TAU-LIMIT AND HIERARCHY NECESSITY — builder-support probe (PROBE 5)

Status: BUILDER-SUPPORT PROBE. Not a verdict. Nothing here moves a P-level, changes a frozen verdict, or creates a claim. P8 remains FULL_PARITY = false; the first unsatisfied level remains P4 transfer. No P-level is raised or lowered by this document.

Question. Is the hierarchy (Lmotor-5 population prior over log-shape, Lmotor-4 per-motor latent eta_m integrated out) doing work, or does the fitted model sit at its non-hierarchical limit? The recorded F-side contrast against M1_WEIBULL is +0.0006145100976836924 nats with percentile half-width 0.01109232367002231 — the tightest contrast in the F-side set and centred essentially on zero. This probe makes that a formal question with real computations.

Split boundary: TRAIN_ONLY (duration-only). Every computation in sections 1–4 uses the 80 training motors / 793 training events of the frozen derived_eligible_1_to_8 cohort. No holdout field is read by either script. The held-out numbers in section 5 are reused verbatim from the landed F-side artifact, not recomputed. nextStateN, direction and jump were not read at any point (D5 firewall).

Units. tau here is the population SD of the log-shape — dimensionless. It is not a torque, and tau * delta_theta thermodynamic work does not appear anywhere in this document. The profile axis is training negative log-likelihood in nats over 793 events; the held-out axis in section 5 is motor-equal NLPD in nats per event on the SECONDS scale. These two axes are different quantities and are never added.

Artifacts and commands

cd hierarchical-aif
python scripts/probe_tau_profile.py        # 687 population-LL calls, 119.5 s wall
python scripts/probe_tau_shrinkage.py      # 30.3 s wall

Both reuse hierarchy.population_log_likelihood, hierarchy.posterior_motor_shape and hazard_survival unchanged. No module was rewritten. No test was touched.

1. Profile log-likelihood over tau

Method. tau grid = exp(linspace(log(1e-4), log(5.0), 61)) — transcribed verbatim from the frozen cell_C11 U2 grid in audits/phase-b/b4-identifiability-robustness-runner.py. The grid endpoints coincide with fit.TAU_MIN = 1e-4 and fit.TAU_MAX = 5.0. At each grid tau, mu was re-optimised by bounded Brent on [-3.0, 3.0], xatol = 1e-5. tau is fixed at each point: this is a profile, not a fit. All 61 points returned finite values; mu_hat never touched its bound.

Curve (selected grid points; full 61 in the JSON).

Consistency against the landed fit. The grid maximum 575.7067888179371 sits 0.036682402574911066 nats above the recorded free-fit trainNLL = 575.6701064153622 at tau_hat = 0.18372082607308418. That is the correct sign — tau_hat is not a grid point, so the constrained grid optimum cannot beat the free optimum. This is a cross-check, not a new result.

The tau -> TAU_MIN limit. At tau = 1e-4 the profile NLL is 583.5855952300938, i.e. a drop of 7.878806412156678 nats from the grid maximum. Per training event that is 0.009935443142694423 nats/event over 793 events. The likelihood-ratio statistic is 2 * 7.878806412156678 = 15.757612824313355.

Independent check that the tau -> 0 limit IS the non-hierarchical model. Fitting a single shared mean-one Weibull shape to all 793 pooled training events (one free parameter, no hierarchy) by an independent 1-D bounded Brent gives k = 0.6250888368122796, NLL 583.5856062645817.

• Against the frozen M1_WEIBULL fitted k = 0.6250888335850175: absolute difference 3.227262013183463e-09, relative 5.162885400902826e-09.
• Against the profile at tau = 1e-4 (583.5855952300938): difference 1.1034487897632062e-05 nats, consistent with the xatol = 1e-5 Brent tolerance on mu.

So the tau -> 0 corner of this hierarchy is M1_WEIBULL, reproduced to 8 significant figures by a separate implementation. That is the anchor the rest of this probe hangs on.

Flatness criterion — DESIGN_ONLY. The frozen B4C11 U2 cell declares a flat set {tau : profileNLL(tau) - min <= 1.9207} where 1.9207 = 0.5 * chi2_{1,0.95}, normalises the log-span by log(5.0) - log(1e-4) = 10.819778284410281, and fires UNIDENTIFIED_FLAT_PROFILE at normalised log-span >= 0.50. Applying that offset here:

0.0667 is 7.5x below the C11 fire rule, and the drop to the tau -> TAU_MIN corner (7.8788 nats) is 4.102049467463257x the 1.9207 offset.

