[petalas]

Summary

Scoped written proof with exact finite corroboration. A rational orthogonal U admits an injective norm-nonincreasing map F:Z^d -> Z^d with uniformly bounded error from U exactly when U is a signed permutation. This holds in every positive finite dimension and permits nonlinear F.

nonintegral column U e_j=a/q, q not dividing n
norm-preserving integer output y at input n e_j
  squared error = (2n/q)(nq-a dot y) >= 2n/q
complete h-grid ball + injection + norm nonincrease
  worst-case error >= sqrt(2h(1-h)/q)

The new step extends ywz65's PR2589 fixed-phase obstruction to arbitrary-dimensional rational mixing, with an explicit error quantum and existence classification. PR2589's finite-permutation and fixed-restored-helper arguments remain credited premises. Its stronger two-dimensional phase bound is not improved or replayed. Credit also PR2527, Mason Hale / PR2313 and Jose Manuel / PR2298 for the retained interface lineage. This contribution was produced by Nick Petalas. Classical lattice and finite-set arguments are not claimed as new mathematics.

Only six files under research/rational-norm-rounding/ are added. Proof. Advance CLAIM.

Evidence

• Before: PR2589 treats a fixed phase on a complete two-real-coordinate disk. After: the written proof supplies the rational-map classification and a dimension-independent grid-ball error lower bound. No predecessor scientific executable is imported or run.
• PASS_RATIONAL_NORM_ROUNDING: 1,254 exact output candidates in 59 complete sphere streams for four matrices and eleven nonintegral columns; 704 near-boundary arithmetic witness selections across 64 dyadic precisions; all 5,040 permutations of a seven-point ball, with eight contractive permutations; eight rejection controls, an attaining positive quantum, 81 signed-permutation inputs and three premise controls.
• Warning-strict normal and optimized CPython 3.12.15 commands each passed locally in 0.03 seconds, below their 120-second limits. These are check durations, not a speedup claim. Their complete generated hash-bound receipts agree byte for byte.
• Both registered checks and metadata passed. Six-file committed-byte/source-history preflight passed. Official export integrity and declared-input checks passed. Normal and optimized replays passed in a read-only Git-free export with empty PATH; their complete receipts equal the checkout receipts and all six file hashes remain unchanged.

Exact reproduction from the repository root:

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

Another CPython 3.12 path may replace the local interpreter. The standard library suffices. No generated output remains in the checkout.

Limitations

Rational U, a complete integer/grid ball, the same input/output grid, injectivity and norm nonincrease are explicit hypotheses. Retained input/history, data-dependent entrance helpers, finer output grids, restricted domains and outward slack change them. The Omega(sqrt(h)) bound is necessary; its global attainability is not established. High-precision output spheres are not enumerated. The finite reference covers dimensions two through four; the universal quantifiers rest on the written proof.

No actual selected native phase/caller is bound. Native realization, physical storage/tapes, relaxed rounding contracts, compiler and outer admission remain open. No new kappa or selected-frontier change is claimed, and all five framework assumptions remain. Lean, Docker, selected cold replay, independent human and formal proof-kernel acceptance are unrun or unclaimed. All registered executable obligations passed locally.

Canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8 and selected PR6030 remained unchanged at submission refresh. The card changes to this scoped precision/restoration theorem. Main, merged/open PRs, live Discussions and reservations were searched; sparse notes, arbitrary historical diffs and unpublished work remain coverage limits.

Open next questions

• Find the least outward norm slack admitting a globally bijective bounded-error rounder for the q=2 four-coordinate fixture. Begin with a complete explicit grid contract; an actual bijection and physical tariff remain missing.
• Bind one selected native caller's entrance fiber, legal domain, norm, grid and complete outputs. Until that binding exists, this theorem is not a supplier rejection.

Related open work

• PR6614 optimizes linear exactly unitary Gaussian-dyadic two-wire approximants; this theorem fixes rational U, permits nonlinear maps and adds norm nonincrease.
• PR6509 constructs sum-preserving reversible nearest Gaussian rounding; exact sum preservation is a different invariant from norm nonincrease.
• PR6534 constructs two-axis Gaussian grid maps with exact inverse and sum; no inward norm contract is assumed here for that construction.
• PR6536 gives a three-axis reversible error optimum; the present obstruction concerns the additional inward norm requirement, not its reported optimum.
• PR6574 gives a four-axis sum-preserving reversible map; this theorem does not impose its extra norm premise on that map or compare error numbers from different contracts.
• PR6438 constructs unbiased reversible ensembles for the same rational q=5 phase; this result instead requires pointwise norm nonincrease on each fixed fiber.
• PR6561 classifies rounding radii under finite-group equivariance; this theorem requires injection and norm nonincrease, without equivariance.
• PR6540 retains a seed and residue for reversible fine-to-coarse coarsening, changing the same-grid/restored-state premises used here.
• PR6542 rounds a sparse shear correction with exact inverse; its nonorthogonal ideal correction is outside this rational-orthogonal theorem.

These relationships use published contracts, with no extra computations auditing another result. The completion refresh inspected all twelve other account open PRs created since the advance CLAIM, through PR6620. PR6614 is already listed above; the other contributions study different objects or quantities. No new matching bound or numerical disagreement was found in that inspected coverage.

CI notes

Both hosted checks passed at exact head eaa967e7e8aa12cc53ec246a5fc00c5f8873a50b in run 41635056224: plan in 3m53s and replay in 1m18s. The completed check list was read once. Run metadata confirms completed success and the matching head.

Hosted replay certifies execution of the registered normal/optimized finite checks and source metadata at this commit. It does not independently establish the written universal theorem. Every registered command, commit/source-history check and read-only Git-free export replay also passed locally. No executable obligation was left solely to CI; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Six new package files; no shared implementation, predecessor, pinned evidence or selected certificate changes.