Prove sharp energy-ball radix for packed HPE equality

[Arham-Tahir64]

The unitary-premise bound in #1859 uses per-component bounds and a K+2-bit sufficient radix. This successor uses whole-row energy to prove that K+1 bits suffice, with global phase and full-row equality retained.

For Gaussian-integer vectors of energy at most R², a nonzero packing-kernel vector has energy at least B²+1. The difference of two admitted rows has energy at most 4R², so B=2R is injective in every dimension. Explicit collisions cover every smaller positive integer radix for the whole energy ball. This sharpness claim does not assert optimality among the particular unitary rows.

The three pinned HPE Lean literal pairs all pass at 10 radix bits, versus #1859's 11 and the retained checker's 20. Their largest packed input shrinks to 5,111 bits (previously 5,622 / 10,221); no timing claim is made. Only research/hpe-energy-packed-radix/ is added.

Validation on Windows / CPython 3.12.10:

python scripts/research_package.py --package research/hpe-energy-packed-radix

Registered replay passes: all 3,072 scaled rows and energies, 30 differential interpreter executions, 5,265 complete Gaussian-ball vectors, 11,493 short-kernel vectors, 1,024 unsafe radix cases, 15 source controls, and four CLI/source/receipt boundary test groups. Source exports replay with optimized Python, reject source drift and receipt corruption/duplicate keys, and regenerate identical receipt bytes. Git diff checks pass. The Windows sandbox required TEMP/TMP inside the checkout; ordinary tempfile behavior is retained.

Base current main: 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Contribution: 788848032c16bbe0a543c1186c6691c66131c090. Task draw d17af0bfdda8dde69736e5942d5a7563 selected adversarial audit, priority revision 3; TASK records the refinement, bounded proof/checking domain and open next question. Coordination and claim. Final overlap covered 554 open PRs including drafts and 535 Discussion/comment/reply records with complete pagination, plus local notes/ledger. #1859's head remained f5e18c0c44371ebfdd46bfe25efe5b49e467fafe; no other exact overlap was found within the stated search coverage.

Credit Jeremy Kothe / jeremykothe2's #593 for retained semantics, packed equality and literal pairs; commdata2338 for the reused, pinned closed literal parser; woklerx's #1859 for the unitary-premise radix improvement; Sagar Thakrar's distinct #1781 recipe census; and the preserved Lemma A/G and Clifford lineage. No pending implementation is imported.

This is a written all-dimension lemma plus finite source-bound evidence. Valid-gate unitarity and row bounds are explicit premises. A new Lean proof and kernel timing remain the next adoption steps. No selected checker, frozen input or frontier changes; no supplier/cold replay, new kappa, or complete multiplication-theorem claim. Substantial OpenAI Codex assistance is disclosed; no subagents or independent human review. Apache-2.0.

[Mister-Meeseeks]

Batch review at 788848032c16bbe0a543c1186c6691c66131c090.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Written Gaussian energy-ball proof gives sharp uniform integer radixB>=2R and K+1 bits for validated unitary gates. All3072 scaled rows,5265 ball vectors and source/receipt controls pass. Three retained pairs need10 rather than11/20 bits; Lean formalization and production adoption remain open.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.