Scoped negative result: no frame reassignment of the selected X4 complex word admits the next tick (exact duals, no new kappa)

[navinor]

Scoped negative result. No new κ. Only research/complex-x4-frame-bound/ is added. The selected record, upstream/ and every supplier file are unchanged.

Claim

Take the selected complex word, PR 799's X4 word (content 1e1d99ba…, all 72 complex calls), and fix its scalar program, start and end frames (so all widths), cut, centres, target levels, precondition, FULL cleanup and early-cleanup membership. Let each of its 6,364 A, B and early-cleanup gate frames be any F₂⁹ subspace allowed by the frame contract that research/gauge44-frames-x/independent/audit_local.py checks: nested frames along every register, centre and target levels at the start of B, and the inverse-row and dirt-column cone conditions of all 126 low ends.

No such assignment admits b_next = 0.001664400278850482327914, the next 10⁻²⁴ tick above the selected complex endpoint. The pinned outer.py gives κ < b/(1+b) (compact_phase_layer with complex_above_bit), and b_next/(1+b_next) lies below the next κ tick. Frame reassignment alone cannot raise the selected κ.

Method and evidence

• Forward joins and backward meets give complete frame intervals: 5,163 gates are forced and 1,201 have 2, 5 or 16 choices. The verifier enumerates every interval and checks each size against the Gaussian-binomial count.
• 140 components, with a local-polytope LP of 9,483 rows and 12,019 columns in total. Dual entries are integer linear forms in C₂…C₉ (C_d = M_d − d·M₁). Of the reduced costs, 8,582 are identically zero and 3,437 are strictly positive under outward bounds from both pinned engines (moment.py and base_two_moment.py). Each component's dual objective equals the selected cost as an exact coefficient identity.
• Global step: 72 × local + {1:1, 3:1, 44:1, 48:2396} equals the pinned global profile. Both engines put the moment above 1 at b_next and below 1 at the selected endpoint.
• The start-of-B centre equality is relaxed to a charged climb, and native-chain conditions beyond audit_local.py are not imposed. Both only enlarge the feasible set.
• Cross-check without duality: 12 random legal reassignments (40 frame changes each) pass the pinned audit_local.py unchanged, with exactly the profile the model predicts, and each is strictly costlier at b_next. The auditor refuses a frame below the forward join and a mismatched profile.
• Six rejection controls: a dropped component, an inflated dual, an omitted frame choice, the accepted endpoint in place of b_next, a legal but costlier baseline, and a changed word.

Reproduce

python3 -B research/complex-x4-frame-bound/verify.py --controls

Expected final line: PASS_X4_FRAME_ASSIGNMENT_EXCLUSION controls=6/6. This takes about 5 s, uses the standard library only and refuses -O. Optional: crosscheck.py (about 1 min), and verify.py --word <fresh X4 FULL-WORD.json.gz> for a rebuilt word. gen_duals.py (HiGHS) only nominates the duals and reproduces data/DUALS.json.gz byte for byte.

Validation run

• Windows 11, CPython 3.12.14 and 3.14.7: verifier with all six controls passes.
• A minimal closure of 21 committed files, extracted with git archive into an empty directory, passes the verifier and the cross-check.
• scripts/verify_selected_result.py passes, and so does the staged whitespace check.
• Not run: full make verify, the native cold replay make frontier799-verify, Linux CI and Lean. The word comes from the retained pr799-replay.tar.gz, whose hash the integration receipt records; its content hash matches RECIPE-X4.json and INDEPENDENT-LOCAL.json.

What it leaves for the complex track

Per call, 1,745 of the 2,548 rank-1 children, all 56 rank-5 children and the eight centre descents lie on edges between forced frames. A κ gain from this supplier needs a scalar change (read sets, order, tail membership), new start or end frames, or an owner and hosting change.

Scope

Not covered: scalar rewrites, reordering, read-set exchanges, early-cleanup membership, entrance promotions or other width changes, owners, hosting and the invoice, the bit supplier, and other outer theorems. The exclusion holds at b_next for this fixed word. Inherited all-size, realization, hosting, precision, tape and analytic hypotheses remain. This is a finite certificate, not a Lean proof.

Prior and concurrent work

The method is chafreaky's PR 557 (PR 551 word). ntedvs's PR 776 applies it to PR 626's jp8 and ck words without tail frames. Rohansguliani's PR 815 solves PR 699's SW4 and PR 626's words with floating-point LP and CP-SAT, with tail frames fixed. JointXF's FrameCoal report in PR 799 (X4 frame-optimal for its moves) is a search report in the PR 799 review; this package corroborates it with an exact certificate. New here: the selected X4 word in native gauge44 format, free early-cleanup frames and cone ceilings derived from the exact prefix map. Related: PR 850 (owner packing at the current inventory) and PR 775 (partial read exchanges on P6k4). PR 858 (closed) changed owners, which is outside this fixed-owner scope. Overlap was refreshed through PR 863.

Construction credit for the X4 word belongs to Jeremy Kothe (PR 799), chafreaky (PR 741, PR 685), BRoute, BlockInvert, icekylinx (PR 645) and the lineage in research/gauge44-frames-x/README.md.

Provenance

Base f567884aeb8851b6e794af1184119a3bfbe1e631. Complex-supplier track, OS-random seed 28. No pinned coordination issue existed, so CLAIM.md records the claim and the two other candidates checked (residual low-end capacity: none on the fixed program; promotion pricing: needs the packing model).

Prepared for @navinor with substantial assistance from Claude Code. Apache-2.0. No independent human review.