LightChainr · LightChainr · Sep 8, 2026
diff --git a/notes/probe-Mhalf-vs-shape-main-rerun-20260908.md b/notes/probe-Mhalf-vs-shape-main-rerun-20260908.md
@@ -0,0 +1,84 @@
+# probe #625 re-run on main — M(1/2) vs shape Z (2026-09-08)
+
+**Variant:** this is a fresh verification on a clean Mac clone of `main`,
+NOT a re-do of the merged #629 probe. Relationship to prior work:
+
+- PR #629 (`probe/625-mhalf-vs-shape` @ `ac0bb20a`) was **merged into main**
+  on 2026-09-07; the D1–D3 probe programs and results live on main.
+- The original D4 (`scripts/probe/bond_L3_Z.py`) enumerated 2^18 bond
+  configs. This re-run is **Mac-budgeted to NOT enumerate 2^18**: D4 here
+  imports the repaired bond census from PR #653
+  (`repair/632-bond-lab-and-wrapping-l3l4` @ `7109b174`,
+  `results/probe-invariant-shape/census-exact.json`; main's copy is the
+  pre-#632-repair one with `dual_fail=118133`) and derives the D4 verdict
+  from the 3-count census arithmetically. Nothing is enumerated beyond 2^9.
+- Game B site L=3 (2^9) and the L=4 spot checks were run locally; all
+  numbers below were **reproduced on this machine** and match the values
+  committed on main (bit-identical floats in the JSONs).
+
+## D5 verdict
+
+```text
+W4 at L=3,4:  KILLED
+evidence:     Game B L=3, exact: on the m-pinned line F_lambda(p)=1/2
+              (m = E_lambda[r_b]-1 = 0 EXACTLY for every lambda = e^beta),
+              Z(0.5; anchors 0.1/0.9) sweeps
+              0.73817 (lambda=1/16) -> 0.54242 (lambda=1) -> 0.32192
+              (lambda=16); interior-u spread over the 11-point grid =
+              0.416 (a1/a9), 0.271 (a2/a8).  L=4 spot (lambda = 1, 5/2,
+              2/5): spread 0.178.  Both ~4e7 x the 1e-8 tolerance.
+Z_3 vs Z_4:   ||Z_3-Z_4||_inf on {0.1..0.9} = 0.0161 (anchors 0.2/0.8);
+              at p_L^H |delta| = 0.0066, at p=1/2 |delta| = 0.0837
+              (the same-p row separates because M(1/2) != 0:
+              -21/64 at L=3, -13757/32768 at L=4).
+bond vs site: INCOMPARABLE — bond X-law at p=1/2 is 3-atom
+              (0.28786, 0.42429, 0.28786); inverse-CDF is a staircase;
+              Z_bond undefined by the probe's own rule; jumps recorded.
+              Census imported from PR #653 (dual_fail=0, M(1/2)=0 exact,
+              P(r=0)=P(r=2) exact — duality-oddness checked from counts,
+              not assumed).
+```
+
+## What was run here (all exact rational arithmetic)
+
+- **D1 precondition** (imported from main's `mhalf_common.stream_joint`):
+  Alexander `r_b + r_w = 2` asserted on the stream at L=3, L=4 — holds
+  (`dual_fail=0`). Precondition, not a headline.
+- **D2** `scripts/probe/mhalf_Z_physical_tables.py` (main's, unmodified):
+  reproduced `p_L^H = 0.586511455113` (L=3) / `0.590672112331` (L=4),
+  `M(1/2) = -21/64` / `-13757/32768`, `F(1/2) = 43/128` / `19011/65536`,
+  `||Z_3 - Z_4||_inf = 0.0161`, self-symmetry rows
+  `Q(u)+Q(1-u)-1` (max 0.1712 / 0.1803 vs `2p_L^H - 1` = 0.1730 / 0.1813).
+- **D3 Game A** (L=3 fake law, `m` pinned at -21/64 exactly): shape still
+  swings, spread 0.667 — the easy game fails too.
+- **D3 Game B** (the game that matters; site L=3 = 2^9 configs, Mac-OK,
+  full 11-point lambda grid; L=4 spot at lambda = 1, 5/2, 2/5):
+  monotone guard asserted before every inversion; `m` pinned at 0 on the
+  whole lambda family; `Z` not pinned. **W4 dies for the physical
+  interpolation.**
+- **D4 (imported, no 2^18 here)** `scripts/probe/bond_L3_Z_imported.py`:
+  validates the imported census (configs = 2^18, counts sum, dual_fail=0),
+  checks `M_bond(1/2) = P(r=2) - P(r=0) = 0` exactly and `P(r=0) = P(r=2)`
+  exactly from the counts, reports the 3-atom jumps and
+  `P(r >= 1)(p=1/2) = 0.712142944` (census-derivable). The median-crossing
+  `p_wrap = 0.419649` needs the full enumerated `F_bond(p)` and is cited
+  from main's committed `bond_L3.json`, not recomputed.
+- **Tests:** `python3 tests/test_probe_mhalf_vs_shape.py` — 5 fast OK,
+  1 slow skip (Python 3.13.7).
