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

.github/workflows/shape-lineage-review.yml

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+name: Existing shape-lineage review
+on:
+  pull_request:
+    branches: [main]
+    paths:
+      - 'scripts/shape*.py'
+      - 'tests/test_shape*.py'
+      - '.github/workflows/shape-lineage-review.yml'
+permissions:
+  contents: read
+jobs:
+  review:
+    runs-on: ubuntu-latest
+    timeout-minutes: 5
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: '3.11'
+      - run: python -m pip install numpy scipy mpmath
+      - run: |
+          python -m unittest discover -s tests -p 'test_shape_lineage_review.py'
+          python -m unittest discover -s tests -p 'test_shape_common_chart.py'
+          python -m unittest discover -s tests -p 'test_shape_reflection_commutator.py'
+          python -m unittest discover -s tests -p 'test_shape_reflection_order.py'
+          python -m unittest discover -s tests -p 'test_shape_transport_order.py'
+      - name: Full-profile transport order; prior executed tests remain archived
+        run: python scripts/shape_transport_order.py

notes/common-chart-commutator-20260912.md

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+# Common symmetrizing coordinate: an exact necessary condition and an unresolved data test
+
+2026-09-12. Follow-up to #706. The real-data calculation IS NOW COMPLETED;
+it was not delegated as another production task. Its result is negative in
+information: the tested commutator is unresolved, not evidence for a chart.
+No novelty claim is made for the algebra.
+
+## 1. Object and conditional lemma
+
+Let Q_i be increasing quantile functions, F_i their inverses, and
+
+    R_i(p)=Q_i(1-F_i(p)),       R_i o R_i=id.
+
+For this application Q_i is the ALREADY COMBINED spin0 quantile function.
+F_i is its inverse, not an orientation-weighted mixture of physical CDFs.
+A nonlinear map need not commute with orientation weighting; do not transfer
+the outcome to a different observable without a separate map.
+
+Suppose one increasing coordinate psi symmetrizes all three laws, possibly
+about distinct centers c_i: psi(R_i(p))=2*c_i-psi(p). Then R_i R_j is
+conjugate to translation by 2(c_i-c_j), and translations commute. Therefore
+
+    (R1 R2)(R2 R3)=(R2 R3)(R1 R2),
+    R1 R3=R2 R3 R1 R2.                                (1)
+
+Composition acts rightmost first. This proves the necessary identity without
+fitting a polynomial or exponent. The evaluated residual is
+
+    C(p)=R1(R3(p))-R2(R3(R1(R2(p)))).                 (2)
+
+Conjugating all maps preserves equality (1), not the numerical magnitude of
+(2). Zero is necessary, NOT sufficient. If R1=R2, (1) holds for ANY R3, so
+a null can be structurally uninformative. Three zero evaluations also cannot
+prove a functional identity on an interval.
+
+## 2. Mathematical and numerical controls
+
+Shared logit coordinates give R_k(p)=k(1-p)/(p+k(1-p)). For k=1,2,3 and rational
+p=.3,.5,.7, (1) holds exactly with Fraction arithmetic. Replacing R3 by
+(1-p^3)^(1/3) preserves its involution property but gives C(.5)=
+0.01884049384875336739614... . Individual symmetrizability is not common
+symmetrizability. Three mathematical controls cover these cases.
+
+The data inverse is cheaper than a nested quantile solver. If
+Q(u)=w1*F1^-1(u)+w2*F2^-1(u), with positive weights summing to one, solve
+
+    F1(x)=F2((p-w1*x)/w2),
+    max(0,(p-w2)/w1) <= x <= min(1,p/w1),
+
+and set F(p)=F1(x). The objective is increasing. This is exactly the inverse
+of the quantile combination; it is NOT the CDF mixture. Positivity is checked.
+Component CDFs use the beta-mixture formula of the birth-rank histograms;
+an independent 55-dps mode-anchored Bernstein sum with rational histogram
