitude exists in the stated observable over the stated band, during the interval observed — and every qualifier is load-bearing: a signal below the limit, in a channel not measured, in a band not examined, or absent during the epoch of the archive, is untouched. That is the logical position of every radio search ever published.

What the nulls achieve is to convert speculation into bounded speculation. Before this work, “the Sun’s output carries structure” had no attached number. It now has twelve, eleven injection-verified, and any future version of the claim must place the signal below those limits, outside those channels, or outside those bands — each a substantive and costly restriction.

5.2 Coverage is the binding constraint, and what it actually costs

The natural objection to enumerating the space — of order 1,300 pair tests and 12,000 triple tests under the admissible forms — is cost. The statistical and computational prices scale differently and must be quoted separately.

The statistical price is logarithmic. The Bonferroni threshold falls linearly in the number of tests, but enters the amplitude through the inverse normal:

Trials penalties are widely feared and, in this regime, mild: a fiftyfold larger search costs 25% in amplitude, against the 245× same-instrument gain available from cadence alone in §4.3.

The computational price is quadratic, and we established this the hard way. Significance is assessed by counting exceedances among M circular shifts, so the smallest attainable p is 1/(M+1); for that floor to reach a Bonferroni threshold of α/N the shift count must satisfy M > N/α, and total work is N × M = **N²/α**.

We report this because a run of ours failed on it. The thirty-channel sweep’s 1,074 tests have a Bonferroni threshold of 4.66×10⁻⁵, while the 6,000 shifts inherited from the 229-test design floor p at 1.67×10⁻⁴ — 3.6× above the line. The run returned “0 of 1,074 survive”, which reads as a clean null and is nothing of the kind: no test could have survived whatever the data contained. It is the mirror of the no-op controls of §3.4 — those could not fail, this could not fire — and it is invisible unless the floor is checked against the threshold explicitly.

Runtimes at a measured 3.0×10⁴ statistic evaluations per second on 80 cores of one machine. The quadratic term is invisible at 229 tests and dominates by 12,000.

The obvious remedy is to stop counting exceedances: a generalized Pareto fit to the upper tail of the shift null returns p-values below the empirical floor from a fixed surrogate budget, which would return the cost to linear in N. We tried it, validated it against direct counting, and rejected it. The validation is reported here because the reasons it failed are the reasons to pay the quadratic cost instead:

Agreement in dex between the tail-fitted and counted p. Over the resolved range (n = 1,049) the median bias is −0.000 dex, the interquartile spread 0.005, and the rank correlation 0.9999 — agreement is essentially perfect everywhere the fit is not needed.

The validation passes on its pre-registered criteria and does not certify the regime it is needed for. Agreement is essentially perfect wherever counting also resolves p — precisely where the fit is unnecessary. In the extreme tail, where the two disagree by up to 2.3 dex, no comparison is possible because counting has no answer there; and the two methods disagree on the survivor count, 22 counted against 10 tail-fitted at the 60,000 shifts this validation used — where the deeper 250,000-shift sweep of §4.8 resolves 19. A twin-instrument check on the fit itself settles how far it can be trusted. The identical fit was computed independently on a second machine — different host, core count, SciPy, and random shift draws. In the bulk the runs are indistinguishable: median difference +0.0000 dex, interquartile spread 0.0064, 90% of deviations within 0.026 dex. In the tail they are not: the largest single disagreement is 10.3 dex, and of ten Bonferroni survivors on each machine only seven are the same tests. The extrapolated p in the far tail depends on the surrogate draw at a level that changes which tests are called significant. This is the control of §4.4 applied to a method rather than to data, and it fired.

The tail fit is therefore rejected, and no p-value in this paper comes from it. The deciding test was to stop extrapolating and simply buy the surrogates: the pair sweep was re-run at 250,000 shifts, where the empirical floor is 4.0×10⁻⁶ and sits 11.6× below the family-wise threshold. Counting then resolves 1,055 of 1,074 tests outright, and the remaining 19 are bounded at the floor. The extrapolation proved not merely unnecessary there but unusable: offered all 19 tests that fell below ten exceedances — the only tests it exists to serve — it failed its own Kolmogorov–Smirnov goodness-of-fit criterion on every one and returned nothing. A bound reading p < 4.0×10⁻⁶ with the exceedance count behind it is worth more than a decimal that moves by ten orders of magnitude when the seed changes.

The conclusion survives the correction, with its margin reduced. A complete enumeration of the pair space is not an instrumental, funding or telescope-time problem — it is a few hours on one machine at a surrogate budget deep enough to matter, against archives that already exist; the triple space is 26 h by direct counting, and direct counting is now the only method we are prepared to use. And the case for finishing it does not depend on §2 being correct: the limits stand as limits regardless of why one went looking.

5.3 The recognition problem, and the discipline it requires

The gating argument has an obvious hazard: if a message is withheld until a receiver demonstrates capability, any null can be attributed to the gate rather than to absence, and the hypothesis becomes unfalsifiable in the way that matters — not by predicting nothing, but by accommodating everything.

§4.10 is what this failure looks like when it is not hypothetical. An alignment-specific excess appeared in the self-keyed search, and the first thing we did was set it aside because its sign was unattractive to the hypothesis — the same failure in the opposite direction: discarding a signal on a criterion the framework had no right to supply. It was caught in review and the result run through the four steps of §3.5. The discipline is only worth stating because we needed it.

We therefore do not invoke the gate to explain the nulls in this paper, and we recommend that no one else does either. Each null was run against a stated carrier at a stated sensitivity; the correct reading is that the carrier is absent above that amplitude, not that a gate intervened. A gating argument allowed to absorb negative results has stopped being a hypothesis and become an excuse.

What the gate is legitimately for is generating hypotheses: the dimensionless constraint makes the space enumerable, the relationship prediction directs the search toward combinations, the short pointer directs it toward the fast band — all of which can be looked for and failed to find. The recognition barrier is a real epistemic limit and not a license: it argues for breadth over depth and for reporting coverage, and it has no bearing on whether a given limit is correct.

5.4 Limitations

5.5 What to measure next

The fast band, on channels already recorded. The 245× gap between daily and two-minute sensitivity means most existing high-cadence solar data has never been searched at the cadence at which it is most sensitive. This costs nothing but analysis: SOHO/VIRGO SPM (three-channel photometry at one minute, unbroken from 1996, the obvious first target), PROBA2/LYRA (four bands, nominally 50 Hz, 2010–), SDO/EVE ESP (0.25 s, 2010–) and the RSTN radio network (one second at eight frequencies, 1966–, the longest fast record of any kind). All four were verified fetchable during this work. All four have since been searched, each null — RSTN at full cadence (§4.14), VIRGO SPM (§4.15), and SDO/EVE ESP (§4.18); and PROBA2/LYRA, whose archive — briefly unreachable — later returned. A first pass on LYRA’s Zirconium soft-X-ray/EUV channel at 100 Hz recovers the five-minute p-modes (3.0× the continuum at 3.36 mHz, the positive control) and is null for aperiodic structure, at a narrowband limit of order 100 ppm at one minute — shallow, because the EUV channels are variability-limited in the same way as ESP’s diodes. The one strong feature is a 93-second periodic line, consistent with a known PROBA2 spacecraft signature rather than any aperiodic message; a full-cadence, multi-day treatment with that line cross-confirmed and masked would firm the number, but the channel is no longer unsearched. With that, every high-cadence solar record named here has been examined at the cadence at which it is most sensitive.

A neutrino-line search that nobody has run. A search of the public IceCube and Super-Kamiokande catalogs for a monoenergetic line, framed as a test for a directed artificial source, is defined, cheap and untried. Present capability is far below the level a designer would set — roughly one Glashow-resonance event per decade at IceCube (IceCube Collaboration 2021), and solar and reactor backgrounds dominating the MeV window — and it closes with Hyper-Kamiokande, JUNO, DUNE, KM3NeT and IceCube-Gen2 on a decade timescale: precisely the shape §3.6 expects of a channel gated on a capability the receiver is still acquiring.

Population scale, on an archive that has just begun. The Nancy Grace Roman Space Telescope launched on 30 August 2026; its Galactic Bulge Time-Domain Survey will monitor of order 10⁸ bulge stars at ~12-minute cadence in six 72-day seasons across a five-year mission, under a mission-wide policy of no proprietary period. Every single-channel statistic in this paper — the k = 1 corner of §3.1 — transfers to a stellar light curve unchanged, and the fractional-depth limits of §4.2 set the sensitivity scale such a search inherits. The present work asked its question of one star with thirty channels; Roman permits the inverse experiment — one channel, a hundred million stars — and the two sample the design space along orthogonal axes: a network built for many targets must either repeat its choice across them, which a population search can see, or vary it per target, which no single-star search can exclude.

The stellar array. The general prediction — independent of the census idea — is that a coordinated network writes its signal not on one star but across many, so the observable is correlated aperiodic structure shared by a group of physically unrelated nearby stars: the pulsar-array statistic of §4.8 moved into photometry. Its power comes from number. Spreading a message across N stars lets each star’s modulation sit below its own single-star detection floor while the coherent sum survives, so the search reaches signals no single-star anomaly search — the regime that found Boyajian’s star — could see, and the reach deepens as the square root of the array size. A distance-shell version was run on Kepler and returned null, but Kepler’s pencil-beam geometry (median target ~3,600 ly) leaves the nearby shells the prediction favours nearly empty, so that null is shallow.

A side test, and what it does and does not bound. On TESS, whose all-sky coverage populates the nearby shells, we ran a specific sub-test of §2.6c’s census idea — does the modulation concentrate on nearby planet-hosts rather than matched non-hosts? It is a null, and it took clearing two void nulls to earn the word: a phase-scramble surrogate that fired as hard on the control as on the hosts (residual instrumental common-mode), and — exposed by injecting a synthetic signal into the real light curves — a within-group common-mode removal that was blind, because a coordinated beacon is itself common-mode and the cleaning removes it. The statistic that survives both is the host-versus-control contrast of the raw correlation, whose floor the injection measures at 90% recovery for a coordinated modulation of ~0.12 of each star’s scatter. Against it the result is null: across 83 all-sky sectors, planet-hosts are no more correlated than matched non-hosts (39 of 83 favour the hosts — chance — one-sided p = 0.95), and it holds when the shell is extended to 300 ly. But this bounds only the narrow, census-specific prediction, and with a method structurally blind to the general case: a beacon spread across a wide population would sit in the control stars too and cancel in the contrast. The general stellar-array search — a large-N absolute correlation across the nearby population, separated from instrument by tracking sky position and distance rather than detector geometry, and plausibly entered through the method-nudge of §2.6c rather than blindly — is therefore not excluded by this null. It is the better-motivated search, and its most tractable form — a centroid-directed search, in which conspicuous within-class variability outliers, selected by the aperiodic-inclusive, class-normalized metric of §2.6c, nominate their own neighborhoods to correlate — has now been run. Across 35 such candidates the neighborhood correlation, tested against angular-scale-matched random clusters so that detector-proximity common-mode cancels on both sides, is null (neighborhood median 0.053 against a matched-cluster median 0.050; Mann-Whitney p = 0.33). An earlier apparent excess proved to be exactly that proximity artifact and vanished once the control was matched in scale — a reminder that a control means nothing unless it is matched in the variable that could fool you. What remains genuinely untested is only the limiting case the centroid idea was meant to resolve: a sub-noise array carrying no conspicuous marker at all, which by construction offers the receiver no entry point to find it.

Continuity before novelty. All 53 unreachable pairs are blocked by records that have stopped (§5.7), so the cheapest gain in the pair space is the cross-calibration of an existing successor onto a retired record — Solar Orbiter SWA/HIS onto ACE SWICS above all.

Quantities not measured at all. Four candidate observables have no continuous record over the seventy-year span: core g-modes, the interplanetary electric field, continuous solar polarimetry, and high-latitude solar wind, which existed only while Ulysses flew (1990–2009). These are blank columns, not sparse ones; no reanalysis reaches them, and they are the only part of the program that would require new instrumentation.

Because the enumeration is cheap and its value lies in completeness, it is better conducted as a registered, incrementally reported survey — carriers and thresholds declared before the data is touched, coverage published whether or not anything is found — than as a series of individually motivated searches.

5.6 A note on method: physics-supplied controls, void results, and the error record

Three practices used here have nothing to do with technosignatures and may be the most portable result in the paper.

Physics-supplied controls. Wherever possible the control was a place the signal cannot be, fixed by physics rather than chosen by the analyst: above the acoustic cutoff, at the ecliptic pole, at the anti-sidereal frequency, on the twin instrument. Such a control cannot be tuned after the fact, and controls of that kind removed two apparent detections that had passed every amplitude test — including one at 4.9σ whose anti-sidereal counterpart stood at 6.6σ. A statistical null calibrated on the same data would have passed both.

Reporting void results as void. A null from an uncalibrated detector is a statement about the analysis, not about the sky, and recording it as a null silently inflates apparent coverage. Any survey that measures its own coverage — the figure of merit this paper argues for — needs the category.

