[petalas]

Summary

Scoped written proof with exact finite corroboration: the complete fifteen-chart binary Lagrangian metric has an exact nonnegative cut representation precisely in the following tariff range.

d(L,M) = 2 - dim(L intersect M),   0 < t <= 1
exact cut representation of d^t <=> 2^t <= 3/2
least L1 distortion = max(1, 2^(t+1)/3)

A five-chart separator proves necessity. Explicit line-pencil and singleton cuts prove sufficiency and attain the distortion bound. The witness embeds in every binary chart menu with at least two axes, excluding exact cut coordinates there above the threshold. Only the two-axis distortion upper bound is claimed.

This contribution was produced by Nick Petalas. It builds on David Ganz / Dugongue's #5693 inverse-Lagrangian rank interface and broader coframe optimization direction, with Jerry Chen / CubeJerry's #907 cut-metric precedent and the retained native/Clifford lineage. The unpowered generalized-quadrangle obstruction is classical; Deza and Laurent's L1-rigid graphs, Fact 5.10(ii), is cited. The powered classification and attaining formulas are derived for this repository interface. zxv's #5283 already has the same constant in a different three-terminal topology comparison. No priority for the constant itself or worldwide literature priority is claimed. Predecessor results are premises, not audit or replay targets.

Advance CLAIM. Base main 7c49fe32c55de02c2c873655c804b16e4c3fef0a; publication head 355c22bd493f9f90a2b9654119a682b7e7dee0e1. No new kappa or selected-frontier change.

Evidence and reproduction

Before, the binary cut interface did not classify nonnegative cut coordinates for the complete small chart menu. After, PROOF.md supplies the exact threshold and best possible distortion, with explicit attaining cuts.

CPython 3.12.15 on macOS, standard library only. Both exact VERIFY.json commands passed locally in 0.049 and 0.050 seconds, respectively. These are timeout checks, not speedup claims. The checker covers all 35 binary planes, all fifteen isotropic charts, 225 ordered pair distances by independent intersection/elimination methods, all 16,384 cuts up to complementation, fifteen pencils, symbolic affine coefficient identities, 33 rational tariffs and twelve rejection controls. The continuum theorem rests on the written proof. Normal and optimized complete generated receipts agree and bind all six package files.

From the repository root:

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

Any CPython 3.12 path may replace the displayed interpreter. Expected markers are PASS_TWO_AXIS_COFRAME_CUTS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Registered metadata/pins and both commands pass. Committed bytes and declared source ancestry pass at the publication head. The official committed export passes integrity checks. Both normal and optimized direct replays from a read-only, Git-free export with empty PATH pass, with identical complete receipts and all seven export files unchanged in bytes and modes. Syntax/function-size and whitespace checks pass. The selected source-binding orientation passed; it is not a supplier replay. No generated receipt is handwritten or a trusted mathematical input.

Limitations and unrun obligations

This is a rank-metric representation theorem. It does not exclude auxiliary-variable cut reductions, restricted chart menus, other optimization algorithms or a favorable actual coframe word. It is not a complexity lower bound or a complete multiplication theorem.

No physical chart lowering, phase-aware native word, independent productive target delivery, complete paid caller invoice, restoration, precision/tape/stock or compiler/outer admission is supplied. All five selected assumptions remain. No selected scientific cold replay, Lean, Docker, independent human review or proof-kernel review ran. The sourced classical nonembedding is credited; no global novelty theorem is claimed.

Open next questions

Can an actual caller restrict its legal inverse-Lagrangian menu to a cut-representable subset while preserving its useful transitions and independent targets? First extract its exact chart menu and complete pair-rank table and test the five-chart witness. Source-owned legality, target debits, phases, restoration and the complete price remain prerequisites.

Related open work

• Prove P104 frame ceilings and finite child bounds for fresh-stage callers #6541 studies a fixed P104 source's monotone frame ceilings and finite child-size bounds. This result classifies an unrestricted two-axis inverse-Lagrangian metric; no source invoice, ceiling or kappa is recomputed.
• Prove the sharp paid-moment floor for independent four-cycles. #6514 prices independent chart choices on retained four-contact cycles. This result concerns cut coordinates for an entire chart menu, with no cycle-word price or same-instance number claimed.
• Certify 62 canonical single-warp entrance outcomes (no κ) #6620 certifies canonical completions for later entrances of a fixed staged native word. This result has no staged-word entrance domain or numerical tariff comparison with that certificate.

Live chart/coframe and cut-metric searches found no open PR computing this same threshold or distortion. Main and merged searches included the formula's general chart-menu form. Sparse historical notes, arbitrary diffs and unpublished work remain overlap limits; no extra computations rechecked another contribution's numbers.

The completion refresh listed all 176 account open PRs and screened the titles and bodies of the fifteen other open contributions created since the advance CLAIM. None studies this chart-menu threshold or distortion, so no additional same-object line or numerical comparison was required. No extra computation rechecked another contribution.

CI notes

Both hosted checks passed at exact head 355c22bd493f9f90a2b9654119a682b7e7dee0e1 in run 41649925068: plan and replay (research/two-axis-coframe-cuts). The completed checks were read once; run metadata independently confirms completed success and the matching head. Local registered checks, committed-byte/source-history preflight, export integrity and both read-only standalone modes also passed. Hosted replay certifies execution of the declared finite checks and metadata, not independent mathematical acceptance.

Merge danger

Door: two-way. Blast radius: isolated. Six new files in research/two-axis-coframe-cuts/; shared code, predecessor source bytes and the selected result are unchanged.