POWER ATLAS — MOTOR-EQUAL SCORING (PROBE 2)

Status: BUILDER-SUPPORT PROBE. Moves no P-level, changes no frozen verdict, creates no claim. It answers one design question: at the current motor count and scoring rule, what effect sizes can this design actually resolve, and what would make the assay severe?

Truth label. Every resolve-rate, CI width and power number in the ATLAS section is SYNTHETIC — produced by simulating per-motor NLPD arrays with a known injected effect and running the frozen paired motor-cluster bootstrap on them. Nothing here is OBSERVED. The only recorded quantities are the calibration constants and the real anchors, each carrying its source artifact inline.

• Artifact: hierarchical-aif/results/motor_stack_aif/power_atlas.json (sha256 66311a07866b916c77a44faab4bc22688db1e4e7fd6438bce77b1ba8ffe8976b, 162654 bytes)
• Generator: hierarchical-aif/scripts/power_atlas.py
• Reproduce: cd hierarchical-aif && python scripts/power_atlas.py
• Runtime: 89.5 s, single process, 200 grid cells + 2 special cells.

0. Prediction and falsifier, recorded before the sweep ran

Label note. These atlas predictions are numbered A1–A4. They are not parity-ladder levels. P0–P8 in this document always mean the frozen ladder defined in CLAUDE.md, which this probe does not touch and cannot move.

1. Method (what was and was not computed)

No model was refit. The atlas isolates the scoring rule from the modelling by construction:

challenger_i = mu0 + sigma_b * z1_i
ref_i        = challenger_i + effect + sigma_d * z2_i          i = 1..n_motors
contrast     = S(ref) - S(challenger)        # frozen convention

then motor_stack_aif.score.contrast_with_ci(ref, challenger, n_rep=2000) — the frozen implementation, resampling unit MOTOR, percentile interval — is run on each synthetic pair.

This is exact rather than approximate: the paired motor-cluster bootstrap statistic is mean(d[idx]), a function of the per-motor difference array alone. The score levels never enter the statistic. That is why one Gaussian difference generator covers every model family: a family enters the atlas only through the pair (effect, sigma_d), and the real (effect, sigma_d) of all nine recorded comparisons are tabulated in §3.

Consistency axis. rho is the equal-marginal correlation between the two per-motor score arrays, so sigma_d = sigma_b * sqrt(2 - 2*rho). The recorded contrasts live in rho ∈ [0.9674, 0.99999999999998], which is what the grid spans.

Independent cross-check. Every cell also carries a normal-approximation power value computed by a routine that shares no code with the bootstrap (analytic_resolve_rate). Max divergence over 200 cells: 0.0876 (worst cell n=100, effect 0.02, rho 0.99, 120 sims; that cell's recorded resolveRateMonteCarloSE is 0.03952847075210474); mean 0.0169.

Determinism. Two full executions, canonical JSON bytes compared with generatedUtc and runtimeSeconds excluded (both are volatile by design): identical, canonical sha256 0816896e16ed8c67d48229652df35a3e1e258945308acea014e389b194311bad on both runs. Byte-identity of the raw file is not claimed, because the file embeds a timestamp and a wall-clock runtime.

Calibration constants — recorded, not invented

Axes not swept

2. The atlas at 19 motors — the actual design

Resolve-rate over synthetic repetitions (200 per cell, Monte-Carlo SE ≤ 0.036). Analytic cross-check in parentheses.

Materiality of each column, against the frozen floor, is fixed and independent of the table: 0 = NULL_NEGATIVE_CONTROL · 1e-06, 0.005, 0.01, 0.02 = IMMATERIAL_BELOW_FROZEN_FLOOR · 0.042, 0.08, 0.15 = MATERIAL_AT_OR_ABOVE_FROZEN_FLOOR.

Detectable effect size at 19 motors (80% target — DESIGN_ONLY)

Motors needed to resolve a 0.042-nat effect at 80% (DESIGN_ONLY)

Resolve-rate at the frozen-floor effect across the motor-count axis:

Mean CI half-width (nats) in the same cells — this is what the 0.042 floor should be compared to:

3. Real anchors — where the recorded comparisons actually sit

All values below are recorded, read from perMotorNLPD arrays in F_SIDE_MOTOR_STACK_SCORING_RESULT.json and M4_M6_M7_PER_MOTOR_CONTRASTS_RESULT.json (the F_MOTOR_STACK array agrees between the two files to max |diff| = 0.000e+00). Sizing columns are analytic, at n=19, 80% DESIGN_ONLY.

