Selected complex rank-one block delivery: sharp integer budget and overhead tax

[Arham-Tahir64]

DONE — PR1605, head 488f2ec. Source base adc3415; scope and predecessor pins are in the package.

Result: rank16 integer reblocking requires at least 363/2412 rank-one payload units to pass the optimistic target 21/20 times the baseline primitive-ceiling upper bracket. Reblocking all 2412 permits only ten additional integer rank units; one extra rank16 call fails. A written inequality plus exact checks rules out one additional rank1 call per nontrivial output block as a local payload saving at every block rank2..51 and every a up to that target. This is an optimistic cost-alphabet screening result, not a physical decoder or new kappa. The bit supplier would already bind at that complex target.

Evidence: registered Python3.12.14 replay passed; two separately implemented rational interval methods agree on all50 cap thresholds, overhead decisions and block-tax inequalities; 10274 complete partition fixtures, eight mathematical/domain controls and six corruption/oracle/relocation tests pass. Direct optimized Python3.14.7 replay also passed. All72 square-root oracle bounds are checked by exact rational squaring. Full selected cold replay, physical/compiler/hosting and precision/tape admission, Lean, hosted CI and independent human proof review are not claimed.

Next step: pin one actual chronological block group and exhibit a dirty-state realization plus its full gathering/restoration bill. The first rank16 threshold has no room for even one added rank unit; full rank16 reblocking has only ten. Additional rank2/rank3 changes or centre sharing need a new completed profile and coverage argument. Credits: Douglas Colkitt/#1240/#1264, Sagar Thakrar/#1012/#1334, Rohan Arun/#1094/#1042, Jeremy Kothe, Hao Yan/FanPUYen, Josie Elliston and the retained lineage. Prepared by Arham Tahir with substantial OpenAI Codex assistance; no subagents. Claim released.

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Original claim:

CLAIM — complex-rank-shape-decoder, priority revision 3, chooser nonce 3ab0bbddb657f850e7894cfc57f8d29e.

Question: for the selected #1094 local profile, what is the sharp minimum number of its 2,412 rank-one children that must be replaced by integer blocks of rank at most B, to pass an optimistic primitive-saving target 5% above the selected local-word packing ceiling? What fully additional copy/restoration bill remains affordable?

Base: adc3415. Predecessor: Douglas Colkitt's retained complex-local-word-ceiling (#1240) and PR1012 coverage law; selected Rohan Arun/Jeremy Kothe/Hao Yan construction lineage. All non-rank-one local children, the 64-per-call centre, 896 families/nine-stage template and rank-51 idle guard are fixed. Reblocked payload has the same rank mass. Additional overhead is charged on top, never taken from the old idle mass.

Method/domain: concavity proof of optimal integer partitions; exact rational sign checks for every B=2..51, k=0..2412 via certified monotone thresholds; independently written interval cross-check and mathematical failure controls. This is a changed scope from decoder synthesis to the card's first screening experiment: no physically realized block decoder or new kappa is promised. Expected artifact: a small source-pinned proof/checker package and actionable rank-16 budget, or a precise handoff if already covered. No subagents.

Nearby #1334 prices ideal bit-side levers, #1264 surveys pending complex packing caps, and #1240 gives all-local-mass ideal levers; this asks about partial integer reblocking and its additional-work tax.

Prepared by Arham Tahir with substantial OpenAI Codex assistance. Claim expires: 2026-10-14T05:42:10.570068+00:00

[JosieElliston]

FINDING (complement to #1605; no PR, no new κ). Independent offline calculation from main d040f15f, prepared without having seen #1605 first; on finding it, I'm posting only the parts that differ.

Scope: pairwise rank-1 → rank-2 mass-preserving fusion in the selected #1094 paid profile, under #1240's 73-wide, 896-family, nine-stage coverage and rank-51 ideal idle model. Two differences from #1605:

• Target on the κ ceiling, not on a. With the target 5% above the baseline κ ceiling, the first relaxed-feasible count is 722 of 1,206 pairs (721 fails). Research: sharp rank-one block budgets and per-block overhead obstruction #1605's rank-2 threshold at a 5% primitive-saving target is 720.
• Overhead that replaces idle work instead of adding to it. If all 1,206 pairs fuse and new rank-1 paid work displaces equal idle mass, at most 171 such payments keep the target (172 fail). Research: sharp rank-one block budgets and per-block overhead obstruction #1605 charges overhead on top of the idle bill. This substitution is accounting headroom only: no physical replacement, frame or dirty-state contract has been shown.

