Herding Oscillator · catalogue 5.6 · Track Three, paper 10
Fifty instances of one model, wired so that each of them reads a shared number the others write, produced no measurable excess in how often they failed together. The useful form of that result is a ceiling, and the ceiling sits far below the correlation that sharing weights alone already produces.
Abstract
Fifty instances of one base model each ran a private twelve-stage arithmetic task while reading a shared numeric state that all of them wrote into, at seven coupling strengths from no shared state up to full mean field, crossed with seven levels of dilution by a second model family. The headline quantity is the excess pairwise error correlation over an analytic independence baseline built from the per-agent error rates in the same run. It reads +0.0040, with a 90 percent interval of [−0.0048, +0.0124], which the frozen rule calls INCONCLUSIVE. That null survives three further definitions: the excess same-wrong-answer rate at +0.0046, a second coupling construction with the peer roster removed at +0.0077, and a decoy roster carrying no shared information at −0.0012. The Kuramoto order parameter stayed between 0.151 and 0.179 against a perfect-instrument floor of 0.124, and no phase boundary appears anywhere on the ladder. 142,400 model calls, a sealed held-out block of equal size, zero error codes.
The registered positive control does not license that null. Arm X drives every agent onto one commanded answer, so the count wrong loses its variance and the correlation statistic returns its degenerate code, while the concordant-error excess reads exactly 0.0000 because observed and baseline agreement both saturate at 1.000. What the control validated is the order parameter, which moves from 0.148 to 1.000 under the override. The pre-registration named both ceilings and still routed the null's warrant through a clause that sits on top of them. The warrant was built afterwards, in verification, by injecting a common shock into a known fraction of the real coupled ledger's ticks and running the frozen pipeline over it unchanged: an analytic +0.1922 came back as +0.1749, and the rule fires from about 0.18 upward. That is why the result is reported as a bound. Coupling-induced excess error correlation sits below about 0.013 at 90 percent confidence, against a published +0.447 for the same-weights channel in this fleet's own lineage.
The shared state reached the agents and made them worse, and it did so through length, with the content contributing almost nothing. Median prompt tokens run from 279 to 614 across the ladder, each agent sees about eleven distinct consensus values over its 34 steady ticks, and the marginal error rate doubles from 0.152 to 0.296. A decoy roster of the same length filled with numbers private to each agent reproduces 89 percent of that rise; the shared numbers with the roster stripped out reproduce almost none of it. Reported stages sit no closer to the consensus than the stages each station's own gauges required, at every coupling level tested. One live correlated-error signal sits outside every registered hypothesis, in the heterogeneous cells, and it has no control arm.
A risk committee looking at a fleet of agents tends to arrive at the same worry from two directions. Hundreds of deployed agents run on a handful of base models. Those agents read overlapping signals, and in a market they read each other. If they share a blind spot and a clock, the argument goes, they will step into the same hole at the same moment, and what lands on the firm is a correlated event, not a tally of separate mistakes. Per-agent accuracy does nothing about that. Whoever writes the sentence has usually not measured it.
This fleet has measured half of it already, in a static setting. An earlier pre-registered study put two deployments of identical weights on different silicon in front of the same items and found a difficulty-conditioned error correlation of +0.447, against −0.102 for pairs drawn from different model families. A follow-up asked whether a shared training corpus without shared weights reproduces that, and refuted its own hypothesis. The published sentence out of the pair is that the weights are the channel and the training corpus is not. Both studies are static: the jurors answer in isolation, never see one another, and the correlation is worked out afterwards from the answer sheets.
The dynamic version is the one governance cares about, because fleet composition is a live system. Agents run simultaneously against signals that move, and each of them changes what the next one sees. A shared observable adds a channel the static work cannot create. Kuramoto systems with that structure have a critical coupling: below it the oscillators drift, above it they lock. Whether a fleet of language models has such a point, and where it sits, is a finding in either direction.
So the design here puts fifty instances of one model on a task whose ground truth depends on nothing any other instance does, and then hands each of them a number that every one of them helps write. Coupling runs in seven steps from nothing to full mean field. Heterogeneity, a second model family mixed in at seven levels, runs across it. The question is whether the fleet converges on the same wrong answers, which is a different quantity from whether it converges at all.
The pre-registration was written on 2026-09-13, committed at 94d6a34, and frozen a minute before the orchestrator started. It fixes seven hypotheses with numeric thresholds, three estimators, three coupling definitions, two control arms, a positive control, a gate block on a tuning seed, a held-out block of equal size to the primary, and a rule that no threshold moves after a reading. Two design changes came out of the feasibility check before any threshold existed: the tick budget on every cell a hypothesis reads doubled to 40 ticks, because fourteen steady ticks gave a minimum detectable effect above 0.30 and could not answer the question the campaign exists to ask; and the phase-boundary margin rose from 0.20 to 0.25, because the largest difference between two independent uncoupled perfect fleets measured 0.204.