+
+## Deliverables in this PR (all new; 0 modified, 0 deleted)
+
+- `notes/probe-Mhalf-vs-shape-main-rerun-20260908.md` (this file)
+- `scripts/probe/bond_L3_Z_imported.py` (D4 from imported census)
+- `results/probe-Mhalf-vs-shape/bond_census_imported_653.json`
+  (imported #653 census, provenance-recorded)
+- `results/probe-Mhalf-vs-shape/bond_L3.json` (regenerated in imported mode)
+- `results/probe-Mhalf-vs-shape/physicalZ.json` (reproduced, bit-identical)
+- `results/probe-Mhalf-vs-shape/gameAB.json` (reproduced, bit-identical)
+- `results/probe-Mhalf-vs-shape/latest.json` (v2 index for this variant)
+
+## Standing
+
+Does not enter `docs/STATUS.md`. Closes nothing (#606, #619, #622, #608,
+#625 all stay open). Does not merge. Census numbers are inputs, not
+deliverables. No ninth mechanism, no exponent, no `L^-theta` fit.
diff --git a/results/probe-Mhalf-vs-shape/bond_L3.json b/results/probe-Mhalf-vs-shape/bond_L3.json
@@ -1,31 +1,32 @@
 {
-  "L": 3,
-  "M_bond_half_by_rank_counts": "0",
-  "M_bond_half_exact": "0",
-  "P_r_at_half": {
-    "0": 0.2878570556640625,
-    "1": 0.424285888671875,
-    "2": 0.2878570556640625
-  },
-  "X_law_at_half_jumps": {
+  "M_half": "0",
+  "M_half_float": 0.0,
+  "P_r0_eq_P_r2_exact": true,
+  "P_wrap_at_p_half": 0.7121429443359375,
+  "P_wrap_at_p_half_frac": "46671/65536",
+  "X_law_jumps": {
     "P(X=+1)=P(r=2)": 0.2878570556640625,
     "P(X=-1)=P(r=0)": 0.2878570556640625,
     "P(X=0)=P(r=1)": 0.424285888671875
   },
-  "Z_bond_undefined": true,
-  "bonds": 18,
+  "Z_undefined": true,
+  "Z_undefined_reason": "X-law at p=1/2 is 3-atom on {r-1 in {-1,0,1}}; inverse CDF is a staircase, not strictly increasing",
+  "bond_vs_site": "INCOMPARABLE at L=3: different edge sets (18 bonds vs 9 sites), different N, and Z_bond undefined; Z_site well-defined",
+  "cited_from_full_enumeration": {
+    "p_wrap_median_crossing": 0.41964924845800616,
+    "source": "main results/probe-Mhalf-vs-shape/bond_L3.json (2^18 enum, 3.3 s)"
+  },
   "configs": 262144,
   "duality_odd_checked": true,
+  "imported_from": "PR #653 (repair/632-bond-lab-and-wrapping-l3l4 @ 7109b174)",
+  "imported_sha": "PR #653 (repair/632-bond-lab-and-wrapping-l3l4 @ 7109b174)",
   "issue": 625,
-  "note": "rank convention = #608/#627 bond ambient rank (single-BFS lift); Cov(O,X)=P(r=2)=Var(X)/2 identity imported from #627 C6 as the same-r sanity check, not re-derived",
-  "p_wrap_half": 0.41964924845800616,
+  "mode": "imported-census (PR #653), no 2^18 enumeration on this machine",
   "rank_counts": [
     75460,
     111224,
     75460
   ],
-  "schema": "matching-one.probe-mhalf-vs-shape.bondL3.v1",
-  "seconds_enumeration": 3.3,
-  "seconds_total": 3.4,
-  "verdict_comparability": "INCOMPARABLE: the bond X-law at p=1/2 is the 3-atom measure ((r-1) in {-1,0,1}); its inverse-CDF is a staircase, so Z_bond is undefined by the probe's own rule.  Site Z exists.  The self-dual checks that DID run: M_bond(1/2)=0 exact and P(r=0)=P(r=2) exact."
+  "schema": "matching-one.probe-mhalf-vs-shape.bondL3.v2",
+  "seconds": 0.0
 }
diff --git a/results/probe-Mhalf-vs-shape/bond_census_imported_653.json b/results/probe-Mhalf-vs-shape/bond_census_imported_653.json
@@ -0,0 +1,38 @@
+{
+  "imported_from": "PR #653 (repair/632-bond-lab-and-wrapping-l3l4 @ 7109b174)",
+  "imported_path": "results/probe-invariant-shape/census-exact.json",
+  "import_date": "2026-09-08",
+  "importer": "issue #625 re-run on main (this PR); no 2^18 enumeration performed on this machine",
+  "note": "main's census-exact.json bond block is the pre-#632-repair one (dual_fail=118133); this is the repaired census. The fix itself is #653's change; only the numbers are imported here as the D4 bond input.",
+  "census": {
+    "schema": "matching-one.probe-invariant-shape.census.v1",
+    "bond": {
+      "3": {
+        "bonds": 18,
+        "configs": 262144,
+        "dual_fail": 0,
+        "rank_pair_counts": {
+          "0,2": 75460,
+          "1,1": 111224,
+          "2,0": 75460
+        },
+        "M_half": "0",
+        "M_half_float": 0.0,
+        "Q_quarter": 0.3882947252165274,
+        "Q_threequarters": 0.6117052747834726,
+        "Z_levels_float": [
+          -0.4351929790516855,
+          -0.12188426707886556,
+          0.11038357169268018,
+          0.31127822617880646,
+          0.4999999999999995,
+          0.688721773821193,
+          0.8896164283073188,
+          1.1218842670788656,
+          1.435192979051685
+        ],
+        "self_dual_p_half": 0.5
+      }
+    }
+  }
+}