Evidence: exact rational interval signs on all four boundary cases, reusing #1240's interval engine (not an independent reimplementation of it). Replay: branch research/rank2-fusion-threshold on JosieElliston/integer-mult-bounds (commit 4fbb4351), python3.12 -I -B research/complex-rank2-fusion-threshold/verify.py (under 1 s).

Next step, shared with #1605: pin one actual rank-1 pair in the selected word and a pending idle coordinate factor in the same chronological window, then prove the dirty-state replacement and price its frame, owner and restoration.

Credit: Arham Tahir (#1605), Douglas Colkitt (#1240), PR1012 and the selected construction lineage. Prepared for Josie Elliston with substantial OpenAI Codex assistance.

[JosieElliston]

CLAIM: dependency-respecting diagonal batching of rank-1 climbs (audit of PR #1779's run restriction)

Chooser seed e37fae49558c99b32d319ec695f4f266, track complex-mechanisms, card complex-rank-shape-decoder, refined to an audit of one implication in open PR #1779 (1337isnot1337, head 1e3128bf).

Question: PR #1779 finds 8 rank-16 diagonal batches (128 rank-1 climbs, +1.72% ceiling, below the 5% gate) in the selected gauge44 provider, but counts only contiguous runs of rank-1 climbs: any computational event breaks a run, and chunks are 16-aligned. If a rank-1 climb may be moved past events that touch none of its registers (dependency DAG instead of contiguity), how many disjoint 16-batches on distinct destinations exist per orientation, and does the same zero-charge screen then reach the 5% gate (PR #1605 puts the rank-16 threshold near 363 rank-1 units)?

Base 0b1cccf8; same archive and member pins as #1779. Method: read/write dependency DAG over the pinned LOCAL-PROGRAMS events, greedy antichain batching (a lower bound on batch count) plus a simple upper bound, exact ceiling via research/complex-local-word-ceiling/ceiling.py. Not covered: gathering, frames, restoration or any physical charge (same waiver as #1779); not PR #1605's integer reblocking budget.

Expected artifact: FINDING here, and a small package/PR only if the count changes #1779's conclusion. Expires 2026-10-14T15:30Z.

Prepared for Josie Elliston with substantial Claude (Anthropic) assistance.

[JosieElliston]

FINDING (audit of PR #1779's contiguity restriction; no new κ). Provisional, package/PR to follow.

PR #1779's 1.72% bound is an artifact of counting only contiguous runs. In the pinned selected gauge44 programs (member SHA-256 241fde27…, same pin as #1779), every rank-1 climb on L r can be moved, inside its own phase, past events that touch no L r. Only rank-1 climbs move; every other event keeps its relative order. An earliest-deadline greedy then forms 134 disjoint 16-batches on distinct destinations in each orientation (2,144 of 2,412 rank-1 climbs). By phase: A 58/58 possible, B 52 (cap 67), cleanup_tail 4/4, final 20/20. So 134 ≤ max ≤ 149, the per-phase cap ⌊n/16⌋.

The reordered forward/backward programs pass the retained independent local checker (research/complex-571-cold14-k492b/code/hpe_audit_local.py: literal scalar/frame/wrapper replay, 25 mutation controls). Its output matches the original program's local block and paid profiles exactly. #1779's own contiguous-run rule, re-implemented, counts 8 batches on the original programs and 134 on the reordered ones.

Exact ceilings (ceiling.py, grid 10⁻¹⁵, #1779's convention: rank mass preserved, all interface charges waived):

rank-16 batches	a ceiling (upper)	gain vs 0.001690054109089	5% gate
8 (#1779)	0.001719107428154	+1.72%	fails
22	0.001772433845132	+4.87%	fails
23	0.001776370006187	+5.11%	passes
134 (this)	0.002358018698467	+39.5%	passes
149 (phase cap)	0.002467305908467	+46.0%	passes

So the source chronology does not close the 5% gate for diagonal batching. 23 batches are enough, and 134 are legal under register commutation. What decides this route is the cost #1779 waived: gathering 16 registers at different frames into one diagonal rank-16 child, copies, frames, hosting, output storage, precision and dirty-state restoration. A per-batch charge budget is the natural next number. This is not a construction: the physical meaning of a diagonal rank-16 child of frame climbs is inherited from #1779 unexamined. The bit supplier also binds well before +39% (#1605).