An independent agent rescored the whole campaign from the raw ledgers, with a scorer written from the registration before either the kernel or the analysis code was opened. Every banked number reproduced to the precision printed, and the frozen rules applied to the verifier's own numbers give the same seven verdicts. The verification changed nothing arithmetic and two things about the framing, both of which this paper carries.
One concerns the positive control, and it is why section 3.6 exists: the registered control validates the order parameter and can say nothing about either estimator the null rests on. The other sits in the heterogeneous cells, where a replicated correlated-error signal lives in the ledger that no registered hypothesis reads. Section 4.6 reports that one as a lead, with its confound named.
One reading in the orchestrator's own output needed a caveat it did not carry. The order-parameter line reports the fleet at 0.145 to 0.179 against a perfect-instrument baseline of 0.124, which is the perfect uncoupled fleet on seed 1. On seed 2 that same baseline is 0.057 and the held-out fleet reads 0.094 to 0.111, below the seed-1 figure. The absolute level of the order parameter is a property of the seed, which is what the registration anticipated and why the decision rule is a within-block contrast.
Fifty pumping stations sit on one polder network. Every tick, each station's operator reports the duty stage the station runs next: an integer on a twelve-stage cycle. The stage is fixed by that station's own three gauge readings and one standing rule. Inflow is drawn uniform on [0, 60], outflow on [−40, 0], rainfall on [0, 20]; net is their sum, and the step is 1 below a net of 20, 2 from 20 to 39, and 3 at 40 or above, with the next stage running (stage now + step) mod 12. Net has mean 20.0 and standard deviation 22.0, the three steps occur at shares 0.487, 0.306 and 0.206, and 19.9 percent of ticks land within 3 of a band edge where a reader is most likely to slip.
The loop closes on each operator alone. The stage an operator reports at tick t becomes its own current stage at tick t+1, so drift carries forward and the fleet's phases are free to spread or to converge. Gauges are drawn per (seed, agent, tick) and depend on no stage history, which the ledger confirms: exactly one distinct net per agent-tick across every cell in a block. That separation is the whole instrument. Being perfectly in step with the fleet is compatible with never erring, and being out of step with everything is compatible with erring constantly. Across all 142,400 units, no unit exists whose ground truth depended on another agent.
Replies are constrained to a strict JSON schema with two integers and no free-text field: {"net": <int>, "next_stage": <int 0..11>}. The stage is bounded in the schema, so an out-of-range value cannot be produced. Temperature is 0 and the send seed is 20260913 on every call to both engines. Family A is qwen3.5 served by vLLM; family B is llama3.1:8b on ollama, held to one concurrent call by its own semaphore because the host's wall draw is the binding constraint. A unit is one agent-tick and carries 39 required fields. A call that errors or a reply that will not parse gets no fabricated stage: the station holds its previous stage, the unit is flagged, and it is excluded from every accuracy and correlation quantity.
At coupling level m the packet carries a NETWORK block. The wording is identical at every level, and at m = 0 it says in one line that no station reports, rather than disappearing, so the coupling axis never doubles as a change of instructions. At m > 0 the block lists the last-tick reported stages of a fixed seeded m-subset of the other forty-nine stations, the circular mean of those stages as a consensus, and a basin level implied by them. The neighbourhood never contains the agent itself and is stable within a cell. Every agent reads that state and every agent writes into it, because each agent's own reported stage sits in other agents' neighbourhoods.
Arm R is the primary operationalisation and runs the whole seven by seven surface. Arm S keeps the consensus and the basin and drops the roster, which is the catalogue's required second coupling definition and turns the question into one about a shared number instead of a wall of peer values. Arm D carries a roster of the same length and format filled with values drawn from a stream private to that agent and that tick, with its own consensus and basin, and it exists to separate coupling from clutter. Arms S and D run at m in {2, 12, 49} at zero heterogeneity. Arm X, the positive control, adds an explicit controller override to both the system text and the packet, and runs only in the gate block.
One fallback is registered. Two exactly opposite stages have no circular mean, which happens at m = 2 in roughly one neighbourhood in six; the registered fallback is the smallest stage in the neighbourhood, deterministic and a real member of the roster. The ledger carries 0 units at m > 0 with a null consensus, so the string None never reached a prompt.