+weights checks the pooled compositions and selected deleted compositions.
+Three more tests check the inverse and beta/Bernstein identity.
+
+## 3. Executed existing-data result
+
+Run 34685079362, job 103530556772, head 2f900dd2066bd38a4d53f02e76ab32cb7b32cedf;
+tested merge checkout 55d2b9b5d66918493593f3b20401707ba2bb50a7. The bounded job
+passed all 13 new mathematical tests, the earlier nonlinear shape recheck,
+and `python scripts/shape_reflection_commutator.py` (78.51 seconds). Raw
+histograms were already committed; there was no Monte Carlo or new evidence.
+
+For N145,N290,N725, use the three reference points p_j=Q290(u_j), u=.3,.5,.7.
+When deleting N290, these reference points are recomputed, not held fixed.
+Each independent size is deleted separately, with 100 batches; batch numbers
+across independent seeds are not artificially paired. Both composed maps are
+re-evaluated and their full 3x3 covariance is retained.
+
+| u | C(p_j), high-precision pooled | nonlinear jackknife SE |
+|---|---:|---:|
+|.3|-5.642573955e-10|1.493472756e-9|
+|.5|-2.733621556e-13|4.855386934e-13|
+|.7|+5.482200467e-10|1.450287413e-9|
+
+All three are below 0.6 marginal standard errors. The descriptive full
+Gaussian quadratic form is 0.5746485/3. There is no resolved contradiction
+of (1). The result is **UNRESOLVED_COMMON_CHART**, not a positive model choice.
+The first and third coordinates are strongly anticorrelated; report the matrix,
+not three independent tests. Its condition number is about 4.81e8.
+
+The beta-path Q values agree with the earlier pooled reconstruction within
+2.14e-14. Pooled commutators agree with the 55-dps Bernstein path within
+1.20e-15; selected deleted controls within 8.11e-16. The largest involution
+control is 1.33e-15, or 0.274% of the smallest MARGINAL standard error. These
+are numerical checks, not all-direction error bounds or exact confidence
+certificates. Pooled composition levels remain in [.23047,.76946]; all deletion
+paths remain inside the checked [.05,.95] range.
+
+Result with full covariance/provenance:
+`results/research-control-20260912/shape-reflection-commutator.json`.
+
+## 4. Why the null needs particular caution
+
+Write T12=R1 R2=id+epsilon*u+O(epsilon^2) and
+T23=R2 R3=id+epsilon*v+O(epsilon^2), in C2 on the tested interval. Expanding
+composition gives
+
+    T12 T23-T23 T12=epsilon^2*(u'*v-v'*u)+O(epsilon^3).
+
+Near coincident involutions, the first variation vanishes and the commutator
+is second-order small. A noisy nonzero derivative estimated away from this
+singular null does not by itself justify ordinary Gaussian/chi-square
+calibration. The printed reference tail is therefore DESCRIPTIVE, not a
+calibrated acceptance or confidence statement. Nonlinear jackknife propagation
+and higher precision address different questions and do not cure nonregularity.
+
+Consequently a null cannot be used to pick a higher-degree chart, price a
+sample top-up from a weak point signal, or claim no dynamic finite-size
+correction. A common symmetrizing coordinate is also weaker than full location-
+scale collapse: it imposes parity but does not pin the symmetric profile.
+
+## 5. Scientific consequence
+
+The N145-fixed quadratic chart removes most measured asymmetry in norm but
+fails as an exact full-vector law. The present parameter-free necessary
+condition does NOT rule out a more general common chart. Both statements can
+be true. The right surviving question is a source/geometry-resolved separation
+between analytic-coordinate contributions and genuine finite-size shape
+corrections, with explicit predictions and an informative statistic. No
+successive polynomial-degree search, larger-N purchase, or GPU run follows
+from this delivery. #275's candidate-specific original-U map remains distinct.

