#622: reflection residual and finite-size shape motion on the N=145,290,725 lineage

[LightChainr]

Draft. Does not merge, does not close #622.

Implements the 2026-09-12 re-scope of #622 on the committed same-observable
P50/#612 blocks. C2 reanalysis only: no Monte Carlo, no GPU, no exponent fit,
no docs/STATUS.md edit.

Order of operations (fixed by the ticket)

Each orientation's pooled inverse CDF Q_i is built first; only then is the
declared spin-0 combination Q = sum_i w_i Q_i formed (CDFs are never mixed
before inversion). Weights satisfy sum w = 1 and sum w cos4theta = 0 to
6e-17. Anchors a=0.2, b=0.8; W = Q(0.8)-Q(0.2) > 0;
Z(u) = [Q(u)-Q(0.2)]/W; A(u) = Z(u)+Z(1-u)-1 (the corrected normalized
reflection residual, not M(p)+M(1-p)). Every delete-one batch repeats the
whole Q -> weighting -> W -> Z -> A map, giving a joint covariance. Independent
coordinates u = 0.1,0.3,0.4,0.5 (since A(0.2)=A(0.8)=0, A(1-u)=A(u)).

Result

Lineage: same channel rank-2 cross wrapping, same observable K_plus/K_minus,
sizes N=145,290,725. site_count = |a+bi|^2, so the shortest period holds
sqrt(N) sites. Spin-0 weights (all interpolating): (0.50711806, 0.49288194)
at N=145 and N=290, (0.53827771, 0.46172229) at N=725.

• Reflection residual resolved nonzero at every size. |A| = 0.0305142,
  0.0236407, 0.0168313 (SE ~3e-5); nominal chi2 (4 dof) = 2.27e6, 1.00e6,
  3.56e5; per-coordinate standardized values 400–1350. The A covariance is
  near-collinear (condition number 2.4e8/6.3e7/1.7e7), so the full-inverse
  chi2 is reported next to the correlation-free diagonal chi2 — it is smaller
  at every size, so the rejection is signal, not inversion of a near-null
  direction. No pseudoinverse is used.
• Decreasing on this finite lineage. |A| steps -0.0068735 +/- 3.86e-5
  (178σ) and -0.0068094 +/- 4.52e-5 (150σ). Three sizes cannot establish
  convergence or a limit; no exponent is fitted.
• Adjacent shape change resolved. |dZ| = 0.00537503 (2.1e-5) and
  0.00499749 (2.7e-5); |dA| = 0.00687410 (3.9e-5) and 0.00680943 (4.5e-5);
  zero-displacement tests reject at nominal p below double reference.
  Between the two intervals |dZ| falls by 3.78e-4 +/- 3.45e-5 (10.9σ)
  while |dA| changes by -6.47e-5 +/- 5.95e-5 (1.1σ, unresolved).
• Equal weighting changes nothing qualitative (max |A| difference
  1.6e-5), though it retains a spin-4 residue -1.37e-2, +1.37e-2, -4.11e-2.

Provenance / resource

• Machine XPk2PZ, main base 6edf775e. Inputs are the committed blocks
  results/server-20260829/P50-n145-n290-fullcurve/raw/{n145,n290}_100m.* and
  results/server-20260907/P612-n725-fullcurve/raw/n725_100m.*
  (100 batches × 1e6 paired × 2 orientations).
• Reuses scripts/threshold_quantile_lineage.py blob 3b328f29 (Restore spin-0 N725 Q/Z from the committed histogram (#633) #655 lineage,
  corrected cos 4theta). N=725 pooled Q reproduces the Restore spin-0 N725 Q/Z from the committed histogram (#633) #655 decile values to
  4.1e-9.
• Profiled first: N=725 pooled pass 1.05 s, one delete-one batch 1.02 s;
  full delivery 5 min 23 s single-threaded, ~0.09 CPU-hours (cap 4).
• 10 unit tests in tests/test_quantile_shape_lineage_622.py pass.

Artifacts: results/probe-invariant-shape/quantile-shape-lineage-622.json
(full pooled Q,W,Z,A and 9x9 delete-one covariances per size and weighting,
per-size A=0 diagnostics, adjacent-size dZ/dA with propagated covariance,
declared-norm changes, equal-weight sensitivity),
notes/issue-622-quantile-shape-lineage-20260912.md, and the reusable script.

[LightChainr]

Independent Grok re-check on the committed P50/#612 histograms (not the agent log).

