How Fast Was the Sophon Link? An Upper Bound on Its Delay from Archived Trisolaran Reply Intervals, and What the Bound Leaves Open About the Mechanism

1. Introduction

We study the delay of the sophon link between Earth and Trisolaris using an inequality on archived reply intervals, a device first developed for the command circuits of the Fleet, where it separated the time a message needed to arrive from the time its recipient needed to answer (Harlund, 40 DE). No instrument built for the purpose has ever timed the link. The paper is written in Bunker Era, Year 12 (BkE 12), after the destruction of Trisolaris, so that no new interval can be taken and every record we use closed by Deterrence Era, Year 58 (DE 58).

Trisolaris describes the link as quantum entanglement between paired sophons, which gives it real-time contact with Earth across about four light-years. The description was relayed through the sophon-mediated exchange and has never been verified by a human instrument, so we treat it as testimony and not as a measurement. The testimony meets an obstacle in physics. The no-communication theorem states that whatever is measured on one member of an entangled pair leaves the statistics at the other member unchanged, so that a message needs a classical channel besides the pair, and such a channel cannot outrun light.

Three resolutions are open. The first holds that the link is not plain entanglement and uses the structure of the sophon itself, a proton unfolded from eleven dimensions into two, etched, and refolded. The number matches the eleven dimensions of M-theory, whose extra dimensions are assumed to be compactified, so that a mechanism in the compact directions lies within reach of known physics even though nobody has written one down. The second holds that the link is a channel outside standard quantum mechanics, which would then have to explain why no human experiment has ever registered it. The third holds that instant entanglement is only the name Trisolaris gave the link, so that the mechanism is unknown and the one observable claim is that the delay is short.

Earlier work bears on sophons within Earth and leaves the interstellar case open. Qin and Tao found no measurable lower bound on the relay latency of paired sophons at any terrestrial baseline (Qin and Tao, 152 CE), and the three-party Bell test of Hu restated that finding and showed that a violation can be certified without relying on the declared suspension of sophon activity (Hu, 59 DE). The longest baseline on the surface of Earth is about 20,000 km, a light time of 0.067 s, so those tests show that sophons on Earth coordinate faster than light across that light time. They say nothing about the pairing between Earth and Trisolaris, which might be a different channel or might degrade with distance.

The test needs a longer baseline.

We define the loop delay and the inequality that bounds it, describe the two archives from which we reconstruct the intervals, report the bound and its sensitivity, and state what it does and does not say about the three resolutions.

2. Methods

Our first record is the exchange log of the Secretariat of the UN Planetary Defense Council (PDC), the body that received the sophon-mediated statements of Trisolaris and entered them in its register (UN Planetary Defense Council, Secretariat, 8 BkE, Exchange Series 5). Between DE 6 and DE 58 the log holds 83 questions that Secretariat operators put to Trisolaris. Of these, 61 received a reply and 22 did not. Fifteen of the 61 lack either an entry-open time or a receipt time, which leaves 46 complete pairs. The pairing, the intervals and the sample flow are our own reconstruction from the log, since the log records no interval as such.

One registry clock stamps both events of a pair to the second, so that no synchronisation error arises between them.

We sorted the 46 pairs by what the reply contains, working from the text of the reply and before the intervals were examined. A reply is content-bound if it answers a specific item that the question introduced, such as a figure, a name or a procedural term, and 39 of the 46 meet this test. The other seven give generic acknowledgements which a reply prepared in advance could have supplied, and we exclude them from the bound while reporting what admitting them would change. The test does not show that a reply was composed after its question, but it removes the cheapest way for a short interval to arise without a link.

Let a question be opened at time t₀, let its content reach Trisolaris after a delay a, let the reply be composed in a time p, and let the reply reach the Secretariat after a delay b. The measured interval R is a + p + b, less a logging error of at most the resolution e. Because p cannot be negative, the loop delay a + b cannot exceed R + e. The inequality needs no assumption about the distribution of the composition time, the symmetry of the link or the speed of the signal that carries the content.

It does need one assumption about t₀. Sophons read anything written, printed or spoken, though not thought, so the content of a question reaches Trisolaris from the first moment at which it exists in any such form. We take t₀ from the opening of the log entry, and the bound is valid only for a question whose content was composed at the console after that opening, with no earlier draft, script or conversation. We cannot test this for any pair. The price of the drafting time that the entry already contains is a bound loosened by minutes.

