LightChainr · LightChainr · Sep 8, 2026
diff --git a/notes/probe-invariant-shape-limit-20260908.md b/notes/probe-invariant-shape-limit-20260908.md
@@ -0,0 +1,197 @@
+# Probe re-run — Invariant shape of the threshold law (#622), 2026-09-08
+
+Frontier `claude/matching-one-workspace-pwr5pv` @ `8b5f9d1a`; base `main`
+(at `07350444`). This is the second run of the #622 probe. The first run
+(merged as PR #628, with the W4 kill in #629 and the direction-2 toy
+theorem in #634) defined `S`, adjudicated W1–W5, and separated `g`. This
+run **re-adjudicates those verdicts on the repaired inputs** — PR #653
+(repaired bond laboratory, `dual_fail = 0`, exact `M(1/2) = 0`) and PR
+#655 (corrected spin-0 N725 Q/Z, pooled histogram, correct `cos 4θ`
+weights) — and reports what actually changes. Nothing is re-derived that
+is cited; nothing entered `docs/STATUS.md`; no issue is closed; no
+production, no MC, no exponent fit; N=725 is read as the committed block,
+not scored; 1.55 stays retired; no literature search.
+
+**Verdict in one line: the repaired bond laboratory converts W2 from
+"half-killed" to "split precisely" — duality pins `Z` to the antisymmetric
+class (proved, T1) but pins no member of that class (proved, T2) — and the
+four-size decomposition (T3) locates the surviving shape-limit question in
+the antisymmetric part `A_N` of `Z`, under a now-exactly-stated extra
+hypothesis C3. The north-star verdict stays (C), but the hypothesis is
+sharper than the one #628 named.**
+
+---
+
+## 0. Inputs (cited, imported, not redone)
+
+| Object | Source | Status |
+|---|---|---|
+| Exact site L=3,4 census, rank-pair decomposition | #606 + merged #628 census | cited |
+| Repaired bond L=3 census (`dual_fail=0`, pairs 75460/111224/75460, `M(1/2)=0`, `F(1/2)=1/2`) | **PR #653**, `results/probe-invariant-shape/census-exact.json` | imported; replaces the #628 bond numbers, whose `dual_fail=118133` was withdrawn as a bug compound (#632) |
+| Corrected spin-0 N725 pooled Q/Z, 11 levels, delete-one SEs | **PR #655**, `results/probe-invariant-shape/n725-zflow-corrected.json` | imported; `n725-zflow.json` (#628) stays withdrawn on disk |
+| Toy location/shape separation theorem | #628 `toy-families.json` | cited |
+| Consensus 9-vector `g` | #605 lineage, `type582-residual/latest.json` | cited |
+| Theorem L statement + union-bound silence on rates | #613/#614, #618 | cited |
+
+The two #633 retractions that constrain this run are honoured exactly:
+(1) batch-covariance rank does **not** bound the mean shape's dimension;
+(2) the g-vs-ΔZ affine-removed angle is reported **unoriented (20.4°)**,
+not "159.6° ≈ orthogonal". No sentence below revives either.
+
+## 1. New machine-checked statements
+
+Script: `scripts/probe_invariant_shape/rerun_shape_limit_20260908.py` →
+`results/probe-invariant-shape/rerun-20260908.json`.
+Tests: `tests/test_rerun_shape_limit_20260908.py` (all pass; the pre-existing
+`tests/test_probe_invariant_shape.py` is updated to the repaired census
+numbers, as #653 already did on its branch).
+
+### T1 (theorem). Duality symmetry of the bond laboratory
+
+If every configuration satisfies `r_b + r_w = 2` under the geometric dual
+transport (i.e. `dual_fail = 0`), then, writing `P11, P20, P02` for the
+per-rank-pair count polynomials in the number of occupied bonds `k`:
+
+```text
+P02[k] = P20[NB−k],   P11[k] = P11[NB−k]
+⇒  M(p) = −M(1−p)  (M antisymmetric about 1/2)
+⇒  F(p) = 1 − F(1−p)  (self-complementing; F(1/2) = 1/2)
+⇒  with the symmetric anchors a = 1/4, b = 3/4:
+   Z(u) + Z(1−u) = 1 and Z(1/2) = 1/2 exactly.
+```
+
+Proof: complementing the occupied set swaps `r_b ↔ r_w` (vacant primal =
+occupied dual under transport), giving the pair-count symmetry; the two
+function identities are the reindexed binomial sums (mass
+`P20+P11+P02 = C(NB,·)` is exactly the `dual_fail = 0` condition); the `Z`
+identities are the affine chart applied to `Q(u) = 1 − Q(1−u)`.
+
+Machine-checked on the repaired census: `Z_antisymmetry_max_err ≈ 1e-15`
+(float side of an exact object), `M(1/2) = 0`, `Z(1/2) = 1/2`.
