[petalas]

Summary

Construct K_(n,n) rank-distance metrics for every n inside commuting 4-by-4 integral rational shear frames. This is a scoped written construction proof with exact finite corroboration, produced by Nick Petalas. No new kappa or physical supplier is claimed.

E(t) = [[t,t^2],[1,t]], O(a) = [[a,a^2],[0,-a]]
E(t)-O(a) = column(t-a,1) row(1,t+a)  -> cross rank 1
within-part determinants = ±(parameter difference)^2 -> rank 2
Phi(M) = [[I_2,M],[0,I_2]] -> commuting actual shear operators

Before, Jerry Chen / CubeJerry's PR907 supplied eight actual rational shear frames and an abstract all-size distortion theorem. The new construction realizes every size in the same fixed operator dimension, with entries bounded by max(1,(n-1)^2). Applying the credited formula c_1(K_(n,n))=2(n-1)/n gives distortion approaching two; the countably infinite realization has distortion exactly two. The predecessor's cut proof and finite numbers are cited, not replayed as a deliverable.

Only research/unbounded-shear-bicliques/ is added. Proof and package. Advance CLAIM. Base main is 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. TASK.json records the changed coverage-law-change scope as a contact-geometry side result, not a completed productive invoice.

Credit PR907 for the metric, eight-frame example and abstract distortion theorem; eumemic / PR288 for the rational shear framework; navinor / PR825 and Jeremy Diviney / PR836 for separate obstruction scopes. Elementary matrix identities are standard, and literature priority is unclaimed. No predecessor code is copied.

Evidence

• The all-n statement follows from the displayed factorization and determinants in PROOF.md. Finite extrapolation is not used.
• Normal and optimized warning-strict CPython 3.12.15 replay passes 8,116 ordered label/operator pairs in thirteen balanced integral fixtures and one 9-by-11 rational fixture, 301 factorization/determinant checks, 400 group-product checks and ten rejection controls. Independent rational elimination agrees with determinant ranks. Each registered command took less than 0.3 seconds locally against a 60-second cap. These are timeout measurements, not a speedup claim.
• PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES pass. Full declared source ancestry and executable modes were checked for e833fa110cb6cb878a1e95680d4957b2638b29a8.
• Official committed export and declared-input coverage pass. An optimized Git-free replay with empty PATH and read-only files passes in 0.255 seconds, preserving all seven exported files, including the export manifest. An unexpected CLI argument is rejected. Receipts are generated and bound to all six package hashes.
• Syntax/function-length inspection and staged whitespace checks pass. The selected baseline prints PASS_RESEARCH_BASELINE_BINDINGS_ONLY; that is orientation, not a selected scientific replay.

Reproduce from the repository root:

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

An absolute CPython 3.12 executable on another system may replace the recorded path. Expected direct marker: PASS_UNBOUNDED_SHEAR_BICLIQUES.

Limitations and unrun obligations

The metric uses the declared rational rank charge. No nonlinear circuit, native Clifford realization, productive target, exact caller, dirty restoration, complete cost, precision/tape implementation or compiler closure is supplied. The operators are upper triangular and commuting; the existing common-flag obstruction is not escaped by a non-L1 metric alone. Coefficient storage grows with n. The infinite distortion corollary uses PR907's abstract theorem as an input.

No selected cold replay, full native replay, Lean, Docker, complete multiplication theorem, independent human review or formal proof-kernel review ran. The selected frontier remains unchanged.

Open next questions

1. Can this geometry participate in a changed completed interface with a favorable signed bill? First specify the legal gates, demanded endpoints and paid cleanup, and identify the exact hypothesis escaping the known common-flag obstruction.
2. What coefficient-height growth is necessary for integral bicliques at fixed matrix dimension? Start with the rank-one contact equations under bounded integral entries. The present construction supplies a quadratic-height upper bound only.

Related open work

The initial read of all 319 open PR titles and bodies found none studying this rational shear biclique metric or its distortion. The completion refresh examined the 13 other open account PRs created since the advance CLAIM. None computes a competing value for this metric. Adjacent work includes:

• #6172 optimizes a signed subspace triangle with variable productive boundaries. This package constructs rational shear contacts and computes no signed-triangle optimum.
• #6173 computes two-channel support in a fixed P104 root code. This package supplies no P104 code-support value or word edit.

PR907, PR825 and PR836 are merged cited predecessors. Sparse local notes and unpublished work remain overlap limits. No extra computations were run to check other contributions.

CI notes

Both hosted checks passed at exact head e833fa110cb6cb878a1e95680d4957b2638b29a8 in run 40880126282: plan in 4m22s and replay (research/unbounded-shear-bicliques) in 8m0s. The completed check list was read once, and run metadata confirms the same head and successful completed conclusion.

The registered hosted replay executes both normal and optimized package checks. All local scientific, committed-byte/mode/source-history and read-only Git-free export checks listed above also passed. Hosted success certifies those bounded checks; independent human and formal proof-kernel review of the written theorem remain unclaimed. Physical supplier and complete multiplication obligations retain their stated limits. No CI failure remains.

Merge danger

Door: two-way. Blast radius: one isolated opt-in research package. No shared implementation or selected-frontier file changes.