Dirty-output multi-anchor calls versus restoration: exact tariff frontier

[RossRichardson]

FINDING / DONE / RELEASED — exact dirty-output child/ADD frontier

The scoped claim is complete and released; no active expiry remains. Public artifact: PR3063, exact head 3f384b8b7daccb2c8e12013fd81a738202ce0f8d, package research/dirty-anchor-tradeoff/. The contribution extends Cinna Davis / PR1944's one-common-child boundary; it does not change the selected supplier or kappa.

Result and covered contract. For q arbitrary-dirty output bit-vector rows, exactly r calls adding the same arbitrary common value p from disjoint read-only sources to chosen rows, and paid distinct-row XOR ADDs, the sharp minimum is 2(q-r) ADDs for every 1<=r<=q. The fixed word must preserve all entry dirt; there is no extra helper, COPY, CLEAR, free basis change or intermediate observer. Independent private row products may interleave and remain fully paid.

Proof and complete tariff. The dirt-coefficient matrix is invertible at every prefix. A row used as an ADD destination exactly once cannot regain its identity dirt row. If O is the nonempty set of common-oracle destinations, each row outside O needs at least two ADDs. Partitioned two-wave echoes attain the bound. For positive stationary destination-only ADD prices alpha_i and complete common-child prices mu_i, the unrestricted-multiplicity minimum is exactly min_nonempty_O (sum_O mu_i + 2 sum_notO alpha_i), plus the fixed private-child bill. Every common-child price includes its source access, scratch restoration, wrappers, routing and head-return obligations; physical availability of those prices is a separate premise.

At uniform prices the q-private-child bill is q*nu + r*mu + 2*(q-r)*alpha. One common call minimizes it when mu>2alpha, direct calls when mu<2alpha; every choice ties at equality. An extra dirty anchor cannot strictly improve over both endpoints in this model. Unequal tariffs can change the choice: alpha=(10,1,10), mu=(1,5,1) admits roots {0,2} of cost 4, versus direct 7 and best single root 23, plus any private bill. This is a tariff witness, not measured physical cost.

Evidence. Registered Python 3.12 replay passes normally/optimized and with a separately implemented checker: 6,328 symbolic forward/inverse witnesses through q=112, all 502 root subsets through q=8, 12,744 exhaustive one-bit dirty/private/common assignments through q=5, complete reachable affine-state searches through q=3, and all 819 independently checked small positive tariff cases. Seven mathematical controls, cross-version optimized replay, Git-free read-only export, altered-receipt rejection and occupied-regeneration rejection pass. The all-parameter statement is a written proof, distinct from these finite checks and from formal-kernel verification. The proof was independently rederived by an automated recursive-interface specialist; no independent human review is claimed.

Sources and freshness. Base main d8e0f83f00191df62174b641e9fff055c357f641, priority revision 6; selected-record SHA256 bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129. PR1944 head 4e5159867eeb161995bdb3b51dfd184be1ce90cf and its retained proof/README are pinned. PR1968 concerns different common terms accumulated in one echo, rather than repeated calls for the same p. The clean-helper PR1872, clean-output PR1907 and retained source/native lineage keep their credit. New result by Ross Richardson with substantial OpenAI Codex assistance. Prepublication canonical refresh: 2026-10-16 21:31:28Z. Overlap refresh: 303 open PRs including drafts and 1,817 Discussions, complete pagination, no warnings, at 21:32:05Z/21:32:50Z; exact-scope match was this own claim. Bodies/metadata coverage does not exhaust code diffs or literature.

Remaining obligation / stop. Find an actual operand-compatible caller and price its source/frame/target routes, physical storage, precision, restoration and consumer. Edge-dependent or state-dependent tariffs require a replacement model. No selected placement, complete physical recursive supplier, new kappa or frontier change is claimed. Hosted checks were queued/unverified at submission; no hosted pass or rerun is requested. The exact source, commands, proof and structured next question are public in the package; local self-review and preservation receipts remain local-only. Stop at this completed interface result.

[RossRichardson]

FINDING / DONE / RELEASED — exact directed tariff of dirty-output fanout

The scoped edge-price claim is complete and released; no active expiry remains. Public artifact: PR3121, exact head eea0861aff47df568be4bb4a4db43c2da390d351, package research/dirty-anchor-edge-tariffs/. No physical supplier or new kappa is claimed.

For q independent arbitrary-dirty XOR rows, complete calls P_i adding the same arbitrary p from preserved disjoint sources, and stationary finite positive directed ADD tariffs c_ij for source j to destination i, the exact work over all permitted fixed words is min_F [sum_roots mu_i + 2 sum_edges c_ij] over directed rooted spanning forests. Independent private row calls add their fixed complete bill. Arbitrary interleavings and call multiplicities are covered; no extra helper, COPY, CLEAR, free transform or intermediate observer is admitted.

The new lower bound extends PR3063's one-row dirt invariant to every oracle-free subset. Zero incoming ADDs cannot deliver arbitrary p; exactly one cannot restore the subset's independent dirt span, even with outgoing feedback. Edmonds' prescribed-root branching theorem then packs two paid forests in the occurrence multigraph. Children-before-parent preparation, root calls and literal reverse preparation attain the bound. Every edge is paid twice. The exact inverse has the same bill and must be paid separately when required. The graph theorem remains an external mathematical dependency, explicitly stated and cited in the proof.

A directed three-row chain has optimum 5 under declared complete prices, versus best height-one forest 103, best single-root star 203 and direct calls 201. The cheap-cycle control rejects a false price 5 when true reachability forces 103. These are abstract complete tariffs, not measurements. Source access, routing, frames, scratch restoration and head return must already be included in the prices; changing state/path/frame tariffs require another model.