The error record, collected. The program’s mistakes are collected here because they share one shape — a statistic or a control not matched to what it was testing — and because several are the reason a stated null is trustworthy. A reader is entitled to judge how often it happened. None was found by inspection; every one was found by a control that could fail.

The voids (§4.5, §4.9) are the failure that, caught later, would have been a false limit; the false nulls (§4.14, §5.2) are the same failure inverted. This table is why the paper trusts the limits it does state.

A fourth practice, learned the hard way: an audit must reproduce what it audits.

The three practices above concern controls — a control must be able to return “no”. A separate discipline governs the checks applied to a completed search, and this program learned it by violating it repeatedly in a single afternoon.

The fast-band threshold μ·ln(N/α) was challenged on review. The challenge was legitimate: the control that had certified it tested a single fixed bin against a threshold set for 1.3 million bins, and returned ≈0% whether the pipeline was sound or not. Replacing it with a band-wide exceedance count was right. Everything concluded from the replacement was wrong.

No stable value was obtained, and the reason is structural rather than statistical. A gapped series carries a comb in its window: EUVS Lyman-α is 26.9% absent on a daily pattern, and that mask is multiplicative. Subtracting a mean daily profile is additive and cannot remove it; notching requires a veto wider than a comb whose lines drift with the spacecraft’s eclipse season. Every excess measured localized entirely to that comb, which the search rejects before declaring any candidate.

The limits are therefore left as printed, on three independent grounds: the one channel whose pipeline genuinely removes the daily structure returns a threshold calibrated to 2%; every measured excess sits in bins the search vetoes; and the candidate lists of §A.1 are clean, which a badly miscalibrated threshold could not produce. That is not a verification — it is the absence of a defensible measurement against. The distinction is stated because the alternative was to change a printed number on the first of six attempts, which is what happened, and was reverted.

The generalizable rule. Each of the six failures was the same kind: the audit did not replicate the procedure it was auditing. The surrogate was not the same kind of object as the data; the bin set was not the one the search reports on; the veto was narrower than the structure it targeted. None was a statistical error, and twice a diagnosis of one instance was followed immediately by committing the next. A control must be able to fail; an audit must reproduce what it audits — and the second is harder, because a broken audit produces a number rather than an error.

5.6b What has not been done, and what it would cost

A coverage claim is only meaningful beside a statement of what was left. Three categories, and only the first is a matter of deciding to.

Runnable now, no new data and no significant compute.

Needs a larger compute budget.

The compute argument is not the binding one and this table should not be read as asking for a machine. Two of the four entries above are blocked on a null rather than on cycles, and §4.3’s central finding is that coverage rather than sensitivity binds. An exascale allocation would buy a larger void.

What the compute actually costs, in three machines. Sustained rates: Cray-1 160 MFLOPS (1976), the 88-core workstation used here ≈ 176 GFLOPS, Frontier 1.353 EFLOPS (Rmax).

The whole program is eighty Cray-1 days and under a millisecond of Frontier. T6 — 55 million Cray-1 years in 1976 — is a long weekend on a flagship machine in 2026. Frontier is 8.5×10⁹ times a Cray-1, thirty-three doublings in forty-eight years, almost exactly Moore cadence.

This table is an argument against an allocation, not for one. T6 is not expensive; it is 2.4 machine-days. It is that 2.4 days spent on a question whose null is undefined returns 487,635 keys’ worth of void, and §4.9 establishes that the higher-order space is blocked on the null rather than on cycles. The gate also opens on its own schedule: T3 is already 0.2 s on a flagship and eighteen days on a workstation. Waiting is cheaper than asking.

5.6c A plan for T3, and the prerequisite that is a research project in its own right

T3 — all 27,405 quadruples of thirty channels, four forms — is eighteen days on one machine and 0.2 s on Frontier. Neither number is the obstacle. The obstacle is that a four-body search needs a null, and §4.15–§4.19 show that the one correct null available is rejected by the Sun itself at median z = 45, because solar output has intrinsic higher-order structure. Running T3 today produces void at scale.

Phase 0 — the null. This is the whole problem. Two routes, and only the second is a compute problem:

• Targeted rather than omnibus. Stop asking “is there four-way structure” — the Sun says yes — and ask “is there structure at the specific alignment a sender would impose”, calibrated against decoy alignments. Cheap, and it narrows the hypothesis to something a null can be written for.
• An ensemble null from physics. Below.

Phase 1 — forms, and one is already excluded. §4.16’s theorem: adding cₖ·s(t) to channel k gives a linear four-body form (cₐ − 3_c_ᵦ + 3_c_ᵧ − cδ)s and a pair form (cᵢ − cⱼ)s, so invisibility to every pair forces all cₖ equal, whereupon (1−3+3−1) = 0. A linear four-body form cannot carry a signal pairs cannot also see. Only the multiplicative form is a genuine four-body carrier, which cuts the space by three quarters before any compute is spent.

Phase 2 — the sweep. Eighteen days on 88 cores, or under a second on a leadership machine, once Phases 0 and 1 are settled. It is the cheapest part and should be scheduled last.

The HPC project: an ensemble null for solar higher-order statistics

The question, stated without reference to technosignatures. What is the null distribution of higher-order statistics — bispectra, trispectra, cross-channel phase alignment — of solar output? Nobody knows, and it is not an idle question: it sets the false-alarm rate of every search for non-linear coupling in solar and heliospheric data, and it is the reason three results in this paper are void.

Why observation cannot answer it. We have one Sun and one realization of it. A null distribution requires an ensemble, and the only ensemble available is a synthetic one.

What the project would be. A large ensemble of independent global solar convective-dynamo simulations — Rayleigh, ASH or MURaM class — each integrated over several simulated activity cycles, from which synthetic disc-integrated irradiance and Doppler-velocity series are extracted through a forward model matched to the instruments actually used (VIRGO SPM and TSI, GOLF, GOES EUVS). The ensemble is the null: the distribution of any higher-order statistic across realizations is what “no imposed signal” looks like for a star that generates its own structure.

Scale. At literature cost of 5×10⁵ to 2×10⁶ CPU core-hours per realization, an ensemble of 50–200 members is 2.5×10⁷ to 4×10⁸ core-hours: 0.4 to 6.3 million node-hours on 64-core nodes. The low end fits a single INCITE-class award; the high end needs several. Two caveats bound these figures. First, they are CPU costs, while a leadership machine's capacity is in its GPUs: on Frontier the CPU cores are a percent or two of the peak. A credible allocation therefore needs a GPU-capable code, which could cut the node-hours several-fold. Second, wall-clock time is set by sequential time-stepping, not by machine size. A member cannot be spread across the whole machine: at ~100 nodes per member, one realization takes 3–13 days, and the ensemble runs in waves over weeks to months. (Corrected 23 September 2026: an earlier version gave 0.4 to 1.8 million node-hours, which understated the upper end about 3.5-fold.)

The gate this project must pass, stated first because it decides whether it is worth running. A null built from simulations that do not reproduce the Sun in the statistics being tested is worse than no null: it would license exactly the false confidence this paper spends §5.6 documenting. Global convection simulations are known to struggle with observed large-scale flows — the convective conundrum — and the ensemble must be validated against the real Sun on the same higher-order statistics before it is used as a null, not after. If it fails that validation the project still answers a real solar-physics question — how far do our dynamo models depart from the Sun in their non-linear structure? — which is worth knowing independently and is arguably the more interesting result.

What it unlocks if it passes. The triple and quadruple spaces become searchable, the two void results of §4.9 and §4.15–19 become limits, and any future search for non-linear coupling in solar data inherits a characterized false-alarm rate. That is the sequence: the ensemble is the expensive part, and the sweep that follows it is 0.2 seconds.

The gate, run: a daily-band surface ensemble in five versions

Before asking for leadership-class time, we built the cheapest ensemble that could plausibly pass the gate and ran the gate on it. Each member is a surface flux-transport model of the Sun: stochastic active-region emergence (latitude, longitude, flux, tilt, Hale and Joy's laws) driven by the observed sunspot cycle, transport by differential rotation, meridional flow and diffusion. A forward model turns every member into the 13 disc-integrated daily channels the triple sweep used, with the real gaps. Each version has 400 members, costs about an hour on the 88-core workstation, and was scored against the real Sun with the gate's criteria unchanged: at most 2 of 65 per-channel statistics outside the ensemble's central 99% (G1), and the real Sun's rank among the members uniform over the triple statistics, with at most 4% in the extreme tails (G2/G3).

version	what changed	G1 failures (≤ 2)	triples in the tails (≤ 4%)
v0	baseline	52 / 65 (44 after the fix below)	57%
v1	activity nests, gradual emergence, flares	48 (44)	54%
v2	plage emission linear in flux; noise sized at 2–4 d; power-law region sizes	40	43%
v3	parabolic spot decay; a flux ceiling of 1.5×10²³ Mx	28	43%
v4, design half	plage saturation, a decaying X-ray flare term; tuned on half the data	23	26%
v4, held out	the frozen v4, scored once on the other half	17	34%
v5, held out	per-line emission laws for the six chromospheric channels	18	34%

Three procedural points, because they decide what the table means. A calibration bug was found and fixed mid-way: channels whose records cover only part of 1980–2025 had been scaled against years with no data, which flattened them. Fixed and re-scored, v0 and v1 each drop to 44, so the bug accounts for 4–8 of their failures and the physics for the rest. The gate was seen before each redesign, so from v4 the data were split into alternating two-year blocks. All tuning and calibration used one half. The frozen model was scored once on the other half, which had informed the earlier versions only in aggregate. The halves are not directly comparable with the full-span rows, because each half has fewer overlapping days per triple.

What the ensemble establishes. First, single-channel solar physics generalizes. The held-out half scores better than the half the model was tuned on (17 against 23 failures), so the per-channel statistics are captured rather than fitted. Second, cross-channel structure does not. Triples degrade out of sample (26% to 34%), and almost all of the degradation is in the log-difference forms, which compare channels against one another. Per-line emission laws do not help (v5). The design half says why. The real chromospheric channels are far less correlated than any physics can make them: day-to-day correlation between MgII and Ca K is 0.68, against 0.92 or more for every emission law we could write. Some channels are line-profile indices that anti-correlate with emission: K2V/K3 correlates −0.42 to −0.80 with the others. Much of the real cross-channel structure is therefore instrumental, the drifts and systematics of a dozen different instruments, and no solar model produces it. Gaussian per-channel noise fitted to the missing power makes the ensemble worse, not better.

The consequence for the HPC project, stated plainly. A global dynamo ensemble addresses the physics, and the physics is the part that already generalizes. The gap that remains is between instruments. A null for cross-channel higher-order statistics must therefore model the instruments as well as the Sun, from each mission's calibration history, not from a Gaussian. Without that, even a perfect solar ensemble fails this gate for reasons that have nothing to do with the Sun. The allocation case changes accordingly: a physics ensemble plus an instrument-systematics model, and a gate restated to say which failures each is responsible for. The surface ensemble cost a few days on one workstation, and it is what showed that.

Needs instruments that do not exist. Four columns are blank across the whole seventy-year record: core g-modes, the interplanetary electric field, continuous disc-integrated polarimetry, and high-latitude solar wind, the last of which existed only while Ulysses flew. These are unmeasured quantities rather than unsearched archives, and they are the only lever on the roughly one-in-five chance that we happen to measure whichever combination was chosen — a probability no instrument on the observables we already have can improve.

5.6d A resolved search: localized modulators, and the SDO archive

Every search in §4 but one asks what a distant receiver would see. That receiver cannot resolve the Sun, so disc-integrated channels are the right ones for it. A sender would then have to modulate globally, or narrowly enough in frequency to stand out from a point source. A modulator meant for a receiver inside the Solar System faces no such constraint, and a localized modulator is the case our disc-integrated sweeps are nearly blind to. §4.19 tried to close that gap with twelve LOI pixels and was stopped by processing upstream. This section sizes the version that is not stopped.

Where this sits in the literature. Searches for technosignatures inside the Solar System have so far looked for objects: bodies at stable Lagrange points, artifacts on the lunar surface, radio emission from interstellar objects, anomalous transients in sky surveys. A recent review concludes that they leave only very crude limits (Lazio 2026). Pixel-level periodicity searches of solar EUV imagery do exist, but in solar physics, where they map natural long-period loop pulsations (Auchère et al. 2014; Froment et al. 2015). Their method is this section's technical cousin, and their pulsations are its natural false positives. To our knowledge, no search has treated the Sun's own resolved emission as a possible carrier, or folded resolved imagery on the star's measured clocks. The search reported below is, as far as we can tell, the first directed technosignature search of resolved solar imagery. The claim rests on a literature search, not an exhaustive one.

