[petalas]

Summary

Scoped written proof with exact finite corroboration. For the unchanged 112-owner rounded integer inverse K(z)=z-M round_away(M^T z/52), eight fixed coordinate residues modulo 32 determine every integer input from its output. In particular, requiring divisibility by 32 only on owners 5, 33, 49, 53, 62, 65, 97 and 108 gives an injective domain strictly larger than the uniform sufficient lattice of zxv's #5444.

- Sufficient domain: all 112 coordinates divisible by 32, lattice index 2^560
+ Sufficient domain: eight selected coordinates divisible by 32, lattice index 2^40

These indices measure restrictions on the input lattice. They do not measure physical storage savings. A direct decoder needs eight scalar interval selections. An explicit collision in 16 Z^112 proves the common five-bit threshold sharp for these fixed eight owners. The forty-bit local residue channel uses more bits than Ali Romman's optimal thirty-bit packed quotient tag in #5323; the access and computation contracts differ.

Only six files under research/eight-owner-rounded-injectivity/ are added. The certificate, sparse domain and decoder were produced by Nick Petalas. Credit zxv #5444 for the unequal-lattice question, matrix and fiber method; Ali Romman #5323 for same-grid recovery; David Ganz #4929 and Dillon Amburgey #4933 for the native matrix/caller; and Brian / JoyboyBrian #5035 for the rounded kernel. Their scientific verifiers were not replayed and their files remain unchanged.

Advance CLAIM. Canonical base e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8, selected #7000 unchanged. This is a declared change from uniform-transfer-audit to a local lattice interface.

Evidence

Before, #5444 supplies uniform relative-grid recovery and leaves unequal per-owner lattices open. The new certificate has AH=HA=2I for eight actual owner rows. A common output bounds quotient differences by thirteen, and equal coordinate residues force multiples of sixteen, proving injectivity.

Local CPython 3.12.15, standard library only, at 3cb32200b874692a49e02d874b39ad1b4ac1516c:

• Both exact inverse products, separate rational inversion and determinant -16, and integer spanning of the complete source row lattice pass.
• 1,128 literal forward/recovery roundtrips, 5,525 scalar candidates over 417 projected values, the explicit four-bit collision and strict domain witness pass. Eleven invalid/corrupted fixtures reject for their stated reasons; half-tie convention controls also pass.
• Warning-strict normal and optimized direct checks produce identical generated receipts bound to all six package files. Each registered command took under one second locally, well inside its 120-second bound. These are check durations, not speedup measurements.
• Registered replay, metadata/source pins, all nine committed package/input bytes, executable modes and declared source ancestry pass. AST compilation and whitespace checks pass.
• Official committed export and declared-input integrity pass. Normal and optimized direct replays pass in a read-only Git-free export with an empty PATH; all ten exported files, including the transport receipt, remain byte-identical.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/eight-owner-rounded-injectivity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/eight-owner-rounded-injectivity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/eight-owner-rounded-injectivity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/eight-owner-rounded-injectivity --check-source-history --check-modes

Expected markers are PASS_EIGHT_OWNER_INJECTIVITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Sparse checkouts must materialize this package and research/precision-state-frontier first. The three predecessor input files are pinned at the canonical base. No handwritten receipt is included.

Limitations

The all-input statement rests on the written proof; finite probes corroborate it. The reference decoder has a 4096-bit operational argument limit. No global optimum over owner subsets, unequal bit allocations or nonlinear observations is claimed.

This is the original integer map restricted to an input lattice, or supplemented by explicit residue observations. Nonuniform diagonal input scaling need not commute with its linear part, so this does not implement the original physical inverse on unequal output grids. Residue acquisition, reversible scratch, eventual restoration, native/physical realization and a favorable complete invoice remain open. No supplier, new kappa, selected cold replay, complete multiplication theorem, independent human review or formal proof-kernel acceptance is claimed.

Open next questions

1. Can an actual caller supply the sparse lattice or acquire these eight residue observations at a favorable complete cost? First derive that invocation's actual grid and observations; acquisition, restoration and the physical invoice are missing.
2. What is the least total local-residue budget over arbitrary subsets and unequal bit allocations? First study congruence kernels against the quotient-difference box and prove any required collision attainability. No global allocation search is supplied.

Related open work

• Share all paired residue samplers in 14,886 bytes (no κ) #7187 packages paired denominator-13 samplers. It concerns stochastic sampling rather than deterministic input recovery for this /52 inverse; no sampler or stopping-time result is imported here.
• A complete 490-open-PR metadata snapshot and targeted searches found no other open package studying this fixed sparse input-lattice recovery quantity. Main README searches and relevant merged PR bodies were also checked. Arbitrary diffs and unpublished work remain outside that overlap evidence.

Final own-open-PR check after the CLAIM covered #7189, #7191, #7193, #7194, #7195 and #7196, excluding this PR. They concern a translated reader, polynomial gap insertions, address-twist residues, three-shear COPY floors, cyclotomic precision and coupled Jordan blocks. None studies this fixed sparse input-lattice recovery quantity; no additional computations or numerical comparisons were made.

CI notes

CI did not certify this package. At unchanged head 3cb32200b874692a49e02d874b39ad1b4ac1516c, run 43000790178, plan job 130456104520, failed in Onboarding helper tests. I read the checks once and inspected the failed log. It reports 760 tests, exactly one failure and 24 skips. The failing test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with AssertionError: '`make frontier7000-verify`' not found in the CI verification document. The dependent replay job was skipped, so no hosted package verification ran.

The mismatch is already present on canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both files and the failing test are unchanged by this six-file package contribution. I confirmed their base contents and the exact diff rather than rerunning the repository job.

I also inspected the failed logs of run 42905094314, run 42903313790, #7189 run 43000211746 and #7191 run 43000676071. Each reports the same exact test and missing-command assertion, with one failure among 760 tests and 24 skips. No fix, retrigger commit or CI rerun was made. The local verification above remains the only executed evidence for this package.

Merge danger

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