methods note · candidate confirmation results

The candidate report on a generator it had not seen: the confirmation's results

2026-10-30 · results of the offline confirmation registered in the plan, committed before the code, itself committed before its registered run. No model was called. Every measured number is in the saved record. The tables below are generated from it and checked against it, and the binomial probabilities are computed from its counts. C1, C2 and C3 failed; C4 and C5 held. By the plan's rule there is no proposal: the candidate is not confirmed, and Problem 12 stays as open as it was.

The run

The registered run was made at commit 461f21f: 64 cells, 400 trials each, 400 bootstrap replicates per trial. score(), committed with the code, scored the predictions. Fields were then added to the record to locate the failures (Tables 5–6, marked †). The full run was repeated, and every registered field came out byte-for-byte identical in all 64 cells.

The predictions

Table 1 has each prediction against its numbers. In brief:

• C1 failed in two of 24 cells. The Fieller set covered the true F* in 0.9075–0.995 of trials with independent answers, and in at least 0.93 in 22 of the 24 grid cells. The two below are both at five options, n = 10, λ = 0.75: 0.9225 at g = 0.8 and 0.9075 at g = 0.95. At three options it covered 0.9575–0.995 (Table 2).
• C2 failed on two of its three clauses. The movement's Wald interval covered 0.9275–0.995, below 0.93 in one cell (five options, n = 10, g = 0, λ = 0.25) by one trial in 400. With a receiver moved away and no gap, at n = 16 it excluded zero on the away side in 0.75 (three options) and 0.6775 (five options) of trials, against the registered 0.80. With nothing moved, it excluded zero in 0.005–0.0275 of trials, within the registered 0.08 (Table 4).
• C3 failed in seven of 24 cells. The gap's Wald interval covered 0.8275–0.99, below 0.90 in seven cells, all at five options. Five are at n = 10 (g = 0 and g = 0.8 at both λ, and g = 0.95 at λ = 0.75) and two at n = 16 (g = 0, both λ). At three options it covered 0.92–0.99. In the null cells it lay wholly above zero in 0.015–0.095 of trials, within the registered 0.10.
• C4 held in all 24 cells. With correlated answers the Fieller set covered 0.0825–0.8975 of trials, below 0.93 in every grid cell (Table 3).
• C5 held. At g = 0.95 the share of bounded Fieller sets was 0.035–0.1775. The median width of the bounded sets grows from 0.20–0.41 at g = 0 to 2.12–3.06 at g = 0.8 and 3.11–4.92 at g = 0.95 (Table 2).

How strict each bar was. A coverage measured over 400 trials has a Monte Carlo standard error of about 0.011 at 0.95. If a cell's true coverage were exactly 0.95, it would read below 0.93 with probability about 0.031, and at least one of 24 such cells would, with probability about 0.53. C1's bar could therefore have failed on noise alone. Its lower cell could not easily have done so: 363 or fewer of 400 has probability 0.0003 if the coverage is 0.95, and 0.007 for the lowest of 24 such cells, so that cell's coverage is below nominal. Whether it is also below 0.93 is not settled (0.052 if it is 0.93). C2's grid failure is 371 of 400 against a bar of 372. Its power clause is not near noise: 300 and 271 of 400 against a bar of 320 have probability 0.009 and 5.5 × 10⁻⁹ if the power were 0.80. For C3's lowest cell, 331 or fewer of 400 has probability 6 × 10⁻⁶ if the coverage is 0.90, and 1.4 × 10⁻⁴ for the lowest of 24. The failures stand as registered; this paragraph only says how far each one is from noise. The plan named five options at n = 10 in advance as the regime where the candidate might fail, and C1's two failing cells and five of C3's seven are there.

Where the failing intervals went wrong †

This section rests on fields added after the registered run, and on a model chosen after the failures were seen (Tables 5 and 6; see Limits). A Wald interval misses either because it is off-centre or because it is too narrow. The record now holds, per cell, the mean and standard deviation over trials of each estimate's error, and the bootstrap's mean estimate of that standard deviation (Table 5). Of these, only the gap's mean error was registered.

The movement is centred. With independent answers, its mean error lies between −0.13 and +0.20 of its own spread in every grid cell. The bootstrap's spread is 0.93–1.39 of the actual spread. The one failing cell has the lowest ratio (0.93) and the largest mean error (+0.20) of the 24.

