[petalas]

Summary

- Gamma_r is a sufficient translation lattice for the integer lifting word.
+ Gamma_r is the full constant-translation group at every rank.
+ The rank-seven 343-bit domain is optimal for exact translation-coset splits.

Close PR3035 open question 2. Subtracting the last pair's outputs cancels rounding exactly and forces each independently varied lower-half increment to be constant. Induction and pair rigidity then force every prefix parity condition. The full inverse group is exactly U_r Gamma_r.

Proof, exact sources, evidence and reproduction. Source PR3035 is pinned at 0cc430349a3f3dc200666bc22e997076e0a557ed; canonical main is d8e0f83f00191df62174b641e9fff055c357f641, priority revision 6. Canonical investigation.

The attempted smaller residue domain did not improve: the theorem excludes that route for this fixed word. This is a new scoped necessity/optimality result, not a circuit-size or runtime lower bound. No supplier, new kappa or selected-frontier change is claimed.

Evidence

• Before: PR3035 proves sufficiency and computes the index, while leaving cancellation between stages open. After: a written all-rank induction proves necessity, including rank zero, inverse and Gaussian monomial wrappers.
• A separately implemented model exhausts all 16/65,536 shifts at ranks one/two. Three explicit base points reject all 64,512 rank-two nonperiods; an origin-only test would retain 9,216 extra false candidates. Signed pair and inverse controls pass.
• The byte-identical predecessor modular checker replays the seven index exponents through 343. This is source-bound reuse, not a newly independent index algorithm.
• Registered CPython 3.12.15 replay, seven behavioral test groups, normal/optimized read-only Git-free exports, warnings-as-errors, Ruff and all 13 committed-byte checks pass locally. Separate internal proof and implementation reviews found no substantive issue. No independent human or formal-kernel review is claimed. Hosted CI is separate.

uv run --python 3.12 --no-project python -B scripts/research_package.py \
  --package research/p104-maximal-translation-lattice

Credit Tekalign/PR3025/3033 for the fixed lifting word and scalar split, Nick Petalas/PR3035 for the sufficient lattice and index computation, ywz65 and Arham Tahir for the native/occurrence lineage, and the retained P104 contributors. Substantial automated assistance and review limits are recorded in TASK.json. Apache-2.0.

Next useful work is a complete reducer/consumer cost bill or a specified changed word/noncoset contract. The theorem does not price dirty restoration, physical storage, the exact wide backend, precision/tape or full supplier admission.

Merge Danger

Door: two-way

Blast Radius: isolated

One additive opt-in research package. Frozen predecessors, shared checkers, priorities and the selected result are unchanged.