The bound, generalized from §4.19. A patch of projected area A_mod with fractional modulation δ, measured in a cell of area A_cell ≥ A_mod, contributes δ·A_mod/A_cell. If the local solar variability has rms σ₀, coherence area a₀ and coherence time τ, and averages down independently, then in time T

SNR = δ·A_mod·√(T/τ) / (σ₀·√(a₀·A_cell))

That is a matched filter. The optimal cell is the modulator's own size. A smaller cell discards signal and adds trials; a larger one dilutes the signal as 1/A_cell while the noise falls only as 1/√A_cell. The gain over a disc-integrated measurement is √(A_disc/A_mod), with A_disc = πR☉² ≈ 1.5×10⁶ Mm². For LOI's pixels this is the 3.46× of §4.19. For a 10 × 10 Mm patch it is about 120× in SNR, or ~15,000× in observing time, and ~700× for a patch one AIA synoptic pixel across. A blind search over ~10⁴ positions and a few sizes raises the threshold from about 3σ to about 5σ, which is small against those factors. The assumption that noise averages independently is optimistic. Supergranulation, active regions and low-degree p-modes are coherent over large areas, and chromospheric and coronal variability is larger and more structured than photospheric granulation. The gains are upper bounds, and the limit must be measured by injection, not projected.

The whole disc need not be watched. A modulator fixed to the surface crosses the central meridian every ~27 days, so a narrow strip along the meridian sweeps every longitude once per rotation, like a pushbroom. That strip is also where foreshortening and limb effects are smallest. The price is a duty cycle of Δλ/360°, on top of the far-side half that no Earth-side observer sees. The poles stay hard, since B₀ reaches only ±7.25°.

The data exist, and at a usable level. SDO/AIA (Lemen et al. 2012) has imaged the full disc in seven EUV channels since May 2010. The public synoptic product is 1024² at 2.4″ per pixel (1.7 Mm), one frame per channel every two minutes, level 1.5: registered and centred, but not temporally filtered. That is the property LOI's L2 lacked. A frame is 1.3 MB, so one channel is ~0.9 GB per day: ~5 TB for sixteen years at full synoptic cadence, or ~0.9 TB at twelve-minute cadence. Both were verified fetchable during this work. The two-minute cadence limits the search to periods of four minutes and longer, and to 24 minutes and longer at the twelve-minute cadence.

The method.

• Grid. Reproject each frame to a heliographic grid that follows the differential rotation, so a surface-fixed source stays in one cell.
• Statistic. Compute a periodogram per cell for each disc passage, and stack the passages incoherently.
• Scales. Repeat at several aperture scales, from one pixel to active-region size.

It does not need the ensemble null of §5.6c, which is the decisive difference from T3. The Sun supplies its own ensemble in space. Decoy rotation rates, the same data sampled on a grid rotating at the wrong rate, keep every noise property and destroy a surface-fixed source's coherence. That makes them a native null for each cell.

The gates, stated first.

1. Rotation control (positive). Active regions must be stationary in the heliographic grid and smeared in the detector frame. This is the control LOI failed, and here it must pass before anything else is read.
2. Detector-frame control. An artifact fixed on the CCD appears in detector coordinates, not heliographic ones. Anything significant in both frames is instrumental.
3. Known lines. SDO's 24-hour geosynchronous orbit, the eclipse seasons, the six-month orbital terms and AIA's exposure control are masked in advance.
4. Injection-recovery. Synthetic modulated patches are injected into real level-1.5 frames at each channel, size, period and latitude. The false-positive rate comes from the decoy grids. No limit is quoted without both.

Gate 1, run: the rotation control passes. We ran it on the first ten days of 2014 in the 171 Å synoptic series (cycle 24 maximum). That is forty frames at four per day, compared in pairs 1, 2 and 4 days apart on a heliographic grid (±50° latitude, ±45° from the central meridian, μ > 0.5). The grid was tracked at s × the Snodgrass–Ulrich magnetic differential rotation. s = 0 is the Earth-fixed detector frame, and s = 1 is the Sun. Each frame was first divided by its own median centre-to-limb profile, so limb structure shared by every frame cannot correlate in the detector frame. The criteria were fixed before running: the correlation must peak within s ∈ [0.95, 1.05] at every lag, and must exceed the detector frame's by at least 0.2.

lag	s = 0	0.5	0.8	0.9	0.95	1.0	1.05	1.1	1.2	1.5	pairs
1 d	0.180	0.305	0.493	0.594	0.650	0.684	0.674	0.631	0.531	0.322	36
2 d	0.164	0.139	0.268	0.401	0.487	0.547	0.537	0.476	0.349	0.161	32
4 d	0.157	0.106	0.086	0.216	0.347	0.462	0.466	0.385	0.246	0.269	24

At every lag the correlation peaks at the solar rate: s = 1.0 at one and two days, and 1.05 at four days, where 1.0 is within 0.004 of the peak. It stands 0.31–0.50 above the detector frame and falls away on both sides. The images recover the Sun's differential rotation from nothing but themselves. The level-1.5 product keeps the spatial structure that removed the LOI test (§4.19), and structure fixed in the detector frame is ruled out as the source of the correlation. This gate licenses the rest of the pipeline; it detects nothing. The searches built on it follow.

One cell for the whole record, measured. The most sensitive form of the resolved search follows a single cell through the entire archive, the matched filter taken to its limit. How far that reaches depends on how noisy one cell is, so we measured it rather than assumed it. 400 cells were tracked at the solar rate through a disc passage (μ > 0.5, 1–8 January 2014) in three AIA channels. Each cell's one-day running mean was removed, and the remaining per-sample rms and lag-1 correlation were converted to the smallest modulation reached at 5σ, assuming a duty cycle of one third of each rotation:

channel	cell	rms per 12-min sample	lag-1	5σ, 1 yr	5σ, 16 yr	16 yr, as disc-integrated depth
171 Å (corona)	1 px (1.7 Mm)	0.19	0.90	3.5×10⁻²	8.7×10⁻³	~2×10⁻⁸
171 Å	5×5 px (8.7 Mm)	0.16	0.96	4.4×10⁻²	1.1×10⁻²	~5×10⁻⁷
1600 Å	5×5 px	0.056	0.40	3.5×10⁻³	8.8×10⁻⁴	~4×10⁻⁸
1700 Å (UV continuum)	1 px	0.113	0.29	6.3×10⁻³	1.6×10⁻³	~3×10⁻⁹
1700 Å	5×5 px	0.046	0.31	2.6×10⁻³	6.6×10⁻⁴	~3×10⁻⁸

Three things follow. First, the hot corona is about the noisiest place to look. A 171 Å pixel flickers by ~20% with a correlation time of hours, and averaging 25 pixels barely helps because the variability is coherent across them. Second, the UV continuum reaches about seventeen times deeper: its noise decorrelates within an hour and averages down across pixels. A designer who wanted a pixel-scale signal to be found would choose a quiet channel, and the search should begin where the designer would. Third, even the coronal floor is deep in the units that matter for a localized source. A 1% modulation of one 171 Å pixel is ~2×10⁻⁸ of the disc, and one UV pixel at 1.6×10⁻³ is ~3×10⁻⁹. Both are below the best disc-integrated limit in this paper (2×10⁻⁷, §4.19). These are white-noise extrapolations. Solar noise is redder at long periods, so they are floors, and injection must confirm them.

What "the same cell" means over sixteen years. Surface plasma drifts poleward at ~11 m s⁻¹ (~0.08° per day), so a modulator embedded in the plasma would wander. One that holds position does so in some frame, and each frame is a hypothesis: rigid rotation at a fixed rate, a tracked latitude, or the Earth-fixed detector frame. Holding one pixel for sixteen years needs the rotation rate to within ~2×10⁻⁵ ° per day, about 8×10⁴ trial rates across the solar range. They cost nothing extra. For each latitude, a time–longitude diagram (sixteen years × 360°) turns a source at any rigid rate into a line, and its two-dimensional Fourier transform searches every rate and every modulation period at once. The cost moves from compute to the pipeline. It must carry calibration through sixteen years of AIA's EUV sensitivity loss, eclipse seasons, CCD bakeouts and cadence gaps, and it is the pipeline, not the arithmetic, that has to be built carefully.

Results: thirty-three clock searches, all null. The clock idea of §2.10 turns the resolved search into a short list of folds on measured natural phases. The first three were run on 2014 AIA data: the rotation clocks (Carrington, equatorial and radiative-interior rates, harmonics up to 512, i.e. bits from ~1.2 h to 3.5 d) on the full year in 171 Å and 1700 Å, and the p-mode clock on January in 1700 Å at two-minute cadence. For the p-mode clock, the phase is taken from a second spacecraft: SOHO/GOLF disc-integrated velocity, band-passed to 2.4–3.9 mHz, is a shared stochastic clock that any observer sees identically. Each search covers the disc-integrated brightness and 432 Carrington cells 10° on a side. Significance is Bonferroni over every counted test at family-wise α = 0.01, with the exact F(2, 2n) null for power over the mean of n local decoy bins. Each false-positive rate was measured, on 40 deliberately wrong clocks, not assumed. The rest extend them to the whole mission (2010–2025), to cells down to 2.5°, to a second observable, the magnetic field, and to a third, the Sun's radio flux measured from the ground since 1980 by two independent networks.

search	data	tests counted	wrong clocks that fired	strongest p (threshold)	depth reached, disc
rotation clocks, corona	171 Å, 2014, 41,662 frames	618,350	0 / 40	6.5×10⁻⁶ (1.6×10⁻⁸)	10⁻³ at 5.1 h
rotation clocks, UV continuum	1700 Å, 2014, 41,879 frames	618,187	0 / 40	6.4×10⁻⁶ (1.6×10⁻⁸)	10⁻⁴ at 5.1 h
p-mode clock (GOLF phase)	1700 Å, Jan 2014, 2 min, 21,268 frames	21,387	0 / 40	3.3×10⁻⁵ (4.7×10⁻⁷)	≤3×10⁻⁵ at 21 min; 10⁻⁴ at 1.4–5.5 h
four rotation clocks, incl. each cell's own latitude rate; 10° cells, central disc	1700 Å, 2010–2025, 667,799 frames, 360 cells	692,280	0 / 40	1.3×10⁻⁷ (1.4×10⁻⁸)	cells: 10⁻³ of a cell's brightness (3×10⁻⁴ in 8 of 12)
rotation clocks, disc; SDO orbital signal fitted and removed in fixed 27-day windows	1700 Å, 2010–2025, 667,778 frames	1,352	0 / 40	3.8×10⁻³ (7.4×10⁻⁶)	10⁻⁴ at 5.1 h
p-mode clock (GOLF phase), 10° cells, bits 52 min–13 h	1700 Å, 2010–2022 (while GOLF overlaps), 12 min, 501,081 frames	106,924	1 / 40	2.1×10⁻⁵ (9.4×10⁻⁸)	not yet measured
four rotation clocks; 2.5° cells, central disc	1700 Å, 2010–2025, 667,799 frames, 5,760 cells	11,071,334	9 / 40 at Bonferroni; threshold then set by the wrong clocks (see below)	2.8×10⁻¹¹ (1.15×10⁻¹¹)	cells: 10⁻³ of a 2.5° cell's brightness
four rotation clocks; 10° cells, magnetic field	HMI mean |Blos|, 2014, hourly, 7,617 frames, 360 cells	382,889	0 / 40	8.7×10⁻⁶ (2.6×10⁻⁸)	cells: 3×10⁻² of a cell's field (~0.2 G)
rotation clocks, disc, magnetic field	HMI, 2014, hourly	784	0 / 40	8.6×10⁻⁵ (1.3×10⁻⁵)	3×10⁻³ of the disc mean |B| (~0.03 G)
four rotation clocks; 10° cells, magnetic field	HMI, 2010–2025, hourly, 113,467 frames, 360 cells	387,068	1 / 40	7.0×10⁻⁷ (2.6×10⁻⁸)	cells: 10⁻² of a cell's field (~0.06 G)
rotation clocks, disc, magnetic field	HMI, 2010–2025, hourly	783	1 / 40	3.6×10⁻⁴ (1.3×10⁻⁵)	3×10⁻⁴ of the disc mean |B| (~3 mG)
three rotation clocks, disc; radio flux, 8 frequencies 245 MHz–15.4 GHz, bits 1.3–8 h	RSTN, 4 ground sites, 1980–2026, 1 min, 2.2×10⁸ site-minutes	7,999	0 / 40	5.6×10⁻⁴ (1.25×10⁻⁶)	10⁻³ of the flux at 1.4–15.4 GHz; 3×10⁻³ at 245–610 MHz
three rotation clocks, disc; radio flux, 1, 2, 3.75, 9.4 GHz, bits 1.3–8 h	Nobeyama Radio Polarimeters, 1994–2026, 0.1 s to 1 min, 2.3×10⁷ minutes	3,999	1 / 40	9.5×10⁻⁶ (2.5×10⁻⁶)	3×10⁻⁴ of the flux at 2 and 9.4 GHz; 10⁻³ at 1 and 3.75 GHz (corrected from 3×10⁻⁴ for all four)
p-mode clock (GOLF phase), disc; radio flux, 1–9.4 GHz, bits 10 min–2 h	Nobeyama Radio Polarimeters, 1996–2022, 10 s, 1.1×10⁸ bins	76	1 / 40	4.3×10⁻³ (1.3×10⁻⁴)	10⁻⁴ of the flux at 21 min
three rotation clocks, disc; circular polarization V/I, 1–9.4 GHz, bits 1.3–8 h	Nobeyama, 1994–2026, 1 min	3,999	2 / 40 (threshold set by wrong clocks)	1.5×10⁻⁵ (2.2×10⁻⁶)	10⁻⁵ of the flux at 9.4 GHz; 10⁻⁴ at 1–3.75 GHz
p-mode clock, disc; circular polarization V/I, bits 10 min–2 h	Nobeyama, 1996–2022, 10 s	76	0 / 40	3.3×10⁻² (1.3×10⁻⁴)	3×10⁻⁵ at all four frequencies
three rotation clocks; the p-modes' intensity-velocity response (amplitude and phase), bits 20 min–3.5 d	SOHO VIRGO SPM (intensity) with GOLF (velocity), 1996–2022, 60 s	11,842	0 / 40	1.1×10⁻³ (8.4×10⁻⁷)	response keyed on/off by 10%
p-mode clock; the same response, bits 10 min–2 h	as above	42	0 / 40	1.7×10⁻² (2.4×10⁻⁴)	response keyed on/off by 10%

