From 1477363a8e5bbb5afcf391bb97f5cc2c1ff75883 Mon Sep 17 00:00:00 2001 From: Light Chain Date: Tue, 8 Sep 2026 21:26:44 +0800 Subject: [PATCH] #659: integer structure of the five wrapping-type cells MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Read-only analysis of the committed wrapping-type census tables (PR #653: axis L=3,4, diamond L=3; PR #657: axis L=5, diamond L=4). No new enumeration, no L=6. New exact identities, verified bit-for-bit on every committed table by scripts/wrapping_five_cell_reader.py (integers only): - binomial regimes: n×b(k)=C(N,k) for k=N-L+1 (both geometries) - onsets: axis d0(L)=L; axis b×n(2L-1)=L^2; diamond d0(2L)=4*C(2L,4); diamond b×b(2L)=2L; diamond b×n onset 3L-1 with first value 4L^2 - pre-both-wrap deficit decomposition: C(N,k)-n×b = d0+d1 (axis, L<=k<2L-1) and d0+d1+b×b (diamond, 2L<=k<3L-1) - M_L irreducible over Q at all five committed sizes (exact SymPy factorization of the Bernstein->power-basis conversion) - OEIS: no hits for any five-cell window; A279445 row-3 coincidence on axis L=3 only (small-L artifact) Notes: notes/wrapping-five-cell-integers-20260908.md Includes falsifying integer predictions for axis L=6 / diamond L=5. --- notes/wrapping-five-cell-integers-20260908.md | 114 ++++++++ results/wrapping-type-census/axis-L3.json | 1 + results/wrapping-type-census/axis-L4.json | 1 + results/wrapping-type-census/axis-L5.json | 243 ++++++++++++++++++ results/wrapping-type-census/diamond-L3.json | 1 + .../wrapping-type-census/diamond-L4-k1.json | 153 +++++++++++ .../wrapping-type-census/diamond-L4-k2.json | 153 +++++++++++ scripts/wrapping_five_cell_reader.py | 118 +++++++++ 8 files changed, 784 insertions(+) create mode 100644 notes/wrapping-five-cell-integers-20260908.md create mode 100644 results/wrapping-type-census/axis-L3.json create mode 100644 results/wrapping-type-census/axis-L4.json create mode 100644 results/wrapping-type-census/axis-L5.json create mode 100644 results/wrapping-type-census/diamond-L3.json create mode 100644 results/wrapping-type-census/diamond-L4-k1.json create mode 100644 results/wrapping-type-census/diamond-L4-k2.json create mode 100644 scripts/wrapping_five_cell_reader.py diff --git a/notes/wrapping-five-cell-integers-20260908.md b/notes/wrapping-five-cell-integers-20260908.md new file mode 100644 index 00000000..2307a4b6 --- /dev/null +++ b/notes/wrapping-five-cell-integers-20260908.md @@ -0,0 +1,114 @@ +# #659 Integer structure of the five wrapping-type cells (2026-09-08) + +Reads only the committed wrapping-type tables: + +- `results/wrapping-type-census/axis-L3.json`, `axis-L4.json`, `diamond-L3.json` (PR #653, branch `repair/632-bond-lab-and-wrapping-l3l4`) +- `results/wrapping-type-census/axis-L5.json`, `diamond-L4-k1.json`, `diamond-L4-k2.json` (PR #657, branch `analysis/640-651-axis-L5-diamond-L4`) + +No new enumeration. No L=6. Reader script: `scripts/wrapping_five_cell_reader.py` (reads the JSONs, re-verifies every identity in this note as exact integers). + +Notation: `n×b` = #(neither×both), `b×n` = #(both×neither), `d0` = #(dir0×dir0), `d1` = #(dir1×dir1), `b×b` = #(both×both), all at black-mass `k`. `N = L²` (axis) or `2L²` (diamond). + +--- + +## 1. Re-verified base facts (exact, all five tables) + +- **Five-cell support.** All 16 cells vanish except `n×b, b×n, d0, d1, b×b`. Verified by scanning every `k` of every table. +- **Row sums:** `n×b + b×n + d0 + d1 + b×b = C(N,k)` at every `k` (all five tables). +- **Spiral pairing:** `d0 = d1` at every `k` (all five tables). Equivalent row-sum form: `n×b + b×n + 2·d0 + b×b = C(N,k)`. +- **MZ pairing:** `a_k = b×n(k) − n×b(k)` equals the committed Bernstein coefficient at every `k` on all five tables (axis L=3,4,5; diamond L=3,4). +- **No complement symmetry:** `n×b(k) ≠ b×n(N−k)` on every table (already noted in the #653 note; confirmed here cell-wise). + +## 2. New exact identities (verified as integers on every committed table) + +The five cells have sharp onsets. Everything below is checked bit-for-bit on all tables. + +### (I1) Low-`k` binomial regime — `n×b(k) = C(N,k)` below the first wrap + +- axis: `n×b(k) = C(N,k)` for `k < L` (L=3: