[petalas]

Summary

Scoped proof with exact finite corroboration. For the rational phase (3+4i)/5, changing one half-integer tie rule reduces the sharp all-grid squared error while preserving an exact two-sided inverse:

- final shear: u -= ceil(v/2), sharp squared error h^2
+ final shear: u -= floor(v/2), sharp squared error (4/5)h^2

Both opposite-tie programs attain 4/5; both same-tie programs attain 1. The optimum is restricted to the four fixed tie pairs. A residual lattice 5Z x 10Z reduces the all-grid proof to 50 cases per program. The same proof gives the inverse error bound and a sharp full-grid additive radial allowance 2h/sqrt(5). The program permits outward motion.

This originally posed tie-optimization question builds on ywz65's PR2589 lifting example and disk boundary. Classical three-shear lifting is credited, and PR2589's sufficient bound 153/80 is not disputed. The new program and sharp residue analysis were produced by Nick Petalas. Full predecessor credit is retained in the package. Only six additive files under research/reversible-phase-tie-optimum/ change.

Evidence

Before, the cited example uses the same tie direction in both half shears. After, the mixed-tie program attains squared error 4/5 at input (0,3) and returns exactly in both directions.

Local CPython 3.12.15, macOS 27.0.1 arm64, standard library only:

• All 200 fundamental cases, 26,244 signed-grid cases, 5,600 signed translations through 4,096 bits, eight radial-family instances, eight semantic controls and six invalid tie-rule rejections pass. A separate Fraction/divmod oracle checks the fundamental and large-translation evaluations.
• Warning-strict normal and optimized direct commands pass in 0.074745 s and 0.080320 s. These are replay-deadline measurements, not speedup claims. Both generated receipts are identical and bind all six package hashes.
• Both registered commands, metadata/pins, AST parsing, in-memory compilation, staged whitespace and committed-byte/source-history checks pass at 8053adb606c74353933457a0d0715028cab23e96.
• Official committed export and declared-input integrity pass. Both modes pass from a read-only Git-free export with empty PATH; all seven exported files retain their hashes. Exclusive receipt creation and invalid CLI behavior are checked. No predecessor scientific replay is the deliverable.

From the repository root:

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

Any CPython 3.12 path can substitute. Expected statuses are PASS_REVERSIBLE_PHASE_TIE_OPTIMUM, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The receipt is generated on stdout; none is handwritten.

Limitations and unrun obligations

This is a generic precision interface side result. The catalogue card's actual-selected-caller requirement is explicitly changed and remains unresolved. Optimality excludes other factorizations, adaptive tie policies and arbitrary lattice bijections. The rational phase is not identified with a selected supplier root, and the map does not preserve each closed disk.

Physical quotient workspace, overflow/word widths, phase generation, repeated-word error accumulation, native/caller realization and a complete signed invoice remain unproved or unrun. No selected cold replay, native suite, Lean or Docker was run. All five framework assumptions remain. No kappa gain, supplier admission or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed. Warning-strict execution supplies diagnostics; no external static analyzer was available.

Open next questions

1. Bind a phase actually needed by a rounded supplier, its grid and outward headroom, then test whether the smaller local error lowers the complete precision allocation. Physical quotient/return and integer storage costs remain missing.
2. Determine whether a reversible program outside the four fixed tie pairs attains a smaller all-grid error for this same phase. Start with an explicit finite residue matching problem and inverse; the present proof gives no broader optimum.

Related open work

• PR6258 studies drift of a different rounded native identity cycle. This result bounds one rational phase map and supplies no repeated-cycle error envelope or changed native-cycle ABI.

• PR6293 records exact selected MOVE tensor phases. These are exact Clifford scalar phases, rather than the arithmetic rounding error of this rational phase map; no tensor-phase census is repeated.

• PR6298 excludes a scalar-frame gauge remedy for those MOVE phases. This contribution changes integer rounding and does not alter or settle that gauge problem.

• PR6312 studies coordinatewise output rounding with exact preservation of eight modes in 112 coordinates. This contribution bounds Euclidean error of a two-coordinate rational phase bijection; the metrics, constraints and quantities differ.

• PR6307 proves a coefficient/output error floor for shared translated Fourier convolution. This result concerns one fixed-grid phase map and does not compute that convolution error.

The completion refresh checked this account's 15 other open PRs created since the CLAIM, PR6298 through PR6313 excluding this PR. The new adjacent rounding/error entries are distinguished above. No competing result for this fixed phase and four tie programs, or disagreement on a common computed quantity, was identified. Sparse local-note and unpublished-work coverage remain incomplete. No extra computation was run to compare another contribution's numbers.

CI notes

Both hosted checks passed at head 8053adb606c74353933457a0d0715028cab23e96 in run 41106823961: plan in 4m27s and replay (research/reversible-phase-tie-optimum) in 1m34s. The checks were read once; separate run metadata confirms completed status, success and the matching head SHA.

Hosted replay executes the registered normal and optimized checks. All local scientific, metadata, publication and export checks listed above also passed; no registered command was deferred to CI. These checks certify execution of the scoped automated verification, not independent human or formal proof-kernel acceptance of the written universal argument.

Advance CLAIM: #1166 (comment)

Merge Danger

Door: two-way. Blast radius: isolated research package. No shared implementation, frozen source, predecessor credit or selected-frontier record changes.