Depths are the smallest phase-locked modulation, as a fraction of the series searched (disc or cell brightness, or mean field strength), recovered in every injection trial at the family-wise threshold (10 to 20 trials per depth; 12 random cells for the cell searches). In 2014 the Sun's 171 Å and 1700 Å output carries no signal locked to its rotation above these depths, and in January 2014 none locked to its five-minute oscillation. Over the whole mission, 2010–2025, the same holds at 10⁻⁴ for the disc and at 10⁻³ for every central-disc cell down to 2.5°. The same holds for every 10° cell at correspondingly shallower relative depth. Positive controls pass: the rotation band recovers the Sun's own plage modulation (13.6 d, p = 7×10⁻²⁰ in 1700 Å), and AIA brightness folded on GOLF's measured phase at the fundamental is coherent at p = 9×10⁻¹⁴⁵. So the clock transfers between two spacecraft and two observables. The UV continuum reaches ten times deeper than the corona, as the single-cell measurement predicted.

Three things that imitate a clock-locked signal. The first two were found by the calibration; the third passes it by construction.

1. The observatory's own orbit. The first full-year 1700 Å fold reported a detection: disc-integrated, interior-rate clock, harmonic 29, 24.05 h, p = 2×10⁻¹². It is SDO's geosynchronous orbit, with its harmonics at 12.0 h and 8.0 h close behind. Its wrong-clock calibration fired 51 times in 100, which is what identified it: a line that strong falls near some harmonic of almost any clock. In 171 Å the same line was weak enough to hide, so a clean calibration in one channel does not transfer to another.
2. Periodic structure that belongs to the Sun, not to any clock. 373 coronal cells and 386 UV cells carry fixed-period lines, mostly between 1.3 and 11 h. In the corona these are the long-period intensity pulsations of heated loops, 3–16 h (Auchère et al. 2014), now understood as thermal non-equilibrium cycles (Froment et al. 2015). The first p-mode search failed its calibration (6 of 20 wrong clocks fired) on one such line, 6.88 h in a single mid-latitude cell.
3. The viewing geometry of a cell fixed to the rotating Sun. The first sixteen-year search of the 10° cells (1700 Å, 2010–2025) reported a detection: Carrington clock, harmonic 45, 14.55 h, at latitude +5° and longitude 205°, p = 4×10⁻¹² against a threshold of 1.2×10⁻⁸. The same harmonic was present in 11 of the 36 cells of that latitude band, 2 at −5°, and none elsewhere. A modulator confined to one patch cannot do that. The cause is geometric. A cell fixed in Carrington coordinates is seen at a viewing angle that repeats exactly with each Carrington rotation. Near the limb, where the cell rotates into and out of view, the brightness correction leaves a residual that repeats with it. The cell's viewing angle, folded on its own, carries the same harmonic at p = 10⁻¹⁹; the flag vanishes when only the central disc (μ > 0.6) is used; and it fails the pre-registered halves test (2010–2017 p = 2.6×10⁻³, 2018–2025 p = 1.5×10⁻¹⁰). Wrong-clock calibration cannot catch this class of artifact, because it is locked to the true clock exactly as a real signal would be.

Both are handled by one rule, adopted before any of the reported runs. A signal must be locked to the clock, not merely present at a period some harmonic reaches. SDO's orbital harmonics are masked in advance. Periods where the wrong clocks already find power (p < 10⁻⁴) are removed from each cell's tests, which costs 1% of the rotation tests and 10% of the p-mode tests. The calibration is leave-one-out, so every false-positive rate above is measured on clocks the masks were not built from. The third case needed a rule of its own, added after the flag and before the fine-grid search: cell searches use the central disc only (μ > 0.6), with the brightness corrected for both viewing angle and east–west position (a cell rotating into view and one rotating out are not equivalent at the same angle). Every leading result is reported with a geometry check: the cell's viewing-angle series folded at the same clock and harmonic. The confirmation criteria were amended accordingly, dated and labelled as prompted by this case. A result that is band-wide in latitude is presumed geometric until shown otherwise. The episode is also the strongest argument for writing the criteria down first: the pipeline produced a p = 4×10⁻¹² "detection" out of nothing but geometry, and the criteria took an hour to dispose of it.

A calibration that failed in the tail, and what replaced it. The 2.5° search reported eight detections, the strongest at p = 2.8×10⁻¹¹ against a Bonferroni threshold of 9.0×10⁻¹⁰. All lay between +36° and +49° latitude, and several recurred at the same clock and harmonic across longitudes. Its wrong-clock calibration failed: 9 of 40 wrong clocks produced a test below the same threshold, and the null check was inflated (1.23% of tests below p = 0.01). With 1.1×10⁷ tests the threshold sits far into the tail of the F distribution, and small, gappy 2.5° cells do not follow it that far. The two strongest wrong-clock results, 3.3×10⁻¹² and 1.1×10⁻¹¹, were stronger than anything on a true clock. The pre-registered criteria disposed of the flags before any follow-up. The search was then re-scored with the threshold taken from the wrong clocks themselves: at most one of 40 may cross it, which puts it at 1.15×10⁻¹¹. Nothing on a true clock crosses it. The 10° searches, with sixteen times fewer tests, calibrated at 0 of 40 on the analytic threshold, so the failure is specific to the fine grid. Where the analytic null has not been shown to hold as far into the tail as the threshold, the threshold comes from the wrong clocks, not from the formula.

A second observable: the magnetic field. A modulator need not act on brightness. The same search was run on the hourly SDO/HMI line-of-sight magnetograms (hmi.M_720s, 1024²; Scherrer et al. 2012) for 2010–2025, on each 10° cell's mean unsigned field and on the disc mean. The cell search is null and well calibrated (1 of 40 wrong clocks; 0.999% of tests below p = 0.01). But magnetograms are a weak channel for this purpose. A quiet central-disc cell's mean |B| is about 6 G, mostly measurement noise, and varies ~15% from hour to hour, so sixteen years reach 1% of a cell's field (0.3% in 1 of 12 trials), against 10⁻³ for the UV continuum. A one-year pilot (2014) reached 3%, as the noise predicted.

A fourth imitation, made by our own processing. The first sixteen-year fold of the magnetic disc mean gave six results below the threshold on the Carrington clock, at harmonics 32–42 (15–20 h) and 8, down to p = 4.7×10⁻¹⁰. The calibration rejected them: 5 of 40 wrong clocks fired, and 22% of all tests fell below p = 0.01. The cause was the orbital fit. It had been done separately in each Carrington rotation, so every join between fits fell at Carrington phase 0. On a series whose rotational swing is 11% rms, fourteen times that of 1700 Å, the joins left a residual that repeats every rotation, which is exactly what a harmonic of the Carrington clock looks for. Refitting in fixed 27-day windows, which are locked to no clock we test, removes it: 1.02% of tests below 0.01, 1 of 40 wrong clocks, nothing within a factor of 28 of the threshold, and 3×10⁻⁴ of the disc mean field recovered in 10 of 10 trials. Skipping the fit instead fails the calibration the other way (15 of 40), because the orbital line then leaks into the wrong clocks. The 1700 Å disc search, which used the same Carrington windows, was rerun the same way. Its limit is unchanged at 10⁻⁴, and its null check (1.63% before) became conservative (0.07%), so it carried a trace of the same artifact. The rule: never segment, fit or detrend a series in windows tied to the phase of a clock that will be tested. Like the viewing geometry, this artifact is locked to the true clock by construction. Here the calibration caught it only because the null inflated across every harmonic at once.

A third observable: the Sun's radio flux, from four continents. The US Air Force Radio Solar Telescope Network (RSTN) has recorded the whole-Sun flux every second at eight fixed frequencies, from 245 MHz to 15.4 GHz, at Learmonth (Australia), Palehua (Hawaii), Sagamore Hill (Massachusetts) and San Vito (Italy) since the early 1980s (NOAA/NCEI archive). The frequencies sit in bands protected for radio astronomy, so they were chosen for quiet, not because the Sun favours them: its radio output is broadband. Ground data carry no solar coordinates, so the Carrington phase was computed from the ephemeris and the IAU rotation elements. It matches the phase SDO measured in 667,000 image headers to 0.04° rms. Each site sees the Sun for 9–12 hours a day and its calibration shifts from day to day, so each session's level and drift are removed. That confines the search to bits of 1.3–8 hours; nothing slower survives. The average daily shape is fitted in fixed 27-day windows tied to no clock (the lesson of the magnetic disc search), and harmonics of the solar and sidereal day are masked as SDO's orbit was. The first run was null and calibrated, but its residual scatter was 15–30% per minute, heavy-tailed from interference and small bursts. Dropping residuals beyond five times each site's robust spread (2–3% of points) and cleaning again tripled the depth. Over 46 years there is no modulation locked to solar rotation above 10⁻³ of the radio flux at 1.4–15.4 GHz, or above 3×10⁻³ at 245–610 MHz. Four sites also give a test SDO cannot. Every leading result was refolded site by site, and none holds up: the strongest, 245 MHz on the Carrington clock at 1.32 h, gives p = 0.37, 2×10⁻⁴, 0.11 and 0.41 at the four sites. A signal from the Sun must appear at every site; a site's own systematics need not.

A second radio network, and the daily cycle again. The Nobeyama Radio Polarimeters (Japan) have measured the whole-Sun flux every 0.1 s at 1, 2, 3.75 and 9.4 GHz since 1994, with a cleaner calibration than RSTN: 0.5–0.9% scatter per minute after cleaning, against 2–7%. Its first run passed the null check but failed the calibration: 5 of 40 wrong clocks fired, down to p = 6×10⁻¹⁰. The lines they hit sat 0.07–0.09 cycles per day below the 11th and 17th harmonics of the solar day. That is the second sideband (±2/27 c/d) which refitting the daily shape in 27-day windows leaves around each daily harmonic, and the ±1.5/27 c/d notch did not reach it. A single observatory's daily cycle is sharp; four sites at different longitudes blur theirs. The notch was widened to ±2.5/27 c/d, covering both sidebands, and both radio searches were rerun with it. This change was made after the failure and is labelled as such. Nobeyama then passes (1 of 40, the limit; null check 0.90%), and RSTN improves to 0 of 40. Over 32 years the 1–9.4 GHz flux carries no rotation-locked modulation above 3×10⁻⁴, for bits of 1.3–8 hours. The closest approach of any radio test, 2 GHz on the equatorial clock at 4.92 h (p = 9.5×10⁻⁶), is a factor of 3.8 short of the threshold and weaker than the wrong clock that fires.

The p-mode clock in radio. Nobeyama's 0.1 s cadence also allows the fastest clock. Averaged to 10-second bins and folded on the p-mode phase GOLF measured from 1996 to 2022 (2.6 million cycles, mean period 311.6 s), 26 years of 1–9.4 GHz flux were searched for bits of 2–22 cycles, 10 minutes to 2 hours: a range the rotation-clock radio searches cannot reach. The same cleaning, outlier clip, daily notch, wrong clocks and injection were used. It is null and calibrated (1 of 40, the limit; nothing within a factor of 30 of the threshold). No modulation locked to the five-minute oscillation exceeds 10⁻⁴ of the radio flux (recovered 19 of 20 across the four frequencies at 21-minute bits; 2 and 9.4 GHz partly to 3×10⁻⁵). This is as deep as the 1700 Å disc, in a different physical channel.