notes/issue-622-quantile-shape-lineage-20260912.md

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+# #622 reviewed: finite shape movement and its dominant smooth-coordinate component
+
+2026-09-12. The original #706 source analysis is retained in Git history and
+`results/probe-invariant-shape/quantile-shape-lineage-622.json` (v1). Its pooled
+Q/Z/A and per-size covariance remain inputs. Its W standard errors and the
+independent-error comparison of two adjacent interval norms are superseded.
+Do not run the historical `quantile_shape_lineage_622.py` and interpret those
+v1 fields as the current verdict. Current analysis entry points are
+`scripts/shape_lineage_review.py` and `scripts/shape_lineage_nonlinear_jackknife.py`.
+
+## Executed validation
+
+Run 34684353419, job 103528608026, head eff8c5e158fb9be93ffde27ba80bc958eff34573,
+merge checkout 51f9817c59fcf757858c0338ec724aa4992e0d94. Seven new mathematical
+checks passed. The first command read existing Q/covariance; the second actually
+reconstructed pooled and all 100 delete-one Q vectors from each of the three
+committed histogram sets. It took 94.70 s; all three pooled Q vectors reproduced
+EXACTLY in the executed float path. No simulation or new independent evidence.
+
+Summary: `results/research-control-20260912/shape-lineage-reviewed-summary.json`.
+The scripts produce full covariance/diagnostics; summary fields were extracted
+from the successful job stdout, not invented or inferred from CI colour.
+This bounded job is not a claim that the complete repository suite passed.
+
+## Corrected original conclusion
+
+A=Z(u)+Z(1-u)-1 is resolved nonzero and decreases in norm across N145/290/725.
+This finite-lineage result survives. Individual adjacent Delta Z/Delta A also
+remain nonzero. The width SEs must be multiplied by 99: the original script
+mistook delete-one estimates for independent observations. The corrected
+primary widths and SEs are
+
+| N | W | corrected jackknife SE |
+|---|---:|---:|
+|145|0.1192132991|3.673003868e-6|
+|290|0.0922037501|2.737674335e-6|
+|725|0.0655307632|2.322561571e-6|
+
+Delta1 and Delta2 share N290, so their cross-covariance is -S290 even though
+size blocks have independent random streams. Correctly propagating it gives
+|Delta Z_2|-|Delta Z_1|=-0.0003775392 with SE 0.00004177061 (about 9.04 nominal
+sigma, not 10.9). The analogous Delta A norm change has SE 0.00007209953 and
+is only 0.897 nominal sigma from zero. Full nonlinear deletion independently
+reproduces these corrected uncertainties. No convergence or exponent follows.
+
+## New finding: most asymmetry is consistent in magnitude with a smooth chart
+
+The ratios ||A||/W are 0.2559635, 0.2563966, 0.2568465. The median coefficient
+K_mid=-4*A(.5)/W is approximately -0.511 at all three sizes. This motivates
+an analytic-coordinate control, not another freely fitted exponent.
+
+The conditional normal-form lemma in `notes/shape-lineage-review-20260912.md`
+states: if Q=h(t+s z) with a common increasing C4 h and reflected z, then
+A/W=K*(X^2-1/4)+O(W^2), K=h''/h'^2 and X=(Z-Z_reflected)/2.
+It is a lemma about coordinate transformations, not a percolation theorem.
+
+One exact quadratic chart was fixed from the N145 MEDIAN ONLY:
+
+    phi(p)=p+beta*(p-.5)^2,
+    beta=0.2684067158 +/- 0.0002742089 (nonlinear jackknife).
+
+It is increasing throughout [0,1]. Applied unchanged to N290 and N725, it
+reduces the independent-coordinate asymmetry norms to respectively 0.267081%
+and 0.068014% of their original values. This is norm reduction, NOT explained
+variance or proof that the physical scaling field is quadratic.
+
+The attractive scalar reading is not the full-vector verdict. Including the
+shared source-beta uncertainty, the two target residual vectors have
+D=105.9080 on 8 nominal degrees of freedom (log10 p=-18.5793). The source's
+other three coordinates already fail exact quadratic symmetry. Likewise,
+exact equality of the entire A/W vector across sizes fails (D=309.74895/8).
+Thus neither exact scalar collapse nor a universally exact quadratic chart
+is established. Do not raise the polynomial degree until something passes.
+
+The full nonlinear jackknife confirms the target covariance against the
+linearized calculation in ALL stochastic directions: generalized eigenvalues
+lie between 0.999998211 and 1.000002384. This matters because the target
+covariance condition number is about 1.3e8; agreement of diagonal errors alone
+would not have sufficed. It checks propagation, not exact tail coverage for
+an estimated covariance. All references remain nominal/asymptotic.
+
+Equal weighting has the same qualitative reading but is a sensitivity of the
+SAME blocks. Its quadratic residual ratios are 0.323681% and 0.052247%; its
+full-vector null also fails. No second evidence vote is counted.
+
+## Decision
+
+The raw A signal should not be named a new irrelevant field merely because it
+shrinks. Separate the dominant smooth-coordinate-like contribution from the
+much smaller resolved residual. Rejecting one exact quadratic chart does not
+reject every smooth chart, while arbitrary higher-degree fitting is not an
+identifying experiment.
+
+`notes/common-chart-commutator-20260912.md` gives the next parameter-free
+necessary condition: three reflection involutions from the effective quantile
+laws must have commuting pair compositions if one common symmetrizing chart
+exists. Its real-data evaluation is not done in this delivery. It can use the
+same histograms; no larger-N or GPU purchase is licensed. #275's original-U
+candidate-map question remains separate.