Recorded percentile half-widths at 19 motors, same artifact: M0 0.1072286 · M2 0.0859624 · M3 0.0789979 · M8 0.0744904 · M5 0.0250071 · M1 0.0110923. The frozen 0.042 floor is the narrowest frozen B3 contrast, not a typical one — real half-widths at this motor count span roughly 0.011 to 0.107.

The load-bearing split in that table

The nine anchors fall into two disjoint regimes, and the atlas says they demand opposite things.

• Serious adversaries (M0, M2, M3, M8, M4): sigma_d ≈ 0.15–0.25. Analytic MDE80 at 19 motors is 0.097 to 0.159 nats — 2.3x to 3.8x the frozen 0.042 floor. At the actual motor count this design cannot reliably resolve even a material difference against the models it most needs to beat. Required motor counts to reach the floor at 80%: 102 to 271.
• Near-duplicates of the candidate (M1, M6, M7): sigma_d ≈ 2e-07 to 0.026. These resolve easily — and every effect they could resolve is far below the floor. This is the D10 zone.

4. Resolve-rate and materiality are separate axes (D10, demonstrated)

Two synthetic cells, both at 19 motors, both with the same injected effect 2.506984e-07 nats — approximately 168 000x below the frozen floor. Only the per-motor difference dispersion differs, and both dispersions are taken from recorded arrays.

Same magnitude; resolve-rate 1.0000 versus 0.0850. Consistency, not magnitude, is what the paired motor-cluster bootstrap responds to. A design that resolves at 100% can be resolving something of no scientific size at all, so this atlas reports resolve-rate and materiality as two independent columns everywhere. They are not the same question, and a report that collapses them will present an immaterial difference as a finding.

These cells are SYNTHETIC. They are calibrated to the recorded M7/M3 numbers so the regime is the real one, but they are not a reproduction of the recorded M7 contrast and must not be cited as one.

The point-estimate direction (pointFavorsChallengerFrac, the self-win analogue) is recorded per cell alongside resolve-rate. At 19 motors, effect 0.02 it runs 0.580 / 0.675 / 0.675 / 0.985 / 1.000 across the rho grid while resolve-rate runs 0.090 / 0.105 / 0.135 / 0.605 / 1.000 — a modal win fraction can be near 1 while the CI-bound verdict remains NOT_ESTABLISHED. A point estimate is never a verdict.

5. Adverse finding — the assay is anti-conservative at 19 motors

Negative control (injected effect exactly 0; the frozen alpha is 0.05), pooled over the rho grid:

At the actual motor count the frozen percentile motor-cluster bootstrap falsely resolved 7.30% of synthetic null repetitions against a nominal 5%, about 2.8 Monte-Carlo SE above nominal. This is a property of the percentile bootstrap at 19 clusters, measured on SYNTHETIC Gaussian data.

Scope and limits of this finding, stated precisely:

• It is not evidence that any recorded verdict is wrong. It is a design diagnostic.
• It does not apply to the 48 frozen B3 contrasts as a criticism of their interval choice: those used BCa, which exists precisely to correct percentile bias, and per D7 intervalUsed was bca in 48/48. The F-side contrasts, the M4/M6/M7 extension, and this atlas are percentile only; their BCa companions are NOT_COMPUTED.
• It is measured under a Gaussian per-motor difference generator. Under the real, unknown difference distribution the excess is NOT_MEASURED.

6. Executive summary — better model math, or more/different data?

Answer from the atlas: more and different DATA — specifically more motors, and a second study. Model math alone cannot make this assay severe.

The reasoning is three recorded numbers plus one atlas column.