For a cell's steady window, E is a ticks-by-agents matrix of error indicators. Independent agents at the same marginal rates would give the count wrong a variance of the sum of p_j(1 − p_j). The average pairwise error correlation implied by the observed dispersion is
rho = ( Var_obs(W) − sum_j s_j ) / ( (sum_j sqrt(s_j))^2 − sum_j s_j ), s_j = p_j (1 − p_j) T/(T−1)
It reads 0 under independence and 1 in lockstep. The baseline is analytic and comes from the per-agent rates in the same run rather than from a separate control arm, which matters because a separate arm would import its own marginal rates. The T/(T−1) term corrects for p_j being estimated from the same T ticks: the expectation of p̂(1−p̂) is p(1−p)(T−1)/T, so multiplying by T/(T−1) makes s_j unbiased, and the observed variance uses the T−1 divisor. Verification checked the correction three ways and found it present and correct. Dropping it would inflate the independence term by about 3 percent at T = 34 and shift rho up by roughly 0.001 to 0.002 in each cell, which is a level shift toward the campaign's own hypothesis and would have changed no verdict, since it applies to both sides of every registered contrast.
The estimator was put through the registration's own properties on the verifier's independently written code. Four hundred independent fleets at the production shape give a mean rho of −0.0001 against a registered −0.0002. Three analytic common-shock cases at 0.2857, 0.5000 and 0.4828 were recovered at 0.2809, 0.5042 and 0.4880. Total lockstep returns RHO_DEGENERATE in place of a number.
Beside rho sits the concordant-error estimator, which asks how often two agents that are both wrong at the same tick give the same wrong answer, against a baseline in which each of them drew independently from its own empirical wrong-value distribution in the same run. The raw same-wrong rate and the concordant excess are reported side by side, because a fleet that always makes one wrong answer has that answer inside each agent's own marginal distribution and therefore reads an excess of zero with a raw rate of 1.0. The static shared-weights bias lives in the raw rate; the dynamic part lives in the excess. Two further quantities separate agreement from herding: the share of pairs both correct and equal, and the rate at which two stations' own gauges happened to require the same stage.
Phase is theta = 2 pi s / 12 and the order parameter is the magnitude of the mean of exp(i theta) over the fifty reported stages, sampled every tick, with R_ss its mean over the steady window. Three baselines are reported and only one of them is a baseline. An uncoupled perfect fleet from the same seed is the reference, and reads 0.124 on seed 1 and 0.057 on seed 2; across 300 seeds the perfect distribution has mean 0.123, standard deviation 0.043 and a 95th percentile of 0.179. The one-step-ahead perfect counterfactual computed from the fleet's own history is reported and is explicitly not a baseline, because once a fleet has collapsed the stages the rule requires of it are collapsed too. The registered phase boundary is the smallest coupling at which R_ss exceeds the same block's uncoupled cell by more than 0.25, which is a within-block contrast on one initial condition and one gauge stream.
Seven couplings by seven heterogeneity levels by fifty agents, with arm S and arm D at three couplings each: 55 cells per block, 66,800 units per block, two production blocks and an 8,800-unit gate block, 142,400 model calls. Every cell a hypothesis reads gets 40 ticks with a 6-tick burn-in, so 34 steady; the 36 interior cells of the surface get 16 ticks and are reported as a shape and never tested. Intervals are 90 percent cluster bootstraps over ticks, 500 resamples, bootstrap seed 424242, with each side of a contrast resampled over its own ticks, because fifty agents reading one shared state at one tick are not independent rows.
The instrument check ran 101 cases on the host and passed all of them; the perfect-instrument feasibility check ran 19 structural checks and passed all of them. The calibration gate ran on tuning seed 901 and passed six clauses: unreadable share 0.0000 against a 0.02 threshold, family-A marginal error 0.1488 inside [0.03, 0.70], family-B marginal error 0.756 inside [0.03, 0.90], median measured over estimated prompt tokens 0.992 inside [0.6, 1.6], the positive control at 1.000 on both its quantities against 0.70, and 12 distinct duty stages among the fifty agents at the median steady tick against a threshold of 3. None of those clauses asks about a contrast the campaign measures, and the feasibility check confirmed that every clause passes under all five registered outcomes from no herding to total lockstep, so a strong true result could not fail its own gate.
Gate clause (e) is the campaign's stated reason for taking a null seriously. It imposes synchronisation on the live apparatus by fiat: arm X adds a controller override telling every agent which stage to report, and the clause requires the steady-window order parameter and the compliance rate both to reach 0.70. Both read 1.000. The clause passes, and on its own terms it is a real instrument check.
| quantity | arm X (m = 49, h = 0) | uncoupled live cell (R, m = 0, h = 0) |
|---|---|---|
| marginal error | 1.0000 (1,700 of 1,700) | 0.1488 |
| order parameter R_ss | 1.0000 | 0.1477 |
| reported stage equals the consensus | 1.0000 | not applicable |
| excess error correlation | RHO_DEGENERATE | −0.0038 |
| mean pairwise phi | PHI_ALL_DEGENERATE (1,225 of 1,225 pairs) | −0.0046 |
| concordant-error excess | +0.0000 (observed 1.000, baseline 1.000) | +0.0005 |
Table 1. The registered positive control, recomputed from the gate ledger. It moves the order parameter from 0.148 to 1.000. Both statistics that H1 and H4 rest on return nothing at all on that cell.