notes/shape-lineage-review-20260912.md

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+# Review of #706 and a smooth-coordinate diagnostic
+
+2026-09-12. This note supersedes the width standard errors and interval-to-interval norm errors of `notes/issue-622-quantile-shape-lineage-20260912.md`. The v1 source artifact is preserved. The reviewed reader is `scripts/shape_lineage_review.py`; it needs only stored pooled Q and covariance, not another production or histogram replay. Actual numerical output must be read before a result is claimed.
+
+## Two corrections
+
+For B=100 delete-one widths, the jackknife variance is (B-1)/B times the centered sum of squares, not the independent-sample-mean variance. The v1 width SE is too small by B-1=99. Because W=Q(b)-Q(a), its corrected variance is exactly S_Q[a,a]+S_Q[b,b]-2*S_Q[a,b], using the stored jackknife Q covariance.
+
+For independent size blocks X1,X2,X3, Delta1=X2-X1 and Delta2=X3-X2 have cross-covariance -S2. Let g_i=Delta_i/||Delta_i||. The variance of ||Delta2||-||Delta1|| is g1^T S1 g1+(g1+g2)^T S2(g1+g2)+g2^T S3 g2. Use this for Z and A; the individual adjacent differences and per-size A norms remain valid.
+
+Full inverse chi-square <= the diagonal sum is NOT an inverse-robustness certificate. Marginal standardized coordinates can independently show nonzero response under their own assumptions, but they do not validate a full inverse's rank/calibration. The reviewed reader uses a 60-decimal solve and labels the estimated-covariance reference nominal. sqrt(N) here is a Euclidean period length, not an integer count of sites on a winding path. Square-site tiny-rank controls share the F=(1+M)/2 mathematical observable at their own geometry; bond controls are a different model, not a rung of this site lineage.
+
+## A deeper alternative to fitting another exponent
+
+Observed A decreasing with size need not by itself identify a new irrelevant field. Assume, only for the following conditional lemma, Q_N(u)=h(t_N+s_N z_N(u)), with z_N(1-u)=-z_N(u), z_N(a)=-1/2, z_N(b)=1/2; h is a common C4 increasing coordinate map with h' bounded away from zero, bounded grid z_N, and s_N->0. Set X_N(u)=[Z_N(u)-Z_N(1-u)]/2. Taylor expansion gives
+
+A_N(u)/W_N = K_N [X_N(u)^2-1/4] + O(W_N^2),
+K_N=h''(t_N)/h'(t_N)^2.
+
+Proof: W=h' s+O(s^3); the reflected numerator is h'' s^2(z^2-1/4)+O(s^4); and X=z+O(s^2). Divide by W^2. Uniformity requires the stated derivative/bounded-grid assumptions. This is a conditional analytic-coordinate lemma, NOT a percolation scaling theorem.
+
+The median eliminates K without fitting a grid direction: K_mid=-4*A(1/2)/W. The remaining three independent coordinates test R=A/W-K_mid*(X^2-1/4). R=O(W^2) is allowed, so rejecting R=0 at finite N does not refute all smooth coordinate explanations. A/W exact equality across sizes is separately a finite model, not a theorem.
+
+The reader also fixes ONE exact quadratic coordinate phi(p)=p+beta*(p-.5)^2 using only N145's median symmetry, then applies it without target refit to N290/N725. All target covariance includes shared source-beta uncertainty via the joint Q Jacobian. This is a stronger finite null than the asymptotic lemma. Rejecting it must not trigger an automatic higher-degree rescue. The experiment was already seen: this is C2, not prospective validation, and the physical occupation parameter is not redefined.
+
+Seven local mathematical checks pass: the factor 99, shared-middle scalar variance 6 versus 4, exact quadratic normal form, order-W^2 smooth cubic remainder, and source-parameter uncertainty propagation. The real-data bounded workflow runs the existing-artifact reader only. The input Q covariance is retained in full; new nonlinear quantities use delta-method propagation, with the original nonlinear-jackknife A covariance shown as a comparison. Equal weighting is a same-block sensitivity, never an independent experiment.

