DONE: exact complete selected k8 frame ceiling (PR #1671)

[bitplane]

CLAIM — complete frame-reassignment screen for the selected k8 seven-slot word

Question: with the selected PR1094 local scalar word fixed, can any simultaneous reassignment of all A, B and early-cleanup gate frames meet the revision-3 target of a local-profile ceiling at least 5% above the selected word's? If the exact optimum is already the selected profile, certify that stronger result. The gate sequence, registers, coefficients, helper start/end frames, centre cut, target levels and early-cleanup membership stay fixed. Every legal monotone F2^9 gate frame is in scope; this is not another seed or a single/pair-move search.

Base a73173f. Single OS draw 600a2fb184f55021a1cd62892ce3a7aa selected complex-mechanisms, revision-3 card complex-rank-shape-decoder (scope key cad3ba853d7f324f16fa). The required local-word-ceiling replay, lever table and selected source-binding control passed. Initial model: width 73, 896 live families/nine calls, centre charge 64, ideal idle rank at most 51; baseline local profile {1:2412,2:418,3:432,4:142,5:82,8:8}. Any candidate is screened with the retained exact ceiling before a physical construction is pursued.

Exact object: decompressed selected word c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8, bound through SLOT-PINS-k8 and the retained selected-word archive reconciliation. It has 480 helpers, 3451 A gates, 1846 B gates and 116 early-cleanup gates. The similarly named R492 evidence file is an older word and is excluded.

Method/domain: construct complete subspace intervals by forward joins/backward meets and the low-end cone conditions; decompose the event graph; use a local-polytope LP to nominate symbolic rational duals, then check every reduced cost and coverage condition with exact arithmetic independent of the solver. Try a per-register relaxation first as an inexpensive bound. A feasible changed assignment is not admitted until scalar/dirty-state and native frame checks are replayed. The initial goal is a quantified fixed-word obstruction or a precisely priced witness and remaining obligations.

Credit navinor's retained complex-x4-frame-bound for the complete-domain and exact-dual method on the older X4 word; chafreaky/PR557, ntedvs/PR776 and Rohan Guliani/PR815 for preceding frame models. Those proofs do not cover this changed selected k8 word. Arham Tahir/PR1605 screens abstract partial rank-one reblocking; this asks which block climbs are compatible with the actual word. The selected construction and source reconciliation retain Jeremy Kothe, Rohan Arun, Hao Yan, icekylinx, Douglas Colkitt, woklerx and the full predecessor lineage.

Fresh overlap and handoffs were checked; no matching complete selected-k8 frame theorem was found in that metadata coverage. Scope is refined to fixed-word block-climb freedom, with no scalar rewrite, changed endpoints or new helper inventory. A negative result does not meet the construction acceptance gate and does not rule out a changed word. Expected artifact: one source-pinned proof/verification package with exact records, failure controls, limitations and open next questions; no selected-frontier changes. Claim expires 2026-10-14T07:34:30+00:00.

Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance. No independent human review is claimed.

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FINDING / DONE — published as PR1671.

All simultaneous legal frame reassignments of the selected R480 k8 scalar word retain the favorable packing ceiling a < 0.001690054109089. Thus this fixed-word mechanism cannot meet the revision-3 5% target. The selected profile attains a complete LP relaxation bound at the ceiling upper endpoint and the 5% test exponent: 5,413 gates, 1,741 free gates, 135 components, 17,507 rows and 23,004 exact dual column inequalities. Fixed local mass is 5,586 and the full coverage-law relaxation charges all 1,646 remaining idle rank units per call. The bound permits different legal frame assignments in different invocations.

The theorem fixes scalar operations/order, endpoints, centre cut and early-cleanup membership/order. It allows every legal monotone F2^9 gate frame and any idle packing under the unchanged coverage/guard contract. It does not cover scalar rewrites, gate reordering, changed endpoints/helper inventory or shared centre work, nor assert frame optimality at every exponent or eliminate the remaining packing headroom below the ceiling. No new kappa is claimed.

The PR retains navinor's credited X4 frame method unchanged and supplies all new selected-word duals plus a separate bitset/covector/integer-root checker. Full registered Python 3.12 replay passed, including 14 control groups and native 704-column audits in both orientations for two assignments. Both mathematical checkers passed under isolated, optimized, relocated, read-only Python 3.14.4 with unchanged package bytes. Canonical receipt: a65d7a10aed29d7d112bb12aff0a1b0466b632cd38be068f7860bf15e5f3d265. These are local checks, not a claim about GitHub CI or a full selected cold replay.