Polarization as a switch. A designer could flip the polarization of emission without changing its total brightness, so polarization is a natural place for a switch. The Nobeyama polarimeters record circular polarization (Stokes V) alongside total flux, and the fraction V/I was searched with the same machinery as the flux, under its own registration: the rotation clocks at bits of 1.3–8 hours over 1994–2026, and the p-mode clock at bits of 10 minutes to 2 hours over 1996–2022. Neither shows anything. Circular polarization carries no modulation locked to either clock above 3×10⁻⁵ of the flux for p-mode-timed bits, or above 10⁻⁵ (9.4 GHz) to 10⁻⁴ (1–3.75 GHz) for rotation-timed bits. The rotation search's calibration fell just outside its limit (2 of 40 wrong clocks), so its threshold was set from the wrong clocks, as registered; it is null either way. The five-minute oscillation seen in total flux is absent from polarization, as quiet-Sun emission predicts. Two polarization channels remain unsearched: linear polarization (EOVSA's cross-hand correlations, or HMI's Stokes Q and U), and optical polarimetry of integrated sunlight, for which we know of no long record.

The Sun's response, not its output. In the natural Sun the five-minute oscillation shows in velocity and in brightness with a fixed ratio and phase lag, set by the physics of the photosphere. A modulator that altered the acoustic surface or the light path could change that relation without changing any mode's frequency, so the relation itself is a carrier. It was measured from two instruments on the same spacecraft, SOHO: GOLF's velocity and VIRGO's intensity, both filtered to the oscillation band, over 1996–2022. The relation is measured cleanly: 15.4 ppm of green-light intensity per m/s of velocity, nearly in antiphase (−174°), at 26σ; the red and blue channels agree. Folded on the rotation clocks (bits of 20 minutes to 3.5 days) and on the oscillation's own cycle count (bits of 10 minutes to 2 hours), under its own dated registration, neither its size nor its phase changes in step with any clock. Switching the response on and off by 30% would have been seen on the rotation clocks (10% on the oscillation's own clock); an earlier version quoted 10% for both, from a test recovered in four trials of five. Both searches calibrate cleanly (0 of 40 wrong clocks). A stretch of GOLF data in late 2002 turned out to hold a constant filler value rather than a gap marker; it was caught because it broke the first run, and it is now excluded.

The Sun's own line list as the key. A spread-spectrum receiver recovers a weak signal by correlating many channels against a known code. An optical spectrograph that measures radial velocities does exactly that with starlight: it correlates each spectrum against a mask of thousands of absorption lines. HARPS-N's solar telescope on La Palma does it for the Sun seen as a star, every five minutes through the day (Dumusque et al. 2015, 2021). The resulting line depth, width, asymmetry and velocity are each averaged over thousands of lines at once, so a change in the lines shared coherently across all of them would show there with the full gain of the line list. We folded all four on the rotation clocks, with bits of 20 minutes to 4 hours, over 2015–2025 (108,486 quality-checked exposures on 2,080 days), under its own dated registration. None of the four changes in step with any clock: 15,463 tests, a best p of 1.0×10⁻⁵ against a threshold of 6.5×10⁻⁷, 0 of 40 wrong clocks firing. Test signals keyed on these clocks and added to the raw data are recovered at 3 cm/s in velocity and in line asymmetry, 3–10 cm/s in line width, and 3×10⁻⁶ of the line depth. For scale, 3 cm/s is about the precision the best planet-hunting spectrographs are built to reach. The first run used only a third of the exposures, because a limit carried over from the one-minute radio data discarded every session shorter than five hours. A second error then surfaced: the exposure times had been read in the wrong date convention, twelve hours late, which split every observing day in two during cleaning. It cannot move a clock's power, only the cleaning, and both runs were null. Each was corrected in a dated amendment, and all three runs are reported; the numbers here are from the corrected one.

The Sun's colour, at its Wien peaks. A star's colour is fixed by physical constants: Planck's law uses only h, c and k, and its peak sits at universal numbers, hν/kT = 2.821 per unit frequency and 4.965 per unit wavelength. For the Sun those fall at 884 nm and 502 nm. The red and green sunphotometers on SOHO/VIRGO, at 862 and 500 nm, happen to sit almost on them, by filter choice rather than design. Their ratio is therefore a thermometer for the Sun's colour, and a modulation of it would be a colour key defined by physics alone. (Red, green and blue as primaries belong to human eyes and were not used.) The three colour ratios had been searched before over all frequencies (§4.15), but never locked to a clock. Under their own dated registration we folded them on the rotation clocks (bits of 20 minutes to 3.5 days, 1996–2023) and on the p-mode clock (10 minutes to 2 hours), and HARPS-N's calcium H and K activity index on the rotation clocks. The Sun's colour does not change in step with any clock. The rotation search counted 17,579 tests, best p = 1.9×10⁻⁴ against 5.7×10⁻⁷; the p-mode search 63 tests, best 0.023 against 1.6×10⁻⁴; the calcium index 3,867 tests, best 1.6×10⁻⁴ against 2.6×10⁻⁶. All calibrate within the registered limit. The five-minute oscillation itself appears in every ratio at p ≈ 10⁻¹⁹⁷, as it must, because its amplitude differs by colour. A signal of 0.3 ppm in green is recovered on rotation-timed bits and 0.1 ppm on p-mode-timed bits: through the Planck curve, a change in the Sun's colour temperature of about one thousandth of a degree (0.9 and 0.3 mK).

Arrival times at an interferometer. An interferometer compares the phase of the same radio wave at two antennas, which gives the difference in its arrival time. The Expanded Owens Valley Solar Array (EOVSA) makes that measurement on the Sun at 1–18 GHz, and its raw one-second data are public. A single ten-second sample is uncertain by 1–3 ps; folding months of them on a clock takes a repeating pattern down to a tenth of a picosecond, about 30 micrometres of path. Three registered searches were run on 2019. A one-week pilot (15–21 April) on the p-mode clock: null, to 0.3 ps. The full second quarter (5,546 solar scans, 6.7 million one-minute points from 169 antenna-pair series): on the p-mode clock, 25 tests, best p = 0.007 against a threshold of 4×10⁻⁴, null to 0.1 ps; on six rotation clocks, 9,294 tests, best 7.7×10⁻⁶ against 1.1×10⁻⁶, not repeated in the two halves of the data. The Sun's microwaves arrive at EOVSA's antennas with no clock-locked change in their timing above 0.1–0.3 ps for bits of 1.3–1.7 hours, or 1 ps at two hours. For multi-hour bits the search is far weaker: EOVSA's raw phases drift and must be cleaned in runs of about an hour, which removes most of a slow signal, so the limit at five-hour bits is 3–10 ps, and at eight hours even 30 ps is not recovered. The first report of that search claimed 0.1 ps for every bit length; its test signals had been added after the cleaning instead of before, and the correction is on its registration. A third search fitted the pattern a real shift of the Sun's radio image would leave across the baselines, a matched filter, and scanned every frequency that any solar rotation rate could produce. It too is null: no shift above about 300 milliarcseconds (some 220 km at the Sun) on the p-mode clock. For five-hour bits it excludes nothing below about 2,000 km, and it gained nothing over the simpler sum, because the antenna pairs stable enough to use are short (median 38 m).

Two corrections to the clocks. Preparing these runs exposed an error in the earlier searches. The equatorial and interior clocks had been built from the Earth's mean orbital motion, but the Sun's rotation seen from the Earth follows its true, elliptical orbit, which departs from the mean by up to 2° of solar longitude over a year. Multiplied by the harmonic number, that smears a signal locked to those clocks at harmonics above about 50. The Carrington clock, taken from the ephemeris, was never affected, and every sensitivity quoted above was measured on it, which is why the error went unseen. All clocks from the EOVSA searches on use the true orbit. Those searches also add sidereal clocks, which follow the Sun's rotation against the stars: a beacon meant for any observer, not the Earth in particular, would key on those, and no named-clock search had tested them.

The searches, rerun. Every earlier search that used a rotation clock was run again under a dated registration, with the corrected clocks and the sidereal ones added: twenty searches, from the 2.5° grid of AIA cells (22 million tests) to HARPS-N's line list. The registration says openly that the first results had been seen. All twenty remain null. Twice as many clocks makes every threshold about twice as strict, and nothing comes near one: the Carrington results are unchanged, as they must be, and the corrected and sidereal clocks reach the same depths as the Carrington clock did, with two exceptions. On ground-based data a sidereal harmonic can land on the daily observing comb (at 131-minute bits HARPS-N's sidereal clock sits at 10.99 cycles a day, on the 11-a-day line) and loses about a factor of ten there. And the new clocks open a new way to be fooled: in SDO's disc brightness the strongest sidereal test (p = 2.5×10⁻⁸, below its threshold) falls exactly on the difference between the satellite's 11th orbital harmonic and the Sun's 8th rotational harmonic, a cross-term of two ordinary signals that a sidereal clock happens to hit and the wrong clocks cannot mask.

The surface velocity field. The last new observable is the motion of the Sun's surface itself, mapped every hour through 2014 by SDO's Dopplergrams, the first velocity search of the program resolved across the disc. Its registration used the corrected and sidereal clocks from the start. As a positive control, the fit recovers the Sun's rotation: 1.84 km/s at the equator, falling to 1.48 km/s at 55°, the known rotation of the surface gas (Snodgrass & Ulrich 1990). No velocity is locked to any of the eight clocks (761,691 tests on 10° cells, best p = 10⁻⁵ against 1.3×10⁻⁸; the disc average likewise null), down to 10 m/s in a single cell and 0.1 m/s averaged over the disc.

The spectrum, channel by channel. EOVSA also records each antenna's total power in 451 frequency channels from 1.1 to 18 GHz. The simplest test of those spectra assumes nothing about how a signal might be spread across frequency: fold every channel of every antenna on its own clock and add up the power at each harmonic, so a signal present in many channels accumulates whatever its arrangement. Its registration assumed the channels were independent. A dry run with the clock scrambled day by day showed they are not, since the same Sun is in every channel, and it returned p = 10⁻⁸⁵ with nothing there. An amendment, written before the real data were folded, takes the null from the same sum at nearby off-harmonic frequencies; the dry run then behaves as chance should. On the second quarter of 2019 (50,461 one-minute spectra, 14 antennas, about 4,100 channel series in each polarization) the search is null: 3,998 tests on seven clocks, best p = 0.021 against a threshold of 2.5×10⁻⁶, and no wrong clock firing. It is also weak. The Sun's own variability is common to every channel, so the sum behaves like a single look: a test signal at 10⁻³ of each channel's power reaches only p ≈ 10⁻³ on the p-mode clock and leaves no trace on the rotation clocks, which need 10⁻¹ of each channel's power (10⁻² at bits under two hours). Adding channels without knowing the pattern buys almost nothing; a spreading code is what would make them add coherently.

A spreading code across the spectrum. A designer who wants to be found by a capable receiver, but not by a casual one, might spread a weak signal across frequency with a code that any mathematically literate civilization would arrive at independently, laid out in a physical unit rather than in anyone's instrument channels. We registered such a key space before any spectrum was folded. Frequency is measured in units of the hydrogen line, and the keys are: harmonics of hydrogen, primes, octaves and Fibonacci numbers; quadratic-residue (Legendre) sequences; the binary digits of π, e, the golden ratio, the fine-structure constant, the proton-to-electron mass ratio and other constants; geometric series of the same constants; hydrogen's own recombination lines; and the harmonic series sorted into the families music uses (octaves, pitch classes, just intonation, and the spectra of square, sawtooth and triangle waves). That makes 1,581 distinct keys, each made to sum to zero across the channels so that the Sun's broadband variations cancel. Each key combines the 451 channels of every antenna into one series, which is folded on the seven clocks; 2,000 random keys of the same shapes go through the same pipeline as a control. No key unlocks anything: 6.3 million tests, best p = 6×10⁻⁵ against a threshold of 1.6×10⁻⁹, and neither any random key nor any wrong clock crosses it. The right key makes the channels add coherently. A code spread across the band at 10⁻⁴ of each channel's power is recovered on the p-mode clock (p ≈ 10⁻³⁵), more than ten times deeper than the plain sum reaches at all, and at 10⁻³ on the rotation clocks, at every bit length from 1.3 to 8 hours. Keys that touch only a few harmonics gain less: 10⁻³ on the p-mode clock and 10⁻² on the rotation clocks.

Five more observables. Five further searches, each registered before its data were folded, looked where none of the earlier ones had. Low-frequency radio from the Wind spacecraft at L1 (50 kHz to 14 MHz, 1994–2021), emission from the outer corona that reaches the spacecraft at the speed of light. Soft X-rays from GOES-16 (2017–2025, ten-second data), the hot corona, with flares masked. Lyman-alpha and the extreme-ultraviolet lines of GOES-16's EUV sensor, and their ratios to Lyman-alpha. Limb darkening: how the Sun's brightness falls from disc centre to limb, measured afresh on 668,000 AIA frames from 2010 to 2025, which would change if the temperature gradient of the photosphere were modulated even with the total brightness held fixed. And constant clocks: uniform clocks built from the fine-structure constant (the hydrogen line × α⁶ and × α⁷, the Rydberg frequency × α⁹ and × α¹⁰, the electron's Compton frequency × α¹¹ and × α¹², periods from 43 minutes to 8 days), folded through thirteen data sets. A sixth search took the ratio of a microwave colour to an extreme-ultraviolet colour, so that a change in one layer of the solar atmosphere relative to another would stand out while common variations cancel. All are null. Signals as small as 3×10⁻⁵ of the limb-darkening exponent (about 20 parts per million of the limb's brightness), 10⁻⁵ of the Lyman-alpha ratios on the five-minute clock, and under 1% of the X-ray flux would have been seen.

