Scoped negative result: selected CK B frames are optimal at fixed boundaries

research/complex-ck-b-frame-seed20/CERTIFICATE.json

@@ -0,0 +1,156 @@
+{
+  "complete_local_histogram": [0,2596,552,304,160,70,8,0,8,0],
+  "fixed_outside_histogram": [0,1360,56,104,64,70,8,0,8,0],
+  "baseline_histogram": [
+    0,
+    1236,
+    496,
+    200,
+    96,
+    0,
+    0,
+    0,
+    0,
+    0
+  ],
+  "conditional_profiles": 2641,
+  "domain_sizes": {
+    "2": 418,
+    "5": 56
+  },
+  "evaluated": 4724,
+  "flexible_gates": 474,
+  "gates": 1930,
+  "induced_width": 5,
+  "max_conditional_frontier": 5,
+  "pinned_gates": 492,
+  "power_certificate": {
+    "baseline_power_optimal": true,
+    "exponent_interval": "0 < p < 1",
+    "largest_derivative_upper": "-21/25",
+    "strict_alternatives": 118
+  },
+  "registers": 716,
+  "source_sha256": "ae3018c0646be06f47d345bb8baf0e6ea9220096a949fa30a8f7656aa4ad6f45",
+  "final_frontier": [
+    [0,1236,496,200,96,0,0,0,0,0],
+    [0,1239,493,201,96,0,0,0,0,0],
+    [0,1242,490,202,96,0,0,0,0,0],
+    [0,1243,492,199,97,0,0,0,0,0],
+    [0,1245,487,203,96,0,0,0,0,0],
+    [0,1248,488,196,100,0,0,0,0,0],
+    [0,1251,485,197,100,0,0,0,0,0],
+    [0,1254,482,198,100,0,0,0,0,0],
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+    [0,1257,479,199,100,0,0,0,0,0],
+    [0,1260,480,192,104,0,0,0,0,0],
+    [0,1263,477,193,104,0,0,0,0,0],
+    [0,1266,474,194,104,0,0,0,0,0],
+    [0,1267,476,191,105,0,0,0,0,0],
+    [0,1269,471,195,104,0,0,0,0,0],
+    [0,1272,472,188,108,0,0,0,0,0],
+    [0,1275,469,189,108,0,0,0,0,0],
+    [0,1278,466,190,108,0,0,0,0,0],
+    [0,1279,468,187,109,0,0,0,0,0],
+    [0,1281,463,191,108,0,0,0,0,0],
+    [0,1284,464,184,112,0,0,0,0,0],
+    [0,1287,461,185,112,0,0,0,0,0],
+    [0,1290,458,186,112,0,0,0,0,0],
+    [0,1291,460,183,113,0,0,0,0,0],
+    [0,1293,455,187,112,0,0,0,0,0],
+    [0,1296,456,180,116,0,0,0,0,0],
+    [0,1299,453,181,116,0,0,0,0,0],
+    [0,1302,450,182,116,0,0,0,0,0],
+    [0,1303,452,179,117,0,0,0,0,0],
+    [0,1305,447,183,116,0,0,0,0,0],
+    [0,1308,448,176,120,0,0,0,0,0],
+    [0,1311,445,177,120,0,0,0,0,0],
+    [0,1314,442,178,120,0,0,0,0,0],
+    [0,1315,444,175,121,0,0,0,0,0],
+    [0,1317,439,179,120,0,0,0,0,0],
+    [0,1320,440,172,124,0,0,0,0,0],
+    [0,1323,437,173,124,0,0,0,0,0],
+    [0,1326,434,174,124,0,0,0,0,0],
+    [0,1327,436,171,125,0,0,0,0,0],
+    [0,1329,431,175,124,0,0,0,0,0],
+    [0,1332,432,168,128,0,0,0,0,0],
+    [0,1335,429,169,128,0,0,0,0,0],
+    [0,1338,426,170,128,0,0,0,0,0],
+    [0,1339,428,167,129,0,0,0,0,0],
+    [0,1341,423,171,128,0,0,0,0,0],
+    [0,1344,424,164,132,0,0,0,0,0],
+    [0,1347,421,165,132,0,0,0,0,0],
+    [0,1350,418,166,132,0,0,0,0,0],
+    [0,1351,420,163,133,0,0,0,0,0],
+    [0,1353,415,167,132,0,0,0,0,0],
+    [0,1356,416,160,136,0,0,0,0,0],
+    [0,1359,413,161,136,0,0,0,0,0],
+    [0,1362,410,162,136,0,0,0,0,0],
+    [0,1365,407,163,136,0,0,0,0,0],
+    [0,1368,404,164,136,0,0,0,0,0],
+    [0,1371,401,165,136,0,0,0,0,0],
+    [0,1371,408,159,137,0,0,0,0,0],
+    [0,1374,398,166,136,0,0,0,0,0],
+    [0,1374,405,160,137,0,0,0,0,0],
+    [0,1377,395,167,136,0,0,0,0,0],
+    [0,1377,402,161,137,0,0,0,0,0],
+    [0,1378,404,158,138,0,0,0,0,0],
+    [0,1380,392,168,136,0,0,0,0,0],
+    [0,1380,399,162,137,0,0,0,0,0],