DESIGN_ONLY — this reuse is not evidential. 1.9207 was frozen for the B4C11 M7 cell: a different model (M7_HIERARCHICAL_MOTOR, point-estimate per-motor shapes) with a different likelihood (b3.m7_train_nll) and a different nuisance parameter being profiled out. It was never frozen for the F-side marginal likelihood. Reusing it here is a design-time yardstick so this probe is not inventing a threshold; it is not a frozen criterion for this cell and confers no status. There is no frozen flatness criterion for the F-side profile. Nor is the chi2_{1} calibration itself appropriate at the tau -> 0 corner: tau = 0 is on the boundary of the parameter space, where the reference distribution is a chi-bar mixture, not chi2_1. A calibrated reference distribution for this statistic is NOT_RUN — COMPUTE_BUDGET (see §3).

Reading. The profile is not flat in tau and the fitted tau does not sit at its non-hierarchical limit in training log-likelihood. The training data prefer a positive tau by roughly 7.9 nats over the whole 793-event training set. That is an in-sample training statement about the likelihood surface. It is not a held-out statement, and §5 shows the two disagree in practical size by more than an order of magnitude.

2. Posterior shrinkage of eta_m

Method. hierarchy.posterior_motor_shape(y_m, c_m, mu, tau) at the recorded fit mu = -0.41607215987582913, tau = 0.18372082607308418 (read from F_SIDE_MOTOR_STACK_SCORING_RESULT.json, not refitted), for each of the 80 training motors. A no-pooling comparator eta^np_m was obtained per motor by unpenalised 1-D bounded-Brent MLE of the log-shape on [log 0.05, log 5.0] — the same box the frozen M7 U2 profile uses — with an observed-information SE from a central second difference of the per-motor profile.

Shrinkage factors.

Aggregate: SD(posterior means) / SD(no-pooling MLEs) = 0.24624046242391676. 69 of 80 motors have 1 - B_m > 0.5; 7 of 80 have 1 - B_m > 0.9. The extreme case is the motor with 1 event: eta^np = 1.6094378597456587 (at the box bound), se = 2.8871688630597583, B = 0.004032899513228368, posterior mean -0.37916259362785415 — i.e. it is returned to the population mean almost exactly, which is the hierarchy behaving correctly rather than a defect.

Reading. Posterior means travel roughly a third of the no-pooling distance from the population mean and retain about a quarter of the no-pooling between-motor spread. Motors are not all collapsed onto the population mean — 45 of 80 sit within 0.5 * tau of mu, so 35 do not, and the widest posterior mean sits 0.27704117521657323 from mu, about 1.5 * tau. But the mean posterior SD (0.1500) is 0.816 of the prior SD tau, meaning a typical motor's duration record moves its own belief only modestly off the prior. The hierarchy is doing something and it is small. That is the honest statement; it is not "the hierarchy is doing nothing" and it is not evidence that the pooling is buying prediction.

3. Motor-level heterogeneity evidence

Events per motor, real training cohort (80 motors, 793 events).

The distribution is heavily right-skewed: the median motor contributes 7 events while a single motor contributes 70. Twenty-six motors contribute at most 5 events each.

Test. Statistic = SD across motors of the no-pooling per-motor log-shape MLE. Null generator = the tau -> 0 limit, i.e. one shared mean-one Weibull shape at the pooled MLE k_null = 0.6250888368122796, with each motor's own n_m preserved, 300 replicates, seed 20260717.

These replicates are SIMULATION. They are a Monte-Carlo reference distribution and may never be labelled OBSERVED. The plug-in k_null ignores estimation uncertainty in k_null itself.

Reading — and this is the adverse result of the probe. The observed between-motor spread of per-motor dwell-shape estimates is below the mean spread produced by a generator with zero between-motor heterogeneity. On this statistic, the apparent variation in dwell shape across motors is entirely consistent with within-motor sampling noise at a median of 7 events per motor.

This is NOT_ESTABLISHED, not equivalence. The statistic is badly underpowered for the effect size at issue: if the true between-motor SD were tau = 0.18372082607308418, the expected observed SD would be roughly sqrt(0.4392^2 + 0.1837^2) = 0.47612176605716156 — comfortably inside the null 97.5th percentile of 0.5453959757815685. So this test could not have detected tau = 0.184 even if it were exactly right. It does not contradict §1; it shows that a moment-style statistic built on noisy per-motor point estimates throws away the information the marginal likelihood in §1 keeps.