What the safeguards caught. These searches mattered as much for what they exposed as for their limits. Folding by summing samples into phase bins coarser than the sampling copies the data's strongest lines to clock-specific frequencies: it produced a test at p = 5×10⁻⁹ in the EUV data that a fold with finer bins put at 0.8; the wrong clocks had caught it, and every later search uses bins finer than the samples. The Wind receivers jump by up to a factor of ten every day at 01:00 UTC, which put a comb of lines at every whole number of cycles per day; a 14.4-minute "candidate" at p = 10⁻¹¹⁵ lay exactly on the 100th harmonic of the day, and its phase stayed fixed in UTC rather than in the Sun's frame. With that masked, one line remained, at 156.6 minutes, on a harmonic of the Sun's sidereal rotation. A solar signal of that kind would reach every spacecraft at the same phase, so we registered a test on the twin STEREO spacecraft before downloading their data. STEREO-A does not see it, though test signals of Wind's strength injected into its data are recovered every time; STEREO-B sees the same frequency at a different phase. It is an artifact of the instrument family. The first limb-darkening data set had removed limb darkening before it was stored; it was rebuilt. None of these reached a threshold. Each was found by a control set in advance, and the corrections were dated and published before the reruns. The searches now also fix their multiple-comparison rules in advance: Holm's procedure for whether anything is there, and false-discovery control only for decoding a signal already confirmed.

The last checks. A final registered search folded the linear polarization EOVSA records in its cross-hand correlations, which a screen or a rotating polarizer would change without changing the total flux: null, to 4% of the cross-hand ratio at five-hour bits and 0.4% at twenty-one minutes. Two audits then went back over every earlier search. One asked whether test signals had been injected before all cleaning, as the method requires; where they were not, the overstatement is at most 6%, below the steps in which limits are quoted, except one EOVSA search already corrected. The other asked whether coarse phase binning had hidden anything: refolding the worst case with one sample per bin changed nothing. No earlier limit changes.

A blind test, and what it found. To test the whole chain, including the analyst, the principal investigator generated three data sets from real ones, injecting hidden square-wave signals or nothing, and sealed the answer with a passphrase behind a public commitment hash. The analysis ran the registered pipelines on the blinded files and sealed its report before the answer was revealed. The radio data set had nothing injected and was reported null. The HARPS-N data set carried a 64-minute signal in the line contrast at 1.9 times the stated sensitivity; it was reported on the exact clock, harmonic and series, with the amplitude within 6%. The third, the HMI magnetic field of 2014, carried a 15.6-hour signal on the whole disc at 2.2 times the sensitivity, and the registered pipeline missed it, so the round failed its own pass rule. The signal was there at p = 10⁻³⁴, but in a one-year record a wrong clock 2.4% fast lands its 41st harmonic on the true clock's 42nd, sees the same signal as a "line", and the mask meant to remove instrument lines removed the signal instead. The sealed report had nonetheless named it exactly, from the odd harmonics a square wave leaves at three and seven times its frequency. That became a dated amendment, and every earlier search was re-examined: in all but two, the mask had removed nothing below the threshold, so no strong signal was hidden; the two exceptions are a line at the third sideband of a daily harmonic and three tests sitting under far stronger clock-independent lines. The weakness is real: a strong signal at many harmonics, nearly all of them in records of a year or less, can be masked by its own reflection in a wrong clock, though it still shows in the count of masked tests, which is how we know none was.

Two more blind rounds. The second round added a sixteen-year record (the AIA disc) and placed half of its signals on exactly the harmonics where a signal can mask itself. It passed: both detectable signals were found on the exact clock and harmonic, one of them after the mask had removed it, and one at half the stated sensitivity, which shows the quoted limits are conservative at short periods. But no strong signal happened to be drawn, so the new rule was not tested. The third round therefore put only strong square waves on those harmonics. The new rule found two of them exactly, with their amplitudes within 9%, by their odd harmonics. It failed twice. A signal at 33 hours, twice the sensitivity measured at five hours, came out barely significant, because at long periods the noise is far redder and "the sensitivity" quoted at one period does not hold at another; the masked test, its weak third harmonic and its second half each just missed. And a strong signal's yearly sidelobe, four cycles per year away on another clock, passed every rule and counted as a false alarm. Both failures were named correctly in the sealed reports before unblinding, but a sealed note is not the registered rule, and the rounds are scored as failures. The fixes are registered: every search's sensitivity is measured as a curve across period instead of one number, and a sidelobe clause now reports a test lying a whole number of cycles per year or per day from a far stronger one as that signal's sidelobe.

Sensitivity across period. The curves are now measured for every search, at eight periods each across the range each one searched, except the one-year ultraviolet searches of 2014, which the sixteen-year searches of the same channel supersede. The quoted limits hold at five hours, where they were measured, and at shorter periods the searches reach as deep or up to three times deeper. At long periods they are shallower: three times for the sixteen-year ultraviolet disc and for the Nobeyama radio flux above six hours, and ten to thirty times for the magnetic-field discs at one to three days, for the GOLF velocity and for the colour of the Sun's light. The 2014 magnetic-field disc reaches only 10⁻² at 28 hours, which is why round three's 33-hour signal was missed. The HARPS-N line-profile search is flat from 20 minutes to 4 hours. A slow beacon, with bits a day or more long, could therefore be up to thirty times stronger than the headline limits and still have gone unseen in those channels. The later searches follow the same pattern. The radio, X-ray and ultraviolet searches on the GOES and Wind spacecraft and the limb-darkening search lose a factor of 30 to 100 between their shortest bits (minutes) and their longest (three and a half days): the X-ray limit is 3×10⁻⁴ of the flux at a few minutes but 3% at three days. Two quoted limits were corrected: the microwave/ultraviolet ratio's 0.3% was its best series (the median is 0.7%), and Nobeyama's 3×10⁻⁴ held at 2 and 9.4 GHz but not at 1 and 3.75 GHz (10⁻³). The sweep also exposed a flaw in one search's own test signals: the Wind/WAVES injections stopped once the signal was far past an ordinary threshold, which still lay above that search's unusually strict one, so some of its bands were reported as not reached without the deeper signals being tried. Tried properly, all 32 are reached at every period.

A fourth blind round. With both fixes in place, the fourth round scaled every hidden signal to the sensitivity at its own period and spread the periods across each search's full range. Every signal was strong in the data, and the sealed report named all three exactly: a square wave in HARPS-N's line depth, found by the rules with its waveform and an amplitude within 8%, and two signals at 1.3 and 2.5 hours that the registered rules could not report. The round therefore failed. Both missing signals sat on a sidereal clock, which is simply a fixed frequency, and one of the forty wrong clocks happened to have a harmonic a fraction of a resolution step away, so it saw the signal about as strongly as the true clock did and the mask removed it; at those short periods the third harmonic, which rescued earlier cases, lies outside the search. A real signal on the clock, though, peaks exactly at the clock's harmonic. A new rule, registered after the round, scans the frequency finely and reports a masked signal when its peak sits at the true harmonic and clearly away from the masking line, with tolerances set by how precisely a peak of that strength can be located. It recovers both missed signals and every earlier resolvable one, rejects a leak onto another clock, and, in a test with 400 artificial lines placed near tested harmonics in real data, mistook 2 for a clock signal. Applied to the real searches it changes nothing: the only masked tests on the real Sun, three in EOVSA's delays, peak well away from their harmonics. A fifth round, with the rule in force, is the test it has to pass.

A fifth blind round, passed. The fifth round used the same design, with the new rule applied automatically so that no judgement entered the decision. All three hidden square waves were found at the exact clock and harmonic, with amplitudes within 8%: in Nobeyama's radio flux at 2.6 hours, in HARPS-N's line depth at 28 minutes, and in the 2014 magnetic field at 49 hours, the long-bit regime where the third round failed. The fourth data set carried nothing and was reported as null, and a strong daily sidelobe of the radio signal was correctly reported as a sidelobe rather than a second detection. It is the first round to pass with every signal a square wave at two or more times the sensitivity at its own period. The new rule was not needed for any of these three; it acted once, correctly setting aside a leak of the 49-hour signal onto another clock, so its ability to rescue a masked signal has been shown on the earlier rounds but not yet blind.

A sixth round, aimed at the new rule. To test the new rule where it matters, the sixth round put every hidden signal exactly where the fourth round failed: on a sidereal clock, on a harmonic another clock nearly shares, and high enough in the range that its third harmonic falls outside the search; half were pure sine waves, which have no harmonics at all. It passed. All three signals were found at the exact clock and harmonic, with amplitudes within 8%, and the empty data set was reported as null. The AIA signal, a sine at 1.39 hours masked by a stronger-reading wrong-clock line, could be reported by no rule except the new one, which found its peak 0.013 resolution steps from the true harmonic and nine standard errors from the masking line. Over six rounds the record is three failures, each traced to a specific gap and fixed under a registered amendment, and three passes: an easy early one, then the last two, run with every fix in place.

A seventh round. The seventh went back to the full mix, with more empty data sets and weaker signals allowed. Three of the four data sets carried nothing, and all three were reported as null. The fourth hid a sine wave at 6.25 hours in the ultraviolet disc, masked by a wrong-clock line of equal strength; the new rule alone found it, at the exact clock and harmonic, identified it as a sine, and measured its amplitude within 2%. That makes four passes in seven rounds, the last three in a row, and two blind rescues by the new rule without a false alarm. No signal near the stated limit happened to be drawn, so how detection fades just above the limit is still to be measured blind.

An eighth round, just above the limits. The eighth round drew every hidden signal between one and two times the stated sensitivity, the weakest level at which the limits claim anything, and put the magnetic-field signal in a single 10-degree cell. Three of the four were found at the exact clock and harmonic, with amplitudes within 13%, two of them identified as square waves. The fourth, a sine in one cell, stood out strongly but was hidden under a much stronger wrong-clock line, and the new rule has not yet been written for single cells; the sealed report named the exact cell and harmonic, but no registered rule could report it. There was no false alarm. Across all eight rounds, signals below twice the stated sensitivity were found six times in eight: every one in the disc, radio and line-profile data, and neither of the two in single cells, which is the gap the blind tests now point to.

A limit that comes from the calendar. Writing the new rule for single cells showed why it could not have saved them. The cells were searched in one year of magnetic data, and in one year the forty wrong-rate clocks have harmonics so densely packed that one lies, on average, a sixtieth of a frequency step from any harmonic of the true clocks; no measurement of where a peak sits can separate them. In sixteen years the nearest one is typically a third of a step away, and the rule works. So in the one-year searches (the 2014 magnetic disc and cells and the 2014 surface velocities), a strong pure sine on such a harmonic cannot be reported, and those limits carry that caveat; a square wave is still caught through its third harmonic. On sixteen years of magnetic cells the rule recovered 21 of 24 test signals of exactly the kind that had been missed, and mistook 3 of 200 artificial lines for clock signals, so it was registered for multi-year cell searches. A ninth blind round then moved the magnetic data to the sixteen-year record. It passed: two hidden square waves, one in the radio flux at 4.2 hours and one in the ultraviolet disc at 50 hours, were found at the exact clock and harmonic, and the two empty data sets were reported as null. By chance the magnetic data set was one of the empty ones, so the cell rule itself still awaits its blind test.

Every rotation rate. The clock searches tested six rotation rates. To cover the rest, the brightest and longest records (the ultraviolet and magnetic discs over sixteen years and Nobeyama's four radio frequencies over thirty-two) were scanned at every frequency from three and a half days down to 1.3 hours, which includes every harmonic of every fixed rotation rate, and again with the yearly wobble that any rotation seen from Earth carries, for rotation periods from 25 to 35 days. That second scan can see a signal locked to the Sun's rotation at a rate nobody chose, which a plain frequency search would miss. Both are null: no rotation-locked signal at any rate, and only two plain lines, in the magnetic data, which are side-bands of a twelve-hour instrumental line created by the way the satellite's orbit was removed. Preparing the scan exposed something about the data: the ultraviolet disc's variations are dominated by flares, a thousandth of the samples carrying nine-tenths of the variance. Clipping them made the scan ten times more sensitive than the earlier clock search of the same data, 10⁻⁵ of the Sun's ultraviolet brightness at bits of a few hours. The clock search of the same data was then repeated, clipped, under its own registration: it is null, and it now reaches 10⁻⁵ of the ultraviolet brightness at bits of one to five hours, ten times deeper than before (bits of a day or more gain nothing, since flares are brief). The searches of single cells did not have this problem: a flare brightens a small patch, so each cell mostly sees its own ordinary noise, while the whole disc adds every flare on the Sun together.