+    [0,1381,402,157,139,0,0,0,0,0],
+    [0,1383,389,169,136,0,0,0,0,0],
+    [0,1383,396,163,137,0,0,0,0,0],
+    [0,1384,400,156,140,0,0,0,0,0],
+    [0,1386,386,170,136,0,0,0,0,0],
+    [0,1386,393,164,137,0,0,0,0,0],
+    [0,1387,397,157,140,0,0,0,0,0],
+    [0,1389,383,171,136,0,0,0,0,0],
+    [0,1389,390,165,137,0,0,0,0,0],
+    [0,1390,394,158,140,0,0,0,0,0],
+    [0,1392,380,172,136,0,0,0,0,0],
+    [0,1392,387,166,137,0,0,0,0,0],
+    [0,1393,391,159,140,0,0,0,0,0],
+    [0,1395,384,167,137,0,0,0,0,0],
+    [0,1396,388,160,140,0,0,0,0,0],
+    [0,1398,381,168,137,0,0,0,0,0],
+    [0,1399,385,161,140,0,0,0,0,0],
+    [0,1399,392,155,141,0,0,0,0,0],
+    [0,1401,378,169,137,0,0,0,0,0],
+    [0,1402,382,162,140,0,0,0,0,0],
+    [0,1402,389,156,141,0,0,0,0,0],
+    [0,1404,375,170,137,0,0,0,0,0],
+    [0,1405,379,163,140,0,0,0,0,0],
+    [0,1405,386,157,141,0,0,0,0,0],
+    [0,1406,388,154,142,0,0,0,0,0],
+    [0,1407,372,171,137,0,0,0,0,0],
+    [0,1408,376,164,140,0,0,0,0,0],
+    [0,1408,383,158,141,0,0,0,0,0],
+    [0,1409,386,153,143,0,0,0,0,0],
+    [0,1411,373,165,140,0,0,0,0,0],
+    [0,1411,380,159,141,0,0,0,0,0],
+    [0,1412,384,152,144,0,0,0,0,0],
+    [0,1414,370,166,140,0,0,0,0,0],
+    [0,1414,377,160,141,0,0,0,0,0],
+    [0,1415,381,153,144,0,0,0,0,0],
+    [0,1417,367,167,140,0,0,0,0,0],
+    [0,1417,374,161,141,0,0,0,0,0],
+    [0,1418,378,154,144,0,0,0,0,0],
+    [0,1420,364,168,140,0,0,0,0,0],
+    [0,1420,371,162,141,0,0,0,0,0],
+    [0,1421,375,155,144,0,0,0,0,0],
+    [0,1423,368,163,141,0,0,0,0,0],
+    [0,1424,372,156,144,0,0,0,0,0],
+    [0,1426,365,164,141,0,0,0,0,0],
+    [0,1427,369,157,144,0,0,0,0,0],
+    [0,1429,362,165,141,0,0,0,0,0],
+    [0,1430,366,158,144,0,0,0,0,0],
+    [0,1432,359,166,141,0,0,0,0,0],
+    [0,1433,363,159,144,0,0,0,0,0],
+    [0,1435,356,167,141,0,0,0,0,0],
+    [0,1436,360,160,144,0,0,0,0,0],
+    [0,1439,357,161,144,0,0,0,0,0],
+    [0,1442,354,162,144,0,0,0,0,0],
+    [0,1445,351,163,144,0,0,0,0,0],
+    [0,1448,348,164,144,0,0,0,0,0]
+  ]
+}

research/complex-ck-b-frame-seed20/CLAIM.json

@@ -0,0 +1,23 @@
+{
+  "track": "complex supplier",
+  "assignment_seed": 20,
+  "random_source": "Python secrets.randbelow(100), operating-system randomness",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "object": "Selected PR626 CK provider, retained PR622 canonical word, simultaneous B-phase frame reassignment at fixed scalar gate order",
+  "method": "Exact F2 subspace lower/upper propagation and per-register rank-partition bounds on the paid recursive moment; preserve A, cut, tail, cleanup, boundaries and ownership",
+  "candidate_tasks": [
+    "Additional signed K-triangles: covered by PR646",
+    "Coordinate-total factory resynthesis: overlaps PR681 and changed-dirt factory PR645",
+    "Fixed-order CK B-phase frame retiming: selected; seed 20 modulo 2 selects CK rather than JP8"
+  ],
+  "expected_deliverable": "One finite improving frame witness or a scoped negative certificate with a reproducible checker and controls; no unsupported kappa claim",
+  "expected_finish_utc": "2026-10-13T04:33:00Z",
+  "coordination": "No pinned issues and no open sprint/coordination issue found. Local claim, with open PR overlap checks.",
