Stated uncertainty

Every one of the five slices breaks at least one of resubstitution, held-out and the Cover–Hart bracket at some sample size, and none of the breakages is visible from inside the estimator that carries it: each interval is computed the same way whether it is right or wrong, and only the closed form says which. Which of the two point estimators breaks is slice-dependent, and the ordering inverts. At the knife slice resubstitution covers 0.00–0.03 while held-out covers 0.93–0.97; at the widest floor slice resubstitution starts at 0.00 and recovers only at the largest n while held-out falls 0.90, 0.50, 0.25, 0.15. Neither of them is simply the good one, so a practitioner who validated a pipeline on any single slice would carry the wrong conclusion to the next. Coverage alone does not settle it either: both brackets cover at 0.90 or better at every slice and every n, and buy that by containing the no-evidence value along with the truth. And at the knife slice a bracket's coverage measures a cap rather than an estimator, since its upper end is held at 1/2, which no binary Bayes error can exceed, and the truth there is 1/2.

Scope. Computed on the five dial-chosen slices at n = 500, 2000, 8000, 32000, forty trials each, one fixed seed, 0-1 score, uniform prior. Nominal coverage 0.95 throughout. Two of the five slices are interior rather than floor slices, and the four find them no harder: what the window moves is the no-evidence baseline, not what an estimator does. A positive control — a two-fiber Bernoulli problem with R* = 0.15, far from 1/2 — passes through the same sampler and the same estimator code before any grid result is read. Toy scale.

verifier: explore_ceiling_ruler.py

Two of the failures are structural rather than small-sample, and neither is a fact about these rings. At the knife slice resubstitution's deficit is the mean absolute deviation of a fair binomial, ½√(2M/πn), while its own half-bar is the binomial 95% one, 1.96√(1/4n): both shrink as 1/√n, so their ratio √(2M/π)/1.96 is free of n — 1.5766 derived, and −1.48, −1.55, −1.56, −1.59 printed across the four sample sizes. The point estimate is consistent, its bias running −0.0644 to −0.0087; the interval is not, and coverage stays at 0.00–0.03 at every sample size, so more data never repairs it. At the widest floor slice held-out fails in the opposite direction, its coverage falling monotonically as n grows through the series above — and only there, since at the narrow floor slice the same estimator's coverage climbs with n. It scores a fitted rule whose excess over the optimum decays with the probability that some fiber's empirical majority flips, while its reported precision decays as 1/√n. A tightening interval reads as improving evidence, and here it is not.

Both reappear with no ring anywhere. The same estimators through the same code, on M bare Bernoulli fibers, reproduce the ratio at −1.52 to −1.62 and the falling series at 0.88, 0.68, 0.23, 0.25. A dial the ring cannot turn — the distance δ of η from 1/2 — locates the interval failure in the band 2δ√(n/M) ≲ 1: at n = 8000 one sixty-fourth of distance from 1/2 already raises coverage from 0.00 to 0.85, and one thirty-second restores it outright. So what a designed family contributes is not the failure but the placement: one of its floor slices sits at η = 1/2 exactly, inside a band about that wide — the others at 4/7 and 6/11, close to 1/2 and off the band at the larger n — with the truth a rational number. Calibrated position on a thin edge, and not a cause.

Scope. The rule is the ring-free reappearance: derived by hand off the ring and confirmed at the grid's own M, η and n. The two failure mechanisms are observations — closed forms derived asymptotically and confirmed at four sample sizes. The falling series is stated as a direction and not as rates: it survives all three tie-breaks of the fitted rule's arbitrary empty-fiber choice, the two harsher ones falling further, while two random streams of one setting differ by 0.12, which is the scale of forty-trial coverage noise. The band's location was a margin guess in the design and broke: the crossing sits nearer 1/2 than √(M/n). 0-1 score, uniform prior. Toy scale.

