[petalas]

Summary

Prove the exact dyadic Gaussian inverse-input lattice for both original and phase-repaired prefixes in Cinna Davis's #4998. Claim type: scoped all-width proof with finite corroboration.

For D = Z[i,1/2], n = 8^f, a = (4^f+2^f)/2 and b = (8^f-4^f)/4:

coker_D G_f = (D/(3))^a direct-sum (D/(15))^b
safe inverse-input fraction = 9^(-a-b) * 25^(-b)
least odd denominator for unrestricted exact inputs = 15

The safe fraction is exactly 1/164025 at width one. The source already supplies an inverse with denominator 15. This adds necessity, the full quotient and explicit framed input congruences. It answers the exact-grid feasibility part of #4998 Open next question 2. No new kappa, paid-call improvement or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Credit Cinna Davis #4998/#4991, lee-phys #4688, Yiyang #4710, Dugongue #4743, Jacob Alan Sussman #4628, eumemic #288 and Swapnil Jain P1/P2. Classical cyclic-module presentation tools retain their ordinary attribution. Predecessor programs are neither imported nor audited.

Evidence

Before, denominator 15 is sufficient and the exact-grid realization remains open. After, the written module proof gives the complete safe lattice and proves both odd primes unavoidable on unrestricted literal inputs.

• Enumerated cycles at widths 1 through 5.
• Exhausted 50,865 real cycle residues. Safe counts are 5, 75 and 3375 for lengths 1, 2 and 4.
• Four literal Gaussian realifications have ranks modulo (3,5) of (8,14) at width one and (84,104) at width two, for both variants.
• Independent rational inversion checks the sharp denominator, 128 input-membership probes and 32 forward-image columns. Unweighted congruences and single-prime replacements have counterexamples.
• Warning-strict normal and optimized direct replays pass locally in 0.459 s and 0.447 s. These are check durations, not speed comparisons.
• Registered replay and committed-byte/source-history checks pass at 06611c0. The verifier prints fresh package-file hashes and results; no receipt is hand-written.

From the worktree root, CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-image-inverse-lattice/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/two-image-inverse-lattice/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-image-inverse-lattice
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-image-inverse-lattice --check-source-history

Expected markers are PASS_TWO_IMAGE_INVERSE_LATTICE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Any CPython 3.12 path can replace the local interpreter. The direct verifier requires no Git or predecessor files.

Limitations

The all-width claim rests on the written proof. Finite enumeration corroborates it. The obstruction concerns literal D-valued outputs. Rational encodings, odd-denominator arithmetic, rounded outputs and deliberately safe-lattice inputs remain possible. Complete-period density is not an assertion about an actual caller's distribution. Finite powers-of-two precision, physical preparation/storage/routing, Gaussian consumers and supplier admission remain unpriced.

Every registered command ran locally. Lean, Docker and the selected scientific cold replay are unrun. Independent human and formal proof-kernel review are unclaimed. No complete multiplication theorem follows.

Open next questions

• Bind one actual independent-input producer to the safe-lattice congruences and charge its complete preparation/restoration bill. No favorable producer is supplied.
• Implement one rationally encoded or rounded invocation with an explicit return contract and complete numerical/storage/routing costs.

Related open work

• Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 develops the small-unit engine and its lower bounds, leaving five-in-fourteen open. This arithmetic boundary for a derived inverse partner does not settle that construction or compare call-count bounds.

The final scan covered all 331 open PR bodies and found no open PR studying this exact inverse lattice. Main-tree formula searches, merged work and live claims were checked again before committing. Sparse local notes, arbitrary unseen diffs and unpublished work limit overlap coverage. Canonical advance claim.

CI notes

CI pending for 06611c0. Both registered commands and the source-history preflight passed locally. Hosted results will be recorded here after they finish.

The card's historical first control fails on the pre-existing selected-schema mismatch KeyError: 'complex_profile'. The current selected-baseline metadata control passes. No historical checker or selected source was changed.

Merge Danger

Door: two-way

Blast Radius: package

Only the six files in research/two-image-inverse-lattice/ are added.