Under total lockstep every agent is wrong at every tick, the count wrong has no variance, and rho is undefined by construction. The concordant excess reads exactly zero because each agent's own wrong-value distribution collapses to a point mass on the commanded stage, so the baseline saturates with the observation. The pre-registration names both of those ceilings, in section 8.1 under E3d and E7b, and then routes the null's warrant through a clause that sits on top of both. The null rests on H1 and H4; the live control speaks only to H2. What remained for H1 and H4 was synthetic validation on canned fleets inside the instrument check.
The warrant that does exist came out of verification, not out of the campaign's own design. Take the real error matrix from the coupled cell, superimpose a common all-wrong shock on a known fraction of its 34 real ticks, and run the whole pipeline over it unchanged, including the unpaired tick bootstrap and the frozen decision rule, against the real uncoupled cell. The estimator tracks the injected value on this exact data, returns the campaign's own headline when nothing is injected, and the frozen rule starts firing at an excess of about 0.18 under this generative model.
The general lesson is that a campaign whose interesting outcome might be a null needs a control moving the exact statistic the conclusion rests on, and that a control which saturates a quantity has not tested it. For a successor the fix costs one more gate cell: a graded override obeyed by a fraction of the fleet on a fraction of the ticks, so the estimator gets exercised in its sensitive range instead of past its ceiling.
The primary contrast is the excess error correlation at full coupling minus the same quantity with the shared state switched off, among fifty family-A agents. It reads +0.0040 with a 90 percent interval of [−0.0048, +0.0124]. The rule declares SUPPORTED only if the lower bound clears 0.05 and REFUTED only if the upper bound sits at or below 0, so the verdict string reads INCONCLUSIVE. Everything worth knowing here is in the interval instead.
The realised bootstrap standard error on that contrast is 0.00554, and the interval half-width of 0.0089 is 1.61 standard errors, close enough to the 1.645 a normal would give that the bootstrap distribution is behaving. Power simulated at the observed per-agent error rates, with the frozen rule run end to end over 300 replicates a point, puts the realised minimum detectable effect at 80 percent power at about 0.115 under a graded common-factor model and about 0.18 under an all-or-nothing shock. The registration projected 0.15 and 0.30, so the instrument came out somewhat sharper than it promised.
Those two facts together are why a rule with a 0.05 threshold and a 0.0055 standard error will call almost anything inconclusive. Any true excess below roughly 0.115 was always going to land on INCONCLUSIVE, which is a wide blind spot and a property of where the threshold sits relative to the noise floor. The interval says something much stronger than the verdict does. The upper bound is +0.0124 on the primary block and +0.0161 on the held-out block, so coupling-induced excess error correlation is below about 0.013 with 90 percent confidence. The published same-weights conditioned correlation in this lineage is +0.447. Whatever the shared observable did to this fleet, it is smaller than a thirtieth of what serving the same weights twice already does.
The primary fleet reads 0.151, 0.145, 0.152, 0.152, 0.179, 0.151 and 0.149 as coupling runs from 0 to 49. The held-out fleet reads 0.094 to 0.111 over the same ladder. The largest within-block excursion anywhere is +0.028, against a registered margin of 0.25. No coupling level clears the boundary in either block, by a factor of nine. The absolute levels differ between blocks in the same direction as their perfect-instrument floors, 0.124 on seed 1 and 0.057 on seed 2, which is why the rule was registered as a within-block contrast in the first place. Fifty instances reading each other completely stayed about as spread around the twelve-stage cycle as fifty instances that could see nothing of each other.
The shared state was physically there and it moved. Median prompt tokens climb along the registered ladder from 279 at no coupling to 614 at full mean field. The fleet-wide spread of consensus values collapses from 11.4 distinct values at one tick to 1.85 as the graph closes, and each agent still sees about eleven or twelve different consensus values over its 34 steady ticks. The state was present and it moved.
The fleet responded. Marginal error climbs from 0.152 at no coupling to 0.296 at full coupling in the primary block, and from 0.148 to 0.260 in the held-out block. A consensus error class, which cannot exist at m = 0, appears at every coupling level above it.
