[petalas]

Summary

A four-axis Gaussian integer-grid map now attains the sharp uniform block error sqrt(8) h, preserves the exact complex sum, fixes zero and has an explicit inverse. This answers the four-axis part of Ali Romman's PR6536 open question. Claim type: scoped written proof with a finite construction certificate. This contribution was produced by Nick Petalas.

rotate by D
apply one certified real Walsh-16 correction table to each component
rotate by -D
inverse: undo rotations and use the paired inverse correction table

The new step is a sum-preserving matching on 64 real quotient classes. Sharing it between real and imaginary components gives a 3,072-byte packed table including keys and both directions. A half-grid witness proves global optimality among integer-output rules on this sixteen-point block. In an ambient N-entry tensor, one four-axis group has error at most h sqrt(N/2), half the inherited three-plus-one triangle certificate. No physical runtime advantage, new kappa or selected-frontier change is claimed.

Credit Ali Romman for PR6536 and PR6534, David Ganz / Dugongue for PR5815, Ali Romman for PR5821, Brian / JoyboyBrian for PR5485, and the retained PR4743/Gaussian native lineage. Walsh transforms, quotient enumeration and matching are classical. Predecessor theorems are cited context, not replay or audit deliverables. Only ten new files under research/four-axis-reversible-grid/ are added.

Evidence

• Before: the stated larger-block question remained open after the three-axis construction.
• After: a deterministic constructor and separate predicate checker certify all 64 correction rows, their complete quotient permutation, both weighted sums and both error bounds. All 4,096 real-class pairs plus 68 signed, large-word and sharp-witness fixtures pass in both directions. There are 4,164 endpoint fixtures, ten rejection controls and maximum squared error times sixteen exactly 128.
• Warning-strict CPython 3.12.15 normal and optimized direct replays pass, locally timed at 1.40 s and 1.45 s. These are replay-deadline measurements, not speedup claims. Both registered checks pass; the complete registered command takes 2.97 s. Normal and optimized generated receipts agree.
• Exact committed bytes, all declared source ancestry, official ten-file export integrity and declared-input coverage pass at 48e312ff18d5b66b0969148f615fdf5ce0f92edd. Both direct modes pass in a read-only, Git-free, empty-PATH export; all eleven exported files retain bytes and modes. The verifier emits source-bound receipts to stdout and trusts no handwritten receipt.

Exact reproduction from the repository root:

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

Any CPython 3.12 path may substitute. Standard library only. Expected scientific marker: PASS_FOUR_AXIS_REVERSIBLE_GRID. The source head for the question is 83fee59d692b0764d716e8356a7d991e4d503868; it is documented rather than used as an unavailable runtime dependency. Canonical main remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, selected PR6030, priority revision 8. Live claims, open bodies, local reservations, canonical-main formula/object searches and merged PR searches were refreshed before committing. The overlap cache covered 521 open PRs and 6,526 Discussion records; sparse notes, arbitrary historical diffs and unpublished work remain limits.

Limitations

The written proof quantifies over all Gaussian integers; the executable API caps each component at 4,096 bits and documents round-trip headroom. This is a changed local precision contract, not an actual-caller assumption audit. Phase covariance, exact fourth power, pointwise nearest output, optimal whole-call error, a complete dirty caller, reversible Boolean/tape costs, accelerated tariffs and supplier/outer admission are unproved. All five selected framework assumptions remain. No selected cold replay, native builds, Lean, Docker, independent human acceptance or formal proof-kernel review ran.

Open next questions

• Bind this block to one literal native caller's entry/exit charts, adjoint order, dirty banks and full error budget.
• Price transforms, classification, shared-table access and restored auxiliary workspace in a complete reversible Boolean/tape program.
• Add exact fourth-power or phase-covariance constraints to the finite quotient problem, if a caller requires them.

Related open work

• PR6536 supplies the three-axis predecessor and poses the larger-block question. This result supplies a four-axis block with a shared real table and an explicit inverse; its fourth-power contract is not inherited.

• PR6534 supplies optimal two-axis fusion. Its different block width and charged additions are not rechecked or claimed here.

• PR6509 supplies conservative one-axis kernels and complete ports. The new block preserves sum but does not claim its phase covariance or complete ports.

• PR6532 gives a selected unbiased native map retaining a seed. This deterministic four-axis bijection has no sampling contract or seed-reuse guarantee.

• PR6540 gives unbiased fine-to-coarse native rounding with a retained residue. This same-grid deterministic map addresses a different information contract.

• PR6546 constructs a complete stochastic caller. This local deterministic block is not installed in that caller and imports no whole-word error coefficient.

• PR6542 rounds a sparse correction operator. Its scalar correction boundary differs from this Gaussian tensor block; no shared optimum is computed.

• PR6523 binds a rounded width-one child to selected MOVE9896. This result supplies a local multi-axis map with no selected-child binding.

• PR6480 addresses rounded spectator padding return. That caller boundary remains a separate obligation here.

• PR6438 constructs a stochastic planar rotation ensemble with retained seeds. This target is a deterministic, sum-preserving native block, without an unbiasedness or random-seed contract.

• PR371 accounts for final-grid error in Gaussian block enclosures. This construction supplies a local optimal block error and makes no change to those outer enclosures.

• PR6573 minimizes stochastic fine-to-coarse native output-prefix variance with a retained residue. This deterministic same-grid four-axis bijection computes a block-error optimum, with no competing prefix-variance value.

The completion refresh screened twelve other account open PRs created since the advance CLAIM, #6571 through #6583 excluding this PR. PR6573 is the additional adjacent rounding entry above. The others concern scalar-correction ADD bounds, phase metadata, atom-power and ADDRESS tariffs, parity fanout, tensor automorphisms, E8 basis/fiber geometry, ternary factor degrees and distinct replay optimizations. No additional computations audited those results, and no conflicting shared block-error value was identified.

CI notes

Both hosted checks, plan and replay (research/four-axis-reversible-grid), passed at head 48e312ff18d5b66b0969148f615fdf5ce0f92edd in run 41616901647. Plan took 4m28s and registered replay took 1m52s. The completed checks were read once; separate run metadata confirms completed success and the matching head. Hosted replay executes both registered normal and optimized commands. Local registered/direct checks, committed bytes/source history and read-only export checks also passed. Hosted execution certifies the declared finite checks and source bindings; independent human and formal proof-kernel acceptance remain unclaimed. Advance CLAIM.

Merge Danger

Door: two-way. Blast radius: isolated. This adds one self-contained research package. It changes no shared runner, frozen evidence, predecessor credit or selected-frontier file.