+  "overlap_snapshot": "501 open PRs through #681; PR661 supplies a related bit-frame barrier, while this object has complex F2-labelled frames and fixed CK tail interfaces",
+  "status": "completed: scoped negative result, no new kappa",
+  "result": "All 474 flexible gates in the selected CK B phase were eliminated over their complete F2 subspace domains. The baseline paid histogram is optimal for every 0 < p < 1 at fixed scalar order and exact end-of-B frames. All 118 distinct retained alternatives are strictly worse.",
+  "result_link": "https://github.com/CrocSwap/integer-mult-bounds/pull/698",
+  "evidence": "README.md and CERTIFICATE.json; reproduce with python3 -B research/complex-ck-b-frame-seed20/check.py",
+  "final_overlap": "513 open PRs through #693, followed immediately by review of #694-#697. #696 lowers scalar primitives with unchanged frames; #697 changes source access and helper count. Neither overlaps this fixed-circuit frame barrier.",
+  "completed_utc": "2026-10-13T04:22:26Z"
+}

research/complex-ck-b-frame-seed20/README.md

@@ -0,0 +1,213 @@
+# A fixed-boundary frame barrier for the selected CK supplier
+
+**Scoped negative result. No new kappa.** The selected CK complex supplier's
+1,930 B-phase scalar gates already minimize their paid recursive moment among
+all simultaneous frame reassignments that preserve their order, operands and
+exact end-of-B register frames. This holds for every power `0 < p < 1`, with
+cost `sum H[r] * r**p`. The allowed frames are all subspaces of `F_2^9`, including
+ones absent from the retained frame catalogue.
+
+This closes a specific optimization class. It does not claim unique frames,
+optimality after reordering or resynthesis, optimality with changed B boundaries,
+or optimality of the JP8 provider. The scalar word and every existing repository
+file remain unchanged.
+
+## Reproduce
+
+From the repository root, using Python 3.11 or later and its standard library:
+
+```sh
+python3 -B research/complex-ck-b-frame-seed20/check.py
+```
+
+The command checks the source pins, reconstructs the complete frame domains,
+recomputes [CERTIFICATE.json](CERTIFICATE.json), and runs the independent
+enumeration checks and negative controls. The core receipt prints:
+
+```text
+PASS_CK_FIXED_END_B_FRAME_BOUND gates=1930 flexible=474 profiles=119 exponent=0<p<1
+```
+
+The core checker can also be run alone as `python3 -B
+research/complex-ck-b-frame-seed20/verify.py`. Neither command writes files,
+needs a solver, or fetches data. They fail if pinned inputs or the receipt differ.
+
+## Object, accounting and boundary
+
+Base commit: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+The selected record at this base is conditional
+`kappa = 51110230636731/31250000000000000 = 0.001635527380375392`.
+Its [maintainer review](../../docs/research/community-k492-review.md) explains
+that #626 retains #622's provider definitions and changes the owner allocation.
+The input here is the freshly replayed #622 CK
+[canonical word](../community-k492-audit/pr622/replay/ck-local/CANONICAL-WORD.json.gz).
+Its decompressed hash matches the CK identity in #626's retained SPEC.
+[SOURCES.json](SOURCES.json) pins both heads and eight source files.
+
+Only B frames vary. A, the center cut, precondition, compensation, cleanup tail,
+FULL cleanup, initial/final adapters, helpers, coefficients and owners are fixed.