1. The entire frozen B3 leaderboard spans 0.1386691245 nats across all nine models, and the top seven span 0.0250191765 nats — below the frozen 0.042 floor. Seven of nine models are mutually immaterial by the project's own materiality standard before any power question is asked.
2. Against the serious adversaries the recorded per-motor difference dispersion is sigma_d ≈ 0.15–0.25 (§3). To resolve the top-seven spread of 0.0250191765 nats at 80% (DESIGN_ONLY) the analytic motor requirement is 288.5 (vs M4) · 399.3 (vs M8) · 444.7 (vs M3) · 497.4 (vs M2) · 764.9 (vs M0) — against 19 available.
3. For a 0.042-nat effect to be resolvable at 80% with the 19 motors that exist, the per-motor difference SD would have to be ≤ 0.065346. The recorded values against M2/M3/M8/M4 are 0.1992 / 0.1883 / 0.1785 / 0.1517 — a 2.32x to 3.05x reduction in per-motor difference dispersion would be required.
4. Better math that reduces sigma_d against near-duplicates buys nothing: that is exactly the D10 zone, where the design resolves effects ~1e-07 nats that no one should report as a win (§4). The recorded M7 anchor is the existence proof.

So the binding constraint is the number of experimental units and the narrowness of the channel, not the sophistication of the density model. A model improvement of 0.02 nats — larger than six of the eight recorded contrasts — is unresolvable at 19 motors against every serious adversary, and would be immaterial even if it resolved.

What would make the assay severe. Severity requires that the design be able to reject. Ranked by how much it moves the atlas:

• More motors. The only lever that moves the serious comparisons. ~100 motors buys 0.883 resolve-rate at the floor for rho 0.99; ~200 buys 1.000 at rho 0.99 and 0.750 at rho 0.97; the rho 0.90 regime stays out of reach at 200 (0.317). Transfer to a second independent study (the P4 level, which no modelling in this repository can close) supplies both motors and the study axis at once.
• A channel where candidate models genuinely disagree. All nine models compress into 0.139 nats on the duration channel; the mark channel is where they would separate, and it is quarantined under D5. intervention required / transfer required is the honest status.
• Pre-declared minimum material effect, reported with power at that effect. The floor exists; what is missing from the reporting surface is the power at it. The columns in §2 supply that.
• Report resolve-rate and materiality as separate columns everywhere. Already the D10 repair by added interpretation; this atlas quantifies why it is load-bearing.
• Reducing the percentile bootstrap's small-n anti-conservatism (§5) improves calibration but does not improve severity — a better-calibrated interval at 19 motors is still ~2.5x too wide for the comparisons that matter.

Which cells would change this answer. The conclusion is not unconditional. It flips if:

• Any new candidate shows recorded sigma_d ≤ 0.065346 against M2, M3 or M8 (not against a near-duplicate) while its mean difference stays at or above 0.042 — then 19 motors would suffice and the answer becomes better math. Cell to check: §3 columns 3 and 6 for the new model.
• The real per-motor difference distribution is far from Gaussian in a way that shrinks the bootstrap width — NOT_MEASURED here, and the shape is unidentifiable from 19 points.
• The events-per-motor axis turns out to dominate sigma_d — NOT_RUN — COMPUTE_BUDGET; would require refitting each family on subsampled event sets. If halving events per motor barely moved sigma_d, more events per motor would be confirmed as the wrong lever and motors the right one; if it moved sigma_d steeply, longer recordings per motor become competitive with more motors.
• Rows at n between 200 and ~500 motors: the atlas grid stops at 200, so the rho 0.90 and 0.97 requirements are analytic extrapolations, not simulated. Simulating them would confirm or refute the 225–765 motor figures.

What this probe does not do. It establishes nothing about biology, mechanism, or any model's correctness. It moves no P-level; the first unsatisfied level remains P4 transfer, unchanged by this probe. P6 for duration-only B3/B4 is unchanged; P6 for C11 U4 remains withdrawn pending the corrected run. No frozen verdict is touched. Every synthetic number here would be worthless as evidence about the motor and is offered only as evidence about the instrument.

NEXT_ACT = Extend the atlas grid to n_motors in {300, 500} for rho in {0.90, 0.97} at the frozen-floor effect to replace the analytic 225.09 / 750.29 motor requirements with simulated resolve-rates, and run it only after B4C11 (PID 26756) and B4C01 (PID 32988) have both landed, so the compute fence is clear.