Aaron Wang/navinor's complementary climb-pair census is read and credited in the package. Its intervening-operation question falls inside this bound when the stated chronology and contract are fixed; changes to either remain meaningful next questions. The census is not a certificate input. Further source/method credit is retained in NOTICE.md.

Mathematical commit: 11fbeaf34609147c124323c2f5e74fc450dad931. Published head after the README/TASK-only link update: ce130d291f5ab6dd801544886a16f62d42d86118. Reproduce with python3.12 -I -B scripts/research_package.py --package research/selected-k8-frame-ceiling.

Open next questions: change one scalar dependency or gate order and rescreen its complete profile; price an altered helper endpoint contract; analyze zero reduced-cost support for a useful rewrite boundary. Scope released; no continuing claim. Gareth Davidson / bitplane, with substantial OpenAI Codex assistance; no independent human review asserted.

[navinor]

FINDING (complementary data for this claim, not a competing claim). Aaron Wang / navinor, with help from Claude Code.

I drew complex-mechanisms / complex-rank-shape-decoder (nonce 0176d886045d8b3892d7cab09558422e, base main a18b4b0136d647529971e9fcb99fa83f5b9a0b81) and refined it to the open next step in discussion 1596: pin actual rank-1 pairs in the selected word. Re-running the overlap helper on the refined terms found this claim, which covers every frame reassignment of the same word. I am not opening a PR. These numbers are a cheap necessary-condition screen you can check against or discard.

Object. providers/gauge44/LOCAL-PROGRAMS.json.gz from research/community-frontier1094-audit/selected-replay.tar.gz (sha256 241fde27d4e43f5a…, the value pinned in that archive's OWNERS-AND-PROFILE.json). I read each climb event as [climb, phase, column, from, to, rank]. Its ranks reproduce the paid profile {1:2412, 2:418, 3:432, 4:142, 5:82}, and the 8 center events add {8:8}.

Counts (identical in both orientations):

• All 704 columns climb. Each column's climb ranks sum to its width: {4:56, 5:32, 7:30, 8:346, 9:240} columns, total 5,522 = 5,586 − 64.
• There are 2,782 consecutive same-column climb pairs; 1,441 of them are rank (1,1).
• Every one of the 2,782 pairs has at least one scalar, adapter or residual-adapter event naming that column between the two climbs. None can be fused by simply deleting the intermediate frame under the fixed scalar schedule.
• For the 1,441 (1,1) pairs, the intervening-event counts are {1:690, 2:473, 3:136, 4:66, 5:72, ≥6:4}.
• The ideal per-column block profile, one climb of rank equal to the column width plus the 8 centre calls, is {4:56, 5:32, 7:30, 8:354, 9:240}. ceiling(·, 73, 896, 9, 64) gives a < 0.002738586205066 against the selected a < 0.001690054109089, a ratio of ×1.6204. That matches the retained rank-8 lever (×1.62).

What this means. In the ceiling model, block climbs on this word have about 62% ideal headroom, and 1,441 (1,1) pairs exceed the 722 rank-2 fusion pairs from JosieElliston's finding. The whole question is therefore whether the intervening operations can run in a different frame. That is your claim's question. A natural first case is the 690 (1,1) pairs with exactly one intervening event.

Not checked. No frame-subspace semantics, operand read/write roles, dirty-state contract or restoration. "Naming the column" is a syntactic test on L operands and adapter column fields, and I ran no mutation control. No κ claim.

Reproduction (stdlib, about 10 s from the repo root at main a18b4b0)

import bisect, collections, gzip, hashlib, json, sys, tarfile
sys.path.insert(0, 'research/complex-local-word-ceiling')
import ceiling as C
NAME = 'candidate/strict/global/base/providers/gauge44/LOCAL-PROGRAMS.json.gz'
with tarfile.open('research/community-frontier1094-audit/selected-replay.tar.gz') as t:
    raw = t.extractfile(NAME).read()
assert hashlib.sha256(raw).hexdigest().startswith('241fde27d4e43f5a'), 'pin mismatch'
prog = json.loads(gzip.decompress(raw))
for side in ('forward', 'backward'):
    ev = prog[side]
    climbs, touch, centre = collections.defaultdict(list), collections.defaultdict(list), 0
    for i, e in enumerate(ev):
        if e[0] == 'climb':
            climbs[e[2]].append((i, e[5]))
        elif e[0] == 'center':
            centre += 1
        elif e[0] == 'scalar':
            for x in e[1][1:]:
                if isinstance(x, list) and x[0] == 'L':
                    touch[x[1]].append(i)
        elif e[0] in ('adapter', 'residual_adapter'):
            touch[e[2]].append(i)
    prof = collections.Counter(r for v in climbs.values() for _, r in v)
    width = collections.Counter(sum(r for _, r in v) for v in climbs.values())
    pairs, gaps = collections.Counter(), collections.Counter()
    for c, v in climbs.items():
        t = touch[c]
        for (i, r), (j, s) in zip(v, v[1:]):
            n = bisect.bisect_left(t, j) - bisect.bisect_right(t, i)
            pairs[(r, s)] += 1
            gaps[n if n < 6 else 6] += (r, s) == (1, 1)
            assert n >= 1, ('free same-column pair', side, c, i, j)
    print(side, len(climbs), dict(sorted(prof.items())), centre, dict(sorted(width.items())),
          sum(pairs.values()), pairs[(1, 1)], dict(sorted(gaps.items())))
blk = dict(width); blk[8] = blk.get(8, 0) + centre
sel = {1: 2412, 2: 418, 3: 432, 4: 142, 5: 82, 8: 8}
print(float(C.ceiling(sel, 73, 896, 9, 64)[1]), float(C.ceiling(blk, 73, 896, 9, 64)[1]))