+
+**This reverses #628's §4.3.** The "bond laboratory loses the duality
+identity" finding was the broken sign, not physics. The site `r_b + r_w = 2`
+identity *does* have a bond carrier, exactly.
+
+### T2 (theorem, by construction). Duality pins the class, not the shape
+
+`F` self-complementing ⟺ density symmetric about 1/2. So the T1 class is
+exactly the symmetric-density class — and inside it `Z` is still free:
+two members, `f ∝ e^{−(p−1/2)²/2s²}` and the same with a `(1 + 3·tanh²((p−1/2)/s))`
+tail-weighting, both satisfy every T1 constraint to integration error and
+have shapes differing by **0.1503** at `u = 0.9` (both exactly
+antisymmetric). Recorded in the JSON with both `Z` vectors.
+
+**Consequence for W2 (final form).** W2 split in two, both halves now
+exact: (i) "self-duality constrains the shape" is *true but only to the
+antisymmetric class* — one linear constraint per reflected pair, no
+member pinned; (ii) "the only finite-size object is shape" survives
+intact. W2's hoped-for mechanism ("self-duality pins the shape") is dead
+as a pinning mechanism; its `p_c = 1/2` freeness was always trivial.
+
+### T3 (observation, four sizes, no fit). The A/S decomposition
+
+```text
+A(u) = (Z(u) − Z(1−u) + 1)/2   (antisymmetric part: A(u) + A(1−u) = 1)
+S(u) = (Z(u) + Z(1−u) − 1)/2   (symmetric part: zero iff T1 holds)
+Z = A + S.
+```
+
+Across site L=3, site L=4, bond L=3 (self-dual), and the corrected spin-0
+N725 block, on the decile grid, `u = 0.1 … 0.5`:
+
+```text
+S(0.1):  site3 −0.0541 → site4 −0.0446 → bond3 0 (exact) → N725 −0.0090
+A(0.1):  site3 −0.4116 → site4 −0.4279 → bond3 −0.4352 → N725 −0.4591
+```
+
+Pairwise max spread over all four objects: **A: 0.0475**, S: 0.0541. The
+ordering at `u = 0.1` is monotone in the sequence (site L3, site L4, bond
+L3, site N725) — recorded as a trend, explicitly **not** claimed as
+convergence, and no rate is fitted (two exact sizes + one self-dual size +
+one committed block do not make a limit).
+
+Reading: the *symmetric* part of the shape — the sector #628 measured as
+the "even deformation" — collapses toward 0 as the objects gain size or
+duality; the *antisymmetric* part is the slowly-drifting survivor. The
+shape-limit question, in this chart, therefore lives in `A_N` alone.
+
+### T4 (conjecture, stated to be killable). C3 — self-matching restoration
+
+```text
+C3:  sup_{p ∈ K} |M_N(p) + M_N(1−p)| → 0  for every compact K ⊂ (0,1).
+```
+
+Equivalent at the level of the grid to `S_N → 0` (both sides exact at
+finite N via `F = (1+M)/2`). Under C3, any shape limit is antisymmetric
+and the open object is `A_∞ = lim A_N`. Status: **open** — not implied by
+Theorem L (whose hypotheses are spent on location), not proved here, not
+refuted here. Its finite-N witnesses are exact: `M(1/2) = −21/64` (site
+L=3), `−13757/32768` (site L=4), `0` (bond L=3), and the corrected N725
+block's `S(0.1) = −0.0090 ± 2.4e-5`.
+
+## 2. Re-adjudicated verdicts (deltas from #628 only)
+
+| Verdict | #628 | This run | Why |
+|---|---|---|---|
+| North-star (3) | (C), extra input "RSW-type shape-side access" | **(C), sharpened**: the concrete, rank-moment-level hypothesis is C3 (T4); the shape-side input #628 named is sufficient-not-necessary language for the same gap | T1–T3 |
+| W1 | killed as question, promoted as theorem | stands unchanged | #628 §3 + #634; T2 is the duality-class refinement: even *within* the antisymmetric class, location + duality leave the shape free |
+| W2 | half-killed ("duality constraint does not transfer to bonds") | **split precisely**: the transfer *does* happen (T1); what fails is pinning (T2) | PR #653's repair removed the false negative |
+| W3 | killed (94°/160° + non-rank-1 covariance) | **downgraded to "undetermined at lab sizes"** per #633: report the unoriented 20.4°, and the covariance argument is retracted; the separation sentence is withdrawn to "no lift exhibited; the lab angle is 20.4° unoriented, which does not separate" | #633 retractions 1–2 |
+| W4 | killed both ways | stands, with the §5 counterexample *downgraded* per #632: `P11 ↦ 3/2 P11` leaves the simplex (mass 593/512), so it is not a probability counterexample; the surviving exact sentence is the weaker "equal M does not determine the full joint `(P11, P20, P02)`" — the W4 kill on "M(1/2) governs" (pointwise value vs functional of the full polynomial) is untouched | #632 table |
+| W5 | resolved (canonical gauge) | stands unchanged | no new input |
+
+## 3. The g-lift question after the corrections
+
+The corrected N725 JSON records the only admissible comparisons:
+
+* tiny-torus `ΔZ` vs `g`: raw 93.9°, affine-removed **unoriented 20.4°** —
+  not a separation;
+* `DZ_Q[g]` vs `Z_725` in the 0.2/0.8 chart: unoriented 81.4° — not the
+  g-vs-ΔZ test (no second committed size), not used as a claim;
+* N725 g-vs-ΔZ: **undetermined** without another committed size. Stopping
+  there, per #633.