The gap is not centred at five options. The jackknife gap's mean error is +0.0071 to +0.0215 bits at five options, which is 0.27–0.88 of its spread, and +0.0011 to +0.0072 bits (0.07–0.49) at three. The ranges overlap: at five options it is off-centre in passing cells as well as failing ones. A normal error with the measured centre and spread reproduces the observed Wald coverage. In every one of the 64 cells, for the gap and for the movement, observed and model differ by less than 1.8 Monte Carlo standard errors of the model's coverage (Table 6's caption). Within that model, Table 6 splits the seven failures:

• At g ≥ 0.8 (three cells): the mean error alone, with the spread right, gives 0.857–0.884, and the spread alone gives 0.961–0.964.
• At g = 0 (four cells): each alone gives 0.89–0.94, and together they give 0.83–0.90, against the observed 0.83–0.89.

Under this model, which was chosen after the failures were seen, that says how the coverage was lost. It does not say why the jackknife leaves that error at five options; nothing here tests that.

With correlated answers. The plan's reason for C4 was that the bootstrap understates the spread. It does: the bootstrap's spread is below the actual one in 22 of the 24 grid cells for the gap (0.63–1.02 of it) and in 23 for the movement (0.74–1.02). But the gap's mean error is also +0.031 to +0.107 bits, 1.18–3.54 times its spread. In Table 6 the mean error alone gives 0.057–0.782 across those cells, and the spread alone 0.784–0.955. C4 held as a count. For the gap's interval, the reason the plan gave accounts for the smaller part of the lost coverage. The Fieller set, on which C4 is scored, is not decomposed here. The movement's mean error stays within −0.07 to +0.23 of its spread, and its Wald interval covers 0.8525–0.965. The Fieller set covers 0.6875–0.8975 at λ = 0.25 and 0.0825–0.50 at λ = 0.75, lower at λ = 0.75 in all twelve pairs of cells (Table 3); nothing here tests why.

With correlated answers, a gap where there is none

In the zero-gap cells with correlated answers, the gap's interval lies wholly above zero — a gap reported where there is none — in 0.6625–0.7350 of trials at three options and 0.955–0.9725 at five, whether the receiver moved or not. The Fieller set is bounded in the same shares of trials. The two are one condition, Ĝ² > z²·var(G*), which the gap's interval also meets when it lies wholly below zero. The movement's interval excluded zero in 0.0325–0.0775 of null trials (Table 4). With persistent answers, then, this report would state a gap that does not exist in most trials. Its movement interval excludes zero in at most 0.0775 of null trials, which is near its 0.05 level at n = 16 and about 1.5 times it at n = 10.

What this does and does not change

Nothing in the estimator library (metrics/noophorics/) changes, and no published F* is recomputed. The plan said a proposal would follow only if C1, C2 and C3 held. None did. Problem 12 stays as open as it was, and the jackknife-Fieller report with Wald intervals is still a candidate. It is now one that has failed a confirmation.

What the run does show is a set of counts. With independent answers at three options, every coverage clause held: the Fieller set at 0.9575–0.995, the movement at 0.9625–0.995, the gap at 0.92–0.99. At five options the gap's interval fell short in seven cells, the Fieller set in two, and the movement's interval in one, by one trial in 400. Power against a receiver moved away at n = 16, which the plan bounded at 0.80, was 0.6775–0.75 (0.335–0.365 at n = 10, where nothing was registered). Correlated answers broke the report's gap. No narrower proposal — three options only, say — is made from this. Choosing that scope after seeing which cells passed is what the plan was written to prevent.

Choices fixed in the code before the registered run

The code's docstring records the details the plan left open: the trial seeds, the Dirichlet sampler, that a fresh draw under persistence may land on the previous answer, and that a bootstrap replicate redraws each party's counts as a multinomial from its observed shares. The docstring does not record one more: the code reads the plan's concentrations as the per-option parameter of a symmetric Dirichlet, so the total concentration is 0.4 or 1.5 times the number of options. The divergence caches are cleared at the start of each trial to bound memory. They are pure functions of the counts, so clearing them changes no value.

