[petalas]

Summary

Scoped proof with finite exact corroboration: determine the complete paid rank frontier for one common internal Clifford coframe under prescribed cubic native transport, with fixed entrance and exit boundaries both charged.

S(x,z) = (z,x+z), a = d(L0,L), b = d(L,S L)
Pareto frontier: (j,m-2j), 0 <= j <= floor(m/2)
min [2 phi(a) + k phi(b)]
  = min(k phi(m), 2 phi(floor(m/2)) + k phi(m mod 2))

Here phi is increasing and concave, phi(0)=0, and k>0. For phi(r)=r^t and even m, the normalized optimum is min(k,2^(1-t)). The proof uses the boundary triangle inequality and the F4 structure of L intersect S L. Explicit paired-plane charts attain every efficient pair. A complete Gaussian-dyadic two-plane circuit retains all phases and both paid boundaries.

This changes the prescribed-native hypothesis excluded by Ali Romman's PR6171, at 343232cdf5273d1f77569de390f0977b46916644. That result is cited, not audited or replayed. This positive-cost, fixed-boundary transported graph differs from PR6172's signed variable-boundary graph. No new gate family, supplier admission, complete multiplication theorem, new kappa or selected-frontier change is claimed.

The new exact frontier and tariff formula were produced by Nick Petalas. Credit Ali Romman, Cinna Davis / PR5140's projection background, icekylinx's canonical frames, Ross Richardson / PR4081's earlier cubic native family and distinct response restriction, Dugongue, Cao Yifan and the retained native/helper/caller/compiler lineage. Classical symplectic, F4 and Bell-basis facts retain their classical status. Independent human and formal proof-kernel review are unclaimed.

Evidence

• Before: PR6171 leaves prescribed native equations outside its tensor reduction. After: this changed native contract has an exact attainable all-width frontier and closed tariff formula.
• All 2,448 Lagrangians in dimensions m=1..4 agree with a separate literal-point-set distance check, invariant-intersection parity and the complete Pareto frontier. All 288 attaining constructions for m<=32 and 200 exact concave-hinge/weight optimum checks pass.
• Full exact 2x2/4x4 matrices verify the actual order-three lift, signed Pauli images, unitarity, monomial internal map, every forward/inverse column and both boundary ranks. Twelve domain/rank/isotropy controls reject. Three disposable mathematical code mutations reject in both normal and optimized modes, including an omitted boundary fee and a wrong native phase.
• Warning-strict normal and optimized direct and registered replay pass locally on CPython 3.12.15/macOS arm64. Each registered command takes below 0.4 seconds locally. These are timeout measurements, not a performance comparison.
• At bedadda56af1cc2c124bec5888eb9f443dc3324e, all seven committed package files and declared source ancestry pass. Official export integrity/input coverage pass. Both modes pass in a read-only Git-free export with empty PATH, preserving all eight exported files' bytes and modes. RESULT.json is generated and bound to all other package files by SHA256.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/fixed-cubic-coframe/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/fixed-cubic-coframe/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/fixed-cubic-coframe
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/fixed-cubic-coframe --check-source-history

Expected marker: PASS_FIXED_CUBIC_COFRAME lagrangians=2448 constructions=288 concave_tariffs=200 rejections=12. A clean Python 3.12 interpreter can replace the local absolute path; standard library only. SOURCES.json pins canonical base a5bff3bf133e482541c09eba4e773884bdb2b1b7, the revision-8 card and citation-only source bytes. No pending source is imported.

Canonical main and guidance were refreshed before committing. Live PRs/claims and distinctive main/merged searches found no same-contract result. Cached overlap covers 338 open PRs and 5,958 Discussion records with complete remote pagination; its sparse local scan is incomplete, so separate searches used main's README tree. This is bounded overlap evidence, not proof of novelty.

Limitations and unrun obligations

The common internal coframe and original native occurrences are fixed. Time-varying charts and unrestricted resynthesis can be cheaper, including cancellation using U^3=I. Rank-zero monomial work has ordinary costs. An actual P104 source interval, compatible scalar contacts, productive delivery, dirty restoration/ownership, precision/tape, stock, compiler, BIT and outer bills remain unpriced. The universal theorem is the written proof, not finite extrapolation. No predecessor audit, selected cold replay, Lean or Docker run was performed.

Open next questions

• Bind one actual supplier interval to the paired charts, retain every scalar/productive contact, and price both boundaries plus all ordinary work. A source-compatible physical word and complete caller bill are missing.
• Determine the optimum for an explicitly specified scalar graph that permits different coframes at different native occurrences. The present common-chart equalities must not be silently removed.

Related open work

The completion refresh found no open account PR created since the advance CLAIM at 2026-10-29T22:08:52Z. The latest 100 open repository PRs were also inspected for this transported cubic-coframe contract. No competing same-contract value was found in that bounded coverage, and no extra computations were run to check predecessors. The three related PRs below are now merged and remain context rather than open work.

• #6192 bounds native rank under a whole-bank odd cyclotomic identity. This result optimizes a conjugated cubic tensor transport and imposes no whole-bank cyclotomic identity on every tensor power; no competing response-cap number is asserted.
• #6208 attains the order-17 response cap using a Bell-spectator construction. This result concerns two identical cubic copies and their complete common-coframe rank bill, without a new order-17 construction or competing rank claim.
• #6207 accelerates a finite Clifford projection kernel. It changes neither this transported graph contract nor this exact optimum; no timing comparison is made.

CI notes

Both hosted checks passed at exact contribution head bedadda56af1cc2c124bec5888eb9f443dc3324e in run 40964393769: plan and replay (research/fixed-cubic-coframe). The completed checks were read once with gh pr checks 6220 --repo CrocSwap/integer-mult-bounds. Run metadata separately confirms this head, completed status and successful conclusion.

Hosted replay certifies the registered normal and optimized checks at this head. The local publication-byte/source-history, mutation and read-only export evidence above also passed. No CI failure remains. Independent human or formal proof-kernel acceptance and physical supplier admission remain unclaimed. GitHub reports the PR merged on 2026-10-30; that repository state does not change this contribution's scoped claim.

Merge danger

Door: two-way. Blast radius: one isolated seven-file research package. Shared code, frozen evidence and selected-frontier files are unchanged.