LightChainr · LightChainr · Sep 8, 2026
diff --git a/notes/issue-635-identity-rewrite-20260908.md b/notes/issue-635-identity-rewrite-20260908.md
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+# #635 identity rewrite after #668 — `both-same` is not rank-2
+
+**Ticket:** LightChainr/Matching-One#672 (parent #650; follow-up to PR #662 from
+#660 and PR #668 from #665). **Date:** 2026-09-08.
+**Mode:** notes-first rewrite. No new census. Axis L=5 joint census is
+**NEED_HUAWEI** and was not run. Integers only. No `docs/STATUS.md` edit;
+no ticket closed or merged; draft PR against main.
+
+This note consumes, and does not re-enumerate:
+
+- **PR #662** — wrapping-form A, written before the #668 dictionary
+  (`notes/issue-635-identity-20260908.md`). The numerical identity there is
+  correct; its dictionary is not. This note replaces that dictionary.
+- **PR #668** — joint `(k, r_b, r_w, wrap_black, wrap_white, label5_black,
+  label5_white)` census at axis L=3 (`2^9`) and L=4 (`2^16`)
+  (`notes/issue-665-joint-rank-wrap-20260908.md`,
+  `results/issue665-joint-rank-wrap/axis-L3.json`, `axis-L4.json`).
+- **PR #646** — probe635 sector map, 5-name labels, two-amplitude decomposition
+  (`notes/probe635-sector-map-20260907.md`,
+  `results/probe635-sector-map/latest.json`).
+- **PR #653** — #632 bond-lab repair; repaired rank lab; wrapping-type 4×4
+  tables (`notes/wrapping-type-census-l3l4-20260908.md`,
+  `results/probe-invariant-shape/census-exact.json`).
+- **PR #657** — axis L=5 (`2^25`) and diamond L=4 (`2^32`) 5-name/coarse
+  censuses; #651 A-continues
+  (`notes/wrapping-type-census-axis-L5-diamond-L4-20260908.md`).
+
+All exact claims below are integers / `Fraction`. Conjectures are marked.
+
+---
+
+## 0. What was wrong with the #662 dictionary
+
+PR #662 wrote wrapping-form A as a difference of `both-same` sector
+amplitudes and mapped `both-same` onto the coarse `both` class, treating
+`both-same` as the rank-2 (cross) class. PR #668's joint census kills that
+identification with exact integers (axis L=3, `2^9`):
+
+```text
+both-same (black side) by rank:  rank-1 (spiral) 6,  rank-2 (cross) 91
+both-same (black side) at L=4:   rank-1 (spiral) 1120, rank-2 (cross) 9045
+```
+
+So `both-same = spirals ⊎ crosses` — the label is a *cluster-level* statement
+(one cluster winds both axes) and a rank-1 spiral cluster satisfies it. PR
+#662's identity `a_k = Δ#(both-same)` is nonetheless bit-for-bit correct at
+every tested size:
+
+```text
+reference bernstein (axis L=3): [-1,-9,-36,-78,-90,-36,36,36,9,1]
+  A_white (m-both-same):        [-1,-9,-36,-78,-90,-45,  0,  0, 0, 0]
+  A_black (p-both-same):        [ 0,  0,   0,   0,   0,   9, 36, 36, 9, 1]
+```
+
+The rewrite below states the same identity in the joint language so a later
+reader cannot confuse the sector label with the rank class, and proves the
+correction is harmless — the spirals cancel, not because every `both-same`
+event is a cross, but because D never fires on a `(1,1)` rank pair.
+
+## 1. The joint language (#668)
+
+For each configuration C of the L×L axis torus with |C| = k occupied sites,
+record the six-tuple of PR #668:
+
+```text
+(r_b, r_w)   ambient H_1 winding ranks of the black NN (primal) graph and
+             the white NN+NNN (matching) graph
+(w_b, w_w)   coarse wrap labels ∈ {neither, dir0, dir1, both}
+(s_b, s_w)   #646 5-names ∈ {none, x, y, both-same, both-two}
+```
+
+and the observable
+
+```text
+D(C) = 1{black NN wraps either axis} − 1{white NN+NNN wraps either axis}
+     = (w_b ≠ neither) − (w_w ≠ neither)
+M(p) = Σ_k a_k p^k (1−p)^(N−k),   a_k = Σ_{|C|=k} D(C).
+```
+
+Structural law (PR #653, certified against an independent incidence-matrix
+elimination): **`r_b + r_w = 2` on every configuration**, zero violations at
+L=3 (512 configs) and L=4 (65 536 configs). Consequence: rank-2 on both
+sides is impossible at any L on site; at most one side ever carries rank 2.
