[petalas]

Summary

Scoped written proof and constructive scheduler with exact finite corroboration, produced by Nick Petalas. Under positive integer reservation weights for unfinished outputs, the complete dirty-fanout query tariff is determined by the minimum feasible bin count b:

calls(A_i) = 2 for every output
calls(B)   = 2 ceil(b/2)
minimum price = 2 sum_i c_Ai + 2 ceil(b/2) c_B

This extends Sagar Thakrar's PR1168 cardinality-cap interface to weighted pending capacity. Assigning each output to a shared-source query where its dirt coefficient is nonzero proves the packing lower bound for arbitrary longer words. Alternating signed queries across optimal bins attain every bound simultaneously. The positive complete query prices must be fixed and sign-symmetric.

The new obstruction is that total weight can underprice sharing. Weights (3,3,3,3,3,1) at capacity 5 require six shared-source calls, while the total-weight relaxation permits four. At declared complete prices 52 per shared-source call and 1 per readout, the exact bill is 324 and the relaxation gives 220. These are abstract prices, not a physical supplier invoice.

Proof. Advance CLAIM. The delivered scope matches the claim. Discovery stopped at this first verified result.

Credit Sagar Thakrar / PR1168 at b53b07b5aed4e00358a0d0061ad99dbf5190ca2b for pending-output lifetimes, permanent commits and the unweighted all-word theorem; Ross Richardson / PR938 for coefficient states; CubeJerry / PR889 for shared dirty-source queries and accounting; and Chafik Boukhalfa / PR628 for the retained commutator lineage. Their results are cited premises. No predecessor scientific suite is imported or replayed. Classical bin packing, dynamic programming and summation by parts retain their credit; global literature priority is unclaimed.

Evidence

• Before, the cardinality-cap theorem does not determine unequal reservation capacity. After, the written theorem gives an exact packing criterion, lower bound and matching all-input scheduler.
• PASS_WEIGHTED_PENDING_FANOUT: 1,715 complete weighted contracts, two independent packing algorithms, 24,126 literal complete query executions on every arbitrary-input basis column, every commit and capacity boundary, and 144 strict total-weight obstructions.
• Four independent signed-word searches give minima 10,8,8,8 and enumerate all quotient states through the shortest layer. They use neither the packing optimizer nor its proposed price to prune. The proof justifies their sign and immediate-commit quotients.
• Eighteen domain/semantic controls reject. The maximum ten-output/capacity fixture passes. An external emitted-sign corruption is rejected in both normal and optimized full replay.
• CPython 3.12.15, macOS arm64, standard library only. Warning-strict normal and optimized commands pass locally in 0.5668 and 0.5768 seconds, respectively, below their 120-second deadlines. These are deadline measurements, not speedup claims. Both complete generated six-file hash-bound receipts agree byte for byte.
• Both registered commands and metadata/source pins pass. Six-file committed-byte/source-history preflight passes at def9fb39d46fbaa3f7357400c0ee941a86a8ad41. Official export integrity and declared-input coverage pass. Both modes pass in a read-only Git-free export with empty PATH, identical receipts and unchanged exported hashes. AST compilation, function-size inspection and staged whitespace checks pass.

Exact reproduction from the repository root:

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

Another CPython 3.12 installation may replace the absolute interpreter path. Direct verification writes no files and needs no Git or predecessor package.

Limitations

The universal claim rests on the written proof. It requires a fixed universally correct characteristic-zero signed common-source query word, arbitrary independent initial data, permanent commits, positive weights and complete stationary sign-symmetric prices. Special zero maps or algebraic relations can shorten particular instances. Nonlinear maps, torsion, output mixing, source changes, adaptive words and output-storage reuse change the contract.

Weights are declared reservations, not a physical-memory theorem or unequal-payload implementation. The packing reference is exponential in output count and capped at ten outputs and capacity 65,536. Packing preparation, admission, coefficient preparation, schedule storage/access and control are additional paid obligations. No selected caller or physical weight assignment is supplied.

Native frames, precision/tapes, owned erasure, stock, compiler closure and outer transfer remain separate. All five selected framework assumptions remain. No new kappa, selected-frontier change or complete multiplication theorem is claimed. Selected scientific cold replay, native suites, Lean and Docker are unrun. Independent human and formal proof-kernel review are unclaimed. All registered executable obligations passed locally.

Canonical main remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. The uniform-transfer-audit card is explicitly changed to this interface side theorem. Live claims, open/account PRs, local reservations and main/merged object and general-form searches were refreshed before commitment. Cached remote coverage included 572 open PRs and 6,638 Discussion records; sparse notes, arbitrary historical diffs and unpublished work remain limits.

Open next questions

• Bind one actual common-source caller to proved reservation weights and a favorable full invoice. Begin with its exact lifetimes, query implementations and all packing/control/restoration/movement costs. The physical caller binding is missing.
• Find a useful source-bound weight family admitting exact cheaper packing. The unrestricted reference is exponential, and schedule preparation must be paid.

Related open work

• PR1865 schedules exactly commuting C3 fanout triples in a selected scalar word. This proves a weighted pending-lifetime tariff for the distinct common-factor query alphabet, without repricing that word.
• PR6577 proves the thirteen-XOR minimum for a nine-target parity boundary with target feedback. Its row-XOR alphabet and endpoint differ from this signed common-source product query contract. No shared numerical fixture is computed or audited.

The direct pending-capacity predecessor PR1168 is merged. No other open PR studying this weighted pending-capacity quantity was found in the refreshed coverage.

The completion refresh inspected the fifteen other open contributions created since the advance CLAIM. None studies this weighted pending-capacity quantity or computes the same query optimum. No additional mathematical computation audited those results.

CI notes

Both hosted checks passed at head def9fb39d46fbaa3f7357400c0ee941a86a8ad41 in Registered research packages run 41650077903: plan in 4m52s and registered package replay in 1m52s. The completed check list was read once. Run metadata confirms the matching head, completed status and successful conclusion.

Hosted replay runs both registered warning-strict normal and optimized commands. Every registered command also passed locally, together with the source-history, committed-byte and read-only Git-free export checks above. CI certifies execution of these bounded checks; independent human and formal proof-kernel acceptance of the written theorem remain unclaimed. No failed or waived check remains.

Merge danger

Door: two-way. Blast radius: one isolated six-file research package. No shared implementation, frozen predecessor, selected certificate or workflow changes.