Research: minimize balanced ternary dirty echo calls at fixed scratch

[dillona]

At fanout three, PR673 sometimes places a larger balanced block in the middle, where it executes four times. This construction refinement and independent verification puts larger blocks at the twice-called ends. For 15 factors, the complete count falls from 208 to 160 calls with the same 5 borrowed arrays.

The note proves the minimum among recursively balanced ternary splits with uniform leaf-call costs. It preserves PR673's echo identity, asymptotic exponent, factor order and scratch requirement. It makes no optimality claim for other fanouts, unbalanced splits or unequal leaf costs, and claims no new κ.

The package independently audits the actual pinned PR673 generator using free nonlinear function terms and a noncommutative linear module. Changing only its partition hook in memory yields exactly the new forward/inverse words. Rounded checks retain PR628's byte-exact leaves, factor fixtures and inherited precision assumptions; the note charges helpers, source/phase access, restoration, preparation and routing obligations. The selected frontier and upstream/ are unchanged.

python3 -I -O -B research/ternary-dirty-composition/verify.py

Expected: PASS_TERNARY_DIRTY_COMPOSITION. See the proof and scope and exact certificate.

Validation:

• 67 refined symbolic words; 69 actual PR673 words with partition-only equivalence; 10,000 resource cases and all 23,328 balanced size placements in that domain.
• 110 rounded generic compositions, 28 certified-kernel compositions, 621 precision-budget cases and 141 rejecting corruption/interface controls.
• Full isolated optimized replay on Python 3.11.12; relocated standalone replay on Python 3.12.10; corrupted vendored source and expected certificate both rejected.
• All 12 vendor files compared to pinned Git bytes; make selected-record-check, Python syntax, local documentation links and staged whitespace checks passed. Two Codex specialist panels were clean across mathematics, implementation/simplification and maintenance.

Focused Linux CI passed on Python 3.11 and 3.14 at the submitted head 0dedef12f05f9a06faf9d4e1c8b1462c0e3bff3a. The broader inherited CI is separate and is not claimed complete.

Full supplier/repository-wide replay was not run locally for this isolated addition. The finite checks support the stated local identities and counts; the all-n result rests on the written induction, not a formal verification of the multiplication theorem.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Dependencies: PR628 0768f97d06c623b8211e651543c66135796b62bf and PR673 0e35f56cab3cae1eb0a99447135569d687c5695a; PR663 e0e546fd01b396478c68b6b6c5a16d87515983ba supplies the credited secondary binary baseline. Exact file pins and preserved source notices are included.

Assignment seed 58, recursive interfaces. The overlap check covered 506 open PRs through #686, followed by an immediate review of #687–#688, with no pinned coordination issue. PR673 is explicitly credited for the general multiway construction; this submission adds the balanced ternary placement minimum and independent symbolic evidence. PR679 concerns selector-volume batching and has a different scope.

Prepared by Dillon Amburgey with substantial OpenAI Codex assistance, under Apache-2.0. Credit to androolloyd, Chafik Boukhalfa / chafreaky, 0xkilgore, and the retained source lineage is recorded in NOTICE.md.

[Mister-Meeseeks]

Integrated at ee055dc44815497927ef618fc985b93c2dcbda2a in the 20-PR pilot, retaining your original commit, source notices and contributor credit.

Balanced ternary echo placement: 208→160 calls at 15 factors, same five arrays. Eight-call identity, monotone recurrence/exchange proof and nonlinear restoration replayed. Only pinned fanout-three dependency from #673 is used.

The scoped package check passed at 0dedef12f05f9a06faf9d4e1c8b1462c0e3bff3a. This is research integration; the selected κ and inherited theorem assumptions are unchanged. Review, evidence and attribution. Thank you for the contribution.