Research: obstruct two simple orders for the coupled T20 packing cap

[commdata2338]

Change and mathematical scope

This scoped proof rules out two simple orders for the fixed PR #3148/PR #3153 coupled inventory. Stage-major and replica-major orders force at least 1,857 and 7,500 capacity-eight bins away from the only tight pattern [8] at T20. The written lower bounds are 78T+297 and 375T for every positive T, including arbitrary replica permutations within complete stage blocks or complete replica blocks, respectively.

Each non-tight capacity-eight bin has reduced cost greater than 3/250 on the inherited cap bracket. At T20 the normalized moment penalties exceed 5571/327040000 and 9/130816; both complete inherited moments exceed one at the bracket's left endpoint. The result is conditional on the pinned inventory, the inherited source-window model (the one PR #3153 used to check PR #3148's T740 allocation), additive bin pricing and the inherited local paid profile. Arbitrary interleaved T20 orders remain open. No supplier or new kappa is claimed.

Artifact: research/coupled-t20-window-audit/, commit 941810a486b6f6b4701194fcf5aead6fbaa74d24. Seven new package files; frozen predecessors, shared tools and the selected result are unchanged.

Task and overlap

The claim is in discussion #3145. The full task card is in research/coupled-t20-window-audit/TASK.json.

• Task/track: complex-new-helper-inventory / complex-mechanisms; single chooser nonce 040d85d315302c1011c104d7d415cc28; scope key 5868e5281a1e63d41a22.
• Actual object: coupled-T20-two-orders-window-obstruction. Scope changed from proposing an inventory to PR test(research): independently audit the coupled-carrier packing cap #3153's explicit smaller-replica handoff with inventory fixed. The third suggestion exposed Discussion HANDOFF: T740 allocation for the rank-7 coupled inventory #3145; PR Realize the coupled-carrier T740 packing cap: κ ≈ 0.001729777494678597 #3148/test(research): independently audit the coupled-carrier packing cap #3153 already completed its original T740 journal task.
• Domain: nine stages, 408 helpers per call, eighteen idle children and 112 ports. Only two T20 orders are constructed for the audit; the general formulas follow from the written proof. Both providers have 99 width-eight roles per call.
• Milestone: audit/scoped obstruction. Stop after the source-window and cost implications; no allocation synthesis. The complete inherited local profile remains {1:2433,2:724,3:228,4:102,5:20,8:8} per call. The inherited stock identity would count 18,429 families at T20, conditional on a complete realization.
• Success evidence: exact source cuts, Hall-count witnesses, positive shipped-order control and strict dual cost penalties. Unknowns: arbitrary interleavings, all-width assignments, native/physical/prefix/outer admission and all-size obligations. The bounded session stopped at the declared obstruction.
• Base and final fetched main: 463113d675af7410ac91d53a4fed51ec6494eb9a, priority revision 6. Refresh at 2026-10-17T17:36:47Z found no changed paths or guidance.

Exact predecessors: PR #3153 c509607d1208396ec0045f0ab79f8392fce40b01 supplies the cap audit, reused arithmetic and handoff; PR #3148 f2927a4b8afcf0701a6b945e14f30de1d94f5176 constructs T740; PR #3143 dcfaf1d1a7cfebaab22b95776a36a543bc684d50 supplies the T74 comparison. PR #3133 supplies the coupled inventory, and PR #2979 be191346391661f4d35fc0d1508d0a6029572576 the allocation order, plan and template read here. All consumed bytes are pinned to the base in SOURCES.json.

Offline overlap queries covered the card, T20 AND coupled, and stage-major AND coupled, main packages and Discussion #3145. Final snapshots report 249 open PRs including drafts at 17:26:19.594925Z and 2,001 discussion records at 17:27:03.117721Z on 17 October, with complete pagination. The refined query found only this claim/package. A later snapshot (253 open PRs at 17:49:11Z, 2,015 discussion records at 17:39:04Z) shows no other T20 chronology work. Open PRs #3184, #3190 and #3198 report higher conditional kappa on PR #3148's T740 fill by moving idle capacity between bins; that lies outside the one-idle-child-per-bin pricing assumed here, and this audit neither covers nor compares them. Unpublished work and arbitrary diffs are outside this coverage.

Validation

CPython 3.12.3, standard library only. The registered replay and all nine behavioral groups pass.

python3 -I -B research/coupled-t20-window-audit/verify.py
python3 -I -B research/coupled-t20-window-audit/test_verify.py
python3 -B scripts/research_package.py --package research/coupled-t20-window-audit
python3 -B scripts/check_research_commit.py --package research/coupled-t20-window-audit

Expected primary output:

PASS_COUPLED_T20_WINDOW_AUDIT stage_major_deficit=1857 replica_major_deficit=7500 baseline_T74_deficit=0 cap8_bad_penalty_gt=3/250

The new forward reconstruction matches all 666 serialized T37 windows. The shipped T74 order matches all 49,728 required bins in the necessary relaxation. A separate exhaustive subset oracle agrees on 5,670 small matching cases. Controls reject incorrect window boundaries, capacity overflow, family aliasing, missing ports, duplicate/noncausal calls, provider/primal mutations and corrupted sources. The full optimized CLI passes. The inherited exact arithmetic replays all 2,551 patterns and all six non-tight capacity-eight patterns; this is authenticated reuse, not a claimed independent arithmetic engine.

Metadata/pins, whitespace checks and all 13 package/input publication-byte comparisons pass. A Git-free export of committed implementation and declared inputs passes under -I -O -Werror -B.

Not run: native compilation, all-width physical allocation, complete selected cold replay, physical stock/prefix/outer admission, hosted CI, proof-assistant checking or all-size/analytic validation. The positive matching control is a necessary relaxation, not a full allocation certificate.

Credit

Nick Petalas retains PR #3153's audit, reused arithmetic and handoff, and PR #3024/#3030's allocation/cap work. Jeremy Diviney retains PR #3148's construction; dcharb78 retains PR #3133's inventory; Gonzalo Mier retains PR #3143's comparison. Credit Marius-Paul Dumitrean/PR #2979 for the allocation order, plan and template read here, Tekalign's owner checks, Arham Tahir/PR #3136, Hiroki Kumazaki's old-journal transport test, and the retained geometry/native/owner contributors. Original notices remain unchanged.

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance. Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

Can a genuinely interleaved T20 order attain the cap? First enforce the three per-owner stage gaps of at least three calls for stage pairs (1,2), (3,4), (5,6), then the complete capacity-eight interval matching constraints. Passing still leaves a tight all-width assignment, physical-family exclusions, bridges/frames, restoration and the complete physical/prefix/outer bill. This result does not prove T740 necessary.

Workflow feedback (optional)

The handoff-first process exposed a concrete remaining question. The chooser still surfaced Discussion #3145's original T740 HANDOFF although later comments record completion. Updating that canonical block to the smaller-replica question would reduce repeated intake of completed work.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG