Research: prove a chronological packing ceiling for every uniform shift of PR #2571's T74 order

[commdata2338]

Change and mathematical scope

For the fixed PR2530/PR2571 422-helper inventory and paid profile, every nonnegative integer uniform initial delay of stages 1–8 has primitive endpoint b < 0.001714127696. This is a scoped proof and exact finite certificate, with no new κ or supplier-admission claim.

A named set of 102 rank-eight groups contains 11,424 bins but can draw width-eight roles from at most 145 calls, each supplying 78. At least 114 bins contain no width-eight role. This does not say that those bins must be empty. An exact dual over all 1,163 whole-helper patterns prices the shortage, including every fixed local child and residual bin. At the threshold its lower bill is 66304.0000001723… > 66304. A separately checked feasible fractional primal shows the window-free LP still permits that same threshold.

The formula for the eligible-call union is 117 + max(0,min(28,69-s)). Seventy-one finite order checks and the written stable-order argument cover every integer s >= 0; s=0 is PR2530 and s=4 is the fetched PR2571 head. The upper bound leaves less than 9.54e-8 above PR2571's pinned primitive endpoint. No packing optimum or fill attaining the bound is claimed. PR2588 changes the local profile and is outside this numerical theorem.

Package: research/t74-diagonal-window-bound (proof in its README.md). The selected record and all predecessor packages are unchanged.

Task and overlap

The claim is in discussion #2608.

• Task ID / scope key: selected-packing / ab5d8f0f0d19bbab8625; refined scope PR2571/uniform-initial-shifts/102-group-window-cut/exact-pattern-dual.
• Selection: priority revision 4, nonce 2ab87a123c529f26e4178a126f56a586, incumbent-optimization, second suggested card. The R439 scalar-to-moment suggestion was passed over in favor of this live, concrete owner-order residual.
• Base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Final canonical-main refresh at 2026-10-15 16:29:48 UTC found no changes to main, guidance or the selected record.
• Exact predecessor/dependency heads: PR2530 b4012098048e12500b2d33d2e84be281915a27ac; PR2571 faff03cf5c29dadd6e0eb51e406efdc921bfa683. The template member is retained verbatim from PR2530's pinned PR2435 archive (origin head 324d7d3590dc21003940a44cb0cbdc32b3620f25), with container/member hashes. PR2442 435e5c412fcad664cef2b52cf6adf1c9f23ffc87 supplies methodological credit, not imported checker code. All three PR heads were rechecked anonymously before submission drafting. There is no runtime dependency on another pending PR checkout.
• Concrete object/question: actual strict source-event windows and the width-eight supply missing from the window-free pattern relaxation.
• Scope status: changed. The historical selected-packing card is applied to PR2571's explicit live residual on the borrowed T74 line. The fetched order clarified the cached PR body's stale description. One bounded extension covers all nonnegative uniform initial shifts.
• Changed premise: retain chronological availability instead of discarding it; keep the inventory, local paid profile, live stock and source-event recipe fixed.
• Method/domain: exact integer source-event reconstruction, independent stage-position windows, one Hall counting inequality, rational cost enclosures and all 1,163 integer patterns for widths 4/5/7/8/9 in bins 8/16/32/48. There is no optimizer-seed novelty claim.
• Full budget and success test: all 666 local profiles plus every residual bin, divided by 66,304 live families. Success is strict moment exclusion with a feasible window-free primal below one, not a scalar count or a new construction.
• Stop rule / resources: one bounded chronological obstruction, exact controls, standard-library single-process replay; no synthesis or global-optimality search.
• Artifact/reproduction: README proof, CERTIFICATE.json, immutable INPUTS.json pins, and the registered command below. The full structured task card is in TASK.json.

Nearest overlap: PR2442 and PR2466 certify older inventories using window-free patterns. PR2530 uses temporal cuts for search nomination; PR2571 and PR2575 give feasible fills. This result proves a necessary priced cut for an entire order family. The concurrent audits in discussions #2605 (local frame charges), #2606 (lender values), and #2616 (actual owner-map dominance) concern different implications. Discussion #2612 searches changed assignments and profiles, outside the fixed-profile numerical theorem. PR2621, opened after this claim, uses the nonuniform half-stage order (8*stage+2*replica+[stage==2],stage,replica) on PR2588's changed words. Under this package's source windows that order gives the 102 named groups exactly 147 eligible calls, so this cut does not apply to it, and its higher κ is outside this theorem.