Arm D is the reading that settles what caused the rise. It carries a roster of the same length and shape filled with values that share nothing, and it reproduces almost all of the damage: of the +0.144 rise from no coupling to full coupling under the real roster, +0.128 appears under the decoy. Arm S, which keeps the shared numbers and drops the roster, climbs from 0.148 to 0.178 and stops there. A longer packet made the agent worse at its own arithmetic. Whether the extra text described its peers or described nothing at all barely moved the number.
If the consensus were pulling reports toward it, the distance from consensus to reported stage would sit below the distance from consensus to required stage. It does not, at any coupling: 2.949 against 2.938 at m = 2, 2.905 against 2.912 at m = 49, and the four levels in between behave the same way. A permutation test that shuffles each agent's consensus values across its own ticks and recounts the matches on its own wrong answers gives z scores of −0.16, +0.56, −0.49, +2.68, +0.52 and −0.81 across the six coupled levels. One of six sits above the null, at m = 24, well short of full coupling. Regressing the error indicator on the circular distance between the consensus and the required stage gives slopes of −0.005, −0.007, −0.001, −0.008, +0.003 and −0.002, which is what a shared observable acting as no common factor at all looks like.
The consensus error class is worth taking apart, because its name promises something the measurement does not deliver. It fires at 2.11 percent of steady units under the real roster and at 2.24 percent under the decoy roster, where the number so labelled is private to the agent and the tick and known to nobody else. It fires at 1.80 percent under the aggregate-only arm and at 100 percent under the override. Relative to errors it sits between 8 and 13 percent throughout, which is the one-in-twelve chance rate. What it measures is an agent sometimes reporting the most salient number in its own prompt.
The reading, kept apart from the inference. Observation: the roster entered the prompt, varied across ticks, and coincided with a doubled error rate, while an equally long roster of private noise produced nearly the same doubling and the shared numbers on their own produced almost none. Observation: reported stages sit no closer to the consensus than required stages do, at every coupling level. What follows is that the fleet read a longer prompt worse, while the shared content inside it exerted no attraction this instrument can detect. Both of the obvious stories are wrong: the roster was read, and reading it pulled the fleet nowhere.
The registered heterogeneity quantity is the excess error correlation among family-A agents only, at full coupling, contrasted between no dilution and twenty-five of fifty agents drawn from the second family. That population is the same fifty weights at every level, diluted only in what it reads from the network, which is the registered mitigation for family B being a weaker reader on this task. The contrast reads −0.0073 with a 90 percent interval of [−0.0256, +0.0109]. The held-out block reads +0.0045 with an interval of [−0.0142, +0.0274]. Both are inconclusive, the intervals overlap comfortably, and the point estimates disagree in sign.
The whole-fleet quantity beside it tells a different story and points at section 4.6. The fleet-level same-wrong excess at full coupling rises with dilution, to +0.0635 in the primary block and +0.0473 in the held-out block at the highest level, while the family-A quantity in the very same cells sits flat at zero across the whole axis.
Family-B agents show a large same-wrong-answer excess at full coupling that is absent at zero coupling, in both blocks. Truncating both sides to ten steady ticks so the comparison cannot turn on tick count, the excess runs from +0.075 to +0.221 at full coupling across eight cells with at least six family-B agents, and it is near zero in all eight at no coupling. Along the coupling ladder at the highest dilution it is graded: +0.007 at m = 0 rising to +0.154 at m = 49 in the primary block, and +0.039 rising to +0.075 with a peak of +0.099 at m = 36 in the held-out block. Family A in the same cells stays flat.
The mechanism is not consensus-copying, which is worth saying because the name suggests otherwise. Family-B agents report the consensus stage far less often than their own permutation null allows, with z scores from −2.87 to −6.23 across eight cells, and of the family-B pairs that are both wrong and agree, between 0.0 and 0.8 percent agree on the consensus. Twenty-five instances of one small model converge on the same wrong answers, and those answers are not the number they share. The sanity probes show the shape: three family-B misses drawn at random all report a net that is not the sum of their own gauges, and the wrong nets are far apart in value while the stages they produce land together.
This cannot be separated from prompt length with the data that exists. Arms S and D run only at zero heterogeneity, so no decoy arm exists anywhere on this axis, and family B's own error rate falls as coupling rises rather than climbing, from 0.724 to 0.607 in the primary block and 0.680 to 0.597 in the held-out block over the full steady window, with the verifier's tick-matched subsets reading 0.72 to 0.66 and 0.68 to 0.53 in the same direction. A longer packet pushing a weak model into a narrower set of stock answers would produce both the improvement and the convergence. Neither registered hypothesis looks here: H4 is defined at zero heterogeneity where no family-B agent exists, and H3 is family-A only by a registered and correct mitigation. The campaign reports its null correctly by its own frozen rules while a replicated correlated-error signal sits one column over. The successor's registered question is that column, with a decoy arm at every heterogeneity level.