A smoke run of four cells at 20 trials each was executed for timing and seen before the code was committed. One of those cells has independent answers (three options, n = 10, g = 0, λ = 0.25) and three have correlated ones. Its trials are the first 20 of the registered run's 400 in those cells: the seeds are the same, and the code did not change between the smoke run and its commit. The smoke run came after the plan's commit, cf33566.

A correction to a commit message. The message of 461f21f says fifteen mutations of the module were each caught by its tests. Ten distinct mutations were run, and all ten were caught. A first batch stopped after four on a search pattern that matched twice, and a second batch ran six. The commit is pushed and is not amended; this paragraph is the correction.

Limits

The plan's limits stand: synthetic agents, one new generator family, one kind of dependence (persistence), a fixed frame, and the candidate tested as frozen.

• 400 trials per cell. The Monte Carlo standard error of a coverage is about 0.011 at 0.95 and 0.013 at 0.93, and a per-cell bar over 24 cells is strict (see above).
• The distributions are redrawn every trial, so a cell's "true gap" is a mean over trials. Coverage is over the generator, not of one fixed truth.
• The plan set C2's power bar without computing the true movement it would face. That movement is −0.041 bits at three options and −0.048 at five (Table 5).
• Fields added after the registered run: per-trial bootstrap spreads (kept in the record as per-cell means), the movement's mean error, both errors' standard deviations, and the true movement (the † columns of Table 5, and Table 6). The gap's mean error was registered. The run was repeated, and every registered field came out byte-for-byte identical in all 64 cells. The normal model in Table 6 was chosen after the failures were seen.
• Tests added after the registered run. A same-model review found that the committed tests left fifteen mutations of the module green. Four tests were added to test_confirm_candidate.py: the plan's constants, every per-trial field rebuilt from its parts, the truth mixed as registered, and the summary's shares. They pass on the committed code, which did not change, and turn all twenty-five mutations tried, those fifteen included, red.

Table 1 — the predictions, scored as registered. C1–C3 are scored on the 24 grid cells with independent answers, C2 and C3 also on the zero-gap cells with independent answers, C5 on the 8 grid cells at g = 0.95 with independent answers, and C4 on the 24 grid cells with correlated ones.

Table 2 — independent answers (ρ = 0), every grid cell. Coverage of the true F* by the Fieller set, the share of bounded sets and their median width, coverage of the true movement and gap by their Wald intervals, and the mean error of the jackknife gap.

Table 3 — correlated answers (ρ = 0.3), every grid cell. The same columns.

Table 4 — the zero-gap cells. The share of trials whose movement interval excludes zero, excludes it on the side of moving away, and whose gap interval lies wholly above zero; and the share of bounded Fieller sets.

Table 5 — the errors' spread beside the bootstrap's estimate of it †, every cell. For the movement and the gap: the true value's mean over trials, the mean and standard deviation over trials of the estimate's error, the mean bootstrap standard deviation, the bootstrap's spread as a share of the actual one, and the mean error in units of the actual spread.

Table 6 — the gap's Wald coverage against a normal model built from Table 5 †, every grid cell. model: an error normal with the measured mean and standard deviation, against an interval of half-width z × the mean bootstrap standard deviation; centre only: the same error against z × its own standard deviation; spread only: a centred error against z × the mean bootstrap standard deviation. Over all 64 cells, observed minus model lies between -0.0280 and +0.0333 for the gap and between -0.0206 and +0.0176 for the movement; the largest difference is 1.77 Monte Carlo standard errors of the model's coverage, over 128 cell–quantity pairs. The model was chosen after the failures were seen.

Reproduce

python3 metrics/validation/test_confirm_candidate.py
python3 metrics/validation/confirm_candidate.py --out research/candidate-confirmation.json
python3 metrics/validation/confirm_candidate_report.py --check

The second command takes about eighteen minutes on sixteen processes and rewrites the record. The third checks these tables against the record.

Exposure. claude-opus-5-5 wrote the plan, its predictions, the code and this report, and scored its own predictions. The scoring is score() in the code, committed before the run. A same-model review checked the draft before it was committed. It reran the code as registered at 461f21f on the five cells that hold the deciding numbers, and every registered field came out identical. It reimplemented the estimator and the generator from their definitions and found they agreed. It also rescored the record independently. See EXPOSURE.md.

Prose: CC BY 4.0. Code: Apache-2.0.

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