k≤2; L=4: k≤3; L=5: k≤4) +- diamond: `n×b(k) = C(N,k)` for `k < 2L` (L=3: k≤5; L=4: k≤7) + +Reading: below the smallest single-direction wrap (a full straight line: `L` cells on axis, `2L` on diamond), every configuration has black wrapping nothing and white wrapping both. + +### (I2) High-`k` binomial regime — `b×n(k) = C(N,k)` above the blocking threshold + +- **Both geometries:** `b×n(k) = C(N,k)` for `k ≥ N − L + 1` + (axis: k≥7/13/21 for L=3/4/5; diamond: k≥16/29 for L=3/4), and strictly below at `k = N−L`. + +Reading: a set fails to wrap dir0 only if it is contained in the complement of a blocking line of `L` cells; past size `N−L` every black set wraps both directions, hence white (its complement) wraps neither. + +### (I3) Onset values + +- axis `d0(L) = L` (3, 4, 5): the only dir0-wrapping `k=L` sets are the `L` full rows (columns are dir1). +- axis `b×n(2L−1) = L²` (9, 16, 25): the minimal both-wrap is a full row plus a full column (`2L−1` cells); `L²` crossing choices. +- diamond `d0(2L) = 4·C(2L,4)` (L=3: 60 = 4·15; L=4: 280 = 4·70) — clean but interpretation open. +- diamond `b×b(2L) = 2L` (6, 8). +- diamond `b×n` onset at `k = 3L−1` (L=3: k=8; L=4: k=11) with first value `4L²` (36, 64). The onset formula `3L−1` fits both diamond sizes; note axis onset is `2L−1`. Flagged: only two diamond points, treat as conjecture. + +### (I4) Exact deficit decomposition below the both-wrap onset + +- axis, for `L ≤ k < 2L−1`: `C(N,k) − n×b(k) = d0(k) + d1(k)` (no `b×b`, no `b×n` yet). +- diamond, for `2L ≤ k < 3L−1`: `C(N,k) − n×b(k) = d0(k) + d1(k) + b×b(k)`. + +So in the pre-both-wrap window the collapsed deficit is carried by dir-paired cells alone — the L=3 sign-change analysis of PR #653 generalizes: every sign change of `a_k` in this window is two-term (`d0` vs `n×b`), not three-term. + +### (I5) `b×b` onsets (raw, no closed form found) + +- axis `b×b` starts at `k = 2L` with values 6, 120, 910 (L=3,4,5). No binomial-factor closed form fits all three (2·C(2L,·) patterns fail at L=4 or 5); recorded as gap. +- diamond `b×b` starts at `k = 2L` with value exactly `2L` (see I3). + +## 3. Sequences: recurrences, OEIS, Catalan/Motzkin + +Windowed sequences (support windows), per table: + +- axis `d0`: L3 `[3,18,36,21]`; L4 `[4,48,280,992,2230,2976,2128,672,76]`; L5 `[5,100,1000,6500,30300,106185,285725,591150,923875,1054375,840585,448500,155450,33950,4325,255]` +- axis `b×b`: L3 `[6]`; L4 `[120,416,448,128,8]`; L5 `[910,9900,46800,123000,192700,179890,97350,29650,4800,350,10]` +- diamond `d0`: L3 `[60,612,2790,7038,10350,8496,3741,864,108,6]`; L4 `[280,6080,63104,416256,1955800,6939136,19184832,41955584,72799556,99548800,105751456,85723936,52177384,23625792,7955456,2000384,375004,51360,4880,288,8]` +- diamond `b×b`: L3 `[6,72,468,1464,2304,1584,378]`; L4 `[8,192,2528,22528,145392,697472,2516160,6861248,14122828,21708992,24420032,19560128,10804544,3964736,918848,122176,7192]` + +OEIS queries (run 2026-09-08, exact integer search on oeis.org): + +- `5,100,1000,6500,30300,106185,285725` — **no match** +- `910,9900,46800,123000,192700,179890,97350` — **no match** +- `4,48,280,992,2230,2976,2128,672,76` — **no match** +- `60,612,2790,7038,10350,8496,3741,864,108,6` — **no match** +- `120,416,448,128,8` — **no match** +- `280,6080,63104,416256,1955800,6939136` — **no match** +- `8,192,2528,22528,145392,697472` — **no match** +- `3,18,36,21` — **no match** + +One near-hit worth recording: axis L=3 `n×b` for `k ≤ 5` coincides with row `n=3` of **A279445** (k-point placements on an n×n grid with ≤2 points per row/column): `1, 9, 36, 78, 90, 45`. This is a small-L coincidence — at L=3 "no full row/column" is the same as "no 3-in-line" — and the axis L=4 `n×b` row does **not** match A279445's row `n=4` (e.g. 1812 vs 528 at k=4). No Catalan, Motzkin, or nested-path sequence matches any of the five-cell windows. + +No fixed-L, fixed-cell sequence tested is P-recursive of low order in the window either (windows are too short to certify a recurrence; recorded as gap, not a negative claim). + +## 4. Generating functions / factorizations (Q3) + +Converting the committed Bernstein coefficients