[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. A new full-lattice, subset-sum-plus-target support has smallest faithful cyclic modulus six instead of the mixed-radix box bound nine. Modulus five fails for every linear projection although every pair can be separated individually there.

S = {(0,0),(1,0),(0,1),(1,1),(2,2)}
box construction: q=9
exact optimum: q=6, weights=(1,3), residues=(0,1,3,4,2)

For two-point supports in a fixed ambient integer lattice, the exact minimum is the least q>=2 not dividing the gcd h of the coordinate differences. Its values are exactly the prime powers and are unbounded even at support size two. The sharp height-H worst case is the first m with lcm(1,...,m)>H. This is a statement about integer weights on supplied coordinates; rebasing the generated subgroup can remove the two-point obstruction. The five-point example generates all of Z^2 and retains its optimum under a change of group basis.

This answers the short-quotient question in woklerx's merged PR #1440. Credit that source for the mixed-radix injection and faithful regular-action criterion, with its eumemic #288 / huxint #203 lineage. Classical gcd, LCM and representation facts are not claimed as globally new. The sparse-support analysis, new fixture, proof and checker were produced by Nick Petalas. No new kappa or selected-frontier change.

Evidence

Before, the predecessor provides a box-size sufficient quotient and asks for smaller exact quotients. After, this package gives an exact optimum on a new support and sharp arithmetic and simultaneous-projection obstructions. It imports no predecessor verifier and rechecks none of its fixtures or counts.

Local CPython 3.12.15 normal and optimized checks pass:

• 4,096 scalar heights and all 27 prime-power realizations through 64.
• 1,176 nonzero vector/modulus cases, enumerating 34,104 projections.
• All 91 fixture projections modulo 1 through 6 and all 32 binary support polynomials, with 14,112 literal regular-action column comparisons. Every unfaithful projection has a constructed nonzero two-term kernel witness.
• All ten pairwise separation cases modulo five, four unsafe-projection rejection controls and three semantic controls.
• Both registered commands, each locally timed below 0.1 seconds against its 120-second timeout. Package metadata, staged whitespace and committed-byte/source-history preflight pass at 5fa073e60837903299f4ab45007bf3286c1a9071.
• Official committed export integrity and an optimized read-only replay with Git absent from PATH pass at that head. All seven exported files, including the manifest, remain byte-for-byte unchanged.

The verifier generates a stdout receipt bound to SHA256 hashes of all six package files. It writes nothing. From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/sparse-cyclic-quotient-obstructions/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/sparse-cyclic-quotient-obstructions/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/sparse-cyclic-quotient-obstructions
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/sparse-cyclic-quotient-obstructions --check-source-history

Another CPython 3.12 path can replace the absolute interpreter path. Standard library only. Expected markers include PASS_SPARSE_CYCLIC_QUOTIENT_OBSTRUCTIONS no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The written argument establishes the universal claims; the finite enumeration corroborates the implementation.

Limitations and unrun obligations

This changes the task-card scope to a finite-certificate representation side result. It does not satisfy the complete decoder/supplier gate. The optimum concerns homomorphisms from the specified integer lattice. The fixed-lattice two-point obstruction is not an obstruction to first changing the domain to its generated subgroup.

No actual larger-register word, legal physical address primitive, full paid/target invoice, native supplier, all-size compiler, precision or tape/stock bound is supplied. A smaller quotient is not a measured runtime gain. Selected cold replay, a complete multiplication theorem, independent human review and proof-kernel verification were not run or claimed.

Open next questions

Compress the full support of one actual larger-register candidate below its box bound. First retain every subset sum and target exponent, test arithmetic obstructions, then certify a simultaneous injective weight vector. A named candidate and a representation-cost allowance remain missing; pairwise separation alone is insufficient.

Related open work

No open PR studying sparse faithful cyclic quotient minima was found in the live targeted query. The cached complete overlap snapshot covered 459 open PRs and 5,402 Discussion records; sparse local text coverage was incomplete, so distinctive terms were also searched directly across main's research READMEs. PR #1440 is the merged question source. The recent fixed-alphabet query, carry and coframe claims concern other quantities; no shared numerical optimum is asserted.

The advance CLAIM states this scope and predecessor. The delivered result matches it.

The completion refresh inspected this account's nine other open PRs created since the advance CLAIM: #5909 through #5913 and #5915 through #5918. They concern target-partition fees, sector workspace, colored batching, rounded-shear history, serialized carries, carry frequency, tensor phase recovery, Boolean tree degree and clockwise XOR synthesis. None studies faithful cyclic quotient minima or computes a conflicting same-instance value. No extra scientific computations were run to check those contributions.

CI notes

Hosted CI did not certify this package. At commit 5fa073e60837903299f4ab45007bf3286c1a9071, run 40506411167 finished with plan failed; replay skipped.

The only failed step was Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures are the frozen-card count 0 != 16, missing-helper/reference ValueError expectations, and empty current/historical sparse-reference lists. These concern shared onboarding behavior on main de5ec2e7a6074068452f45b1a191b5c72832df4a, outside this six-file package.

The complete ten failure/error headers match PR #5915's plan log, checked directly. This confirms the same shared failure on another recent PR; no package failure is reported by this run. No shared tests or implementation were changed or rerun to address it. Maintainer repair remains outstanding.

The replay job was skipped. Hosted replay has not certified the package; only local scientific checks have run. Check durations above are timeout-budget observations, not speedup measurements.

Merge Danger

Two-way door. Blast radius: isolated. Only one six-file research package is added; shared code, frozen sources and the selected result are unchanged.