NOT_RUN — COMPUTE_BUDGET. The properly matched test is a parametric-bootstrap calibration of the §1 profile statistic itself: simulate under tau -> 0 at k_null preserving each n_m, and for each replicate refit the full 2-parameter hierarchy and recompute the drop to tau = TAU_MIN, giving a boundary-correct null distribution for 2 * 7.878806412156678 without assuming chi2_1. Each replicate costs one fit_motor_stack call; the measured cost of a single profile point (11 population-LL evaluations) was ~1.65 s, and a full fit is nfev = 165 LL evaluations, so ~2.1 s per replicate and ~10.5 minutes for 300 replicates in one process. That exceeds this pack's compute fence and would contend with the two in-flight long runs. It is not run, and its absence is why §1 reports a statistic and no p-value.

4. tau-at-boundary

fit.py bounds tau to [TAU_MIN, TAU_MAX] = [1e-4, 5.0] and flags tauAtBoundary = (tau <= TAU_MIN * 1.01) or (tau >= TAU_MAX * 0.99). The module comment states the reason explicitly: tau -> 0 collapses the hierarchy to a single shared shape, which is a different model, so the boundary is reported rather than silently crossed. §1 confirms that comment numerically — the tau -> 0 corner reproduces M1_WEIBULL's k to 3.2e-09.

So the frozen M7 U1 verdict and the F-side fit agree: tau is interior, on both the point estimate and the whole DESIGN_ONLY flat set.

tau being interior is NOT evidence of a mechanism. It is a statement about where a two-parameter likelihood surface peaks on 793 dwell durations. It licenses no claim about stators, CheY-P, load, torque, or any physical process. A hierarchy diagram is a parameterisation, not a mechanism, and this probe exists partly to prevent that inference.

5. Comparison to M1_WEIBULL — recorded, not recomputed

Reused verbatim from the landed F-side artifact. No new contrast was computed.

The frozen resolution floor is a motor-equal half-width of 0.042 nats over 19 holdout motors. This contrast's half-width is 0.0111, i.e. 3.715827656374414x narrower than the floor — so even the whole interval lies inside what the cohort treats as scientifically immaterial. Per D10, the floor states what is scientifically material, not what the bootstrap will call; a narrow interval that still contains 0 is NOT_ESTABLISHED and remains so.

The number that matters for this probe. Training gain from the hierarchy over its own tau -> 0 limit: 0.009935443142694423 nats/event (§1). Held-out gain over M1_WEIBULL: +0.000615 nats/event, interval crossing 0. The training gain is 16.155192101942152x the held-out point estimate, and the held-out point estimate is itself ~68x below the 0.042 materiality floor. The hierarchy is preferred in training and its held-out benefit over the non-hierarchical limit is NOT_ESTABLISHED at this sample size.

Note the structural relationship this sits next to: the F-side hierarchy coincides with M7_HIERARCHICAL_MOTOR to numerical precision (exp(mu) = 0.659632669755436 vs M7 k = 0.6596322379287862; tau = 0.18372082607308418 vs M7 tau = 0.18372185667134974), which is why the recorded F-vs-M7 contrast is +2.506984e-07 nats — RESOLVED_ABOVE but recorded as SCIENTIFICALLY_NULL. The F-side candidate is bracketed on one side by a model it nearly is (M7) and on the other by its own degenerate limit (M1), and it separates from neither.

6. When the hierarchy is justified, and when it is decoration

Builder-facing conclusion. On this cohort the hierarchy is load-bearing for inference and not demonstrably load-bearing for prediction.

Where it is justified — keep it:

1. Small-n motors. 26 of 80 training motors have <= 5 events; one has a single event. Its no-pooling MLE runs to the box bound with se = 2.887. Pooling returns it to mu with 1 - B = 0.996. Any no-pooling alternative would have to special-case these motors, and a per-motor free-shape model would fit 80 numbers to a median of 7 events each — the exact failure the hierarchy.py docstring says the design refuses.
2. Parameter economy under the frozen scoring rule. The hierarchy carries 2 free parameters regardless of motor count because eta_m is integrated by 33-node Gauss-Hermite, never estimated. Against M1_WEIBULL's 1 parameter, the training likelihood buys 7.88 nats for that one extra parameter. The structure is cheap.
3. It gives a per-motor posterior at all. posterior_motor_shape yields q(eta_m) with a real SD (0.078–0.183). M7_HIERARCHICAL_MOTOR reports point estimates only. If the next step is a per-motor prediction with uncertainty, only the hierarchy can supply it.