+The center phase acts on private copies, so A's ending live frames are B's
+starting frames. Each scalar gate uses one common frame for its operands.
+Each operand climbs monotonically to it, paying one child whose rank is the
+dimension increase. See the retained
+[emitter](../complex-k492-owner-repair/code/hpe_emit_local.py) and
+[native auditor](../complex-k492-owner-repair/code/hpe_audit_and_transport.py).
+
+For each register touched in B, its last gate is pinned to the original frame.
+There are 492 distinct pinned gates. This preserves every end-of-B frame exactly,
+including the boundary used by the cold-helper row/column cone check. All later
+climbs and all endpoint wrappers therefore stay fixed. The scalar circuit itself
+is unchanged on arbitrary input and helper columns, in either orientation.
+
+The baseline B histogram is
+
+```text
+H_B = {1:1236, 2:496, 3:200, 4:96}, rank mass = 3212.
+```
+
+The checker separately recounts the fixed A, center, tail and final climbs:
+
+```text
+H_fixed = {1:1360, 2:56, 3:104, 4:64, 5:70, 6:8, 8:8}.
+H_total = {1:2596, 2:552, 3:304, 4:160, 5:70, 6:8, 8:8}.
+```
+
+Their sum matches the canonical word's complete local profile. The native
+[moment engine](../complex-multicold/engines/moment.py) multiplies
+`sum H[r]*r**(1-a)` by the positive factor `m**(a-1)/W`. Thus the result covers
+its recursive moment for every `0 < a < 1`. No global assembly or new exponent
+is inferred from this local certificate.
+
+## Complete finite domain
+
+Forward propagation takes the join of the current operand frames. Reverse
+propagation takes their intersection. Both passes inject the exact last-gate
+pins. Induction along each register proves that every feasible frame at gate
+`i` lies between the resulting subspaces `L_i` and `U_i`.
+
+The quotient dimensions `dim(U_i/L_i)` are 0, 1 or 2. Their complete subspace
+intervals contain 1, 2 or 5 frames respectively. The checker enumerates an
+interval by repeatedly adjoining each missing vector. Every intermediate
+subspace has a basis extending the lower subspace, so this reaches all of them.
+
+| Gate domains | Number of gates |
+| --- | ---: |
+| One frame | 1,456 |
+| Two frames | 418 |
+| Five frames | 56 |
+
+The unrestricted product is `2**418 * 5**56` assignments before shared-register
+constraints. This is not a sampled search. A factor for every operand transition
+checks subspace inclusion and charges its exact rank difference. These factors
+and the pinned boundaries are equivalent to the entire monotone common-frame
+constraint in the stated class.
+
+Min-fill variable elimination exhausts that factor graph. A table entry retains
+all paid histograms not excluded by the exact dominance rule below. Eliminating
+a gate combines all incident factors and enumerates every gate-domain value for
+each separator assignment. Discarded histories have no future constraints except
+that separator, which is the usual variable-elimination invariant.
+
+The largest separator has five gates. The replay evaluates 4,724 local domain
+assignments, retains at most five histograms per conditional table entry, and
+ends with 119 profiles. This count is work after exact elimination, not the
+number of feasible full schedules. The stopping rule is elimination of every
+one of the 474 flexible gates; there is no timeout or incomplete branch.
+
+## Why the pruning and final comparison are exact
+
+For a histogram `H`, let `M(H)=sum r*H[r]` and
+`S_k(H)=sum min(r,k)*H[r]`. At a fixed separator, transition ranks telescope, so
+all alternatives have the same mass. The implementation asserts this invariant.
+It discards `G` only when another profile `H` has equal mass and
+`S_k(H) <= S_k(G)` for all integer `1 <= k <= 9`.
+
+To justify that rule, set `f(r)=r**p`, `f(0)=0`, and
+`d_k=f(k)-f(k-1)`. For `0<p<1`, the `d_k` decrease and
+
+```text
+f(r) = d_9*r + sum_{k=1}^8 (d_k-d_{k+1})*min(r,k).
+```
+
+All coefficients on the right are nonnegative. Dominance therefore holds for
+every power in the claimed interval and remains valid when adding other factors.
+Incomparable profiles are retained rather than resolved by floating arithmetic.
+
+Concave dominance alone leaves 119 profiles. One is the baseline. Every other
+profile differs only at ranks 1 through 4. Write its cost minus the baseline as
+
+```text
+D(p) = a + b*2**p + c*3**p + d*4**p.
