[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. Decide whether a supplied rational strict affine parameter region has a dyadic point at any precision, and construct one when it does.

E x=b, A x<h; supplied Smith charts and strict rational witness
  odd Smith divisor fails to divide transformed b -> no dyadic point at any precision
  otherwise -> exact equality precision p_eq
            -> explicit free-coordinate rounding bound preserving every strict margin

The permanent obstruction is oddpart(d_i) ∤ (Ub)_i. When none occurs, the exact equality threshold is the largest binary denominator exponent of (Ub)_i/d_i. For each strict row, the construction preserves its margin once sum_free |(AV)_ij| < 2^(p+1) delta_i. The strict sign is necessary for the stated rounding guarantee. This extends the finite precision cap in Ali Romman's #5794, rather than auditing that search. No new kappa or selected-frontier change.

Evidence

• Before: Construct minimum-precision dyadic parameters with complete exclusion proofs #5794's infeasible_through_limit leaves all finer grids open. After: the supplied strict affine model has a complete all-precision criterion and a constructive sufficient precision. The equality threshold is exact; the whole-region precision bound is sufficient.
• Warning-strict normal and optimized local replays pass with the same generated counts: 2,080 scalar equalities; 5,568 coupled regions with 3,712 constructed points and 1,856 odd obstructions; 216 complete bounded grids containing 62,424 enumerated points; eleven rejected inputs and three semantic controls.
• Each registered direct command was timed locally at 1.51 seconds, below its 60-second timeout. Both registered commands and metadata validation pass. Committed-byte, mode and source-history checks pass at 8811e913cef13d3d2d7d0941c791d58c9be78e3b. Official five-file export and offline integrity checks pass.
• Normal and optimized read-only Git-free exported replays pass with an empty PATH; all six export files retain their bytes and modes.
• Generated stdout receipts bind all five package files by SHA256. No receipt is hand-written, and normal verification writes no files.

From the repository root, CPython 3.12 and the standard library:

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

Another CPython 3.12 path may replace the absolute path. The standalone output contains PASS_DYADIC_AFFINE_FEASIBILITY and the registered runner ends with PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations

The theorem requires a supplied Smith decomposition and a rational witness strict on all remaining inequalities. It does not extract an affine hull, compute Smith charts, find the witness or minimize the precision of the inequalities. A general closed region needs its forced equalities exposed before this interface applies. The executable helper is bounded to six equality rows/coordinates, 32 inequalities, 64-bit input rational components and 4,096 precision bits; unsupported inputs raise an error. The proof has no dimension or precision cap.

No actual supplier parameter proof, nonlinear feasibility, physical arithmetic/storage invoice, native caller, compiler admission or analytic recovery is supplied. No selected scientific replay, Lean, Docker or proof-kernel check was run. Smith normal form, localization and nearest-grid error bounds are classical; no literature-priority claim is made for those ingredients.

Open next questions

• Bind one actual supplier's literal equalities, strict margins, Smith charts and witness, then price the resulting format. Source validity and physical arithmetic remain blockers.
• Construct a checked affine-hull extractor for general closed rational regions. Start with equalities forced by paired inequalities; complete forced-face identification remains open.

Related open work

• Construct minimum-precision dyadic parameters with complete exclusion proofs #5794 constructs minimum-precision dyadic points within a bounded grid range. This contribution adds a permanent obstruction and an explicit sufficient precision for supplied strict affine regions, without rerunning its search.
• Construct exact optima and dual bounds for bounded parameter models #5792 constructs exact rational optima and dual bounds for bounded linear parameter models. Rational feasibility does not ensure dyadic feasibility; this contribution addresses that representation question and does not optimize the source objective.
• Prove the sharp R-unit Gaussian bulk denominator (no new κ) #5890 proves the sharp Gaussian denominator debt of a central spectral bulk. This contribution instead decides dyadic parameter feasibility for supplied rational affine regions; it computes no common Gaussian denominator or physical precision invoice.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-27T11:15:23Z were refreshed. The other new PRs are #5886, #5887, #5888, #5891, #5892, #5893 and #5895. Their bounded NTT storage, signed moments, coframe optimization, two-FULL obstruction, coupled bills, resource migrations and pebble storage quantities do not study dyadic affine feasibility. No common computed number disagrees, and no extra scientific computations were run to check those contributions.

CI notes

At commit 8811e913cef13d3d2d7d0941c791d58c9be78e3b, run 40495794184 failed in plan only at Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py. The exact final summary is FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card counts, missing-helper/reference rejection and sparse-reference receipts. All ten failure/error headers match #5890's failed plan step exactly. This confirms the shared onboarding failure on canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a. This PR changes only its isolated research package; no shared tests or implementation were changed to address CI.

replay was skipped. Hosted replay has not certified this package; only the local scientific and publication checks listed above have run. No hosted scientific success is claimed.

Merge Danger

Door: two-way. Blast radius: isolated. Only research/dyadic-affine-feasibility/ is added.

This contribution was produced by Nick Petalas. Credit Ali Romman #5794 at b67e65c5fcc313f3fcb1e92b199025a63cc1dec7 for the bounded dyadic interface and explicit finite-limit distinction. The selected supplier and its credited lineage are historical context only. Independent human and formal proof-kernel review are unclaimed.