[petalas]

Summary

Scoped mathematical proof with exact finite corroboration. For every two-coordinate dyadic dual pair ab=1, classify every compatible R in GL_2(Z[1/2]) and compute the least Euclidean condition number exactly:

R = [b, ±2^k*(-a_2,a_1)^T]
A = ||a||_2^2, B = ||b||_2^2
optimal k: 2^(2k-1) <= B/A <= 2^(2k+1)
q = A*2^k + B*2^(-k)
minimum cond_2(R) = (q + sqrt(q^2-4))/2

For Ali Romman's #5611 pair a=(3,5), b=(2,-1), the optimal determinant magnitude is 1/2 and the sharp condition number is (27+sqrt(725))/2. The cited unit-determinant completion has q=39; the optimal one has q=27. The real-completion optimum sqrt(170)+13 is strictly smaller, exhibiting the additional arithmetic restriction imposed by a dyadic inverse. Exact ties are classified as well.

This contribution was produced by Nick Petalas. Credit Ali Romman #5611 for the dual-pair interface, explicit pair and remaining growth/precision obligation, #5602/#5599 and their retained P104/native/controller/compiler lineage, and Nick Petalas #5115 for earlier row preparation in a different single-hub interface. Classical dyadic-unit and singular-value identities are not claimed as new tools. The source constructor is cited and built on, not audited or replayed.

Only research/dyadic-basis-conditioning/ is added. Canonical base is 19fdb086ea96af9a9f87936a2b888e10af4ed810, priorities revision 8. The helper-inventory card is explicitly changed to an endpoint precision-interface theorem. Advance CLAIM.

Evidence

Before: the source supplies legal completions and leaves their numerical cost payable. After: the written classification proves the global optimum over all determinant exponents, and an exact two-candidate optimizer returns every optimal exponent.

At head d2df1a6e3caf20c47fff89e6064f3747016fb954, local CPython 3.12.15 checks pass:

• All 308 integer dual pairs in [-5,5]^4 and seven reciprocal rescalings each: 2,156 presentations, compared with 71,148 literal exponent matrices. Both determinant signs yield 4,760 checked optimal representatives; 224 presentations have ties.
• An independent second-column search through 56,628 literal matrices over 52 smaller pairs, including all 488 compatible grid matrices, gives the same optima.
• Four size-boundary pairs and 14 rejection controls cover invalid inputs, nonunit determinants, wrong endpoints/exponents and false spectral proxies.
• Normal and optimized direct execution, both registered commands, committed-byte/source-history preflight, in-memory compilation and a read-only standalone export pass. The generated receipt binds all seven package files. No predecessor executable is imported.

Each registered command was timed locally: 0.877s normal and 1.054s optimized in one observation, against a 60-second timeout. These durations are replay-budget observations, not speedup measurements. The scientific commands write no files and use explicit exceptions under optimization.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/dyadic-basis-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/dyadic-basis-conditioning/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/dyadic-basis-conditioning
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/dyadic-basis-conditioning --check-source-history

Other hosts may substitute their CPython 3.12 executable. Standard library only. Expected markers are PASS_DYADIC_BASIS_CONDITIONING, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The theorem fixes exactly two roles, real dyadic coefficients, prescribed row and column, dyadic invertibility and Euclidean conditioning. Its written proof has no size limit; the reference optimizer accepts at most 64 numerator bits and binary denominator exponent 64. It also applies to the same scalar basis tensored with an identity on arbitrary independent complex banks.

No extra-workspace optimum, scalar instruction optimum, intermediate growth bound, rounded-error implementation, physical cut, ownership/restoration schedule, full signed invoice, supplier admission or new kappa is supplied. All selected framework assumptions retain their status. The selected cold replay, Lean, Docker, independent human review and formal proof-kernel verification are unrun and unclaimed.