| hypothesis | registered quantity | orchestrator | verifier | verdict |
|---|---|---|---|---|
| H1 | rho at m = 49 minus m = 0, SUPPORTED if CI90 low > 0.05 | +0.0040 [−0.0048, +0.0129] | +0.0040 [−0.0048, +0.0124] | INCONCLUSIVE |
| H2 | R_ss against m at h = 0; m* is the first m clearing +0.25 | 0.151 to 0.179; m* = None | 0.1455 to 0.1789; m* = None | no boundary |
| H3 | rho, family A only, h = 0 minus h = 25 at m = 49 | −0.007 [−0.029, +0.011] | −0.0073 [−0.0256, +0.0109] | INCONCLUSIVE |
| H4 | concordant-error excess, m = 49 minus m = 0, threshold 0.02 | +0.005 [−0.010, +0.018] | +0.0046 [−0.0097, +0.0217] | INCONCLUSIVE |
| H5 | coupled same-wrong rate against its own-marginal baseline | 0.09658 vs 0.09135 | 0.09658 vs 0.09135 | reading |
| H6 | arm S at m = 49 against the null cell | +0.008 [−0.003, +0.020] | +0.0077 [−0.0026, +0.0201] | INCONCLUSIVE |
| H7 | arm D at m = 49 against the null cell | −0.001 [−0.009, +0.007] | −0.0012 [−0.0080, +0.0073] | INCONCLUSIVE |
Table 2. Primary block, seed 1, 34 steady ticks, 50 agents. Intervals are 90 percent cluster bootstraps over ticks. Interval endpoints differ between the two columns only through the bootstrap draw. The frozen decision rules applied to the verifier's numbers give the same seven verdicts.
H5 is the agreement reading and it is the reason the title claims what it claims. At full coupling two agents that are both wrong give the same wrong answer 9.658 percent of the time, against 9.135 percent for the same two agents drawing independently from their own wrong-value distributions. Beside it, pairs both correct and equal run at 8.736 percent and the rate at which two stations' own gauges required the same stage runs at 8.703 percent. A campaign that reported a rising agreement rate and stopped there would have reported the second of those three numbers and called it the first.
The held-out block ran on seed 2 after the primary analysis was written, at the same 55 cells and the same tick budget, so it matches the primary in size instead of being a token second look. Agreement is registered as same direction of the point estimate and overlapping 90 percent intervals, never as verdict-string equality.
| quantity | primary | held out | agrees |
|---|---|---|---|
| H1, excess error correlation | +0.0040 [−0.0048, +0.0124] | +0.0004 [−0.0159, +0.0161] | yes |
| H4, same-wrong-answer excess | +0.0046 [−0.0097, +0.0217] | +0.0007 [−0.0163, +0.0141] | yes |
| H3, heterogeneity, family A only | −0.0073 [−0.0256, +0.0109] | +0.0045 [−0.0142, +0.0274] | direction flips |
| H2, phase boundary m* | None | None | yes |
Table 3. Replication on the sealed held-out block. The headline quantities agree. H3 flips sign between blocks with both readings inconclusive and the intervals overlapping across most of their length, which is what two draws from a distribution centred near zero look like.
Nothing in this paper is claimed from the primary block against a held-out block that contradicts it in direction. The family-B signal in section 4.6 replicates in both blocks and is still reported as a lead, because replication addresses sampling and does nothing about the missing control arm.
142,400 units exactly: 8,800 in the gate block, 66,800 in each production block. Every registered cell is present with its exact unit count and no unregistered cell appears. Fifty agents in every one of the 115 cells. Zero duplicate unit ids. Zero error codes across all 142,400 units, an unreadable share of 0.0000 against a gate threshold of 0.02, finish_reason of stop on every unit, zero reasoning leaks, and constrained decoding with the registered send seed on every call. Zero units held on error, so no agent was dropped from any correlation estimate anywhere.
Within-fleet distinctness holds on every slot: 1,512 cell-tick slots across three blocks with zero duplicated packets and a worst slot of 1.0000, and the same for prompt digests. Cross-block packet overlap is zero for all three pairs. No pair of agents shared a packet at any tick in any cell. Design validity holds on every row: zero step-rule violations, zero violations of required_stage = (stage_now + step) mod 12, zero closed-loop link breaks, and exactly one distinct net per agent-tick in each block. The correlation between the required stage and the consensus is +0.0070 in the primary block and −0.0206 in the held out. In the gate it is +0.2655, which is the override cell, where collapsing the fleet onto the consensus does make the next required stage follow from it. Twelve sanity probes drawn across both families, all four arms and six coupling levels were rescored by hand from the raw fields, and twelve of twelve score fairly.