The final shared snapshot had complete pagination: open PRs at 16:37:18 UTC (823 records), Discussions at 16:27:10 UTC (1,276 records). Queries included the selected card, diagonal order, source-partner, Hall bound, initial shift, chronological packing, diagonal-window and 114-bin. Coverage is explicitly cached and not a novelty guarantee.

Validation

Python 3.12.3, Linux, standard library only:

python3 -B scripts/research_package.py --package research/t74-diagonal-window-bound --check-only
python3 -B scripts/research_package.py --package research/t74-diagonal-window-bound
python3 -I -O -B research/t74-diagonal-window-bound/verify.py

Passed metadata/source provenance, registered replay, and isolated optimized replay. Expected mathematical output:

PASS_T74_DIAGONAL_WINDOW_BOUND shifts=71+stable_tail groups=102 deficit>=114 patterns=1163 b<0.001714127696 controls=10 no_new_kappa

The checker binds five retained inputs; reconstructs 1,332 windows per order; independently checks all 444 rank-eight windows per order; verifies exact bin/width equations, complete rank mass and deficit; and checks every pattern inequality. Logarithm and exponential bounds use exact rational series with explicit tails. Linear, square-root and constant identities control the arithmetic. The default replay is read-only and compares the complete regenerated certificate with type-preserving canonical JSON.

Ten controls reject source mutation, reversed replica stages, an inclusive bridge endpoint, an omitted source bridge, duplicated cut groups, a claimed deficit of 115, unbound width-eight inventory 79, an inflated bin price, an inflated per-defect penalty, and an unbalanced relaxed primal. The predecessor endpoint is not rejected. No check relies on Python assertions being enabled.

Not checked: full scalar/native word replay, physical realization of a new owner fill, complete cold supplier admission, arbitrary/nonuniform schedules, alternative inventory or live stock, split helpers, sharpness of the final ceiling, all-size/compiler, precision/exact recovery, outer transfer, bit interface, Lean, or the full multiplication theorem. The finite certificate and written stable-tail argument establish only the stated packing obstruction.

Credit

Rohan Arun (PR2571), zxv (PR2530), Jeremy Kothe (PR2435 and the T74 supplier lineage), Ali Romman (PR1129), Jeremy Diviney (PR2360), and Sagar Thakrar (PR2387/PR2442) supplied the question, source construction, pattern model and prior methods. Jeremy Kothe's PR2575/PR2588, Philippe Van Velzen's PR2559, Tekalign's PR2436, Vlado Vojdanovski's PR2322, Cinna Davis's PR2226 and the retained supplier authors are credited for related mechanisms and constructions. The new chronological cut, uniform-shift proof, strengthened dual and checker are prepared for @commdata2338. Retained source bytes and the new work preserve Apache-2.0 provenance.

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

1. Use a nonuniform stage shift to give the named groups at least 147 eligible calls, then check all other window and inventory constraints before building a physical fill. Every uniform initial shift retains this cut. PR2621's half-stage order already reaches exactly 147 on PR2588's changed words; for the fixed PR2530 profile, the open step is a fill or a new priced cut under such an order.
2. Couple the width-seven scarcity of early rank-16 groups to the width-eight cut. A rank-16 bin can replace its width-seven role with two width-eight roles at zero reduced cost, so the width-seven count alone is not a priced obstruction. The next step is one exact coupled dual, not another root calculation.

These are structured in the package's TASK.json; neither is claimed solved here.

Workflow feedback (optional)

Checking active claims before computation exposed an overlapping source-lifetime audit (discussion #2606), so a distinct packing question was chosen. The largest detour was that PR2571's cached description and fetched README/build/order disagreed about the schedule. Including the actual order formula and exact artifact link in each current-status summary would make subsequent task selection more reliable.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG