The recorded eigengap reproduces the audit's association — and stratifies the orientation shortfall

Run date: 2026-09-02 Code revision: working tree of the record-latent-config-spectrum change, parent 84061bdec824da9c33865d52e59a13b3750fb9fb (the merge of the geometry audit) Configuration: examples/trajectory_power_study/angle_pivotality_diagnostic.json — byte-identical to the 2026-09-01 diagnostic run (same SHA-256, 2dfa5cc3…) Records: results/latent-spectrum-2026-09-02/merged.jsonl (500 replicates, 0 failures, 199 permutations each, 500 of 500 carrying the configuration-spectrum block) Tables: results/latent-spectrum-2026-09-02/report/ — pivotality_spectrum.csv, eigengap_stratified_power.csv, config_spectrum.csv, plus the pre-existing pivotality and study tables Provenance: results/latent-spectrum-2026-09-02/PROVENANCE.txt Change: openspec/changes/record-latent-config-spectrum Answers: Phase 5 readiness items 1 (left-open gap), 3, and 4 (stratifying covariate); geometry audit plan item P1 Environment: Python 3.11.15, numpy 2.3.5, pandas 2.3.3, scikit-learn 1.8.0, scipy 1.16.3; no R at runtime

Summary

The pivotality report established that an orientation replicate's outcome is decided by the width of its own angle permutation null, and stated that "nothing the harness records today identifies which replicates will be resolvable". The geometry audit (finding F1) named the missing quantity — the relative eigengap (λ₁−λ₂)/Σλ of the centered latent stage-mean configuration — by regenerating replicates from persisted seeds. This run persists that quantity per replicate and re-measures the association from the record alone.

Three results, all from the same 500 records:

The audit's association reproduces. In the orientation cell, Spearman(pooled eigengap, own angle null q95) = −0.749 [−0.824, −0.648]; log–log Pearson −0.622. The audit measured −0.75 and −0.62.
It is specific to angle. In the same cell the association with the delta and shape null widths is +0.093 [−0.105, +0.284] and −0.032 [−0.227, +0.166] — both intervals cover zero.
It stratifies the shortfall. Orientation angle power rises monotonically across within-cell eigengap terciles: 0.44 → 0.76 → 0.85. The overall 0.68 is an average over a covariate the harness now records.

And the change is inert: every rejection rate and every pivotality slope reproduces the 2026-09-01 run exactly.

1. The association, measured from records

pivotality_spectrum.csv, null_target = q95, n = 100 per cell. Spearman with a Fisher-z 95% interval:

Cell	`delta`	`angle`	`shape`
`none` (zero effect)	−0.092	−0.814 [−0.871, −0.735]	−0.108
`translation` @1.00 (`type_i_baseline`)	+0.019	−0.740 [−0.817, −0.636]	+0.163
`translation` @1.00 (`power_primary`)	+0.053	−0.836 [−0.887, −0.765]	+0.009
`magnitude` @1.00	−0.086	−0.701 [−0.789, −0.586]	−0.038
`orientation` @1.00	+0.093	−0.749 [−0.824, −0.648]	−0.032

Every angle interval excludes zero by a wide margin; no delta or shape interval does. Log–log Pearson tells the same story for angle (−0.61 to −0.71 across cells; −0.622 in the orientation cell).

Read this as a property of the geometry, not of the effect. The association is present in the zero-effect baseline and in both translation controls at essentially the same strength. That is what the audit's mechanism predicts: PC1's estimator variance scales like noise over the eigengap, so a near-isotropic stage configuration yields a wide angle null whether or not a group difference was constructed. The recorded eigengap therefore predicts resolvability, and it does so without seeing the group labels.

2. It stratifies the orientation shortfall

eigengap_stratified_power.csv, orientation @1.00, α = 0.05, within-cell terciles of the recorded pooled eigengap:

Stratum	eigengap range	mean eigengap	n	`angle` rejections	rate	MC SE
bottom	0.0057 – 0.0359	0.0243	34	15	0.44	0.085
middle	0.0364 – 0.0552	0.0451	33	25	0.76	0.075
top	0.0556 – 0.1173	0.0739	33	28	0.85	0.062

A 3× separation in eigengap maps to a 0.41 separation in power (SE of the difference 0.105). The cell's headline 0.68 is an average over this spread: replicates whose latent stage configuration has a dominant axis clear the 0.80 floor; those whose configuration is near-isotropic do not, and no amount of signal in a direction the configuration cannot resolve will move them.

The recorded distributions (config_spectrum.csv) show how tight the regime is. Pooled eigengaps per cell:

Cell	mean	median	min	max
`orientation` @1.00	0.0475	0.0444	0.0057	0.1173
`magnitude` @1.00	0.0422	0.0406	0.0028	0.1612
`translation` @1.00 (`power_primary`)	0.0431	0.0404	0.0080	0.1126
`translation` @1.00 (`type_i_baseline`)	0.0532	0.0537	0.0030	0.1670
`none`	0.0450	0.0409	0.0022	0.1248

Every cell sits near the isotropic end (a straight trajectory would read 1.0), and the per-group entries agree with the pooled ones — consistent with the audit's account that the independent-indicator baseline, not the trajectory surgery, sets the eigengap. That is the lever readiness item 4 hands to baseline continuity (plan item P4), and it is now measured per replicate rather than inferred.

3. The change is inert

The config is byte-identical to the 2026-09-01 diagnostic, so the operating point is directly comparable. Every published number reproduces:

Quantity	2026-09-01	this run
orientation `angle` / `delta` / `shape` power	0.68 / 0.65 / 0.99	0.68 / 0.65 / 0.99
magnitude `delta` / `angle` / `shape`	1.00 / 0.00 / 0.00	1.00 / 0.00 / 0.00
Type I `none` (delta/angle/shape, combined)	—	0.04 / 0.05 / 0.01, 0.09
`angle` null-tracking slope: `none` / `translation` (t1b, pp) / `magnitude` / `orientation`	0.931 / 0.925, 0.960 / 0.874 / 0.811	0.931 / 0.925, 0.960 / 0.874 / 0.811
orientation `delta` / `shape` slope	0.018 / 0.058	0.018 / 0.058

Reproduction to three decimals on every slope, and to the replicate on every rate. The spectrum consumes no randomness: it is computed from the LS-mean vectors estimate_difference already fits, after the permutation index is drawn. Unit tests pin the permutation draws and null distributions as byte-identical with the recording flag on and off, serially and in parallel (tests/test_trajectory_spectrum.py).

Cost. One SVD of a k×d matrix per configuration. On this design point (4 stages, rank-3 PLS latent space) recording the per-permutation eigengap adds ~12 ms to a 199-permutation RRPP — under 0.1% of a ~14 s replicate, which is dominated by the PLS double cross-validation.

4. What this does and does not change

Nothing statistical. No statistic, p-value, permutation draw, generator behaviour, integration method, or latent-space sizing changed. The eigengap is a recorded covariate and a reporting qualifier — never a filter, weight, or test modification.
The audit's caution stands. The spectrum is only observed. Any group-aware sizing or supervision would void the fixed-latent-space RRPP conditioning; nothing here reads the group labels.
Resume policy changed deliberately. parameter_signature now includes config_spectrum_version, so a shard written before this change refuses resume rather than producing a merged set in which only some records carry the covariate. Already-merged pre-change record sets still load (empty block) and their committed reports are unaffected — verified by reporting a pre-change record set under the new code and finding specificity_matrix.csv, power_curves.csv, type_i_table.csv, acceptance_report.csv, and acceptance_report.json byte-identical to the pre-change code's output.

Reproduction

# 5 cells x 100 replicates = 500 work units; ~50 minutes on 16 local shards.
for i in $(seq 0 15); do
  uv run python scripts/run_study_shard.py \
    --config examples/trajectory_power_study/angle_pivotality_diagnostic.json \
    --out-dir results/latent-spectrum-2026-09-02 \
    --shard-index "$i" --n-shards 16 --error-policy record &
done
wait
uv run python scripts/motco_study.py merge --out-dir results/latent-spectrum-2026-09-02
uv run python scripts/motco_study.py report \
  --config examples/trajectory_power_study/angle_pivotality_diagnostic.json \
  --out-dir results/latent-spectrum-2026-09-02
uv run python scripts/angle_null_pivotality.py \
  --merged  results/latent-spectrum-2026-09-02/merged.jsonl \
  --out-dir results/latent-spectrum-2026-09-02/report

Do not pass --n-jobs: it is part of the cell parameter signature. The shard and merged JSONL are gitignored as regenerable; report/ and PROVENANCE.txt are committed.