[petalas]

Summary

Scoped proof with exact finite corroboration: for supplied heterogeneous Smith factors C_a=[[i,1],[1,(1+i)^a-i]], sorting valuation gaps descending in the triangular gather and ascending in delivery minimizes the complete two-pass ADD-precision mass. The proof gives the exact optimum and every equality case, including zero precision and gaps clipped at the raw modulus.

gaps (1,4), raw precision 3, same four-coordinate target
  gather (1,4), delivery (4,1): 11+11 = 22
  gather (4,1), delivery (1,4):  9+ 9 = 18

This contribution was produced by Nick Petalas. Credit David Ganz / Dugongue #5815 at f0fcda576df410dc15cd5b8183d4f74d284eba2b for the integral Gaussian Smith circuit, propagated moduli and two-pass accounting. Tensor factorization, valuation arithmetic and adjacent-exchange arguments are classical. The new result changes identical Gaussian gaps to unequal supplied gaps and proves the optimal order class. No literature-wide priority is claimed; no predecessor result or verifier was audited or replayed.

Evidence

Before, the identical-gap word supplies no heterogeneous ordering theorem. After, PROOF.md gives a legal arbitrary-input modular word for every gather/delivery permutation and an exact adjacent-exchange argument. Equal Gaussian gaps tie, so no ordering gain is claimed for the existing identical Gaussian tensor.

At 376db3265f7c0418f6d95209fbf4eee5700417b0, local CPython 3.12.15 checks pass:

• 153,405 physical-axis orders across 2,259 profile/precision cases; 593,730 exchange identities and exact optimal-order equality checks.
• 18,648 basis endpoints, 4,320 arbitrary-input endpoints and 357 complete finite-ring inputs against independent dense tensor and Gaussian parity-lattice oracles. Twelve additional endpoints execute both strict-fixture words at the claimed precision 3.
• Thirteen malformed-input rejections and six semantic controls, including early truncation, omitted delivery, clipped ties and corrupted outputs.
• Warning-strict direct normal and optimized checks take 7.94 s and 7.96 s. Both registered checks pass; the complete registered invocation takes 16.18 s. Each check stays below its 180-second limit.
• Package metadata, seven committed files, file modes and source-history preflight pass. The official export passes integrity checking. Its optimized read-only Git-free replay with empty PATH passes in 8.08 s and preserves all eight exported files and modes. Generated receipts bind all seven package hashes.
• Python diagnostics and formatting checks pass. No shared implementation or frozen evidence is changed.

Exact reproduction from the repository root:

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

Any CPython 3.12 path can replace the interpreter shown. Direct replay ends PASS_SMITH_AXIS_ORDER; registered replay ends PASS_REGISTERED_RESEARCH_PACKAGE. The standard-library verifier writes no files by default. An optional --receipt path saves a generated hash-bound receipt.

Limitations

The theorem minimizes ADD-precision mass among the declared fixed-coordinate two-pass permutations with equal per-axis ADD prices. It is not a minimum over other algorithms or a runtime result. Phases, scales, physical access/routing, initialization, truncation, cleanup and complete compiler costs require separate charges. Owned truncation is allowed; reversible truncation is a different contract. No actual heterogeneous supplier is instantiated.

The executable accepts 1–8 axes, gaps/precision 0–64 and input component height at most 10^6. The written proof covers every finite nonnegative integer profile; the finite census covers the smaller ranges specified in README.md. Endpoint tests use optimal/opposite pairings, rather than every independent pair of orders. No Lean, Docker, supplier cold replay, rounded multiplier or complete multiplication theorem was run. Independent human and formal proof-kernel review are unclaimed. No new kappa or selected-frontier change is claimed; all five framework assumptions remain.

The formal-interface card is explicitly changed to this algebraic side result. The advance CLAIM, source pins, scope and bounded overlap coverage are recorded in the package. Refreshed canonical main remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8.

Open next questions

• Derive an ordering criterion with unequal physical axis prices, starting from the weighted two-axis exchange and testing context dependence.
• Supply one actual heterogeneous factorization and price its complete routing, scale, truncation, cleanup and consumer bill. Identical Gaussian gaps offer no ordering gain.

Related open work

• Construct exact Gaussian corrections with pruned and mixed precision circuits #5815 supplies the original integral Gaussian Smith word and two-pass precision accounting. This changes the identical-gap hypothesis and optimizes axis scheduling; its predecessor checks are not replayed.

• Execute Gaussian Smith circuits and prove sharp precision requirements #5821 executes the identical Gaussian circuit and studies sharp cut/output precision. This supplies a different heterogeneous endpoint and an ordering optimum, without auditing its precision claims.

• Construct Gaussian spectral and precision compilers from the ETH preprint #5813 supplies the central Gaussian spectral bulk and precision compiler. This optimizes a heterogeneous triangular tensor representation rather than that polynomial bulk.

• Execute pruned Gaussian corrections and bound the standalone Newton-W route #5825 supplies pruned spectral correction and a standalone Newton-W work obstruction. Its spectral graph and shared-bank bill are separate from this two-pass order class.

• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 studies initially clean storage for reversible Gaussian overwrite. Storage capacity and reversible truncation are different quantities from the declared ordinary ADD-precision mass; no common number is compared.

• Prove the sharp R-unit Gaussian bulk denominator (no new κ) #5890 proves sharp denominator debt for the central Gaussian bulk. Its fixed Boolean-degree response and numerator-precision threshold differ from this heterogeneous tensor order optimization; no extra computations were run to check its values.

• Prove one-axis transported coframe optimization is NP-hard #5888 studies one-axis transported Gaussian coframe rank tariffs; its rank-transition optimization differs from this ordinary heterogeneous ADD-precision bill.

Completion refresh inspected this account's nine other open PRs created since the CLAIM: #5886, #5887, #5888, #5889, #5890, #5891, #5892, #5893 and #5895. The Gaussian denominator result #5890 is related above. The other seven concern bounded NTT storage, factorized invoice envelopes, dyadic affine feasibility, scratch-reading query obstructions, observed recurrences, resource migration and reversible dependency scratch. None computes the heterogeneous tensor order quantity or the 22-to-18 fixture; no additional mathematical computations were run to compare them.

CI notes

At commit 376db3265f7c0418f6d95209fbf4eee5700417b0, run 40496993476 finished with plan failed; replay skipped. The only failed step is Onboarding helper tests. All ten failure/error entries point to unchanged tests/test_research_current_assignment.py; the log ends FAILED (failures=8, errors=2, skipped=22) and exit code 1.

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failures concern frozen-card count 0 != 16, expected missing-helper/reference rejection, and sparse-reference receipt lists. All ten failure headers and their exception/assertion messages match PR #5890's plan log after timestamp removal; both logs were inspected. This confirms the reported shared onboarding failure on canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a. Maintainers will fix it; no shared implementation or tests were changed.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific checks listed above have run. No package failure was reported.

Merge Danger

Door: two-way.

Blast Radius: package.

Only research/smith-axis-order is added. Shared code, frozen predecessors and selected certificates retain their bytes.