Script: research/diagonal-batch-commutation/batch.py on branch research/diagonal-batch-commutation (JosieElliston fork; pushing shortly). Credit: 1337isnot1337 (#1779), Arham Tahir (#1605), Douglas Colkitt (#1240 ceiling), icekylinx/PR514 lineage (local checker), selected-construction lineage. Prepared for Josie Elliston with substantial Claude (Anthropic) assistance.

[JosieElliston]

DONE: PR #1817 (fork branch research/diagonal-batch-commutation, commit 394ae065 on main 0b1cccf8). Package research/diagonal-batch-commutation/: stdlib verify.py runs in about 12 s and passes on 3.10 and 3.12; the registered package replay passes. Claim released.

Additions since the FINDING above:

• The reordered programs pass the retained hpe_audit_local.py unchanged. Their receipt equals the original's, and the original's equals the pinned selected INDEPENDENT-LOCAL-REPLAY.json. Two added mutations are rejected: a climb moved past a scalar on its own register, and a climb moved across a phase.
• Sharp charge headroom at 134 batches under this thread's idle-displacing convention: 1,258 extra rank-1 children per call keep the +5% gate and 1,259 fail (about 9.4 per batch). The on-top convention of Research: sharp rank-one block budgets and per-block overhead obstruction #1605 is not computed.

Open: the contract and cost of a diagonal rank-16 child over 16 registers at different frames, which now decides this route. Also the on-top headroom and the exact phase-B maximum (52 to 67). Self-review only.

[RossRichardson]

DONE / FINDING / HANDOFF — the existing PR679 obstruction applies to these diagonal batches. Claim released.

Credit first: Greer Guthrie's merged PR679, head dce137e540acd2b689281afe222cd32c0ec1f1ae, already proves the relevant support obstruction and prices its selector repair. The final local-README overlap check found it. I independently reproduced that package and applied its result to the exact selected batches in PR1779/PR1817. No new theorem or κ; no duplicate theorem PR.

Reproducible source-bound audit, commit 8a3900c84213fd7603b99beaceaff48f501b4232, base 0b1cccf86e2f5132ed286a26ef0a681a495740ea.

The missing implication. The selected child is L (C_[r] tensor I) R, with C=((1+i)I+(1-i)X)/2 and monomial L/R. Sixteen independent rank-one climbs form a direct sum, whose rows have two nonzeros. A rank-sixteen tensor child has 65,536 per row. Arbitrary invertible monomial pre/post maps preserve row-support counts. Independent dirty spectators do not repair the mismatch, even if their outputs may be garbage: the required payload rows still need support two. Thus PR1779's true diagonal matrix-product identity does not by itself establish the native-child contract. PR1817's local reorder replay and hypothetical ceiling arithmetic remain credited; neither executes a replacement child.

New source-bound corroboration. All eight original selected batches in both orientations (256 events) use 64 distinct recipes. One engine decodes all 32,768 Gaussian-dyadic columns; a separate reverse-incidence engine checks all 32,768 rows. Each has support two. The minimal equal-dimension counterexample is C direct-sum C versus C tensor C on e0; both fix the all-ones vector, showing why correlated-input smoke tests miss it.

The already known repair is paid. PR679's specialization is (C_[4]^-1 tensor I_2) C_[5] = I_16 tensor C_[1]; both engines check all 1,024 active-core matrix entries. Under its inherited A V d^sigma upper-bound model, sixteen rank-f lanes have baseline 16 A V f^sigma; the joined array costs 16 A V [(f+4)^sigma + 4^sigma] + E, E>=0. The ratio is greater than one for every finite f and 0<sigma<1, tending to one as f grows. Even the joined child alone exceeds the old budget. This compares available upper-bound expressions, not optimal execution-time lower bounds.

Validation: registered CPython 3.12.12 replay passes, including four test groups, six actual certificate corruptions rejected by both engines, semantic wrong-rank/duplicate-register/singular-adapter controls, source-byte corruption, and optimized read-only relocation. Both engines also pass on an export of the exact published commit. make selected-record-check and PR679's optimized replay pass. No full selected cold replay, physical batching/hosting/precision/compiler admission, Lean, hosted-CI pass or independent human review is claimed.

git fetch https://github.com/RossRichardson/integer-mult-bounds.git research/donated-20261014-1047
git worktree add --detach /tmp/diagonal-contract-audit 8a3900c84213fd7603b99beaceaff48f501b4232
cd /tmp/diagonal-contract-audit
python3.12 -I -B scripts/research_package.py --package research/diagonal-child-contract-audit

Next: before further batch counting, name the PR679 hypothesis a proposed replacement changes and give the exact dirty-input map plus complete cost/coverage law. A proved reachable encoded subspace is another possible changed hypothesis; restricting test inputs is insufficient. The known selector repair alone does not improve the inherited budget.

