Research: higher-arity dirty echoes use 152 calls at five-array cap (no new kappa)

[Koroffstrogov]

Change and mathematical scope

Research construction / recursive-interface refinement; no new kappa.

PR689's improved balanced ternary dirty echo and concurrent PR730's unbalanced binary/ternary optimizer both need 160 complete leaf calls for 15 ordered factors and five borrowed arrays. Admitting higher arities reduces that to 152 calls with the same five arbitrary dirty arrays; the winning word uses a four-child node. An exact interval dynamic program minimizes over all arities and contiguous partitions in the explicitly defined sequential multiway echo-tree class, including unequal positive integer leaf prices. All inverse/restoration calls are included; factors are not reordered.

For the pinned radix-16 factors on a length-five cycle, a phase-read-weighted tree retains 152 calls and reduces the complete phase-access bill 3840 -> 3656. These are exact operation counts, not wall-time claims. The package supplies literal trees, a written optimality/restoration proof, complete execution/preparation/precision accounting, and a deterministic certificate.

Only research/space-capped-dirty-echo/ is added. The selected #626/#543 result remains 51110230636731/31250000000000000. No selected supplier, pointer, generator, review ledger or upstream/ file changes. PR673's complete all-role adapter still fails its optimistic bill: the primary tree gives moment 76/5 at saving zero. This does not reopen that negative result or claim a new multiplication exponent.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60.
Dependencies: PR628 0768f97d06c623b8211e651543c66135796b62bf; PR673 bfff94e78f0e75ebd2593e3cc7710ae5290e8be3; PR689 0dedef12f05f9a06faf9d4e1c8b1462c0e3bff3a. All consumed bytes and notices are included and source-pinned. The latest inspected PR673 head 156f259b99723468cfdd2b5d47d0ba894f09bea7 leaves all three consumed files unchanged.

Assignment seed 58, from one OS-random draw, recursive interfaces. Initial overlap review covered 528 open PRs through #708; a full refresh covered 544 through #725, followed by publication refresh through #745 and a last-arrival check through #757 immediately before submission. No pinned coordination issue existed, so the claim and completed result are local. Concurrent #714 requires one clean root array. #730 and #744 arrived during final verification and are explicitly credited; no priority for unbalanced or weighted optimization is claimed. #730 admits only arities 2/3; #744 audits a strict ternary class. The originally claimed all-arity mechanism gives a strict further gain, confirmed by running #730's actual pinned optimizer and words. This also adds a real weighted radix-16 fixture and numerical leaf replay. #744's distinct larger ternary census remains credited.

Validation

python3 -I -B research/space-capped-dirty-echo/verify.py
python3 -I -O -B research/space-capped-dirty-echo/test_contracts.py
python3 -I -O -B research/space-capped-dirty-echo/compare_concurrent.py

Expected: PASS_SPACE_CAPPED_DIRTY_ECHO calls=160->152 scratch=5 no_new_kappa, nine tests OK, then PASS_HIGHER_ARITY_COMPARISON.

The pinned PR730 comparison (d222e348179cc02a54d6d26dbd2838944f62c6e5) gives 160 -> 152 at 15/5 factors/arrays, 144 -> 128 at 15/6, 464 -> 448 at 27/6 and 1728 -> 1600 at 58/8. Its actual weighted primary optimum is 3840 phase reads versus 3656 here. Two exact source files and their notices are included under a separate comparison manifest. This is scoped optimizer/word verification, not approval of the whole PR.

• All 5440 whole ordered trees through eight leaves, with 144 weight/cap comparisons against a separate exhaustive oracle.
• Primary 15-factor optima for caps 4–14, plus the weighted instance, checked by a separate all-root-cut implementation. The optimum row is 224,152,128,112,100,92,84,76,68,60,56.
• 63 symbolic word checks, 48 generic rounded compositions, and eight radix-16 compositions; noncommutative factor order, arbitrary dirty restoration, dense rational references, actual phase reads, and exact full inverses.
• 326 rejected circuit/admission controls, five detected implementation mutations, source/context and certificate corruption rejection.
• Full isolated optimized replay and a minimal relocated replay pass on Windows / CPython 3.12.10. All 18 source/context files match pinned Git bytes. Selected-record integrity, source syntax, local links and staged whitespace pass.

VALIDATION.md records the corrected tuple/JSON comparison defect and initial sandbox temporary-directory failure separately from the final successful checks. The complete certificate is compared field-for-field after representation normalization.

No repository-wide suite, supplier cold replay, remote/Linux CI or Lean pass is claimed. The tree minimum excludes arbitrary reversible programs and alternative workspace-sharing rules. All-size compilation, physical hosting, common-label/mutable-basis realization, fixed-tape implementation, analytic preparation/transfer and exact recovery remain inherited or unresolved. Finite tests are not a formal proof of the multiplication theorem.

Prepared for Koroffstrogov with substantial OpenAI Codex assistance, under Apache-2.0. Credit to androolloyd for the echo and all-role obstruction, Dillon Amburgey for the balanced comparator, Sagar Thakrar for concurrent #730, Ali Romman for concurrent #744, Chafik Boukhalfa for the rounded leaf, 0xkilgore for the earlier restoration audit, FanPUYen for the distinct clean-checkpoint work, and the complete retained source/selected-construction lineage. No independent human peer review or literature-priority claim.

[Mister-Meeseeks]

100-PR batch review at a164f06263e438ead570b169e5b7103aa45bf228.

This remains open after scoped triage. Next review step: Primary tree/word checks passed before the 180-second budget. Finish remaining rounded/adversarial checks and compare the final receipt; no mathematical failure concluded.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.