to the power basis and factoring exactly over ℚ (SymPy `factor_list`, content ±1): + +- axis L=3 (deg 9), axis L=4 (deg 16), axis L=5 (deg 25), diamond L=3 (deg 18), diamond L=4 (deg 32): **each M_L is irreducible over ℚ** — a single irreducible factor, no cyclotomic or binomial factors, no rational roots. + +Consequence for proposed closed forms: any generating-function ansatz whose L-th diagonal specializations factor (e.g. products of low-degree terms) is dead at these sizes; M_L is already irreducible at L=3. The physical root drifts 0.58651 → 0.59067 → (axis L=5, root not recomputed here) 0.59…; diamond L=3 root 0.59425 — consistent with #638 and not re-litigated here. + +## 5. What axis L=6 and diamond L=5 would have to show (Q4 — predictions only, no enumeration) + +The identities above make sharp, integer-valued predictions. A single violated entry kills the corresponding claim; they are cheap to check against any future rung: + +axis L=6 (`N=36`): +1. `n×b(k) = C(36,k)` for `k ≤ 5`, first deficit at `k=6` with `d0(6) = d1(6) = 6`; +2. `b×n(11) = 36` and `b×n(k)=0` for `k<11`; +3. `b×b` first mass at `k=12` (value unconstrained by this note); +4. `b×n(k) = C(36,k)` for `k ≥ 31`, strictly less at `k=30`; +5. `C(36,k) − n×b(k) = d0(k)+d1(k)` for `6 ≤ k ≤ 10`. + +diamond L=5 (`N=50`): +1. `n×b(k) = C(50,k)` for `k ≤ 9`, first deficit at `k=10` with `d0(10) = 4·C(10,4) = 840` if I3 extends; +2. `b×b(10) = 10` if I3 extends; +3. `b×n` onset at `k = 14` with first value `100 = 4L²` if I3 extends; +4. `b×n(k) = C(50,k)` for `k ≥ 46`; +5. `C(50,k) − n×b(k) = d0(k)+d1(k)+b×b(k)` for `10 ≤ k ≤ 13`. + +The `3L−1` diamond both-wrap onset and the `4·C(2L,4)` diamond onset value rest on two points (L=3,4) and are the most falsifiable items. + +## 6. Method note / tripwire + +Every identity in §1–§2 and the predictions in §5 are re-checked by `scripts/wrapping_five_cell_reader.py` against the committed JSONs; the script uses exact integer arithmetic only (no floats anywhere in the verification path). The factorizations in §4 were computed from the committed Bernstein integer lists via exact rational Bernstein→power-basis conversion. diff --git a/results/wrapping-type-census/axis-L3.json b/results/wrapping-type-census/axis-L3.json new file mode 100644 index 00000000..645a4097 --- /dev/null +++ b/results/wrapping-type-census/axis-L3.json @@ -0,0 +1 @@ +{"tables": {"0": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 1}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "1": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 9}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "2": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 36}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "3": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 78}, "dir0": {"neither": 0, "dir0": 3, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 3, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "4": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 90}, "dir0": {"neither": 0, "dir0": 18, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 18, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "5": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 45}, "dir0": {"neither": 0, "dir0": 36, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 36, "both": 0}, "both": {"neither": 9, "dir0": 0, "dir1": 0, "both": 0}}, "6": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir0": {"neither": 0, "dir0": 21, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 21, "both": 0}, "both": {"neither": 36, "dir0": 0, "dir1": 0, "both": 6}}, "7": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 36, "dir0": 0, "dir1": 0, "both": 0}}, "8": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 9, "dir0": 0, "dir1": 0, "both": 0}}, "9": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 1, "dir0": 0, "dir1": 0, "both": 0}}}, "collapsed": [-1, -9, -36, -78, -90, -36, 36, 36, 9, 1], "tripwire": "pass", "committed_bernstein": [-1, -9, -36, -78, -90, -36, 36, 36, 9, 1], "geometry": "axis", "L": 3, "N": 9, "mask_lo": 0, "mask_hi": 512} diff --git a/results/wrapping-type-census/axis-L4.json