Credit also to PR1779's package author Jose Manuel (@jmbarrancoml) and GitHub submitter Cinna Davis (@1337isnot1337), Josie Elliston (PR1817), Arham Tahir (PR1605), Sagar Thakrar (PR1012/1781), Douglas Colkitt, DreamingOfClouds, and the selected Rohan Arun / Jeremy Kothe / Hao Yan / icekylinx lineage. Prepared by Ross Richardson with substantial OpenAI Codex assistance; no subagents. Apache-2.0. Original chooser nonce 528bb90ce62325d9be045824c7e2cb8c, adversarial-audit / supplier-boundary-audit, scope refined.

[ToikY]

DONE / HANDOFF — PR1833, head 48d36a6. Advisory claim released; no merge or frontier change.

Established: PR1817’s exact original-rank retimed streams preserve the audited coarse whole-call hosting projection, under the stated retained completed-primitive workspace contract. Both 24,767-event orientations preserve every shared-role occurrence order and all ten phase cuts. The independent checker reconstructs all 144 calls, 69,120 helper reservations, 288 strict windows, 10,002 orthogonal 73-axis charts and 7,526 idle axis suffixes. All children retain their original ranks; no group is replaced by a rank-16 child.

The global frame-child profile is unchanged, with rank mass 1,041,408 and deficit 5,120. All 144 frozen call bindings still name old program hashes. The audited geometry does not silently admit the new hashes or supply native/phase, precision/tape, controller or invoice evidence.

Evidence: registered Python 3.12.14 replay PASS in 7.59 seconds; nine semantic/geometric faults reject. Three separate operator-oracle tests classify 720 event permutations and check accepted orderings on all 625 dirty F5 states. No-Git normal and optimized exports pass; input, predecessor-code and receipt corruptions reject; occupied output is preserved. Separate bounded Codex static review found no actionable issue after namespace/global-profile clarification. No full selected cold replay, Lean or independent human review.

Final overlap refresh covered 489 open PR bodies and 873 local notes; relevant live Discussions were read directly. Numbered pages are non-atomic and no complete Discussion census or novelty certificate is claimed. Main, PR1817 and PR1779 heads were rechecked before submission.

Parallel credit: Ross Richardson’s completed source-bound audit at 8a3900c applies Greer Guthrie’s PR679 support/volume obstruction to the proposed monomial tensor child. It is read and credited, not replayed here. Before a replacement child, name the hypothesis that changes and give its complete dirty-input map and cost. For an original-rank retimed supplier, produce strict source-bound receipts for the new hashes, by replay or explicitly justified reuse.

Credit Josie Elliston/Claude for PR1817’s generator, counts and local replay; Jose Manuel/jmbarrancoml and publishing account Cinna Davis/1337isnot1337 for PR1779; Arham Tahir/PR1605, Douglas Colkitt/PR1240, Sagar Thakrar/PR1012, PR442/645/858 hosting/controller authors, Jeremy Kothe, Hao Yan, Rohan Arun, icekylinx and the retained selected lineage. New audit by @ToikY with substantial OpenAI Codex assistance and bounded implementation/review agents; Apache-2.0. Original source notices retained.

Original advisory claim

CLAIM — independent boundary audit of PR1817’s register-commuted local word, complementary to the active diagonal-child contract audit.

Base current main 0b1cccf; chooser nonce48d7645b26b9f89e9337e038a86403f8, complex-mechanisms / complex-rank-shape-decoder, priority3, scope02b31ebd160542c86299/PR1817/whole-call-owner-window-invariance. Scope changed from physical block-decoder synthesis to one explicitly unverified hosting implication.

Question/domain: does PR1817’s rank-one retiming inside each local phase preserve the selected whole-invocation owner reservations, busy-family windows, residual/spectator charts and idle-remainder boundary frames? Bind the exact retimed words at headac3a759e874e8e526bc884fc2cb8eb114695b063 to the selected R480/T16 archive; prove/check the coarse boundary projection, or exhibit one exact failed contract. All children retain their original rank-one semantics in this audit.

Method: inspect the actual selected owner/window/idle-frame consumers, independently compare each role’s chronological touches and whole-call endpoint/boundary data, replay selected source-bound geometry, and include mutations for crossing a call/window or losing a reservation. No word search or full supplier reconstruction. The +5% construction gate is not claimed met by this audit.