Our second record is the Council's register of public announcements, which we use here as the paper's own source (UN Planetary Defense Council, Secretariat, 8 BkE, Register R-2) and which lists 214 announcements over the same years. In 11 of them the exchange log carries an unsolicited Trisolaran statement, within 24 hours, that cites the content of the announcement without ambiguity. We report these intervals from the release time, stamped by the register to the minute, but we do not use them as a bound. A Council announcement is drafted, circulated and approved in writing before release, and a sophon reads each draft, so the content reached Trisolaris before the release time and the interval can be shorter than the loop delay. The record shows how soon Trisolaris answered a public act.

The bound for a set of pairs is its shortest interval plus e, and we add three summaries to show how much rests on that interval. The first is the bound recomputed after dropping the one, two and three shortest pairs. The second is a one-sided 95% Clopper-Pearson lower limit on the share of replies whose loop delay was at most a stated time, which treats each pair as an independent trial, an assumption we cannot test and use only for scale. The third concerns a scheduled uplink: the log dates the changes in the stated conventions of the Trisolaran side, and a batch update would show as a regular rhythm in the gaps between them, which we measure by their coefficient of variation and compare with 12 gaps from a memoryless process, taking the null distribution from a fixed-seed simulation of 200,000 replicates with continuous gaps.

We convert each bound to a fraction of the light round trip, taking the distance as 4.0 light-years, so that the round trip is 8.0 years or 2.52 × 10⁸ s. We also convert it to cT/2, the longest path in each direction that a signal moving at the speed of light through a shortcut in the compactified dimensions could have taken. The distance is rounded, and a value up to 4.4 light-years would change the fractions by 10%.

3. Results

The 39 content-bound intervals ran from 131 s to 4,533 s, with a median of 928 s, or 15.5 minutes. The shortest is 2.2 minutes. The loop delay was therefore at most 132 s, which is 5.2 × 10^-7 of the light round trip, or one part in 1.9 million. If the link was symmetric, each direction took no more than 66 s.

Nothing hangs on one record. Dropping the shortest pair raises it to 162 s, dropping the two shortest raises it to 206 s, and dropping the three shortest raises it to 246 s, within a factor of two of the headline value. Admitting the seven generic replies lowers it to 68 s, a value we do not use.

Eleven unsolicited statements followed their announcements by 2,835 s to 40,910 s, which is 47 minutes to 11.4 hours, with a median of 10,578 s. These intervals are descriptive. Their start point is the release, not the first moment at which the content existed in writing, so they give no bound on the loop delay and we do not compare them with the first record.

All 39 content-bound pairs closed within 4,534 s, which gives a 95% lower limit of 0.926 on the share of replies whose loop delay was at most 4,534 s. Thirty-seven closed within an hour, with a lower limit of 0.847. These limits matter only if the delay varied between exchanges, since for a fixed delay the shortest interval already bounds every pair. Table 1 gives the three records side by side.

The first row of the table is the bound we report, the second is shown for sensitivity only because it admits replies that could have been prepared in advance, and the third gives no bound, for the reason stated in Section 2.

Converted to a length, the first bound is 2.0 × 10¹⁰ m, or 0.13 AU. A signal at the speed of light through a shortcut in the compactified dimensions travelled no farther than that in each direction, against an ordinary separation of 3.8 × 10¹⁶ m. The ratio of the two is the same 5.2 × 10^-7 as the time fraction, by construction.

The log dates 13 changes in the stated conventions of the Trisolaran side, such as the record format, the hours of the exchange and the topics accepted. The 12 gaps between them run from 1 to 9 years, with a mean of 4.0 years and a coefficient of variation of 0.62. For 12 gaps from a memoryless process the median coefficient of variation is 0.91, and a value of 0.62 or less arises in about 5% of cases. The rhythm is more regular than chance and far from periodic.

4. Discussion

The archive shows that, if each question was composed at the console, the loop delay of the sophon link was at most 132 s, about one part in 1.9 million of the light round trip, and we find it likely that the interstellar pairing did not operate at the speed of light. That statement is as far as the data go. The bound constrains how long the link took and says nothing about how it worked.