+```
+
+The checker establishes `b<=0`, `d>=0` and `a+2*b+3*c+4*d=0` for all 118
+alternatives. Thus `D(1)=0`. It also certifies `D'(1)<0` exactly:
+
+```text
+D'(1) = (2*b+8*d)*log(2) + 3*c*log(3).
+```
+
+Eight terms of the rational atanh series, with its geometric remainder, prove
+`69/100 < log(2) < 7/10` and `109/100 < log(3) < 11/10`. Substituting the
+appropriate endpoint for each coefficient gives a strictly negative upper
+bound in every case. The largest is `-21/25`.
+
+If `c>=0`, then `D'(p)/2**p` is nondecreasing in `p`, because its nonconstant
+terms have nonnegative coefficients and bases greater than one. If `c<0`, use
+`D'(p)/3**p`: the term with coefficient `b*log(2)` has base `2/3`, so it is
+nondecreasing because `b<=0`; the `d*log(4)` term has base `4/3` and is also
+nondecreasing. The remaining term is constant. In either case `D'(1)<0` implies
+`D'(p)<0` for `p<1`. Integrating to `D(1)=0` gives `D(p)>0` for `0<p<1`.
+
+Every feasible profile is therefore no better than a retained profile, and
+every distinct retained profile is worse than the baseline. This proves weak
+optimality of the baseline paid moment in the stated finite class. It does not
+prove uniqueness of the frame assignment.
+
+## Sprint choice, provenance and limits
+
+The operating-system assignment seed was **20**, selecting the complex-supplier
+track and CK via `20 modulo 2`. The algorithm itself is deterministic. Three
+candidates and the local claim are recorded in [CLAIM.json](CLAIM.json). No
+pinned coordination issue existed. The initial overlap review covered 501 open
+PRs through #681, plus the maintainer ledger and relevant prior negative results.
+
+Related #661, by @yuj1r0, proves an all-frame barrier for the R420 **bit** B phase.
+This complex instance has coupled frame choices that its per-register argument
+does not settle. The contribution here is complete finite-domain elimination
+for CK with its cold-helper boundary, plus the four-rank derivative certificate.
+No code was imported from #661. #651 studies target hosting and raw-face sharing,
+#646 changes the K-triangle skeleton, and #681/#645 change factory circuits or
+dirty-input propagation. Those mechanisms are outside this fixed-circuit class.
+The final refresh covered 513 open PRs through #693. New #683 concerns mixed
+K/M bypass carriers and #687 enumerates one-read relocations under minimal join
+frames. Neither changes this result's fixed-order, all-frame scope or supplies
+a dependency. The pinned-issue query remained empty.
+Immediately before submission, #694 through #697 were also reviewed. #696
+lowers complex scalar primitives while preserving frames; #697 changes source
+access and helper count. Both lie outside the fixed-circuit class here.
+
+Credit goes to Jeremy Kothe's #622 K-triangle construction and schedule;
+Chafik Boukhalfa's #626 owner package; DreamingOfClouds' cold-helper, emitter and
+native-verification work; icekylinx's coordinate-total/direct-delivery lineage;
+Dominik Scholz's prefix work; the frame-coalescing, recurrence and remaining
+contributors retained in the selected review. The unchanged bit supplier is
+eumemic's #543. Their results are inputs, not new contributions of this note.
+
+Prepared for **@thakrarsagar**, with substantial **OpenAI Codex** assistance in
+research, implementation, proof development, independent checks and writing.
+New files use the repository's Apache-2.0 license. No predecessor file is copied
+or relicensed.
+
+The finite certificate does not reprove the inherited scalar construction,
+uniform compiler, common-label or mutable-basis realization, whole-call hosting,
+normalized residuals, precision/tape/exact recovery or analytic transfer.
+It is not a Lean proof or a full multiplication-theorem audit. Reordering,
+different helper/source interfaces, changed boundary frames, and JP8 remain
+outside this result. The expensive full supplier/global replay is not needed to
+establish this isolated barrier and was not rerun; the selected-record check was.
+The complete package command passed on macOS arm64 with Python 3.11.17 and
+3.14.8. [VALIDATION.json](VALIDATION.json) records the checks and their limits.
+The same command passed from a clean archive containing only this package and
+the eight pinned inputs. [PR #698](https://github.com/CrocSwap/integer-mult-bounds/pull/698)
+is the completed sprint handoff.