Where it is decoration — do not sell it:

1. Held-out separation from its own tau -> 0 limit is NOT_ESTABLISHED (§5). A builder may not describe the hierarchy as predictively necessary on this cohort.
2. The heterogeneity it models is not independently detectable by a per-motor-estimate statistic on this cohort (§3) — and that statistic is underpowered, so this is NOT_ESTABLISHED in both directions, never "no heterogeneity" and never "the motors are equivalent".
3. A hierarchy diagram is not a mechanism. Nothing in §1–§4 identifies a physical source of between-motor variation. Between-motor differences in dwell shape, if real, are confounded on this dataset with load, stator number, PMF, temperature, and cell-to-cell state, none of which is a covariate here. Attributing tau to any of those would be an unsupported inference.
4. The Lmotor-2 kinetic-mode layer stays uninstantiated. If a 2-parameter hierarchy cannot separate itself from a 1-parameter model on held-out data, adding a further latent layer adds unidentifiable capacity, not testability.

The receipt that would make the "hierarchy is necessary" lane true. Held-out motor-equal separation of the hierarchy from M1_WEIBULL with an interval that excludes 0 and lies outside +/- 0.042 nats. At the observed per-event effect size (~0.0006 held-out, ~0.0099 training) that is not reachable by re-analysis of these 19 holdout motors: it needs either far more motors, or motors with genuinely larger dwell-shape dispersion, or a covariate that predicts eta_m — all of which require transfer (P4) or intervention (P6), neither of which this repository's passive dataset can supply. The action set is empty; that is structural, not sample-size-limited.

The receipt that would kill the lane. A boundary-correct null calibration of the §1 profile statistic (the NOT_RUN — COMPUTE_BUDGET item in §3) in which the observed 2 * 7.878806412156678 = 15.7576 falls inside the tau -> 0 null distribution. That would mean the training-side preference for a positive tau is itself an artefact of the boundary geometry rather than evidence of dispersion, and the honest description of the hierarchy would collapse to decoration in both directions. That computation is specified, cheap in principle (~10.5 min), and should be run before anyone describes tau > 0 as supported.

Limitations and things this probe did not compute

• NOT_RUN — COMPUTE_BUDGET: boundary-correct parametric-bootstrap null for the §1 profile statistic (~10.5 min, one process). No p-value is attached to 7.8788 nats anywhere above.
• NOT_COMPUTED: BCa intervals for any F-side contrast. Per D7, BCa exists only for the 48 frozen B3 contrasts; the F-side contrasts are percentile only. No BCa was computed or estimated.
• NOT_COMPUTED: implied excess between-motor SD in §3 — the variance excess is negative (-0.01599464546541901).
• NOT_CHECKED: anything about the mark process (nextStateN, direction, jump). Would require holdout access; the D5 firewall was not crossed.
• Right-censoring is unexercised. The frozen cohort excludes right-censored events entirely (compare.Inputs asserts this), so the censoring branch of the F-side likelihood is untested by every number here. Sections 1–3 all pass all-False censoring arrays.
• The §3 null is a plug-in. k_null was fixed at the pooled MLE; uncertainty in k_null is not propagated, which if anything makes the null spread slightly too narrow — and the observed SD is already below the null mean, so correcting it would not change the direction.
• mu optimiser tolerance. xatol = 1e-5 on mu leaves ~1.1e-05 nats of slack in the profile (measured against the independent pooled fit). Every reported drop is >= 0.72 nats at the reported grid points, so this slack does not affect any statement made.
• Only one cohort, one species, one study. derived_eligible_1_to_8 from Wadhwa 2022, E. coli behavioural evidence. Nothing here transfers to Salmonella or Bacillus structural evidence, and no cross-species statement is made.
• Two long runs (B4C11, B4C01) were in flight throughout. Neither was read, cited, disturbed, nor written to. No counter from either progress log appears in this document.

NEXT_ACT = run the boundary-correct parametric-bootstrap null for the §1 profile drop (300 replicates under the tau -> 0 generator at k_null = 0.6250888368122796, each replicate refitting fit_motor_stack and recomputing the drop to TAU_MIN), as a single ~10.5-minute process scheduled AFTER B4C11 and B4C01 have landed, so that tau > 0 is either supported by a calibrated reference distribution or withdrawn to decoration.