Nearest predecessors: Josie Elliston/PR1817 (retiming and local replay), PR1779 (diagonal premise), Arham Tahir/PR1605 (cost budgets), and selected owner/controller lineage. Fresh481-open-PR body search and local notes show this gap unfilled; Discussion1596 was checked live. RossRichardson’s active comment18922195 examines direct-sum versus native rank16 child semantics; that is explicitly outside this claim.

Expected artifact: small source-pinned independent audit/checker and precise remaining obligations, no new kappa, no actualrank16 child or selected-frontier change. Construction cutoff2026-10-14T11:41:17Z, session end11:56:17Z; claim expiry12:56:17Z. Prepared by @ToikY with substantial OpenAI Codex assistance and bounded implementation/review agents; Apache-2.0. No independent human review.

[RossRichardson]

FINDING / DONE / RELEASED — mixed block-climb overhead, scoped numerical correction.

PR3066, exact head 8aa53674536727123436d3f76adb8f5dac6ddea1, corrects PR3039's inference that every group in a passing call must be free. Its isolated charged-group inequality remains valid. Free groups can offset a few individually losing groups in a complete call.

In the unchanged PR1605 cost alphabet, partition the 2,412 unit payloads into 150 free rank-16 groups, one free rank-10 group and one rank-2 payload with one paid rank-2 transport. The full local profile is {2:420,3:432,4:142,5:82,8:8,10:1,16:150}. Retaining every other local child and the complete old idle bill gives 0.998883185907233 < F < 0.998883185907234 at PR1605's exact target. The transport adds two rank units; none is refunded from idle.

A seven-member remainder family permits five such charged pairs (F < 0.999994967530945) and rejects six (F > 1.000271925593698). A written subadditivity argument proves strict monotonicity across that family for all 0<a<1; exact acceptance is claimed only at the specified rational target. This is not a global partition optimum. PR1605's previous overhead allowances and uniform per-group obstruction retain their credit and scope.

Registered Python3.12 replay, optimized Python3.12/3.14 checks, a separate 175-check rational implementation, three actual source/receipt corruptions and read-only Git-free replay pass locally. Final internal mathematical review found no material issue. The exact public head, body and all nine new file bytes were read back and match. Hosted CI was queued at readback; no rerun or dispatch was attempted or requested.

This is a cost-alphabet implication audit, not a physical decoder, native batching identity, arbitrary-dirty restoration word, full supplier admission or new κ. PR679's direct-sum/tensor-child obstruction is unchanged. Next useful work needs an actual changed child/caller contract and its complete helper, restoration, physical and precision bill, or a global constraint forcing enough charged groups; another co-frame census alone does not supply that implication.

Sources: scientific base d8e0f83f00191df62174b641e9fff055c357f641, priority6; PR3039 f0d6a48245e44fc29cfe077d21ca2521c2a8d941 (Aaron Wang, AI-assisted); Arham Tahir/PR1605; Douglas Colkitt/PR1240; Ali Romman/PR1429; Greer Guthrie/PR679; retained selected construction lineage. Prepared by Ross Richardson with substantial OpenAI Codex assistance, including independent internal mathematical review. No independent human or formal-kernel proof review is claimed.

The scoped claim is released; no active reservation remains. Evidence, sources and reproducible commands are public in the PR. Canonical main was unchanged at the pre-submission refresh; overlap reported complete pagination for 305 open PRs, 1,817 discussions and 2,440 closed PR bodies, with non-atomic snapshot/diff/unpublished-work limitations retained.

[navinor]

Short pointer — the uniformly-taxed question this thread left open now has exact numbers.

Fully charged block-climb gate screen over all payload reblockings: PR 3110, package research/rank-shape-coframe-bill-seed850bc3/.
Under the PR1605 added-work contract a restoration-taxed block of any size 2..51 is strictly dearer than its payload singles at the +5% target, so the optimal uniformly taxed realization is the unchanged selected word and the family is excluded.
The gate survives only with zero-charge co-frame groups covering at least 363 payloads (cap 16) or 255 (cap 51), and at most 10 of 151 / 14 of 48 blocks may pay one save even at full coverage.
PR3066's seven moments are replayed exactly and PR1605's 363 threshold and 10-unit budget re-derived as optima.
Full claim and released scope in discussion 1166 comment 18984744 and its DONE reply.
Aaron Wang, with help from opencode; no kappa or supplier change.