+
+## 2. Wrapping-form A′ — the identity in the joint language
+
+**A′ (rewrite of PR #662's wrapping-form A).** At every finite L on the
+axis torus, in the (rank pair × wrap × 5-name) joint language:
+
+```text
+a_k = Σ_{|C|=k} D(C)
+    = Σ_{|C|=k} [ 1{(w_b, w_w) = (both, neither)} − 1{(w_b, w_w) = (neither, both)} ]
+```
+
+i.e. **the D-carrying cells are exactly the two exclusive-cross cells**, and
+within each of them the 5-name pair and rank pair are constant:
+
+| cell `(w_b × w_w)` | 5-name pair | rank pair | D | axis L=3 total | axis L=4 total |
+|---|---|---|---|---|---|
+| `neither × both` | `none × both-same` | `(0, 2)` | −1 | 259 | 36 559 |
+| `both × neither` | `both-same × none` | `(2, 0)` | +1 | 91 | 9 045 |
+| `dir0 × dir0` | `x × x` | `(1, 1)` | 0 | 78 | 9 406 |
+| `dir1 × dir1` | `y × y` | `(1, 1)` | 0 | 78 | 9 406 |
+| `both × both` | `both-same × both-same` | `(1, 1)` | 0 | 6 | 1 120 |
+
+These five cells are the entire support at L=3 and L=4 (PR #668 §"Cell
+support"); no sixth coarse cell is populated, and within each populated
+coarse cell the 5-name pair is constant. The name A′ carries the two
+corrections to #662's A:
+
+1. **D-carrying cells are named as exclusive crosses.** `(neither × both)`
+   is the white side cross-wrapping alone (white rank 2, black rank 0) — a
+   single cross cluster on the matching lattice, by definition of the rank.
+   `both-same` there is doing cluster-level work (one cluster, both axes);
+   the *rank* statement `r_w = 2` is the cross statement. A′ writes D on the
+   rank/exclusive-cross cells, not on the `both-same` label.
+2. **`both-same × both-same` is not rank-2 and carries D = 0.** It is
+   exactly `(r_b, r_w) = (1,1)` — 6 configs at L=3 (k=6), 1 120 at L=4
+   (k=8: 120, k=9: 416, k=10: 448, k=11: 128, k=12: 8). Both sides wrap,
+   neither side cross-wraps: these are the spirals.
+
+**Per-k form at axis L=3** (PR #668 `axis-L3.json`, integers): D fires only
+at `neither×both` (k=0:1, 1:9, 2:36, 3:78, 4:90, 5:45) and `both×neither`
+(k=5:9, 6:36, 7:36, 8:9, 9:1), plus the D=0 spiral cell at k=6 (6) and the
+D=0 `x×x`/`y×y` cells (k=3:3+3, k=4:18+18, k=5:36+36, k=6:21+21). Collapsing
+over the coarse 4×4 with the D above reproduces the committed Bernstein
+integers `[-1,-9,-36,-78,-90,-36,36,36,9,1]` bit-for-bit — the same
+integers PR #662 accepted, now with the correct cell bookkeeping. The
+L=4 joint table (`axis-L4.json`) reproduces the committed L=4 rung the same
+way (`collapsed_D` field, bit-for-bit with `[-1,-16,-120,-560,-1812,-4272,
+-7448,-9424,-7874,-2896,1720,2832,1660,560,120,16,1]`).
+
+## 3. The spiral-cancellation lemma
+
+**Lemma (spiral cancellation; exact at L=3,4; conjecture for all L).** For
+every configuration C with `(s_b, s_w) = (both-same, both-same)` —
+equivalently (at every L where the census has been run, and structurally at
+all L) `(r_b, r_w) = (1,1)` — the contribution of C to `a_k` is zero:
+`D(C) = 0`. Hence the rank-1 spiral mass inside `both-same` on either side
+contributes nothing to `Δ#(both-same)`, and
+
+```text
+a_k = Δ#(both-same)(k)  still holds,  a_k = Δ#(exclusive cross)(k)  is the
+rank-honest form,  and the two agree because the spirals cancel.
+```
+
+**Proof at L=3 and L=4 (from the #668 integers).** By definition
+`D(C) = (w_b ≠ neither) − (w_w ≠ neither)`. On the cell `both × both` both
+indicators are 1, so `D = 0` — this is arithmetic, not a census fact. The
+census facts are: (i) the coarse `both × both` cell contains exactly the 6
+(L=3) / 1 120 (L=4) configurations tabulated, all `(1,1)`, all
+`both-same × both-same`; (ii) no `both-same` configuration occurs outside
+the cells `both × neither`, `neither × both`, `both × both`; (iii) the
+`x`, `y` cells (`dir0 × dir0`, `dir1 × dir1`) are diagonal and carry D = 0
+the same way. Summing D over the full per-k joint table therefore equals
+summing D over the two exclusive-cross cells alone, and the collapse
+reproduces the committed Bernstein integers bit-for-bit at both sizes
+(tripwire 1 of PR #668). ∎ (at L=3,4)
+
+**Conjecture 3.1 (all L).** On the axis torus at every L: the support of
+the joint distribution is contained in the five cells above; every
+`(both, both)` configuration has `(r_b, r_w) = (1,1)`; every `(neither,
+both)` configuration has `(0,2)` and every `(both, neither)` has `(2,0)`;
+and `both-two` has zero mass at every k. Evidence: exact at L=3 (2⁹), L=4
+(2¹⁶); the 5-name/coarse projections of the same statement hold at axis L=5
+(2²⁵) and diamond L=4 (2³²) (PR #657: five cells, `both-two` mass 0, no
+sixth cell) — but the joint rank-splitting of PR #668 has **not** been run
+at L=5 (axis L=5 joint census is NEED_HUAWEI).