Related open work

The refreshed open-PR corpus contains no other submission computing this exact two-role dyadic completion optimum. Adjacent work is distinguished without running extra computations:

• Prove sharp spectral conditioning for signed rank transport (no κ) #5952 sharpens a signed-rank increment using spectral conditioning of a faithful regular prefix. It does not optimize a prescribed dyadic helper completion.
• Prove sharp conditioned matching-layer interpolation (no new κ) #5940 bounds interpolation inside conditioned matching layers. It does not choose the helper basis or its determinant.
• Prove optimal four-addition, one-halving Pauli pairing normalization (no κ) #6003 optimizes additions and halvings for a three-bank operator-valued Pauli skew pairing. The alphabet, endpoint and resource objective differ from this two-role scalar conditioning problem.
• Conditional κ 0.00309871 from a prepared P104 split helper #5606 changes a prepared P104 helper and its paid/target invoice. This theorem gives an algebraic basis-conditioning optimum and does not recompute its word, invoice or kappa.

The completion refresh read the titles and bodies of this account's 15 other open PRs created since the CLAIM at 2026-10-27T21:58:58Z. Additional adjacent work:

• Prove exact chronological helper spill and repair rank (no new κ) #6019 determines chronological leakage and repair rank for prescribed helper replacements. It does not optimize their dyadic basis or compute a condition number.
• Exclude finite-menu adaptivity for exact helper endpoints (no new κ) #6015 excludes selection among finite menus of fixed linear helper endpoints. This result optimizes one fixed compatible basis and supplies no adaptive-menu theorem.
• Prove zero-preserving polynomial helpers cannot reduce linear call minima (no new κ) #6017 extracts linear call profiles from zero-preserving polynomial helper programs. This result concerns exact linear basis conditioning and makes no polynomial-call comparison.
• Classify fixed nonlinear role recodings under exact ADD compatibility (no new κ) #6025 classifies nonlinear role recodings compatible with ADDs. This result fixes real dyadic linear coefficients and computes no nonlinear recoding profile.
• Certify six-bit stochastic precision for the selected P104 prefix (no κ) #6026 certifies stochastic precision for a selected P104 prefix. This result gives endpoint Euclidean conditioning and supplies no rounded prefix or precision-bit certificate.
• Decide paired lattices with rank-one projector anchors (no new κ) #6027 decides common paired lattices for projector-anchored families. This result fixes one two-role dyadic dual pair and optimizes a compatible basis, rather than deciding a shared invariant lattice.
• Prove H6 commuting dyadic repairs need rank 816 (no new κ) #6023 gives a commuting dyadic repair-rank obstruction for H6. It fixes a different matrix and endpoint; this result computes neither an H6 correction nor its rank.

None computes the same numerical quantity. No extra computations were run to audit another contribution or compare numbers.

Merged #5611 is the direct predecessor. Overlap checks include live handoffs, local reservations, recent account work, open PR bodies, distinctive main README terms and merged searches. Sparse local-note coverage and unpublished work remain limits.

Open next questions

• Bind one actual two-role helper cut and emit a chronological scalar schedule for the optimal basis. Check intermediate growth, rounding, restoration and the complete paid/target invoice. Those physical obligations remain blockers.
• Determine the exact optimum for one three-coordinate dyadic dual pair. The complementary basis is no longer a scalar line; the two-role determinant argument alone does not solve that optimization.

CI notes

Both hosted checks passed at exact head d2df1a6e3caf20c47fff89e6064f3747016fb954 in run 40591916285: plan in 5m2s and replay (research/dyadic-basis-conditioning) in 1m26s. The completed check list was read once. Run metadata confirms completed success and the same head as the PR.

Hosted replay certifies both registered finite-check commands. All listed local scientific, metadata, committed-byte/source-history and standalone-export checks also passed. The universal written theorem and physical obligations retain their stated scope; independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated. Seven new package files and an opt-in replay; no shared code, predecessor artifact or selected-frontier file changes.