+
+So the merged "g is separated from the shape tangent" headline of #628 is
+**withdrawn to**: *no lift with a transformation law has been exhibited;
+the one lab-size angle compatible with the repaired inputs is 20.4°
+unoriented, which neither supports nor refutes a lift.* The main result of
+this run does not depend on that question either way.
+
+## 4. What this run contributes, in priority order
+
+1. **T1**: the repaired bond laboratory is an exactly self-complementing
+   census — `M` antisymmetric, `F` self-complementing, `Z` antisymmetric,
+   all proved from `dual_fail = 0` and machine-checked. This is the clean
+   laboratory the issue asked for, restored.
+2. **T2**: duality pins `Z` to the antisymmetric class and nothing more
+   (spread 0.15 inside the class). W2's mechanism dies a *precise* death.
+3. **T3 + T4**: the shape-limit question is localised to `A_N` under the
+   explicit conjecture C3 (`M_N + M_N(1−·) → 0`), with exact finite-N
+   witnesses at four sizes and no fitted rate. This is the concrete form
+   in which the next probe (or a production ticket, if one is ever
+   funded) should attack the limit: measure `S_N`'s decay and `A_N`'s
+   drift on the committed blocks that already exist.
+
+## 5. What was *not* done (per standing)
+
+No `docs/STATUS.md` edit; no issue closed; no merge; no production or MC;
+no second scale (1.55 retired); no N=725 scoring — the block is read once,
+through PR #655's corrected object; no λ-sweep, no jackknife invention; no
+literature retrieval (no `BLOCKED_ON_#620` marker was needed: every
+theorem used is elementary combinatorics on the census itself); no Cardy
+citation as finite-L shape; GOVERNANCE §2E honoured (site and bond remain
+two experiments; the T3 comparison is cross-model and labelled as such).
+
+## Handoff
+
+* To #619: the bond pair-count symmetry `P02[k] = P20[NB−k]`,
+  `P11[k] = P11[NB−k]` is an exact ledger fact with `dual_fail = 0`.
+* To #620: unchanged single question — is there a theorem from torus
+  crossing probabilities to the inverse-CDF shape? Now with a sharper
+  target: the *antisymmetric part* `A_∞`, and the testable conjecture C3.
+* To any future shape probe: `S_N`'s decay is measurable on existing
+  committed histograms (`S(0.1) = −0.0090 ± 2.4e-5` at N=725); `A_N`'s
+  drift across L=3,4,N=725 spans only 0.048 — both are cheap re-reads, and
+  the next size is the only thing that can turn the T3 trend into a
+  verdict.
diff --git a/results/probe-invariant-shape/census-exact.json b/results/probe-invariant-shape/census-exact.json
@@ -97,31 +97,26 @@
   "3": {
    "bonds": 18,
    "configs": 262144,
-   "dual_fail": 118133,
+   "dual_fail": 0,
    "rank_pair_counts": {
-    "0,2": 51704,
-    "1,2": 49263,
-    "0,1": 4397,
-    "1,1": 41839,
-    "1,0": 5633,
-    "2,2": 12183,
-    "2,1": 46657,
-    "2,0": 50468
+    "0,2": 75460,
+    "1,1": 111224,
+    "2,0": 75460
    },
-   "M_half": "-309/65536",
-   "M_half_float": -0.0047149658203125,
-   "Q_quarter": 0.06398471681373033,
-   "Q_threequarters": 0.0756899475848746,
+   "M_half": "0",
+   "M_half_float": 0.0,
+   "Q_quarter": 0.3882947252165274,
+   "Q_threequarters": 0.6117052747834726,
    "Z_levels_float": [
-    -0.7416617264257065,
-    -0.1881488746822507,
-    0.15741237889336662,
-    0.41262227554113784,
-    0.6164158893134861,
-    0.7867630751111613,
-    0.9335085336027731,
-    1.062655985722729,
-    1.1781480664979216
+    -0.4351929790516855,
+    -0.12188426707886556,
+    0.11038357169268018,
+    0.31127822617880646,
+    0.4999999999999995,
+    0.688721773821193,
+    0.8896164283073188,
+    1.1218842670788656,
+    1.435192979051685
    ],
    "self_dual_p_half": 0.5
   }