Research: optimize ordered dirty echoes at fixed scratch (no new kappa)

research/ordered-dirty-echo/CLAIM.md

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+# Ordered dirty echo under a scratch budget
+
+- Track: recursive interfaces; OS-random assignment seed **56** (`secrets.randbelow(100)`).
+- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Claim time: 2026-10-13 04:42 UTC; expected finish: 05:17 UTC.
+- Object: the ordered multiway dirty echo from PR #673 at
+  `bfff94e78f0e75ebd2593e3cc7710ae5290e8be3`.
+- Method: exact interval dynamic programming for ordered binary/ternary
+  composition trees, with arbitrary positive integer leaf costs and a hard
+  peak scratch-array budget. No factor permutation or new leaf operation.
+- Deliverable: written complete cost recurrence, a small reproducible optimizer,
+  exact witnesses, and independent nonlinear restoration/linear product checks.
+- Stop rule: one package; finish experimentation by 05:07 UTC and reserve the
+  remaining ten minutes for verification and handoff.
+
+No pinned coordination issue existed at the initial GitHub check. This local
+claim replaces a public reservation. Open PRs were checked through #709.
+
+Three candidates considered: common Fourier boundary fusion (the existing
+`fourier-boundary-fusion/REPORT.md` already supplies a strong obstruction),
+Gaussian RHS boundary batching (#481 already supplies that construction), and
+scratch-budgeted ordered dirty composition. The third was chosen because
+#663 establishes the binary count and #689 optimizes balanced ternary uniform
+costs, but #689 explicitly excludes unbalanced splits and unequal leaf costs.
+Seed 56 selects the binary/ternary mixed-tree class (2 + 56 mod 3 = 4 is the
+first scratch-budget probe); it is not used as evidence of novelty.
+
+The selected conditional result is
+`51110230636731/31250000000000000 = 0.001635527380375392`, as bound by
+`certificates/selected-result.json` to #626 and its maintainer review.
+This work does not change that record or `upstream/` and does not assert a
+new multiplication exponent.
+
+Prepared for Sagar Thakrar (@thakrarsagar), with substantial OpenAI Codex
+assistance. Apache-2.0. Predecessor attribution and exact source pins will
+accompany the completed result.
+
+## Result update — 2026-10-13 04:57 UTC
+
+Completed early and submitted as [PR #730](https://github.com/CrocSwap/integer-mult-bounds/pull/730).
+The experiment stopped before the allotted cutoff; the final phase was devoted
+to verification, overlap review and packaging. No pinned coordination issue
+existed at the final check, so this is the single local result update.
+
+The optimum within the stated all-dirty reserved-connector tree class is
+464 calls at 27 factors / six borrowed arrays and 1,600 at 56 factors / eight
+borrowed arrays. The improved balanced baseline from #689 uses 512 and 2,112
+respectively. Four exact witnesses, a written optimality proof, complete
+accounting and independent checks are packaged here. All 642 exhaustive
+cost/budget cases, 207 tree shapes, 64 predecessor-word comparisons, four
+mathematical mutations and two external package-corruption controls passed.
+The selected-record check and standalone isolated replay passed. Full scope
+and skipped checks are in README.md and VALIDATION.json; no new κ is claimed.
+
+The final pre-submission refresh covered 547 open PRs through #728. #689
+remained at `0dedef12f05f9a06faf9d4e1c8b1462c0e3bff3a`. #673 advanced to
+`156f259b99723468cfdd2b5d47d0ba894f09bea7` with separate bit-role sharing
+research; all three consumed source files remain unchanged from our pin.
+Concurrent #714 (`efad8a40094ab453c0525efef0b0eff2a5d8b850`) uses one clean
+root checkpoint with balanced binary children. Our borrowed arrays remain
+arbitrary/dirty, and the entire ordered binary/ternary tree is optimized.
+That clean-checkpoint result is a distinct storage contract and is not an
+input to this contribution. The base and current upstream main remained
+`66849e1475c930fd7732b1a0067ffee7e1e95f60` at the publication check.

research/ordered-dirty-echo/NOTICE.md

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+# Attribution and license
+
+Prepared for **Sagar Thakrar (@thakrarsagar)** with substantial **OpenAI Codex
+assistance**, including separate workers for the written proof and independent
+verification. New code and documentation are Apache-2.0 under the repository's
+root [LICENSE](../../LICENSE). No independent human review is claimed.
+
+- **androolloyd**, [PR #673](https://github.com/CrocSwap/integer-mult-bounds/pull/673):
+  the multiway dirty echo, its nonlinear restoration identity, recursive
+  instruction generator, and precision/resource argument. Its full-transform
+  experiment supplies the scoped negative result for a direct all-role adapter.
+  `vendor/pr673_echo.py` retains the exact source, including its AI disclosure.
+  `optimize.py` adapts that emitter to an explicit ordered tree; this package
+  contributes the interval optimization and its independently checked witnesses.
+- **Dillon Amburgey (@dillona)**,
+  [PR #689](https://github.com/CrocSwap/integer-mult-bounds/pull/689): the minimum
+  among recursively balanced ternary splits with uniform leaf costs, and the
+  corrected baseline used here. The result at 15 factors is credited to #689;
+  this package's savings at 27 and 56 factors use a larger, explicitly scoped
+  tree class. No claim of priority over general alphabetic-tree optimization
+  algorithms is made.
+- **Chafik Boukhalfa (@chafreaky)**,
+  [PR #628](https://github.com/CrocSwap/integer-mult-bounds/pull/628), head
+  `0768f97d06c623b8211e651543c66135796b62bf`: the rounded cyclic-inverse leaves,
+  balanced binary dirty commutator, and inherited preparation/precision
+  contracts. Those analytic and numerical implementations are not re-audited
+  or copied here. Their cited mathematical predecessors retain their credit.
+- **0xkilgore**, [PR #663](https://github.com/CrocSwap/integer-mult-bounds/pull/663),
+  head `e0e546fd01b396478c68b6b6c5a16d87515983ba`: independent nonlinear
+  restoration audit and the exact count for the balanced binary dispatcher.
+
+[SOURCES.json](SOURCES.json) records exact source commits, SHA-256 hashes and
+Git blob identifiers. The checker needs no network or pending-PR checkout.
+Only the vendored echo is an executable predecessor dependency. The other
+pins document comparisons, mathematical dependencies, and the selected-result
+context at the base commit.
+
+The accepted #626 construction remains credited to Jeremy Kothe's K-triangle
+construction and the complete supplier/owner-allocation lineage in the
+[maintainer review](../../docs/research/community-k492-review.md). This local
+interface research neither alters that construction nor establishes a new κ.