[petalas]

Summary

Scoped all-size proof with exact finite corroboration: the reserved-scratch dirty-echo lower bound is attained exactly on the following maximal interval.

- sufficient: 3*2^(s-2) <= n <= 2^s, s>=2
+ exact: ceil(2^(s+1)/3) <= n <= 2^s, s>=0
  min N = 2^(s+1)n - 4^s

The exact slack identity also characterizes every equality tree: binary nodes have tight children; ternary nodes have tight children and capacity-saturating endpoints; arity four or higher has positive slack. At s=4 the theorem includes n=11 and 96 completed calls. PR784 already tabulates that finite optimum; this contribution is the general maximality and shape theorem.

This contribution is produced by Nick Petalas. Credit Alexander Schwartzman #784 for the lower bound and sufficient family, Gareth Davidson #887 for its scoped formalization, androolloyd #673 for the echo identity, Koroffstrogov #758, Sagar Thakrar #730, Dillon Amburgey #689 and the retained Chafik Boukhalfa #628 lineage. No predecessor implementation is copied or imported. Canonical CLAIM.

Evidence

• Before: Research theorem: sharp scratch-dependent call bound for dirty echoes #784 and [Lean] Prove reserved-scratch echo bounds and equality family (no new kappa) #887 explicitly leave maximality unclaimed.
• After: PROOF.md proves the exact interval and complete recursive equality classification. The independent all-arity recurrence checks 2,304 states with 51,374 actual transitions. Literal enumeration checks 5,440 ordered trees through eight leaves and 48,960 tree/budget pairs. The constructor checks all 175 equality sizes through budget eight.
• Local CPython 3.12.15 checks pass: 22 invalid-input rejections, two structural controls, four actual behavioral mutations, four source corruptions, three receipt corruptions, exclusive output and optimized standalone replay. Mutation rejections bypass receipt comparison and therefore exercise scientific predicates.
• Generated RESULT.json binds the proof and executable file hashes. All nine committed files and the canonical-ancestor source history pass publication preflight. No receipt is handwritten.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/dirty-echo-maximal-equality
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/check_research_commit.py --package research/dirty-echo-maximal-equality --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/dirty-echo-maximal-equality/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/dirty-echo-maximal-equality/test_package.py

Replace the absolute interpreter path with a local CPython 3.12 executable for portable replay. Set TMPDIR to a private external directory for temporary controls. The direct scientific check took 0.15 seconds, optimized controls 2.00 seconds, and the complete registered replay 2.33 seconds locally. Expected outputs include PASS_DIRTY_ECHO_MAXIMAL_EQUALITY, PASS_DIRTY_ECHO_EQUALITY_CONTROLS, PASS_RESEARCH_COMMIT_BYTES and PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

The theorem concerns unit-price completed leaf calls in static ordered trees with parent-reserved dirty scratch. Connector reuse, clean checkpoints, arbitrary reversible programs, weighted prices, physical delivery and complete supplier pricing are outside it. The written proof supplies the universal quantifiers; the finite census does not. The predecessor bound and echo semantics are cited premises, not audit deliverables. No predecessor replay, Lean extension, physical caller or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed. No new kappa or selected-frontier change.

Open next questions

• Count equality trees at each (s,n) using the disjoint binary/ternary shape recurrence.
• Derive a useful analytic unequal-price bound while retaining factor order and every restoration multiplier.

Related open work

• Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 studies a P104 dirty-restored gather/channel and cites the echo lineage. Its paid frame profile is a different quantity; this static-tree theorem neither validates nor reprices that construction.
• Prove exact canonical-checkpoint costs for cyclic output frames (no new κ) #4860 studies canonical checkpoint costs for fixed cyclic-equivariant children in an echo boundary. It changes output-frame restoration prices, while this theorem keeps the original static tree grammar and counts completed unit-price leaves. No competing numerical quantity is compared.
• The pre-DONE refresh inspected all 51 open PRs on this account, including ten created since the CLAIM. None of the other nine studies this reserved-scratch affine equality interval. No related numerical disagreement was found in that coverage, and no additional predecessor computation was run. Search coverage excludes unpublished work and arbitrary diffs.

CI notes

Both hosted checks passed on 1603c6736c1077119a7f9f77801d8eb7ba804262 in run 39864537692: planning in 3m20s and registered package replay in 1m21s. The run is completed/success and its head SHA matches. Registered replay covers the warning-strict scientific verifier and optimized standalone/corruption controls. Both registered commands also passed locally, each below its 120-second manifest timeout. This is automated finite replay evidence; no independent human, formal-kernel or complete multiplication-theorem acceptance is claimed.

Merge Danger

Door: two-way. Blast radius: isolated package. Only research/dirty-echo-maximal-equality/ is added; no shared executable or selected result changes.