[petalas]

Summary

Scoped written proof with exact finite corroboration. Equal-energy Gaussian-integer rows admit a smaller sufficient even packing radix. At R=512, use 726 instead of 1024; the largest input at 512 columns shrinks by 254 bits. No new kappa is claimed.

- Whole-ball sufficient radix: B = 2R
+ Equal-energy sufficient even radix: B^2-B+1 > 2R^2
- bit_length(1024^511) = 5111
+ bit_length(726^511) = 4857

The new ingredient is the sharp inequality E(d)>=2(B^2-B+1) for every nonzero even-energy Gaussian packing-kernel vector at an even radix. Polynomial division and an exact discrete energy identity prove it in every finite dimension; (B,B-1,-1) attains it. Equal row energies force the difference energy to be even. The resulting comparator criterion is sufficient, not claimed optimal.

Only five files under research/equal-energy-radix-parity/ are added. The proof and verifier were produced by Nick Petalas. Credit Arham Tahir / Arham-Tahir64 #1906 for the whole-ball predecessor and its excluded equal-energy scope; woklerx #1859, Jeremy Kothe #593, commdata2338, Ross Richardson #1934, ywz65 #2499 and their retained lineage. Their circuit inventories and numeric receipts are cited, not audited or replayed.

Evidence

CPython 3.12.15, standard library only. Both registered warning-strict commands pass locally, normally and under -O, in about 1.5 seconds each:

• 6,200 quotient/kernel identities, including 3,112 even-energy kernel vectors; 256 sharp witnesses.
• 3,051 real and 1,875 complex direct digit words; all ten nonzero even-energy kernel cases satisfy the bound.
• 19,800 complete equal-energy shell vectors at energies 2,6,10 and complex lengths 1..4, from 6,299,180 enumerated digit words, with no admissible packing collision.
• Thirty new small scaled-unitary fixtures, 840 Gram entries, 300 normalized matrix comparisons and thirty absolute-phase rejections.
• Four discriminating hypothesis controls, thirteen invalid-input rejections and exact criterion arithmetic for M=1..4096.

Metadata/provenance checks, exact committed bytes and declared source-history preflight pass. The official five-file export passes integrity and input-coverage checks. Optimized, read-only, Git-free replay passes with all six export files unchanged. Receipts are generated and bound to the five source hashes.

From the repository root, exact local reproduction commands:

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

Expected markers are PASS_EQUAL_ENERGY_RADIX_PARITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. A different host can substitute its CPython 3.12 interpreter path. Add --receipt /absolute/path/outside/checkout.json to the direct verifier for an actual hash-bound receipt.

Limitations

The sharp claim concerns even-energy kernel vectors, not an optimal radix for unitary or actual Clifford rows. Equal energies, even radix and the complete scaling contract are essential. Radix 726 needs general integer multiplication during packing; the smallest power-of-two radix from this criterion remains 1024. No runtime saving, production comparator change, new Lean proof, supplier replay, physical cost result or complete multiplication theorem is claimed. The selected frontier and all five framework assumptions are unchanged. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine the least common energy of a genuine collision at a fixed even radix, then test which pairs extend to scaled Clifford matrices. The sharp kernel witness alone does not settle this.
• Formalize the quotient energy identity and path variation bound, then connect the equal-row-energy guards to an actual comparator. Formal proof and implementation-cost obligations remain unrun.

Related open work

The closeout refresh found 14 open account PRs created since the CLAIM at 2026-10-27T14:39:11Z, including this PR. The other 13 stated scopes report no competing packed-row radix or common radix value. No additional mathematical computations were run for comparison. Adjacent Gaussian, Clifford and norm work is distinguished below.

• [Lean] Packed Clifford representation bridge and a source-bound CK child #872 proves a packed/vector Clifford representation bridge for a historical CK child. This result changes a sufficient integer evaluation radix and supplies no replacement for that formal bridge.
• research: bind selected HPE tail premise #3912 binds selected HPE tail records to retained pair literals. This result uses the cited abstract row-energy contract and adds no selected-word semantic coverage.
• Prove exact tensor annihilator coefficient heights (no new κ) #5934 determines Gaussian tensor-annihilator coefficient denominators and heights. This result bounds the energy of a Gaussian polynomial that vanishes at an integer packing radix; no common coefficient-height or radix value is compared.
• Prove sharp-order Clifford row-sum convergence (no new κ) #5939 studies full-projective-Clifford absolute Gram row sums and their dimension-dependent convergence. This result uses exact energy of individual scaled-unitary rows; it does not compute that Gram row-sum quantity.
• Prove sharp conditioned matching-layer interpolation (no new κ) #5940 proves conditioned matching-layer norm interpolation. This result concerns exact Gaussian packing-kernel parity under equal row energies, with no shared numerical fixture compared.

The overlap search also checked the current open PR list, recent contributions from this account, prospective claims, local reservations and distinctive all-main/merged searches. Coverage limits are recorded in the package; a search is not proof of novelty.

CI notes

Both hosted checks passed for exact head 2ae6151ba75698220722433ed2d3a5a7b10bf898 in run 40525068631: plan in 4m38s and replay (research/equal-energy-radix-parity) in 1m30s. The checks were read once at closeout; run metadata confirms this commit and completed success. The package replay executes the registered warning-strict normal and optimized commands.

The same registered checks, metadata/provenance, committed-byte/source-history and read-only Git-free export checks also passed locally. No required check was left solely to CI. Hosted replay certifies the bounded registered checks; it does not provide independent human or formal proof-kernel acceptance of the written theorem.

Prospective CLAIM. The delivered mathematical scope matches it.

Merge danger

Two-way door. Blast radius: isolated research package. The new replay is opt-in; no shared checker, predecessor file or selected record changes.