notes/shape-transport-order-20260912.md

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+# Full-profile common-coordinate collapse has a noncrossing transport test
+
+2026-09-12. Mathematical extension of #706; the real-data result must be read
+from the executed job before a verdict is claimed. Fixed points remain
+p=Q290(.3,.5,.7), the same central set used in the reflection controls.
+
+Suppose ONE increasing psi and ONE profile z satisfy psi(Q_i(u))=a_i+b_i*z(u),
+b_i>0, for i=145,290,725 on a common domain. Define A=Q145 o F290 and
+B=Q725 o F290, where F290 is inverse of the already combined spin0 Q290.
+Then psi A psi^-1 and psi B psi^-1 are positive-slope affine maps. For any
+affine maps a(x)=s*x+t and b(x)=r*x+v,
+
+    a(b(x))-b(a(x))=(s-1)*v-(r-1)*t,
+
+which is independent of x. Thus A(B(p))-B(A(p)) is either identically zero
+or has one strict sign throughout the common domain. Increasing conjugacy
+preserves sign, not magnitude. Opposite signs rule out common affine
+conjugacy, and hence the specified exact common-coordinate location-scale
+collapse. This is stronger than mere simultaneous reflection symmetry,
+which need not pin the symmetric part of the profile.
+
+IMPORTANT: nonzero alone does NOT reject. Aff(1) is nonabelian; two affine
+maps with distinct fixed points need not commute. This would be the same
+mistake as using symplectic orthogonality as Euclidean orthogonality: the
+correct group identity, not a familiar label, carries the inference.
+
+The script uses separate-size nonlinear deletion, including randomness in
+the Q290 reference points, full stored covariance, and 55-dps pooled
+Bernstein checks against a fast beta-mixture path. A fixed Bonferroni set
+of three two-sided nominal intervals (total alpha=.0027) tests whether
+opposite signs are resolved. It uses no inverse covariance, no fitted
+chart, no covariance-selected contrast, and no search over points. The
+entire composition and every deletion must remain in central quantiles
+[.05,.95]. A failed domain or precision condition is a limitation, not
+permission to choose nicer points. The result is C2 on existing data.
+
+No resolved crossing does not prove a common coordinate. A future power
+claim needs a specific alternative, not repeated null-compatible tests.
+Three mathematical controls cover affine noncommutation, nonlinear
+conjugacy, and an actual crossing of two increasing non-affine maps.