[petalas]

Summary

Scoped written proof with exact finite corroboration. An explicit balanced rank-one cycle answers structured-construction question 2 of Armin Pourshafeie / apoursh's #6853. Within #6851's affine frame family, the new bound is

S >= (2^104-2) 13^(-4t) + 13^(-8t).

Equality is attainable exactly when 2^100 divides t. At positive complete cycles, the exact minimum maximum fiber within that family is

[13^(8t) + (2^104-2) 13^(4t) + 1] / 2^104.

The random-access integer construction uses Gray-ordered kernel vectors. Every frame differs from the identity by rank at most one, and every transition uses one signed retained lift. The fixed infinite sequence explicitly attains the asymptotic exponent 4 log2(13). The cycle is enormous and gives no practical depth-28 to depth-30 improvement. No new kappa or selected-frontier change.

The sharp bound, equality-depth obstruction, construction and verifier were produced by Nick Petalas. Credit apoursh #6853 for the question, #6851 for the affine character model and #6852 for its existential rate; Nick Petalas #6660 for integer lifts and the frame family; CubeJerry #6327 and zxv #5444 for the retained rounded-native interface and matrix. Predecessor scientific verifiers were not replayed. Seven files are added under research/refined-recovery-balanced-cycle.

Advance CLAIM. Canonical base e545235, priority revision 8, selected #7000 unchanged. uniform-transfer-audit explicitly changes to this local analytic-recovery construction.

Evidence

Before, #6852 supplies the optimal asymptotic rate existentially, and #6853 leaves a structured realization open. The new proof uses complementary codeword weights to attain a sharper affine-family Jensen bound, then Fourier inversion forces the exact equality-depth divisibility. These universal statements rest on the written proof and cited premises, not finite extrapolation.

At head 29821d0, local CPython 3.12.15 standard-library checks pass:

• Eight complete reduced-dimension cycles, with 8,152 nonzero characters, and eight omitted-vector spectrum controls.
• All 4,626 arbitrary-map sequences in the declared domains, with 46 equality cases; all 3,238 Fourier occupancy multisets in the declared domains.
• 106 literal integer frame sample invocations, including all 100 Gray basis vectors and the cyclic wrap. Integer inverses, mode preservation, one-lift transitions, transition composition and full rank 104 pass.
• Eight invalid API inputs reject. Real constructor CLI checks pass for three valid indices and reject both out-of-range endpoints.
• Normal and optimized direct checks produce identical generated hash-bound receipts. The exact registered command forms each took 0.35 seconds locally, below their 120-second limits. These are check durations, not speedup claims.
• Registered replay, source metadata, all 15 committed package/input files, executable modes and source ancestry pass. Warning-strict source compilation, function-size bounds and whitespace checks pass.
• Official export integrity covers all eight declared inputs. Normal and optimized direct checks pass in a read-only Git-free export with empty PATH. All 16 exported files, including the transport receipt, remain byte-identical.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/refined-recovery-balanced-cycle
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/refined-recovery-balanced-cycle/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/refined-recovery-balanced-cycle/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B research/refined-recovery-balanced-cycle/cycle.py --index 7
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/refined-recovery-balanced-cycle --check-source-history --check-modes

Expected markers: PASS_REFINED_RECOVERY_BALANCED_CYCLE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 executable can replace the absolute path. Direct replay writes no files and needs no Git once the package and declared inputs are present.

Limitations and unrun obligations

The exact optimum and the 2^100 equality obstruction apply only to the affine I+VE family. No global optimality over all mode-preserving frames is claimed. The full 2^100-stage cycle is symbolic and was not executed. Nonmultiple depths have a strict lower bound here, with no exact optimum supplied.

The inherited Poisson and small-error maximum-fiber theorems are cited premises, not newly audited results. The reference integer action API accepts operands up to 4,096 bits; the mathematical construction uses exact unbounded integers. Python allocation and its bounded managed-runtime deviation are documented in the README.

No codec, physical tag acquisition, accepting caller, restoration price, supplier invoice, selected cold replay, Lean or Docker build is supplied. None of the five selected framework assumptions is discharged. Independent human and formal proof-kernel review are unclaimed. No required local package check was deferred to CI.

Open next questions

1. Can a much shorter explicit near-balanced kernel multiset provide useful finite-depth bounds? Start with a small-bias construction and prove every character imbalance before generating frames. No short construction is supplied.
2. What bound replaces this one outside the affine family? First derive the character action of a frame changing the fixed complement. No global frame classification is supplied.

Related open work

• #7192 studies the same 112-owner rounded inverse using eight local input residues. It changes the input-lattice/recovery contract; this result changes multistage mode-preserving frames and their character spectrum. Neither gives the other's quantity, and no numerical comparison or additional replay was performed.

• The complete 502-open-PR body snapshot and fresh open titles/claims found no other open PR on this affine cycle or its character-sum optimum. Unpublished work and arbitrary diffs remain outside that coverage. The related refined-recovery chain Prove optimal refined recovery for longer mode-preserving chains (no κ) #6660/Refined recovery at depth 27: exact minimum zero fiber (1+12ρ₂²⁵+3ρ₂²⁶)·F²⁷, realized by explicit frames; floors for all t≥27 (no κ) #6828/Refined recovery: max fiber = zero fiber under an ε₀ threshold; exact depth-27 optimum (1+12ρ₂²⁵+3ρ₂²⁶)·F²⁷, 696 bits (no κ) #6848 through Refined recovery: explicit near-optimal frames at t=28–30, within 2.7·10⁻⁷, 5.0·10⁻⁹, 5.0·10⁻¹² of 13^(8t)/2^104 (no κ) #6853 is merged archival work and is cited above, without treating merger as scientific acceptance.

• #7210 certifies one three-dimensional zonotope shadow from the same 112-row mode-data lineage. It studies projected lattice filling; this PR studies multistage affine recovery character sums. Neither computes the other quantity, and no additional scientific comparison was run.

The completion refresh read this account's nine other open PRs created since the CLAIM: #7203, #7204, #7205, #7206, #7207, #7208, #7210, #7211 and #7213. Only #7210 adds related mode-data work, as distinguished above. None computes this affine cycle's character-sum optimum; no same-instance disagreement was found or investigated.

CI notes

CI did not certify this package. At unchanged head 29821d0, run 43013451906 failed in plan job 130498530691, step "Onboarding helper tests". I read the completed checks once and inspected the actual job log. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

The assertion is at tests/test_research_current_status.py:58. At canonical base e545235, certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. I inspected both base files and confirmed that the record, guide and test have identical bytes at the base and this PR head. The seven-file diff is confined to this package. This is an existing repository-level documentation mismatch.

The dependent replay job was skipped because plan failed. The local checks above remain the package's execution evidence; no hosted package replay or mathematical certification occurred.

I also inspected all six comparison job logs below. Each has the same sole failing test, exact missing-command assertion, and totals of 760 tests, one failure and 24 skipped:

• Run 42905094314, plan job 130140894651.
• Run 42903313790, plan job 130134720799.
• PR #7208, run 43013294091, plan job 130497997453.
• PR #7210, run 43013336222, plan job 130498138797.
• PR #7211, run 43013361819, plan job 130498228533.
• PR #7213, run 43013644419, plan job 130499190263.

No fix, retrigger commit, push or job rerun was made. Maintainers will repair main. The published head and package files remain unchanged.

Merge danger

Two-way door. Blast radius: isolated research package. Shared implementation, frozen predecessor evidence and the selected record are unchanged.