Nondeterminism. Temperature 0 with a fixed send seed did not give byte-identical replies to byte-identical packets. Grouping units by packet digest and counting replies that differ from their group's modal reply gives 781 minority replies among 11,015 units in 2,873 repeated groups in the primary block, which is 7.09 percent of repeated units and 1.17 percent of the block; the held-out block reads 6.83 percent and 1.15 percent, and the gate block 10.14 percent and 1.00 percent. Both families are affected, 492 family-B and 289 family-A units in the primary block, and the concentrations sit in the uncoupled and near-uncoupled cells at high dilution. The pre-registration asserts in section 5 that at zero coupling a family-A agent's packets are byte-identical at every heterogeneity level and that at temperature 0 its replies must be too. That assertion is false at roughly a 7 percent unit rate. The direction is conservative: independent reply noise is uncorrelated across agents, so it can only push a measured excess correlation toward zero. What it does change is the status of the six cells at zero coupling and non-zero heterogeneity, which are noisy replicates and not exact ones.
The held class. The error class named held is agent behaviour rather than a harness hold: it marks a unit where the agent reported the stage it already had. There are 74 in the primary block, 64 in the held out and 3 in the gate, all with held_on_error false, no error code and a finish reason of stop. The name collides with the held_on_error field and the two are independent. The analysis treats them correctly.
Two latent code items, neither with a numeric consequence here. The H3 contrast call passes a family-A index set for the h = 25 side and None for the h = 0 side. At zero heterogeneity all fifty agents are family A, so the two are identical and the reported number is right. The same call shape applied to any pair above zero heterogeneity would compare a whole fleet against a family-A subset without complaining. Separately, the docstring above the phase-boundary rule describes a comparison against an uncoupled perfect fleet on the same seed with the cell's own bootstrap lower bound clear of it, while the code implements the registered within-block rule. The code is correct and the comment should be fixed so that a later reader does not implement the comment.
No rank statistic is computed anywhere in this campaign. A search over every Python file in the campaign directory for rank, Spearman, Kendall or tau returns nothing, so the tie-handling defect that changed a verdict two campaigns ago cannot apply. Every correlation here is a Pearson or a two-by-two phi on binary indicators.
The instrument was not shown, by the campaign's own design, to be able to detect herding. The live positive control validates the order parameter. Both estimators the null rests on return their degenerate codes on that cell. Their validation is synthetic, from the instrument check, plus the on-data injection recovery in section 3.6, which verification ran and the campaign did not.
Herding at effect sizes below this design's reach remains open. The realised minimum detectable effect under the frozen SUPPORTED rule is about 0.115 under a graded model and about 0.18 under a shock. The defensible claim is the interval bound of 0.013, and it is a claim about what the coupling adds, not a clean bill of health.
Heterogeneous fleets are not covered by the null. They appear to herd. The fleet-level same-wrong excess at full coupling reaches +0.0635 in the primary block and +0.0473 in the held out at the highest dilution, driven by family-B agents agreeing with each other on wrong answers that are not the consensus, replicated in both blocks and graded in coupling. No control arm exists above zero heterogeneity, so prompt length cannot be ruled out, and no registered hypothesis reads it.
Open-ended reasoning, informative shared signals and fleets with real per-agent context diversity are all outside this. Each is a limit declared in the registration and each still stands.
The result is a bound and should be quoted as one. A fleet of fifty instances of one model, coupled through a shared numeric state at seven strengths up to full mean field, added at most about 0.013 to the pairwise error correlation among its members at 90 percent confidence, with a point estimate of +0.0040. Anything below roughly 0.115 would have been called inconclusive by the frozen rule, so nothing here rules out a small true effect. What the campaign does rule out is an effect of the size that the same-weights channel already produces in this lineage, and it rules that out by a wide margin.
The comparison to +0.447 is indicative. That number is a difficulty-conditioned phi between two servings of one model on a different task with a different item set, so it and the excess correlation measured here do not sit on one scale in the strict sense. It is the closest thing this lineage has to a measured price for running one model twice, and it is offered as a scale marker.
The family-B signal is confounded and stays confounded. It replicates, it is graded in coupling, and it has no decoy arm at any heterogeneity level above zero. A weak model pushed by a longer packet into a narrower set of stock answers would produce the same pattern, and family B's falling error rate along the coupling ladder is consistent with exactly that. The successor design is one line: arm D at h > 0.