verifiers: explore_ceiling_ruler.py, explore_ruler_boundary.py

The two bracket estimators are not two bounds. Once n ≫ M every test point has an exact residue match, so the 1-nearest-neighbour rule returns a random same-fiber training label and its error tends to Σ_r P(r)·2η_r(1−η_r) — 1/2, 24/49 and 60/121 on the slices run, with measured gaps at most 0.0007 at the largest n. The Henze–Penrose divergence the spanning-tree estimator targets has the exact value E[(2η−1)²] here, and (1 − that)/2 is the same quantity again, so the two are one bracket read at two statistics. A second estimator family therefore buys a different way of estimating the bound and never a second opinion on it. Its equality case is this family's own window: the lower endpoint's shortfall is the Jensen gap of |2η_r−1|, which vanishes exactly when that is constant across fibers — when t is divisible by M, or inside the odd-c middle window. Off the window the endpoint is strictly conservative, by at most 0.0178 across the thirty thresholds run.

That one bound is also too weak to certify this family's own lift, and that too is a fact about the bound rather than about either estimator reading it. Read exactly, with no sampling anywhere, the bracket excludes the no-evidence value at 2 of the 14 in-window thresholds and contains it at the other 12; add the sampling half-bar at n = 8000 and it excludes it at none. The family's own exact lift there is 7/105, and the sampled divergence bracket is still 0.0838 wide at the largest n. A bracket can be correct at every slice and unable to certify that the evidence is worth anything, with the emptiness located in the bound rather than in the sample.

Scope. The identity of the two brackets and the tightness set are derived by hand and confirmed in exact rational arithmetic on all 35 (slice, threshold) pairs the file touches, and both are the rule the badge states; the certification counts are observations, computed on the 30-threshold profile of the N = 105 cell at n = 8000, forty trials. Its high coverage has two independent sources and neither is accuracy: the truth sits at the exact bracket's lower endpoint on all five slices, so containing it costs nothing at the top, and the stated bar is binomial where a tree's edges are dependent, which overstates the statistic's real dispersion by 1.75 to 2.75 times at every slice and sample size. Two estimator families, 0-1 score, uniform prior. Toy scale.

verifiers: explore_ceiling_ruler.py, explore_ruler_divergence.py

A spanning-tree estimator needs a distance, and the only one the evidence offers — how many shown residues two points differ in — puts every pair of points in the same fiber at distance zero. So any spanning tree of a fiber's points is a genuine minimum spanning tree, and the statistic is not a function of the sample. Fed the rows in class order — what concatenating two label arrays hands a deterministic routine — each fiber contributes exactly one edge joining opposite labels, so the statistic is a function of the fiber count alone: the cross-label fraction reads 0.0005 at n = 32,000 at all four slices with M = 15, with across-trial spread 0.0000, identically where the truth is 3/7 and where it is 1/2, and the bracket it maps to — [0.0001, 0.0007] — covers at 0.00. It converges on the assertion that the labels are perfectly separable. Shuffling the rows, which changes no datum, moves the same statistic to 0.4897.

Those two are not the extremes. A fiber's zero-weight edges admit a spanning tree of the complete bipartite graph between its two label classes whenever both are present, and every edge of that tree joins opposite labels, so across genuine minimum spanning trees of one sample the reported bracket runs from [0, 0] to [1/2, 1/2]. The identified set — the values the data leaves possible once the estimator's arbitrary choices are allowed to range — is therefore the whole interval a binary Bayes error can occupy. That is not a perturbed answer but the absence of one, and it is a question of whether the estimator is defined on data of this shape rather than of how well it does. What the repair needs is not randomness but a tree whose shape does not consult the labels; the labels are then exchangeable across its edges and the expected cross-label count is the population quantity whatever the tree, so a deterministic order is fine and class order is not. An exact Euclidean minimum spanning tree on a jittered embedding, built by geometry and blind to labels, agrees with the shuffle — 233.1 same-fiber cross edges against 239.8 at n = 500.