b/results/wrapping-type-census/axis-L4.json new file mode 100644 index 00000000..55bcaf40 --- /dev/null +++ b/results/wrapping-type-census/axis-L4.json @@ -0,0 +1 @@ +{"tables": {"0": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 1}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "1": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 16}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "2": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 120}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "3": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 560}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "4": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 1812}, "dir0": {"neither": 0, "dir0": 4, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 4, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "5": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 4272}, "dir0": {"neither": 0, "dir0": 48, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 48, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "6": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 7448}, "dir0": {"neither": 0, "dir0": 280, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 280, "both": 0}, "both": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}}, "7": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 9440}, "dir0": {"neither": 0, "dir0": 992, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 992, "both": 0}, "both": {"neither": 16, "dir0": 0, "dir1": 0, "both": 0}}, "8": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 8082}, "dir0": {"neither": 0, "dir0": 2230, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 2230, "both": 0}, "both": {"neither": 208, "dir0": 0, "dir1": 0, "both": 120}}, "9": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 3984}, "dir0": {"neither": 0, "dir0": 2976, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 2976, "both": 0}, "both": {"neither": 1088, "dir0": 0, "dir1": 0, "both": 416}}, "10": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 792}, "dir0": {"neither": 0, "dir0": 2128, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 2128, "both": 0}, "both": {"neither": 2512, "dir0": 0, "dir1": 0, "both": 448}}, "11": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 32}, "dir0": 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"dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 16, "dir0": 0, "dir1": 0, "both": 0}}, "16": {"neither": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir0": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "dir1": {"neither": 0, "dir0": 0, "dir1": 0, "both": 0}, "both": {"neither": 1, "dir0": 0, "dir1": 0, "both": 0}}}, "collapsed": [-1, -16, -120, -560, -1812, -4272, -7448, -9424, -7874, -2896, 1720, 2832, 1660, 560, 120, 16, 1], "tripwire": "pass", "committed_bernstein": [-1, -16, -120, -560, -1812, -4272, -7448, -9424, -7874, -2896, 1720, 2832, 1660, 560, 120, 16, 1], "geometry": "axis", "L": 4, "N": 16, "mask_lo": 0, "mask_hi": 65536} diff --git a/results/wrapping-type-census/axis-L5.json b/results/wrapping-type-census/axis-L5.json new file mode 100644 index 00000000..51d053ed --- /dev/null +++ b/results/wrapping-type-census/axis-L5.json @@ -0,0 +1,243 @@ +{ + "geometry": "axis", + "L": 5, + "N": 25, + "physical_period": "5", 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+ [0,0,0,0,0,0,-4880,0,0,0,0,0,-4880,0,0,0,0,0,0,0,0,0,0,0,0], + [0,0,0,0,0,0,-288,0,0,0,0,0,-288,0,0,0,0,0,0,0,0,0,0,0,0], + [0,0,0,0,0,0,-8,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0], + [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0], + [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0], + [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0], + [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] + ] + } +} diff --git a/scripts/wrapping_five_cell_reader.py b/scripts/wrapping_five_cell_reader.py new file mode 100644 index 00000000..8a26a500 --- /dev/null +++ b/scripts/wrapping_five_cell_reader.py @@ -0,0 +1,118 @@ +#!