[petalas]

Summary

Prove the exact minimum peak clean scratch for a five-node reversible dependency path with arbitrary positive real widths:

C(w) = w5 + max(w1+w2, w3+w4, w2+w3+min(w1,w4)).

The endpoint starts empty and retains only node 5. Arbitrary finite recomputation is allowed. A suffix argument gives the lower bound, and one of two explicit schedules attains it for every width tuple. This is a scoped mathematical proof with finite corroboration, not a new kappa or supplier admission.

Credit Ali Romman #5780 for the declared pure compute/erase model and scratch-budget application question, and classical reversible pebbling for the game. The predecessor supplies a bounded per-instance planner; this adds a width-parametric theorem without importing or replaying predecessor code. The formula, proof and finite corroboration were produced by Nick Petalas.

Advance CLAIM. Base de5ec2e, priority revision 8. The selected-stock card's changed scope is recorded explicitly as an analytical side result.

Evidence

Before, the cited interface searches a supplied numerical instance. After, the five-width capacity and an attaining schedule follow directly from the formula.

Local CPython 3.12.15 checks pass:

• All 16,807 tuples in {1,...,7}^5 agree with a complete 32-state bottleneck oracle.
• Literal set-based replay attains the bound; separate capacity-bounded reachability excludes every strictly smaller capacity.
• Three separating-state checks corroborate the universal proof.
• Nine invalid width and seven invalid schedule controls reject. Visiting node 5 requires three unit pebbles, while clean return requires four.
• Both weighted schedule branches and the 10^12 width limit pass.
• Warning-strict normal and optimized commands pass, timed individually at 0.504 and 0.505 seconds, below their 120-second limits.
• Registered replay, committed-byte/source-history validation, official export integrity, and an optimized read-only replay without Git all pass. The export receipt equals the local receipt and binds all five package files by SHA256.
• Source compilation, function-length bounds and diff whitespace checks pass. Ruff is not installed in the recorded environment.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/weighted-five-node-chain/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/weighted-five-node-chain/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/weighted-five-node-chain
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/weighted-five-node-chain --check-source-history

Expected result: PASS_WEIGHTED_FIVE_NODE_CHAIN with width_tuples 16807, followed by PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES for the corresponding commands. The direct verifier needs only the standard library and works without Git. Its integer interface is bounded to widths 1..10^12; the written proof covers all positive real widths.

Limitations

This optimizes peak stored clean node data in the declared atomic pebble model. It does not minimize operation charges or cover arbitrary productive dirty helpers. External inputs and the internal workspace of compute/erase routines are outside this scratch count. An actual source DAG, native legality, dirty restoration, precision, routes and complete caller charges remain open. All five framework assumptions remain.

Independent human and formal proof-kernel review are unclaimed. No Lean, native replay, selected-supplier cold replay or complete multiplication theorem was attempted.

Open next questions

• Bind an actual five-node pure supplier chain, its widths and legal compute/erase operations, then price every move of the attaining schedule.
• Minimize unequal compute/erase charges at the proved optimal scratch capacity on the same 32-state graph. Neither supplied schedule is claimed to minimize that charge.

Related open work

• Construct optimal reversible scratch schedules with replay certificates #5780 constructs minimum-cost reversible pebble schedules for supplied numerical DAGs and scratch budgets; this gives an exact universal space threshold for one five-node topology, without checking its algorithm or fixtures.
• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 computes clean information capacity for a destructive Gaussian overwrite; it studies a different state/operation contract and does not compute this pure-DAG pebble threshold.
• Reduce bounded NTT temporary storage from 17,920 to 1,282 (no new κ) #5886 reduces a bounded NTT program's temporary arena under fixed closed lifetimes and unchanged instructions; this permits recomputation in an abstract reversible path and computes no NTT storage quantity.
• Exclude scratch-reading two-FULL completions (no new κ) #5891 bounds projected rank when two FULL shears may read dirty scratch; this studies clean pure-node storage and supplies no two-FULL completion.
• Prove sharp migration tariffs for two reusable resources (no new κ) #5893 prices migrations between two reusable resources with fixed consumer lifetimes; this optimizes peak stored node width under compute/erase moves, a different capacity and operation contract.

Completion refresh checked the titles and bodies of this account's eight other open PRs created since the CLAIM at 2026-10-27T11:19:38Z: #5886, #5887, #5888, #5889, #5890, #5891, #5892 and #5893. The other five concern power-invoice envelopes, transported-coframe optimization, dyadic feasibility, Gaussian denominator debt and observed coupled bills. None computes the same five-node scratch threshold or supplies a common numerical fixture. No extra scientific computations were run to validate or compare those contributions.

The advance searches of all main package READMEs and merged PRs used the topology, weighted scratch terms and formula shape. The sparse overlap report is incomplete, so no exhaustive novelty claim is made.

CI notes

At head 9dca6f3bc26bced432de014f84c0d4790a0d121e, run 40496362171 finished with plan failed; replay skipped. The only failed step is Onboarding helper tests. All ten failure/error headers are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22) and exit code 1.

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen-card transformation, missing current/historical references and sparse omission lists: 0 != 16, ValueError not raised, and empty omission lists where source paths are expected. All ten failure/error headers and exception/assertion messages match PR #5893's plan log. This confirms the same shared onboarding failure on base de5ec2e7a6074068452f45b1a191b5c72832df4a.

Replay was skipped. Hosted replay has not certified this package; only the local scientific checks listed above have run. The package and commit are unchanged. No shared source or tests were modified or rerun; maintainers will handle the shared repair.

Merge Danger

Door: two-way

Blast Radius: package

Adds only research/weighted-five-node-chain. No shared verifier, frozen predecessor, selected result or frontier is changed. New material is Apache-2.0.