Scope. Computed on the five slices at four sample sizes, forty trials, with the mechanism and the identified set derived rather than measured; the label-blind control is a single configuration at n = 500, its agreement inside a conservative Poisson bound. Categorical evidence is what forces the non-identification: the residues carry no order and no size, so every within-fiber distance is zero and the tree has nothing to minimize. 0-1 score, uniform prior. Toy scale.

verifier: explore_ruler_divergence.py

What fails is the statement rather than the estimator. A stated interval is computed from the estimator's own model, so it can price only error that model has a coordinate for; error outside the model is uncounted and reaches the comparison against truth without ever entering the arithmetic. Two properties follow, both measured against exactly known truth — each world’s own Bayes error, a rational number — across six arms differing only in what their statement knows. Each arm runs on a designed world: fifteen equally likely fibers carrying stated rates, clear where no rate is nearer than 1/10 to 1/2, knife where every rate is 1/2 exactly, and mild where three rates sit just past 1/2 and the other twelve well away from it.

The first is an asymmetry. Call a statement a bound when its uncounted term has a known sign: fit the majority rule on half the draw, score it on the other half, state the one-sided upper confidence limit, and the slack carried is the rule's excess over the optimum, which cannot be negative. The bound arm never came out overconfident, on either world or at any sample size: it covers 0.963–0.988 against its own one-sided nominal 0.975 at all eight of its cases — the clear world and the knife, four sample sizes each — while on that knife world a fit statement covers 0.015–0.035. What is forced is that one direction and not its mirror. A fit is merely the only kind that can fail downward, and nothing makes it: the knife fit's own half-bar runs 3.10 to 3.26 times σ, the true spread of its estimate across trials — too wide, not too narrow. The usual corollary is false too. That bound excluded the no-evidence value in every trial on the clear world, so conservatism is forced by the sign and emptiness is not: what the two brackets above show is their own slack.

The second decides whether more data cures it, and it is one ratio — the uncounted term b against the half-bar h the statement does count. Three regimes, all three from the same plug-in code. Where the model contains the truth, b/h falls 0.032 to 0.002 and coverage sits at nominal: 0.955, 0.945, 0.932, 0.953. At the knife the two shrink at one rate, the ratio holds at 1.573–1.595, and coverage is flat at 0.035, 0.015, 0.025, 0.035 — sample size irrelevant. And where the model cannot hold the truth — the mild world's fifteen fibers pooled into three groups of five, each holding four of the twelve rates well off 1/2 and one of the three just past it — the ratio grows 0.259 to 1.781 while coverage falls 0.915, 0.875, 0.555, 0.045, the bias sitting on the exact pooling gap 7/750 from the second sample size on. Nothing there degrades except the claim. The crossing has a location: coverage passes 1/2 where the uncounted term equals the stated half-bar, which holds wherever that half-bar outruns σ, as it does by about a factor of two at every fit arm here.

The two are one fact. Across six arms and both statement kinds — seven arm-and-world pairs, since the bound runs on two worlds, at four sample sizes each — measured coverage is what the pair of ratios b/σ and h/σ predicts and nothing else enters — largest deviation 0.0133, mean 0.0049 over all twenty-eight. Mechanisms that read as distinct act only by moving those two numbers, and the direction a statement fails in is set by which region of that plane its construction can reach.

Scope. Observation throughout, at 400 trials each and n = 500 to 32,000 over M = 15 uniform fibers with exact rational targets. Each arm's regime follows from its construction; its trajectory does not. The two-ratio formula follows from approximate normality of the estimate, and what is measured is that the approximation holds at every one. A control is read before any of it: the knife arm reproduces the constant derived above, √(2M/π)/1.96 = 1.5766, at 1.573–1.595 — a differently written estimator on a differently built world landing on the same closed form. The third regime is read on the small pooling gap because a large one leaves no trajectory to see: the same pooling on groups that straddle 1/2 gives a gap twenty times wider and coverage 0.000 at every n. Two overconfidence mechanisms are instantiated — a bias omitted at the same rate as the noise that is counted, and a model class that cannot contain the truth — plus the bound kind. Two others recorded elsewhere were not run: a precision decaying faster than the fitted rule's excess over the optimum, and a continuous class fitted to an atomic truth. Every arm's bias has a known cause, so nothing is said about whether an uncounted term is detectable from inside. 0-1 score, uniform prior. Toy scale.

verifier: explore_uncounted_term.py