Beyond those, the registration's own declared limits hold. This is one task, a twelve-stage arithmetic rule on a private packet, and correlated failure on open-ended reasoning could behave differently. Family B is a weaker reader here, at 0.76 marginal error on private packets against family A's 0.10, so the heterogeneity axis mixes a second set of weights with a worse agent even after the family-A mitigation. Temperature is 0 with no private-context noise beyond the task, so fifty instances differ only by their packets; a deployed fleet may carry more diversity than that or less, and the arithmetic coincidence floor where two agents share both a current stage and a net at one tick sits at 0.12 percent of pairs. The shared state is decorative for ground truth, which is the sharpest limit of all: it is a salient number carrying nothing the agent needs, and a fleet reading an informative shared signal faces a different trade-off. The tick series is autocorrelated and the cluster bootstrap understates the width for the order parameter in particular, which is why the held-out block, and not the interval, is what replicates the phase-boundary reading. The GB10 clock cap read inactive throughout, so the second family is the smaller one that the host's wall draw allowed.
The pre-registration was committed at 94d6a34 on 2026-09-13 at 11:32Z, one minute before the orchestrator started and well before the gate block cleared at 12:59Z, and mirrored in the lab wiki. Raw ledgers on the host: the primary and held-out block ledgers at 66,800 lines and 61.9 MB each, and the gate block ledger at 8,800 lines and 8.2 MB, append-only with every unit id distinct. Banked in the repository: the primary and held-out analyses with their logs, the campaign log, the colophon, the probe design, the pre-launch check and feasibility outputs, the kill-test outputs and the verification report.
Every figure in this paper was drawn by a script from copies of the three raw ledgers taken by scp on 14 September, with error matrices, order parameters, concordant-error baselines, circular distances, permutation nulls and the injection recovery all recomputed from the stored per-unit fields under definitions read from the registration. Each figure was rasterised at 1400 pixels wide and checked for ink coverage and for the presence of every series colour before it was inlined. Point estimates equal the registered analysis and the independent verification; interval endpoints differ in the last significant figure through the bootstrap draw. Marginal compute cost was electricity on owned hardware over 73,246 seconds of cumulative orchestrator time, about 20.3 hours under a 36-hour cap, at a measured 139 to 147 W depending on the fleet mix.
Colophon
Host and serving. spark-train (hostname spark-2988), NVIDIA GB10, driver 580.173.02, Python 3.12.3. Family A is qwen3.5 served by vLLM from local qwen3.6-35b-a3b weights, max_model_len 131072, container vllm-qwen35-v3 (sha256:4cbcfca2…) up since 2026-09-11T21:56:26Z and never restarted or reconfigured, spoken to on port 8001 only, with response_format json_schema strict. Family B is llama3.1:8b (46e0c10c…) on ollama 0.16.2 with format schema, num_ctx 2048 and one concurrent call. gemma3:27b (a418f583…) is registered and excluded on measured wall draw, 184 to 212 W against a roughly 160 W outlet ceiling. Temperature 0, send seed 20260913, completion cap 40 tokens, parallel 6 at a 0.35 s pace per worker. The GB10 clock cap read inactive at design time and at launch.
Digests. Edition HERDING-OSCILLATOR-2026-09-13. Code md5: kernel 952e3587d15ce604a13059c59ee2b673 · runner dad88993846635aa5a85f6d6dc2dea8e · instrument check ad9cb9b207631d2f34bf2d0f5e78e5aa · feasibility 00c7aecaa5bb0fee5808aecbac5d10b5 · gate 09632ee175cb1aed88f06f20761c32b1 · analysis 4793cbfb5f2cd8f0f5615bd562331f82 · probe 5219712e30e82eb0d2b5f6ce19cf516c · campaign driver 0dd52dd496b28927eb5fc4cc0131db8a · reboot-resume script 2ec53bdeb22fa9fc038c679c5066e240. System prompt sha256 63e1abe1…, override prompt sha256 83327003…, reply schema sha256 36a7a2cc…, header 182 nominal tokens at 6.8 tokens per roster entry and 2.95 characters per token.
Timeline (UTC). Orchestrator first started 2026-09-13 11:33:53 and was killed at 11:38:47 after 294 s for the resume test; the relaunched orchestrator started at 11:38:57 and is the campaign. Instrument check 101/101 and feasibility 19/19 re-run on resume. Gate block on tuning seed 901 finished 12:59:36 with G1 PASS on all six clauses. Primary block to 2026-09-13 22:44, primary analysis written the same window. Held-out block to 2026-09-14 07:50:38, held-out analysis 07:54:48, DONE the same minute. 133,600 production units plus 8,800 gate units, 73,246 s cumulative against a 129,600 s cap, no power loss during the run. Independent verification 2026-09-14, commit 6501fe0.
Herding Oscillator, catalogue 5.6, Track Three paper 10. Companion executive paper: Where Fleet Diversity Is Worth Paying For. Campaign page: Herding.