[petalas]

Summary

Scoped optimality proof with a finite integer certificate. With nineteen native leaves, PR6348's four-field merge and its three-query root, topology alone cannot improve the complete recursive target-minus-paid power sum for any H(r)=r^s, 0<s<1. The 4,7,8 forest attains the optimum, with the unique optimal internal-node size histogram:

size          2  3  4  7  8
multiplicity  9  1  4  1  1

The proof covers every positive three-part root partition and every full binary topology. Fifteen exact hinge certificates prove the stronger statement for every nonincreasing convex node reward. The power-tariff consequence follows by differentiation and a finite-difference decomposition. This adds a topology obstruction to the predecessor's construction; its module identity and native realization are cited premises, not audit deliverables. No new kappa or selected-frontier change is claimed.

The contribution was produced by Nick Petalas. Credit Ali Romman for PR6348's merge, complement, caller, invoice and attaining forest; Cinna Davis for the thin-query/path interface; DreamingOfClouds for the preceding native word; and the retained P104/helper/native/compiler/outer lineage. Dynamic programming and finite differences are classical. Predecessor bytes and credits are unchanged.

Advance claim. Canonical base 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, refreshed before publication. The scoped overlap search included current main, merged PRs, open PR bodies, recent claims and the cached Discussion corpus. Sparse notes and unpublished work remain coverage limits.

Evidence

Before, PR6348 supplied one tree and explicitly claimed no global caller/tree optimum. After, a certificate checker separate from the optimizer establishes every required upper bound and constructs an attainer. Each normal/optimized check reports:

PASS_QUERY_TREE_TOPOLOGY_OPTIMUM
split_inequalities = 2040
root_inequalities = 2295
basis_identity_checks = 320
rejection_controls = 6

Local CPython 3.12.15 on macOS arm64 passes all three registered commands. Individual local times are 0.031112 seconds for normal verification, 0.034514 seconds for optimized verification and 0.017598 seconds for deterministic certificate regeneration. These are CI-budget observations, not a speedup claim. The full registered runner, metadata/pins, committed bytes/modes and declared source ancestry pass. No predecessor scientific replay was run as this contribution.

The official export passes integrity and declared-input coverage. All three commands pass from a read-only Git-free export with empty PATH; all ten exported files retain their hashes and modes. Actual corruption of either predecessor text input rejects in normal and optimized modes, four checks in total. Receipts are generated and bind source/package hashes.

From the repository root, reproduce with the actual local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/query-tree-topology-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/query-tree-topology-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/query-tree-topology-optimum/generate.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/query-tree-topology-optimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/query-tree-topology-optimum --check-source-history

Another host may substitute its CPython 3.12 executable. The standard library is sufficient. The registered runner additionally prints PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

The leaf count, merge, three-query root and native local profiles are fixed. The theorem concerns the complete recursive moment in that contract; it does not optimize ordinary scalar work, routing, storage, ownership or compiler parameters. The broader reward theorem assumes nonincreasing convex node rewards, not an arbitrary concave tariff without that condition. Other leaf counts, merge modules and native profiles remain open.

The induction, derivative and equality arguments are written proofs outside a formal kernel. Independent human and formal proof-kernel review are unclaimed. All five framework assumptions retain their status. Source/selected cold replay, physical caller execution, Lean, Docker and a complete rounded multiplier were unrun. No supplier admission is claimed.

Open next questions

• Can a different leaf count improve the fully charged supplier invoice? Start with complete local, thin-query, whole-idle and root profiles, including compiler and outer charges.
• Can a changed merge or root contract beat this topology optimum? State arbitrary independent inputs and every paid complement before searching; native realization and restoration remain required.

Related open work

• Compose native input lifts with the query tree at conditional κ 0.00314980 #6350 composes input lifts with this same fixed tree; this result optimizes topology at a fixed local profile.

• Contract nine native helper outputs at conditional κ 0.00315057 #6353 contracts native helper outputs while retaining the tree; its local-word gain is a separate mechanism.

• Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372 changes a two-anchor native boundary and retains the same caller; no new boundary is constructed here.

• Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375 changes shared-carrier input lifts and keeps the caller; this proof does not recount its invoice or compare its kappa.

• Independently audit PR6353 rational joint basis and public boundary #6355 studies the rational boundary of Contract nine native helper outputs at conditional κ 0.00315057 #6353, a different proof obligation from topology optimality.

• Independently recount PR6353 recursive invoices and fixed-saving margin #6359 studies Contract nine native helper outputs at conditional κ 0.00315057 #6353's frozen formal invoice; this contribution certifies an entire topology class rather than recounting that invoice.

• Independently check four frame1094 native matrices #6360 studies four fixed native matrices; native realization is a premise here.

• Independently verify the fixed balanced caller Laurent endpoint #6361 studies the fixed balanced caller endpoint; the new result bounds all binary topologies at its leaf count without re-verifying that endpoint.

• Independently audit PR6372 two-anchor scalar boundary and endpoint #6376 studies Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372's scalar boundary and restoration; those obligations are separate from the tree reward.

• Certify PR6375 matched profiles and unique relative-tariff threshold #6378 studies Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375's relative tariff switch; the new result compares topology rewards at every strictly concave power exponent.

• Prove a fixed-schedule obstruction for PR6345 helper respanning #6381 proves a fixed-schedule helper-respanning obstruction in the native word; this contribution fixes that word and optimizes a separate caller topology.

• Construct transported-query substitution for noncommuting coded jobs (no κ) #6382 constructs a transported-query substitution for noncommuting coded jobs; it changes the merge contract and performs no comparison of the nineteen-leaf topology reward proved here.

• Prove a strictly weaker mirror-capacity criterion (no κ) #6384 weakens a capped-flow mirror-capacity criterion; compiler funding is distinct from the fixed-tree recursive moment optimized here.

• Prove a paid-rank floor of 22 for three-carrier rank-two tails #6386 bounds a three-carrier rank-two tail in a local P104 transport model; that paid-rank bound is separate from this caller's power-tariff topology optimum.

• Prove exact tariffs for carriers visiting distinct entrance lines (no κ) #6388 prices carriers visiting distinct entrance lines; that transport tariff does not optimize the binary query forest.

• Prove the sharp direct higher-moment integer split charge (no κ) #6390 sharpens a higher-moment integer splitting coefficient; this theorem fixes the caller's recursive invoice and supplies no new splitting or completion coefficient.

CI notes

Both hosted checks, plan and replay (research/query-tree-topology-optimum), passed in run 41510715469 at exact head f1b5c2847793e99c19a934ff7a625ce7c4e72c18. The completed checks were read once; separate run metadata confirms the matching head and successful conclusion. Hosted replay certifies the registered finite checks, source bindings and deterministic certificate regeneration. It does not supply independent scientific acceptance or a formal-kernel proof of the written argument. All registered checks also passed locally. Related open work was refreshed before completion; no conflicting computation of this same topology quantity was identified, and no additional computations audited another contribution.

Merge Danger

Door: two-way. Only one new research package is added.

Blast Radius: isolated. Shared code, predecessor artifacts and the selected record are unchanged.