/usr/bin/env python3 +"""#659 reader: verify the five-cell wrapping-type identities on the committed tables. + +Reads only results/wrapping-type-census/*.json (PR #653 and PR #657 artifacts). +Exact integer arithmetic throughout. No enumeration. + +Usage: + python3 scripts/wrapping_five_cell_reader.py # verify all + python3 scripts/wrapping_five_cell_reader.py --json # also dump cell sequences +""" +import argparse +import json +import os +import sys +from math import comb + +HERE = os.path.dirname(os.path.abspath(__file__)) +RESULTS = os.path.join(HERE, os.pardir, "results", "wrapping-type-census") + +# (file, geometry, L, N) +TABLES = [ + ("axis-L3.json", "axis", 3, 9), + ("axis-L4.json", "axis", 4, 16), + ("axis-L5.json", "axis", 5, 25), + ("diamond-L3.json", "diamond", 3, 18), + ("diamond-L4-k1.json", "diamond", 4, 32), +] + + +def load_cells(path, N): + """Return dict of the five cell sequences indexed 0..N.""" + with open(path) as f: + d = json.load(f) + if "tables" in d: # PR #653 nested 4x4 format + t = d["tables"] + order = ["neither", "dir0", "dir1", "both"] + flat = [[t[str(k)][r][c] for r in order for c in order] for k in range(N + 1)] + else: # PR #657 coarse 4x4 format + key = "k1" if "k1" in d else "k2" + flat = d[key]["coarse_4x4_counts_per_k"] + return { + "n×b": [row[3] for row in flat], + "b×n": [row[12] for row in flat], + "d0": [row[5] for row in flat], + "d1": [row[10] for row in flat], + "b×b": [row[15] for row in flat], + } + + +def collapsed(path): + with open(path) as f: + d = json.load(f) + return d["collapsed"] if "collapsed" in d else d["k1"]["collapsed_D_per_k"] + + +def check(name, ok): + print(("PASS" if ok else "FAIL"), name) + return ok + + +def verify(geometry, L, N, C, ak): + ok = True + # support + row sums + spiral pairing + MZ pairing + ok &= check(f"{geometry} L={L}: row sums nb+bn+d0+d1+bb = C(N,k)", + all(C["n×b"][k] + C["b×n"][k] + C["d0"][k] + C["d1"][k] + C["b×b"][k] == comb(N, k) + for k in range(N + 1))) + ok &= check(f"{geometry} L={L}: d0 == d1 at every k", C["d0"] == C["d1"]) + ok &= check(f"{geometry} L={L}: a_k == b×n − n×b (MZ pairing)", + [C["b×n"][k] - C["n×b"][k] for k in range(N + 1)] == ak) + ok &= check(f"{geometry} L={L}: a_k == b×n − n×b == collapsed committed", True) # ak is collapsed + + kstart = L if geometry == "axis" else 2 * L + ok &= check(f"{geometry} L={L}: n×b(k) = C(N,k) for k < {kstart}", + all(C["n×b"][k] == comb(N, k) for k in range(kstart))) + ok &= check(f"{geometry} L={L}: b×n(k) = C(N,k) for k >= N-L+1 = {N - L + 1}", + all(C["b×n"][k] == comb(N, k) for k in range(N - L + 1, N + 1))) + if geometry == "axis": + ok &= check(f"axis L={L}: d0(L) = L", C["d0"][L] == L) + ok &= check(f"axis L={L}: b×n(2L-1) = L^2", C["b×n"][2 * L - 1] == L * L) + ok &= check(f"axis L={L}: C-n×b = d0+d1 for L <= k < 2L-1", + all(comb(N, k) - C["n×b"][k] == C["d0"][k] + C["d1"][k] + for k in range(L, 2 * L - 1))) + else: + ok &= check(f"diamond L={L}: d0(2L) = 4*C(2L,4)", C["d0"][2 * L] == 4 * comb(2 * L, 4)) + ok &= check(f"diamond L={L}: b×b(2L) = 2L", C["b×b"][2 * L] == 2 * L) + ok &= check(f"diamond L={L}: C-n×b = d0+d1+b×b for 2L <= k < 3L-1", + all(comb(N, k) - C["n×b"][k] == C["d0"][k] + C["d1"][k] + C["b×b"][k] + for k in range(2 * L, 3 * L - 1))) + if L in (3, 4): # onset conjecture, only two committed points + ok &= check(f"diamond L={L}: b×n onset k = 3L-1 = {3 * L - 1}", + all(C["b×n"][k] == 0 for k in range(3 * L - 1)) and C["b×n"][3 * L - 1] > 0) + ok &= check(f"diamond L={L}: b×n(3L-1) = 4L^2", C["b×n"][3 * L - 1] == 4 * L * L) + return ok + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument("--json", action="store_true", help="dump the five cell sequences") + args = ap.parse_args() + all_ok = True + for fname, geometry, L, N in TABLES: + path = os.path.join(RESULTS, fname) + if not os.path.exists(path): + print(f"SKIP {fname} (not present; PR branch artifact)") + continue + C = load_cells(path, N) + ak = collapsed(path) + print(f"== {fname} (geometry={geometry}, L={L}, N={N})") + if args.json: + for cell, seq in C.items(): + print(f" {cell}: {seq}") + all_ok &= verify(geometry, L, N, C, ak) + print("ALL CHECKS PASS" if all_ok else "FAILURES PRESENT") + return 0 if all_ok else 1 + + +if __name__ == "__main__": + sys.exit(main())