A by-product: the Sun's global oscillations in its microwave output. The five-minute oscillation itself was never counted in the search, because physics puts power there. It was followed up because nobody appears to have seen it in whole-Sun microwave flux: the closest precedent searched the same four frequencies in 1977–78 and found nothing, with an upper limit of 2.7 K at 9.4 GHz (Morita 1979). Folded on the phase GOLF measures from a million miles away, Nobeyama's 9.4 GHz flux shows it at p = 1.7×10⁻⁴, with an amplitude of 1.3×10⁻⁵ of the flux (about 0.2 K). None of 1,000 time slides, which shift GOLF's clock against the radio data by a day to five years, comes close. An independent network, RSTN's four stations, weighted by each station's noise and tested at the phase Nobeyama predicted, finds it again at 4.995 + 8.8 GHz (Z = +2.77; 6 of 1,000 slides as extreme). Together that is about 4σ. It appears at all eight frequencies from 1 to 15 GHz across both networks, at 1–2×10⁻⁵ and at a consistent phase (89–152°).

Is anything riding on it? Before examining its structure we registered four tests (the file is dated and checksummed). It carries no circular polarization, as quiet-Sun emission should not. It is constant over 331 rotations (χ² = 553 for 660 degrees of freedom) with no phase flips. Its fractional amplitude falls as the Sun grows more active (−2.0σ) and its absolute amplitude does not rise (−1.7σ), as known for the p-modes themselves and unlike the oscillations above sunspots. It is broadband and smooth. No keying by a rotation clock was found, but that test cannot see even full on/off keying of a carrier this faint, as the registration stated in advance. So this is the Sun being the Sun: to our knowledge the first detection of its global oscillations in whole-Sun microwave flux, and a positive control showing the method finds a real 10⁻⁵ signal locked to a natural clock in noisy ground data. That makes its nulls elsewhere mean something. Two later checks sharpen it. Refolded with exact Fourier sums, the 9.4 GHz detection strengthens (p = 4.8×10⁻⁵), and with each series' noise measured from its own 1,000 time slides it stands at 4.2σ. A third instrument, EOVSA's raw total power over three months of 2019, was registered as a test but is too noisy to decide (its 95% upper limit, 1.4×10⁻⁴, lies eleven times above the signal): about 4σ from two independent networks remains the claim.

What remains. Gate 2, the detector-frame control, has not been run. Nothing reached a threshold, so it had nothing to examine, but any future candidate must pass it. So must a second vantage point: STEREO/EUVI (from 2006) and Solar Orbiter/EUI see the same Carrington cell at a different viewing geometry, which is the direct test for the artifacts above. The UV continuum is now searched over the whole mission, on the disc and in cells down to 2.5°; the corona only for 2014. The sixteen-year p-mode search has no measured depth yet. Not yet searched: the other six EUV channels, HMI velocity and continuum intensity, cadences faster than two minutes, cells finer than 2.5°, radio modulation slower than 8 hours or faster than 1.3 hours on the rotation clocks, radio bits shorter than 10 minutes, and linear polarization.

What it costs. At synoptic resolution this is not an HPC problem. Reprojecting ~7×10⁵ frames per channel is tens of core-hours, the periodograms are cheaper, and one channel over the full mission fits on one workstation's disk. The HPC-scale version is full resolution: 4096² at 0.6″ every 12 s in all seven EUV channels, petabytes in total, reaching sub-megametre patches and periods of tens of seconds. That version is a better use of a leadership-class allocation than brute-forcing T3–T6: it tests a hypothesis none of our searches can reach, and its null comes from the data rather than from a simulated Sun that must first pass a gate.

5.7 The receiver’s own position, and why coverage is not simply a matter of time

A gate keyed to capability invites an obvious thought about our own position: we have monitored the Sun for about five decades, the fast band for less than one, and the compute for a combination sweep has existed for perhaps fifteen years. Our capability is still rising steeply, and a null obtained early on a rising curve is worth less than the same null obtained later.

That reasoning is comfortable, and the data does not support the comfortable version of it. Coverage is not a monotonic function of elapsed time. Of the 53 pairs the sweep cannot reach, we asked how many would close if we simply waited:

The blocking channels are the Sac Peak Ca II K indices, which end in 2015 and appear in eleven blocked pairs each, and the ACE SWICS charge-state and abundance ratios, which end in 2011 and appear in five each. Successors exist — Solar Orbiter’s SWA/HIS has returned heavy-ion charge states since 2020, and Ca II K synoptic programs continue — but a successor is not a continuation: closing these pairs requires cross-calibrating a new record onto a retired one. (Several other channels are flagged as ended only because our retrieved copy stops — total irradiance, F10.7 and the neutron monitors are still produced. Refreshing those would not change the 53.)

Our capability is not rising in every dimension: it is rising in compute and cadence, and falling in continuity. The last 12% of the pair space is waiting on somebody flying a solar-wind charge-state spectrometer again. The binding constraint on this space is instrument continuity — a decade-long gap removes pairs from the reachable set until somebody does the cross-calibration. The same arithmetic leaves 1,126 of 4,060 triples unreachable.

****The compute dimension, quantified.** The paper measured 3.0×10⁴ statistic evaluations per second on 80 cores (§5.2), which with the 26-hour triple sweep implies about 3×10⁷ FLOP per evaluation on daily-cadence series and about 10⁹ on one-minute series. Six tiers of the combination space follow, using direct exceedance counting with the surrogate budget the paper’s own rule requires, M > N/α — the conservative design, since a designer sizing a gate cannot assume the receiver has the tail-fit screen of §5.2, which brings the larger tiers down by 10³–10⁴ but must not be used as a verdict. Sixty observables is the thirty of §3.2 plus the thirty candidates named in §5.5 and §3.6; ten forms is the four of §3.3 plus arrangement, timing, profile-ratio and polarization forms. T6 is everything we can currently name, not a claim that the space ends there.

Machine rates are TOP500 Rmax, or FP64 peak for GPU clusters, × 0.3 for an FFT-heavy statistic (× 0.5 of peak for the Cray era). GPU clusters are quoted at double precision; the FP8 and BF16 figures used for AI training are 30–100× larger and do not apply to this workload. FLOP per evaluation is calibrated from one workstation and one statistic; a third-order surrogate of the kind §4.9 calls for could cost 10× more per evaluation, and the tiers move together. Compute is necessary and not sufficient: every error in §5.6 was made with adequate compute, and the gate is passed by a receiver that can search and validate.

None of this is offered as an account of the nulls, and §5.3 forbids using it that way.** What it bears on is the weight a reader should give a coverage figure: a coverage fraction that cannot be improved by waiting is a different quantity from one that can, and this paper should not have quoted the first as though it were the second.

5.9 Scored against the nine axes of merit

Sheikh (2019) organizes technosignature searches along nine axes — four functions of us, five of the technology sought — and the framework was developed at the same 2018 workshop whose taxonomy §1.1 sets this work against. Scoring against it is the honest way to answer where does this sit, and two of the answers are “worse”.

The two concessions, stated plainly.

Inevitability is the weakest axis and cannot be argued away. This architecture requires a civilization to want to signal, to prefer a capability gate to a broadcast, and to choose a combination we happen to measure. §2 argues each is the efficient choice; efficient is not inevitable. Radio leakage and waste heat score higher because they require no intent at all, and the arithmetic of §B.1 puts the chance that we measure a chosen pair at about one in five. What is offered in exchange is that the search costs nothing — the archives are already collected — so a low prior is affordable here in a way it would not be in a proposal for observing time.

Ambiguity is limited by the analysis, not by the physics, and §5.6 is the evidence. The claim that a dimensionless carrier is hard for nature to imitate is true and is not the binding constraint. In the course of this program four distinct artifact families were identified — instrument lines carrying sidebands sixty times wider than the veto written for them, solar-cycle harmonics, harmonics of the analysis’s own detrend window, and the sampling Nyquist — and three were unanticipated. Every control that failed was a control incapable of returning “no”. The pipeline generates candidate structure faster than the sky does, and a search of this kind is bounded by how well that is policed rather than by how exotic the carrier is.

6. Conclusions

We have asked what a technosignature search looks like when the channel is not known in advance — no dedicated radiating apparatus assumed, the carrier possibly a relationship among observables. Two constraints make the question finite: a sender sharing no units can encode only in dimensionless quantities, and a sender minimizing energy gates the message on demonstrated capability. Against the resulting combination space we ran 30 searches of public archives, none of it data collected for SETI.

The one detection is a re-detection: solar p-modes in GOES EXIS Mg II irradiance, comb spacing 135.1 and 135.0 µHz on the two spacecraft against an accepted 134.9. Priority belongs to Eden et al. (2024), and we claim none. Its value is as a positive control whose answer was fixed in advance and corroborated by an independent group — a stronger check on the pipeline than any self-designed injection, and it correspondingly strengthens the twenty-six nulls.

We do not claim that the framework of §§2–3 is supported by these results, and we do not invoke it to explain them. A gating hypothesis that absorbs negative results has stopped being a hypothesis. The nulls are reported as nulls: each carrier was searched at a stated sensitivity and was not found. What the framework is credited with here is generating the searches, not surviving them.

The limits stand independently of the argument that motivated them. A reader who rejects §§2–3 entirely is still left with twelve injection-verified bounds on dimensionless modulation of solar output, at cadences and in combinations not previously examined — and with the observation that the archives are far more informative at two minutes than at one day, which is a fact about solar physics and not about SETI.

Three things would advance the problem, in ascending order of cost. Search the fast band on channels already recorded, in the order §3.6 gives: VIRGO/SPM and TIM at minute cadence, first as a transit-timing-sequence search after Arnold (2005) and then coherently at 10⁻⁷; sub-minute EUV from PROBA2/LYRA and SDO/EVE ESP; RSTN spectral indices at one second; and BiSON and GOLF mode frequencies. Complete the pair enumeration and begin the triples, pre-registered — completeness is only credible when it is declared in advance. Measure the five quantities that have no continuous record at all: core g-modes, the interplanetary electric field, continuous solar polarimetry, high-latitude solar wind, and a disc-integrated irradiance monitor off the Earth line (§4.12).

Wright et al. (2018) established that the value of a SETI result lies in the fraction of a defined space it excludes; this paper applies that standard to a space whose axes are quantities rather than pointings. By that measure the honest summary is not that nothing was found, but that the pair space is now essentially closed and the triple space was attempted and returned void — closing the pairs required no new instrument, only seventeen archives that already existed — that the first viewpoint test bounds the mechanism §2.7 proposes rather than modulation in general (§4.12), and that the framework’s own ordering names where the next searches belong: in archives that exist, at sensitivities already achieved, and not yet examined (§3.6).

7. Addendum (September 2026): the neutrino control channel, measured

Twice in this paper (§2.5, §3.2) the solar neutrino flux is set aside as the one control channel no mechanism proposed here can touch: nothing at the photosphere or corona can modulate a flux produced in the core. A control that cannot be read, however, is an argument rather than a measurement. Following a data request to the Super-Kamiokande collaboration — answered by its spokesperson within a day, with a pointer to a public dataset finer than the one we asked for — the channel has now been read, in three searches run after this paper closed. They are numbered N1–N3 and are reported here as an addendum; they do not alter the counts of §4 or Appendix A.

Data. Two independent detectors, two flux components. Super-Kamiokande: the ⁸B flux over 5,804 live days, 1996–2018, in 1,343 bins of ~5 days — the series the collaboration recommended in response to our request (H. Sekiya, private communication, 2026). Borexino: the ⁷Be-window event rate at 8-hour binning, 9,055 bins, 2011–2021, from the collaboration’s open-data release accompanying its orbital-parameters analysis (Astroparticle Physics 145, 2023). Both are released binned; per-event data remains collaboration-only in both cases, which caps the fastest accessible timescale at 16 hours. Every published time-variability search on either dataset is a periodicity search; the aperiodic question this paper asks had not been put to either.

Method, in this paper’s terms. A battery of nine statistics — band-summed Lomb–Scargle power, autocorrelation, runs, and block variance at five scales — on per-phase standardized residuals, with the surrogate matched to the statistic as §3.4 requires: within-phase (SK) or within-segment (Borexino) permutation, valid here because the null hypothesis is a constant flux with independent measurement noise, so residuals are exchangeable — the property whose absence voided the TSI runs recorded in §5.6. A max-statistic correction covers the nine-fold multiplicity. Every limit is injection-verified; none is analytic. False-positive rates were measured on cross-model surrogates: 0.005 and 0.013 at a 0.01 threshold. The physics-supplied control (§5.6) is the annual eccentricity modulation of each rate, ±3.4%: recovered in Super-K with response 0.77 ± 0.19 against 1.0 expected, and in Borexino at 5.6σ with maximum on Dec 31 against a true perihelion of Jan 3 — where it also calibrates the solar fraction of the total rate (27.1 counts/day per 100 t), so the Borexino limits are quoted in fractional-solar units with no assumed background split.

One artifact, attributed. The only statistic to approach significance anywhere in the campaign — Super-K block variance at the 120-day scale, p = 0.0125 before correction — lives entirely in the SK-III detector period (2.76 against 1.04, 0.72, 1.26 in the other three periods). Structure that respects detector-phase boundaries is systematics, not signal; it is consistent with the ~2% slow scatter expected from the released errors being statistical only.

