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

[thakrarsagar]

An exact interval optimizer reduces the complete dirty-echo leaf bill from 512 to 464 calls at 27 factors / six borrowed arrays, and 2,112 to 1,600 at 56 factors / eight borrowed arrays, compared with PR689's improved balanced ternary schedule. Every inverse and restoration call is included, and factor order is preserved.

Status: recursive-interface research / finite construction refinement. No new κ.

The optimizer admits unbalanced contiguous splits, mixed binary/ternary nodes and positive integer leaf prices at fixed common precision. It proves and implements the minimum over that precise tree class and PR673's reserved-connector allocation discipline. The 27-factor root splits into 9,7,11; this is outside PR689's balanced class. The package also retains the 15-factor / 160-call result as predecessor work.

Only research/ordered-dirty-echo/ is added. It contains the written identity and complete cost/precision recurrence, a small optimizer and streaming word emitter, frozen exact witnesses, source pins, an independent verifier, and a local validation receipt. upstream/ and the selected frontier are unchanged.

python3 -I -O -B research/ordered-dirty-echo/verify.py

Expected: PASS_ORDERED_DIRTY_ECHO: 642 exhaustive cases; 207 ordered trees; 64 pinned predecessor words; 4 witnesses; 4 rejected mutations.

Validation passed on macOS arm64 / Python 3.14.8:

• Independent exhaustive postfix-tree enumeration for all {1,3} price vectors through six factors and every scratch budget from zero through n-1; separate path-based price/scratch accounting.
• Exact noncommutative symbolic products and dirty restoration for every enumerated tree and both signs; four larger witnesses; nonlinear integer restoration and a dirt-dependent-output counterexample.
• Exact complete-word comparison against the unchanged pinned PR673 generator; rejected omitted restore, reversed factor order, undercharged middle child and insufficient scratch.
• Complete certificate replay under ordinary and isolated optimized Python, standalone package relocation, certificate integer/boolean corruption rejection, source-drift rejection, selected-record binding and staged whitespace checks.

The optimum is local to the stated tree/allocation class. The uniform factor counts are interface fixtures, not claimed PR628 prime-power inversion instances; unequal prices are abstract scheduling inputs, not measured supplier costs. No lifetime-aware ancestor reuse, higher-fanout optimum, source-access realization or practical speedup is claimed. The inherited direct all-role adapter still has optimistic moment at least 6-6/n >= 3, so this scheduling change does not establish a new multiplication exponent. Analytic, compiler, physical layout, precision, tape and supplier obligations remain inherited. The full repository suite, selected cold replay, remote CI, Lean and PR628 numerical leaf replay were not run.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. OS assignment seed 56, recursive-interfaces track. No pinned coordination issue existed, so the claim is local. Three candidate tasks, scope, stopping rule and overlap evidence are recorded in CLAIM.md.

The final overlap refresh covered 547 open PRs through #728. Concurrent PR714, head efad8a40094ab453c0525efef0b0eff2a5d8b850, optimizes a root split with one clean checkpoint and balanced binary dirty children. This contribution keeps every borrowed array arbitrary/dirty and changes the entire ordered binary/ternary tree. No PR714 code or result is imported; its different storage contract is credited explicitly.

Credit: androolloyd's PR673 (bfff94e78f0e75ebd2593e3cc7710ae5290e8be3) supplies the echo identity and source; Dillon Amburgey's PR689 (0dedef12f05f9a06faf9d4e1c8b1462c0e3bff3a) supplies the corrected balanced comparison; Chafik Boukhalfa's PR628 supplies the inherited leaf contract; 0xkilgore's PR663 supplies the prior independent binary audit. PR673 advanced to 156f259b99723468cfdd2b5d47d0ba894f09bea7 during this run with separate sharing-cost work; all three consumed source files remain byte-identical. Exact blob/SHA-256 pins are in SOURCES.json.

Prepared for Sagar Thakrar (@thakrarsagar) with substantial OpenAI Codex assistance, including separate proof, independent-checking and adversarial-review workers. Apache-2.0; no independent human review or full formal theorem claim.

[Mister-Meeseeks]

100-PR batch review at d222e348179cc02a54d6d26dbd2838944f62c6e5.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact ordered binary/ternary dirty-echo optimization with reserved connectors and weighted leaves; nonlinear restoration does not imply scratch-independent rounded products.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

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