+
+**Conjecture 3.2 (structural half of 3.1, provable-looking).** The rank
+statements of 3.1 do not need enumeration to fail: `r_b + r_w = 2` (proved
+at L=3 over all 2¹⁸ bond configurations in PR #646's repaired lab and
+certified at site L=3,4 in PR #653/#668) forces at most one rank-2 side,
+so rank-2 × rank-2 never occurs at any L, and if the cell support of 3.1
+holds then the rank exclusivity `(0,2)`/`(2,0)` on the D-cells is forced.
+What enumeration is actually needed for is the *wrap-cell* support claim
+(no sixth coarse cell, `both × both` always `(1,1)`), which is genuinely
+per-L. Marked conjecture because `r_b + r_w = 2` itself is certified at
+finite L only (it is a consequence of exact bond duality on the L=3 square
+bond torus, PR #646; its all-L status is not claimed here).
+
+## 4. A-continues survives the rewrite
+
+**Question 3 of the ticket: did A-continues (#651: axis L=5 / diamond L=4,
+`both-two = 0`) secretly depend on the wrong dictionary? No.**
+
+A-continues is a statement about the 5-name projection alone: the `both-two`
+cell has zero mass at every k at axis L=5 (2²⁵) and diamond L=4 (2³²), and
+the coarse 4×4 support stays the five Mertens–Ziff cells (PR #657). None of
+its inputs is the identification `both-same = rank-2`:
+
+- The five-cell support claim lives at the level of `(w_b, w_w)` pairs and
+  5-name pairs, which PR #668 showed are constant within each populated
+  coarse cell at L=3,4 — the projection discards exactly the rank column
+  that the wrong dictionary misidentified, so the projection is insensitive
+  to the misidentification.
+- The wrong dictionary entered only in PR #662's *interpretation* of the
+  `both × both` cell (it called the spiral pairing a cross-pairing
+  implicitly, by treating `both-same` as the rank-2 class). The verdict
+  letter (A at finite L, wrapping-form) and the A-continues caveat-upgrade
+  never used that reading: they used D, which is defined on the coarse wrap
+  labels, and the bit-for-bit Bernstein collapse, which is rank-blind.
+- Under A′, A-continues restates cleanly: at axis L=5 and diamond L=4 the
+  five cells of §2's table keep their D signs (the two exclusive-cross
+  cells carry ±1, the three diagonal cells carry 0), `both-two` never
+  appears to add a sixth row, and the collapse still reproduces the
+  committed Bernstein rungs (PR #657 tripwire vs PR #649, pass on both
+  diamond L=4 kernels).
+
+So A-continues survives verbatim; what changes is only that its "five
+Mertens–Ziff cells" language should be read with §2's table — the
+`both × both` row is the spiral row, `(1,1)`, D = 0.
+
+**What A-continues does not settle** (unchanged from PR #662 §4, and still
+not reachable without Huawei): the joint rank-splitting at axis L=5
+(NEED_HUAWEI — comment and stop if a future ticket needs that rung), and
+the Jacobsen-map gaps (order of limits; one pTL operator vs two lattices;
+novelty boundary vs Mertens–Ziff 2016 / Jacobsen 2015). None of those is
+affected by the dictionary rewrite.
+
+## 5. Boundary: this rewrite still does not fund #636
+
+PR #662's §5 recommendation stands unchanged, and the rewrite strengthens
+it slightly: the sector amplitudes of PR #646 are **not rank-pure** (PR
+#668), so any future sector-level transfer-matrix reading would first have
+to split `both-same` by rank to be measuring one homology class — an
+additional cost the old dictionary hid. The wrapping-form of verdict A
+remains published (Mertens–Ziff 2016); the unpublished piece remains the
+conceptual sector identification; degeneracy, if exact (Conjecture 3.1),
+still makes sector machinery carry less structure than its cost. Nothing
+here promotes #636.
+
+**Tripwire standing rule (unchanged):** any claimed decomposition must
+reproduce `[-1,-9,-36,-78,-90,-36,36,36,9,1]` at axis L=3 bit-for-bit,
+integers / Fraction only, or it is unused.
+
+---
+
+**Claim boundary.** Integers only in exact artifacts; the per-k and
+per-cell integers above are quoted from `results/issue665-joint-rank-wrap/`
+(PR #668, axis L=3,4), `results/probe-invariant-shape/census-exact.json`
+(PR #653), and the PR #657 committed tables — none re-enumerated here; the
+only new computation on the Mac for this note was none (notes-first; the 2⁹
+occupancy lists were not needed, the #668 JSON carries the per-k cells
+already). Conjectures 3.1/3.2 are marked. Do not cite #628's 118133 bond
+`dual_fail` as physics (repaired compound artifact, PR #653/#646).
+`M(p) + M(1−p) ≠ 0` on the site side stands as in PR #662.
+
+Related: #635, #636 (not funded), #650, #651, #660, #665; PRs #645, #646,
+#649, #653, #654, #657, #662, #668.