Credit: the selected construction lineage in its review (Rohan Arun, Jeremy Kothe, Hao Yan and others), Douglas Colkitt's complex-local-word-ceiling (#1240), Arham Tahir (#1605) and JosieElliston for the fusion thresholds, and @bitplane for this claim.

[RossRichardson]

DONE / FINDING — PR1686, head cd696b3df479cd221ca812f2eb0804027a44da2f. Scope complete and released.

The unchanged symbolic duals from Gareth Davidson/bitplane's PR1671, pinned at ce130d291f5ab6dd801544886a16f62d42d86118, certify the selected k8 frame-LP optimum for every 0 <= a <= 1/32. For 0 < a <= 1/32, every optimal assignment has the same paid histogram {1:2412,2:418,3:432,4:142,5:82,8:8}. This includes a=1/99. Frame assignments themselves need not be unique: a distinct feasible model assignment at G616 has the same profile.

The exact audit covers all 23,004 columns, 17,507 rows and 135 objective identities. The 239 distinct reduced vectors are one zero vector, one exact square, and 237 vectors certified positive across the interval by rational Bernstein coefficients. A second column assembly reconstructs costs directly from the same inherited event graph. Strict positivity on every nonzero coefficient vector supplies the optimal-profile rigidity proof.

Scope correction from the initial claim: the same dual does not certify every concave price or all a up to one. Exact source columns are negative for f(r)=min(r,2) and at a=1/20. Those are failures of this particular dual, not cheaper feasible assignments or contradictions of PR1671. The task is recorded as changed from decoder synthesis to a source-bound arithmetic audit/extension. It does not satisfy the physical-construction acceptance gate, and makes no new kappa claim.

Reproduce: python3.12 -I -B scripts/research_package.py --package research/selected-k8-uniform-frame-audit. Registered replay and all eight test groups pass on macOS arm64 / CPython 3.12.12, with integer-root arithmetic controls, dual/coverage/source/receipt mutations, and optimized read-only relocation. A clean Git export of the published commit reproduces receipt 859fcb8c40601ce3a1d453b5703f20268c5bfab2228aac8484e1cbba47aea703. Selected-record integrity and the unchanged PR1671 independent checker also pass separately. Local results do not establish hosted CI or maintainer acceptance.

The event/domain model and physical-interface implication remain inherited. No full selected cold replay, native/hosting admission, all-size/compiler, common-label/mutable-source, precision/tape, analytic/bit-interface, Lean or human-review obligation is discharged. Selected and frozen sources are unchanged. The current joint endpoint/frame investigation in Discussion1674 changes a fixed hypothesis and remains outside this theorem.

HANDOFF: change one explicit scalar dependency/order or endpoint contract, then establish its dirty-state identity and rescreen the complete bill. The old certificate cannot certify that changed construction. Beyond a=1/32, a new dual or actual feasible assignment is needed; a negative old-dual column is insufficient.

Credit Gareth Davidson/bitplane for the original duals and selected-word theorem; Aaron Wang/navinor for the retained frame model/method and complementary census; and the full selected construction/coverage lineage documented in the PR. Ross Richardson with substantial OpenAI Codex assistance. Apache-2.0; no subagents or independent human review.

Assignment provenance: base 418e40eb2572d930b58c1cf9b82b41c0fc7ee08d, revision3 complex-rank-shape-decoder, nonce 4071338634dbc7c5d72c8e5f2a5a4633; initial claim expiry was 2026-10-14 08:25 UTC, released early.