Estimator. Used scripts/threshold_quantile_lineage.py blob 3b328f29 only for per-orientation inversion + spin-0 weights. Built Z and A myself: anchors 0.2/0.8, A(u)=Z(u)+Z(1-u)-1. Delete-one repeated the whole map (100 batches/size). Did not call quantile_shape_lineage_622.analyse_size for the numbers below.

Pooled, my numbers:

N	W	A(0.1,0.3,0.4,0.5)	|A| (SE)	spin-0 w
145	0.11921330	−0.0204652, 0.0093537, 0.0138721, 0.0152436	0.0305142 (2.57e-5)	(0.50711806, 0.49288194) interpolating
290	0.09220375	−0.0159133, 0.0072317, 0.0107147, 0.0117706	0.0236407 (2.88e-5)	same
725	0.06553076	−0.0113451, 0.0051435, 0.0076198, 0.0083705	0.0168313 (3.49e-5)	(0.53827771, 0.46172229) interpolating

These match the note to the printed digits. Identities hold: Z(0.2)=0, Z(0.8)=1, A(0.2)=A(0.8)=0, A(1-u)=A(u). w·cos4θ = 0 to 1e-16. Channel in the N=725 metadata is rank-2 cross wrapping; seeds differ across sizes.

Adjacent motion (pooled L2 on independent coordinates): 145→290 \|dZ\|=0.00537503, \|dA\|=0.00687410; 290→725 0.00499749, 0.00680943 — same as the note. \|A\| steps 0.0068735 / 0.0068094 with independent-size SEs 3.86e-5 / 4.52e-5 → 178σ / 151σ.

Order of operations. Mix-then-invert vs invert-then-weight differs by at most 2.5e-6 (N=145), two orders below the A signal; the ticket-correct order was used. Tests in tests/test_quantile_shape_lineage_622.py are tiny algebra locks (anchors, no pseudoinverse), 10/10 pass; they are not the lineage run.

Verdict I will keep: on this finite same-observable lineage the normalized shape is not reflection-symmetric, \|A\| decreases, and adjacent non-affine motion is resolved. No exponent, field, or p_c. Do not merge. Do not close #622 / #650.

[LightChainr]

Control review in progress, 2026-09-12. The pooled Q -> orientation weighting -> Z/A pipeline and same-lineage comparison are useful. Two concrete uncertainty corrections are required before the reported derived numbers are used:

1. analyse_size treats the 100 delete-one widths as 100 independent measurements: sqrt(sample_variance(width_deleted)/B). The delete-one jackknife variance is (B-1)/B * sum((w_deleted-mean)^2), so the printed W standard errors are too small by B-1=99. The exact linear correction is already recoverable from stored Q covariance: Var(W)=S_Q[b,b]+S_Q[a,a]-2*S_Q[a,b]. No histogram rerun is needed.
2. finite_movement adds the two interval norm variances as if independent. Even if Q145,Q290,Q725 are independent, Delta1=Z290-Z145 and Delta2=Z725-Z290 satisfy Cov(Delta1,Delta2)=-S290. For h=||Delta2||-||Delta1||, with g_i=Delta_i/||Delta_i||, Var(h)=g1^T S145 g1+(g1+g2)^T S290(g1+g2)+g2^T S725 g2. The same correction applies to A. The individual adjacent displacements and individual ||A|| changes are not invalidated by this shared-middle error.

Also withdraw the general claim that full chi-square <= diagonal sum proves absence of near-null sensitivity: that implication is false. Large individual standardized coordinates can establish nonzero response under their own assumptions, but are not a validation of the full inverse's calibration. I am directly implementing a read-only correction and further shape analysis from the stored Q/covariance. Keep the historical artifact; do not commission another production or overwrite the old result.

[LightChainr]

Review of later files on this PR (head 59fff91c), not a new histogram job.

Keep the finite-lineage findings I independently recomputed on the original blocks: A≠0, ||A|| decreasing, adjacent non-affine displacement resolved. Do not merge until the opening note matches the later corrections already in notes/issue-622-quantile-shape-lineage-20260912.md: width SE was 99× too small; adjacent-interval ||ΔZ|| comparison must keep the shared-N290 covariance (later text: about 9.04 nominal σ, not 10.9); ||ΔA|| change about 0.897 σ, still unresolved.

A source-fixed quadratic from N145 absorbing most of the N290/N725 residual norm is not a physical coordinate. Transport-order job 34685638146 is archived in PR #707 as unresolved (composition differences ~-4.3e-6, SE 1.8–1.9e-6, no sign crossing). Do not re-dispatch #622 as the same analysis. Do not close.