DONE: arbitrary selected k8 helper terminal frames retain the ceiling (PR #2013)

[bitplane]

CLAIM — arbitrary terminal helper frames for the selected k8 word

Question: can nonnested rotations or enlargements of the selected word's 78 restricted helper endpoints, jointly with every A/B/early-cleanup gate frame, reach a 5% gain over its favorable packing ceiling?

Base main 1c47876ef742ba64bf9140b0527d63acf656a3b4. Single OS nonce 5f5984c68e5381ec12276b69c721f48c, revision-3 complex-mechanisms / complex-new-helper-inventory, scope key f3b0895a4f9f1cc62e75; no reroll.

Exact domain: selected R480 word with decompressed SHA256 c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8. Fix scalar operations/order/coefficients, entrances, centre and target levels, and early/full cleanup membership/order. For each originally restricted helper q, allow every end E_q containing its entrance, all the way to FULL. The old end H_q is no longer an upper cap. All 402 helpers touched by full cleanup still end FULL. Gate frames remain monotone along each register and common to all gate operands; low-end cone constraints must track each new E_q. Released or added local rank mass is charged through the complete coverage-law/idle bill.

Changed premise: PR1995, pinned head d47b4b6, only permits nested contractions E_q <= H_q. Arham Tahir's PR1583 fixes gate frames; PR1671 and Ross Richardson's uniform extension fix endpoints. PR1996's coordinate-plane rail census also fixes endpoints. Marius-Paul Dumitrean's PR1186 provides per-instance frame solving and changed-word search; it does not assert this joint unrestricted-end ceiling. Alexander Schwartzman/PR1030 and Cinna Davis/PR1090 retain credit for particular smaller ends, and navinor for the complete frame LP method.

The first source-bound baseline, ceiling and lever controls passed. Waiving the baseline climb costs of all 78 affected helpers and recharging the released mass gives a planning profile with headroom above the declared target; it is not a realizable word. A preliminary domain-size calculation finds 78 free endpoint nodes, 2,589 free gate nodes and largest quotient dimension five. The old enumerator intentionally stops at quotient dimension four, so it is not used to claim complete enumeration of this larger class.

Method: complete subspace interval enumeration through quotient dimension five, coupled zero-charge cone edges, and an exact dual certificate for the full variable-mass moment. Use a separately implemented domain/graph/arithmetic replay and actual scalar/frame audits for a candidate. A solver proposes data only. Stop after one complete terminal-frame screen, or record a precise interface/certificate obstruction; do not expand to changed entrances, scalar rewrites, gate order or cleanup changes. No selected frontier update or full native/global construction is presumed.

Overlap entry scan: 413 open PRs and 623 Discussions with complete pagination, the terminal/helper-end search returned all 167 matches, with relevant scopes and heads inspected. This is not a novelty certificate. Expected artifact: one scoped proof or candidate/replay package, with explicit unrun hosting, native phase/wrapper, all-size, precision/tape and outer obligations. Claim expires 2026-10-14T18:43:26.337577+00:00.

Gareth Davidson / bitplane, with substantial OpenAI Codex assistance; no independent human review.

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FINDING / DONE — published as PR2013.

The complete unrestricted-terminal domain has the same certified ceiling bracket as the nested-end result: 0.001690198050302 < best ceiling < 0.001690198050303. Arbitrary terminal rotations, enlargements and FULL ends therefore cannot raise the fixed-end ceiling by 5%; the gain is less than 0.00852%. The scalar word, entrances, centre/target pins, cleanup and coverage/guard contract remain the explicit hypotheses. No new kappa or global supplier is claimed.

Exact rational duals check 396,241 columns and 103,563 rows across 129 components. The domains contain 32 terminal intervals with 374 choices and 46 with five choices, plus 2,589 free gate nodes. Every one of the 106 low-end cone relations is present. All possible assignments have nonnegative idle mass, and the variable local mass is fully recharged through the rank-51 idle relaxation.

The known PR1995 four-end profile attains the bound: ends 574,575,580,581; profile {1:2400,2:446,3:416,4:142,5:82,8:8}; local mass 5582, idle mass 1650. The contribution is the quantified larger-domain certificate, not first discovery of this profile. Baseline, credited PR1090 two-end word and the attaining assignment each pass all 704 scalar/frame columns in both orientations.

Full registered CPython 3.12.3 replay and all 20 control groups passed. Both checkers also passed isolated, optimized, relocated read-only CPython 3.14.4 replay with unchanged files. The separate checker reconstructs the graph/domains/cones using bitsets and inverse covectors, and proves dual/moment bounds with integer-root intervals. No primary frame/LP/moment implementation is imported.

Emitted native programs, phase/wrapper admission, a physical owner map, selected cold replay and the global/outer invoice were not replayed. Fractional ideal idle packing need not be realizable. Changed entrances, scalar rewrites/order, cleanup changes, nonmonotone internal transport, new helper registers, shared centres and new coverage laws remain outside scope.

Final overlap covered 381 open PRs and 643 Discussions with complete pagination; nearby PR1995/1186/1996 heads were unchanged. Main b6135f9 preserved all mathematical inputs and the selected record; its new canonical-path validation passed separately. Metadata search is not a novelty certificate.

Mathematical commit 31e788df17a4df614fa5b3605a06d06e60f18bb6; published head a43cc72abcb16ca4d61fc469174a9b971b084d6f. Canonical receipt 1891bcf17e86405d265ed4b51b9de6f6059be95c50a5b6c28b21d2ac8057093e. Run /usr/bin/python3.12 -I -B scripts/research_package.py --package research/selected-k8-unrestricted-ends. Hosted CI is separate from these local checks.

Full predecessor credit is retained in NOTICE.md, including navinor's frame LP method, PR1030/1090's smaller ends, Arham Tahir/PR1583, PR1671/1995, Ross Richardson's uniform audit, Marius-Paul Dumitrean/PR1186, Cinna Davis/PR1996, Douglas Colkitt/PR1240, Sagar Thakrar/PR1012 and the selected construction lineage.

Open next questions: change an entrance or scalar dependency with a complete dirty-state and cost argument; or physically admit and host the four-end word, whose available ceiling gain is tiny. Scope released; no continuing claim. Gareth Davidson / bitplane, with substantial OpenAI Codex assistance; no independent human review.