What the result means. Read as this paper’s control channel: quiet, as the architecture requires — a detection here would have pointed past every surface mechanism this paper considers. Read against the architecture §2.7 rejects on cost, a device modulating the fusion rate itself (the inverse of the Learned et al. Cepheid scheme): these are the first aperiodic limits on it, and the 1–3% amplitude band such a beacon could plausibly occupy remains open in every band measured. Closing it awaits a Super-K release extending past 2018, or the JUNO / Hyper-Kamiokande era. Of the three limits the cross-detector coherence is the cleanest: no instrumental effect correlates across Kamioka and Gran Sasso.

How the deepest register could be written at all. The core is opaque to photons and transparent to neutrinos: energy random-walks out over ~10⁵ years while the neutrino flux reports the fusion rate at 8.3 light-minutes, so this is the one register whose bandwidth is set by the actuator rather than by transport. And the star is its own amplifier: ⁸B neutrino production scales as roughly the 25th power of core temperature, so a perturbation of 4 parts in 10⁴ in the production region is a 1% flux modulation at the detector. What the architecture requires is an actuator embedded where chemistry does not exist — at 15.7 MK thermal energy is a thousand times any molecular bond — which is capability far beyond the coronal swarm of §2, and the energy ledger of §2.7 prices it out for a pointer. It is retained here as the strongest-capability variant of the design, stated as assumed; the control-channel claim of §2.5 stands, because no surface mechanism reaches the core. What matters operationally is that the observable is defined regardless of the mechanism’s plausibility — aperiodic structure in the flux — and N1–N3 have now measured it.

And the receiver is only now acquiring the capability to read it. Hyper-Kamiokande — 260 kton, a fiducial volume ten times Super-Kamiokande’s, photosensors of roughly twice the detection efficiency, operations from 2028 — multiplies the solar event rate by an order of magnitude: from ~15 to ~150 events per day. The per-bin statistical error of the N1 series falls from 16.6% to ~5%, and scaling the injection-verified limits of the table above, the same battery run on one decade of Hyper-K data reaches ≈2%, and on two decades ≈1.5% — inside the 1–3% window the amplifier physics allows, for the first time. Day-scale binning also becomes statistically meaningful (~150 events/day), moving the fast edge of the band from ten days toward two — though whether data is released at that binning is a policy question, and the experience recorded above says to ask early. If the channel is in use, the situation is exactly the shape §3.6 predicts for a gated register: the N1–N3 nulls bound the loud versions, and the receiver is at the beginning of the detection threshold, not past it — the gate opens on detector mass we are still pouring.

On access. Both datasets were public before we asked; one had been public for three years without the aperiodic question being put to it. The Super-Kamiokande collaboration’s reply also stated plainly what will not be released and why — per-event times, because the background rate is not time-stable and only the collaboration can interpret it. That is the correct shape for a data policy: what exists is findable, and the boundary is a physics argument, not silence. We thank the collaboration, and H. Sekiya in particular, for the prompt response and the pointer to the extended series, and the Borexino collaboration for its open-data release.

Appendix A. All 30 searches

Thirty searches, one row each: null (no detection), void (the detector or its null failed a positive control, so the row constrains nothing), detection. Numbering is the order in which the searches were run within the wider program, retained for traceability to code and data; the grouping is by domain.

Why each search ended where it did. “Null” is not one thing, and a reader deciding where to spend effort next needs the distinction. Across the thirty, the reasons sort into six:

Only the first is an astronomical statement. The next three are statements about the analysis, and the last two about the data. That distinction matters more than the tally: twenty-six nulls sound like broad coverage, and the honest reading is that twenty-two channels were searched competently, three searches were too weak to count, two were defeated by their own machinery, and one was defeated by the data.

A.1 Solar, fast band (14)

A.2 Solar, slow band (10)

A.3 Frame & geometry (4)

A.4 Outer solar system (1)

29 rows: 25 null, 3 void, 1 detection. Eighteen of the twenty-six nulls state a sensitivity — 15 injection-verified amplitude limits and 3 measured false-alarm-and-power pairs — and the other eight are enumeration or control results that bound structure without quoting an amplitude. Per-search method, code and data provenance are in the supplementary material.

Appendix B. The datasets

A bird’s-eye inventory of every archive processed in this program: who collects it, when it starts, how many samples it holds, at what cadence, and — where an archive was not used — why.

Every row count, start date, end date and duty cycle below was measured from the files on disk on 2026-09-12, not quoted from documentation. Provenance (instrument, agency, archive) is from the acquisition code and the Data Availability table. Where a figure could not be measured it is left blank rather than estimated.

B.1 At a glance

What has been asked of them, and what came back

Four voids, and they are not all the same failure. Two are detector failures. Three are null failures — the circular shift assumed pairwise structure was irrelevant, the common-phase surrogate assumed stationarity, the envelope surrogate assumed the nonlinearity was pure amplitude modulation, and all three assumptions are false. The distinction matters to anyone picking this up: a void from a bad null is repaired by a better control, never by more data.

B.2 Daily solar and heliospheric channels

These thirty form the grid the combination space is enumerated over. All are resampled to a common daily grid spanning 1980-01-01 to 2025-12-31 (16,802 days); “rows” is the number of days actually present, so the fraction present is rows/16,802.

Why the grid starts in 1980 even though several records run far longer: the grid is the intersection window chosen so that the majority of channels are live. Sunspot area and sunspot number extend to 1874 and 1700 respectively and are truncated here, not lost. Retired records are the binding constraint on pair coverage — ACE SWICS ended 2011, Ca II K 2015, sunspot area 2016, Mg II 2013 — which is why 12% of the pair space cannot be closed by waiting.

B.3 High-cadence solar — the fast band

EUVS bands: 25.6, 28.4, 30.4, 117.5, 121.6 (Lyman-α), 133.5, 140.5 nm. GOES-16 and GOES-17 overlap for 1,121 days, which is what makes the twin-instrument gate possible: a candidate must appear on both spacecraft. EVE ESP carries CH_D, a dark channel processed through the identical pipeline as a null control.

B.4 Helioseismology — SOHO

Two features make this group unusually well controlled. GOLF PM1/PM2 are two photomultipliers inside one instrument, so the twin-detector gate needs no cross-calibration. And GOLF measures Doppler velocity while VIRGO measures photometry — genuinely different observables of the same oscillation, so a candidate can be required to appear in both.

Two caveats that must travel with any limit from VIRGO SPM. The L2 header states a seven-degree polynomial fit and a two-month highpass filter, so this product is blind by construction to modulation slower than ~2 months. It also states correction for orbit, degradation, outliers and “attractors” — and an outlier step can remove exactly the impulsive or quantized structure these searches look for. The first is a hard sensitivity limit; the second is testable by injection and must not be assumed either way.

VIRGO SPM also carries strong instrument lines at exactly 180.0000 s and 360.0000 s (3 and 6 minutes, being 2/3 and 1/3 of the 60 s Nyquist), reaching R ≈ 19,000. They are identifiable a priori as exact multiples of the sample interval and are excluded from the search band.

Rows 6 and 7 both pass all their gates (§4.15, §4.16). The comb came out at Δν = 135.00 µHz against 134.9 predicted from √(M/R³) with r ≈ 0.90 on all three GOLF channels; the known solar-cycle frequency shift came out at 0.584 µHz peak-to-peak (fractional 1.89×10⁻⁴ against a literature ~1.3×10⁻⁴); and PM1 and PM2 agree at r = 0.982 over 104 segments. Both gates initially failed on bugs of my own — a continuum filter of 501 bins = 0.61 µHz against a mode linewidth, which flattened the modes it was meant to normalize against and returned r = 0.004, and an unguarded parabolic interpolation that produced shifts of order 10²⁹ µHz. Row 6 (VIRGO SPM) still fails gate 1 on GREEN and RED for the same reason in a different place: a 2,001-bin envelope filter is 2.3 µHz against an envelope ~1,000 µHz wide. Its search is clean — zero of 7,164,744 bins above threshold on every single channel, gate 2 passing nothing — but no null is claimed until the gate passes.

The failure class is worth naming. Three times in one session: a smoothing or masking window chosen without checking it against the scale of the feature it operates on. §5.6 records one instance (“a re-implementation of the mode-comb detector returning 127 µHz where the original gave 135”) as a single incident. It is a class, and it is the most productive thing the gates have caught.

B.5 Beyond the Sun

B.6 Cross-viewpoint

The level-3b MAVEN product is deliberately not used — it is FISM-M model output, not a measurement. The PDS PPI mirror at UCLA is used rather than the LASP SDC because it serves 9,000 days/hour against 40 — a 245× difference that turned a week of acquisition into an afternoon.

B.7 Archives not used, and why

Stating these matters: a coverage claim is only meaningful if what is missing is named.

The generalizable lesson, which cost four revisions: a dead primary source is not an unavailable dataset. Rows 5, 6 and 7 sat marked “capable but unsearched” across four drafts on the strength of one true fact and one wrong inference drawn from it. Before recording an archive as blocked, check for a mirror.

B.8 Notes for a researcher picking this up

What the coverage actually is. Pair space is 88% closed (382 of 435 pairs); the residual 12% is limited by record overlap between retired instruments, not by sensitivity, and cannot close by waiting. Triple space returned void. Quadruple space returned void for a different and sharper reason.

The three-clause regularizer is the load-bearing constraint on which channels may be combined: a candidate quantity must be dimensionless, independently measured, and must not share an instrument, spectrum or denominator with its partner. Plasma β and the Alfvén Mach number were dropped under this rule despite being dimensionless — they are functions of density, temperature, speed and field, all already in the set, and testing |B| against β tests |B| against a formula containing |B|. Including them produced sixteen spurious Bonferroni survivors. Mg II from EXIS is excluded from EUVS band combinations for the same reason.

Cadence is the single biggest lever on sensitivity. The same instrument, the same days, the same pipeline is 245× more sensitive at two minutes than at one day. Any future work should start at the fast end, which is why the 0.25 s ESP and 20 s GOLF records are the most valuable rows in this table despite being the shortest.

Software

On the scale of the tooling, and its reliability. The program as reported is closer in scale to what a small team would undertake than to one investigator’s hand-work, and a reader assessing this many results is entitled to know how they were produced. Claude Code is a tool and not an author: Robert Griffin is responsible for every number here, and that responsibility is not nominal. Errors made by the tool and recorded in this paper include most of the §5.6 table, a statistic that sat at 0.954 for data and surrogates alike and so could not fail (§4.9), and a re-implementation of the mode-comb detector returning 127 µHz where the original gave 135. Each was found by checking rather than by inspection: a result is worth what its controls are worth, and a pipeline that produces limits faster than they can be checked produces limits that have not been checked. The audit reported here was run against commit 0aa39cd, and it checked 20 quoted numbers against the result files that produced them: 20 reconcile, and the one manuscript error it found — a triple count that gave the reachable figure where the completed one was meant — is corrected in §4.9, restated at submission against whatever commit is submitted; the repository README carries the claim-to-script table, and two of its checks execute directly from a clean clone.

Acknowledgments

The analysis, figures and manuscript were produced using Claude Code (Anthropic); its scope and documented failures are set out under Software. This work used public archives maintained by NOAA/NCEI, NASA GSFC and JPL, the ACE Science Center, LASP, the Royal Observatory of Belgium, NRCan, NMDB, the Wilcox Solar Observatory, the CDAW Data Center and CDS/VizieR, including the US Air Force Radio Solar Telescope Network 1-second data served by NCEI. The Nobeyama Radio Polarimeters (NoRP) are operated by Solar Science Observatory, a branch of National Astronomical Observatory of Japan, and their observing data are verified scientifically by the consortium for NoRP scientific operations. SDO data are courtesy of NASA/SDO and the AIA and HMI science teams. The HARPS-N solar data are from the HARPS-N solar telescope at the Telescopio Nazionale Galileo, La Palma, served through the Data & Analysis Center for Exoplanets (DACE) at the University of Geneva. EOVSA is operated by the New Jersey Institute of Technology as a community facility with support from the US National Science Foundation; its raw data are from the public archive at the Space Sciences Laboratory, UC Berkeley. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium); funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. None of these data were gathered with SETI in mind — which is the point of §1.5.

Data availability

Every result in this paper derives from a public archive. No proprietary, embargoed or author-collected data is used, and no observation was taken for this work.

Per-search provenance — exact file versions, retrieval dates and the quality flags applied — is given in the supplementary material, together with the analysis code and the injection curve underlying each limit.

One dataset was removed. An earlier version included three searches of a privately collected amateur-radio propagation archive (10.1 million spots). They are withdrawn: the archive is not public, so the searches were not reproducible, and the medium is HF ionospheric propagation, whose diurnal and seasonal structure is itself the dominant signal. Removing them costs no measured limit — all three fell among the nulls that carried none.

References

Data references (Solanki 2004; Usoskin et al. 2021; SILSO; NMDB; NOAA/NCEI GOES-R; ACE Science Center; LASP LISIRD) are given in the data availability statement.