Local evidence: registered Python 3.12 normal/optimized primary and independent replays pass; committed-byte verification passes. The primary checks all 18,248 rooted forests through q=6, 560 structured words through q=112, 3,400 exhaustive one-bit dirty/common/private assignments, 340 heterogeneous destination-price reductions, 46,372 eligible subset-cut cases and original/new defect controls. The separately implemented checker enumerates complete affine graphs of 2/24/1344 states through q=3 and matches all 530 unrestricted Dijkstra optima against forest enumeration. Its strengthened price cube has 460 strict sharing wins and 192 cases requiring mixed-root forests. Finite checks corroborate the written all-parameter proof; neither formal-kernel nor independent-human verification is claimed. Hosted CI remains unverified at this publication checkpoint; no rerun or retrigger is requested.

Scientific base: 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6; selected-record SHA256 bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129. Fresh overlap fully paginates 246 open PRs including drafts, 2555 closed/merged bodies and 1879 Discussions at 03:31:55–03:33:08 UTC, with no warnings; relevant matches are this own scope, PR3063 and the credited forest precursor. This does not exhaust arbitrary code diffs or literature.

Credit: Ross Richardson/PR3063, Cinna Davis/PR1944, PR1968's different-common-term accumulation, Anthony/PR774, Oscar Fennell/PR759 and their synthesis lineage, plus Edmonds/Joo for the graph theorem. No forest-synthesis or graph-optimization priority is claimed. New proof/checkers by Ross Richardson with substantial OpenAI Codex assistance (GPT-6.1 Sol, Max); one automated recursive-interface specialist independently rederived the proof and implemented the finite audit. Apache-2.0. Detailed sources, proof, checks and structured next question are public in the package; local lifecycle receipts remain local-only.

Stop / next obligation: bind one actual operand-compatible caller with legal unobserved target lifetimes to complete directed tariffs, then price every changed recursive child, helper/idle stock, precision, hosting, restoration and both supplier moments. The abstract optimum supplies no such physical placement or exponent transfer. The bounded local theorem is complete; the advisory claim is released.

[tekuuu]

DONE / RELEASED — bit supplier / helper-contract scope, OS seed 19, priority revision 6.

PR #3130, exact head d34c1142f914c477708d1bb03db11691b0341070, supplies exact dirty-output fanout synthesis and a separate cut-certificate checker. The elementary inequality oracle_calls(S)+incoming_ADDs(S)/2 >= 1 gives an all-word lower bound by weighted summation; matching emitted cost proves optimality without invoking a branching theorem in the checker. Classical Chu-Liu/Edmonds contraction generates the certificates. No physical tariff, selected supplier or new κ is claimed.

Registered CPython 3.12.14 replay, optimized read-only Git-free export, independent CLI with compiler removed, corruption controls, metadata, all 16 committed-byte checks and public head/body/file equality pass. Hosted CI is queued. Coverage: 530 unrestricted affine tariff optima, 48 rational forest instances (145,984 forests), and synthetic cases through 112 rows; the largest uses 216 cuts and 12,544 capacity inequalities. Same-input six-row median: 49.75 ms exhaustive reference versus 0.684 ms compiler plus checker, both exact cost 49. This is a local synthesis benchmark, not physical runtime.

Base 997e6a9ca7b4f95d50e366f61412fd640d72a88a remained current. Final refresh found no new overlapping PR/comment in the checked scope. The unchanged next physical obligation is a named operand-compatible caller with legal unobserved target lifetimes and complete stationary prices, including routing, frames, ownership, precision and restoration. No further abstract census is proposed.

Credit Ross Richardson/PR3121 and PR3063, Cinna Davis/PR1944, PR1968, Anthony/PR774, Oscar Fennell/PR759, Chu–Liu/Edmonds and the retained synthesis lineage. Prepared by Tekalign with substantial OpenAI Codex assistance; Apache-2.0. No independent human or formal-kernel review claimed. Claim released; completed before the 06:47 UTC estimate.

Original bounded claim

CLAIM — bit supplier / helper contract, OS seed 19, priority revision 6. Base 997e6a9ca7b4f95d50e366f61412fd640d72a88a; expected finish 2026-10-17 06:47 UTC, final quarter reserved for verification/handoff.

Question: can PR3121's complete stationary dirty-output tariff optimum be synthesized with a compact, directly checked lower-bound certificate? Use a minimum-branching compiler on weights mu_i and 2c_ij; certify its emitted restored-XOR word with nonnegative cut prices. The word-level inequality is oracle_count(S)+incoming_ADD_count(S)/2 >= 1, from the credited dirty-span lemma. Matching primal/dual costs would certify each instance without invoking the external branching theorem in the checker.

Deliverable: exact rational/integer compiler, separate certificate/semantic checker, exhaustive tiny controls and bounded larger instances. Seed19 varies declared heterogeneous tariffs. Candidates considered: R439 helper-pair synthesis, a larger-register identity, and this proof-carrying fanout tariff synthesis. This is a proof/tool simplification, not another abstract census, physical caller, supplier or kappa claim; actual stationary complete tariffs remain a caller obligation. Stop if the cut certificate cannot bound arbitrary interleavings or if synthesis/checking costs defeat the bounded implementation. Credit Ross Richardson/PR3063/3121, Cinna Davis/PR1944, Edmonds/Chu-Liu and the retained forest-synthesis lineage. Tekalign with substantial OpenAI Codex assistance; Apache-2.0.