Earlier discussion held that the replies of Trisolaris appeared prompt, and nobody could turn that impression into a number, because every interval in the log contains the drafting time of the question and the composition time of the reply, neither of which is known. Thanks to the inequality of Section 2, the shortest interval becomes a bound that needs no model of composition, though it inherits the console assumption, and the number it yields, 132 s, is large against the 0.067 s of the longest terrestrial test and small against the 8.0 years that light would need. The terrestrial tests already show coordination faster than light between sophons on Earth. The contribution of the bound is to carry that result across four light-years, where nothing before it reached, and it neither contradicts the finding of Qin and Tao or that of Hu nor depends on either, since a pairing at interstellar distance could have been a different channel.

Of the three resolutions, the first is the only one that the bound constrains in a way we can state. A shortcut through the compact directions is a classical channel, since it carries a signal and not a measurement correlation, and so it does not breach the no-communication theorem. It would have to be shorter than 5.2 × 10^-7 of the ordinary separation, a severe demand on the geometry of the compact directions, which the bound neither excludes nor supports. The second resolution, a channel outside standard quantum mechanics, faces the further question of why no human experiment found it. The audit of archived collision cross sections found no sophon effect on near-thermal ion collisions (Rauch-Ibáñez, 11 BkE), which excludes a channel that perturbs such collisions and leaves open one that does not touch ordinary matter. The form of the second resolution that survives is therefore one confined to the sophon and its partner. The third resolution asks for nothing beyond the bound.

The position of the author bears on how the mechanism is judged. We write as a partisan of the Lurker framework, in which the Lurker (also 'Spirit', 'Spirit Mother' in some translations) fixed the form of physical law at the cosmogonic scale, and the Luo-Ji / Ye axioms are treated in this paper as local-domain-only. They constrain no physical property of matter and so say nothing about a link. The framework expects that, if the compact geometry was fixed with the form of law, the delay does not depend on the era of the exchange, whereas an engineered link might have changed with its devices, but we have not divided the 39 pairs by era, and the sample is too thin to test this, and an author within the Luo-Ji / Ye framework would compute the same numbers.

One alternative explanation of the short intervals deserves a direct answer. A sophon held on Earth, carrying a model of Trisolaran replies, could have composed the answers locally, and then the bound would describe Earth-side processing and not a link at all. The account departs from the record of the Deterrence-Era envoy, whose conduct was attributed to remote operation from Trisolaris, and we raise it as our own hypothesis. The data cannot exclude it, and the one observation that bears on it is the rhythm of dated changes on the Trisolaran side, since a model on Earth that needed periodic updates through the link would show a schedule. The 12 gaps have a coefficient of variation of 0.62, which is more regular than a memoryless process in about 5% of cases and far from periodic, so the rhythm lies between a schedule and chance and decides nothing. A change in convention is a policy decision and not necessarily a model update, and a local composer could have been updated continuously with no rhythm at all, so that our preference for the link rests on the testimony of Trisolaris alone and we state it at that strength.

5. Limitations

Everything above holds only if the archive was not altered. A sophon at the Secretariat could have stamped a false time, composed a reply in place of Trisolaris, or removed the longer pairs, and nothing in a record kept by humans can exclude this at any epoch from the Crisis to the Bunker Era. The reply-interval method has no defence against an author of the record who knows the inequality.

The bound also holds only if each question was composed at the console after its entry opened. A question planned in a memorandum or discussed aloud beforehand was readable by sophons before the entry, and for such a pair the interval understates the loop delay. We cannot identify these pairs, and the shortest interval, which is the one we use, is the one that a prepared question would most likely have produced.

These pairs are not a designed sample. Twenty-two of the 83 questions went unanswered, 15 replies lack a timestamp, and operators chose the questions for reasons of their own. The share limit treats the pairs as independent, the content test is ours, and a different sorting rule would move the 39. The distance is rounded, and the inequality bounds the sum of the two directions, so that the figure of 66 s for each assumes a symmetry that we cannot check.

This method reaches only as far as the shortest interval the archive holds. A shorter interval would sharpen the bound, and an archive with no content-bound reply under an hour would have loosened it by a factor of at least 27, so the result stands or falls on a single class of records, those in which an operator waited at the console. The present bound is one step toward a timing of the link, and the next calculation is the same inequality applied to the earlier exchange series, whose intervals have not been transcribed.