Research: certify the staged544 packing ceiling and reach it on PR3734's T360 clock (no new κ; same value as #3802)

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Answers the first open next question of #3734 ("Can a changed source clock attain more of the ideal544 packing gain?") and closes the packing question for its staged544 supplier. Stacked on #3734's head fa041037a3374eaa08feeca0b26a5d6893194147; only research/staged544-clock-ceiling/ is added.

No new κ. #3802 (@hexapode, T385) reached the same conditional κ = 0.001730578437266246818740 − 10^-46 − 10^-70 while this was in progress; its idle histogram is exactly 385/360 times the one here. No priority is claimed on that value.

What this adds

1. The ceiling is a cap. ceiling_certificate.py gives an exact rational dual over all 2551 idle-bin patterns of ranks 8/16/32/48: every owner fill of the staged544 inventory, under any clock and any replica count, has paid-profile root below 0.001730578437266246818741. Neither Stage helper544 after a returning producer excursion; improve conditional kappa #3734 nor Improve conditional kappa beyond T488 with a T385 combined-clock allocation #3802 claims packing optimality; with this, further gains for this supplier need a changed inventory, template or word, not another allocation.
2. Why Stage helper544 after a returning producer excursion; improve conditional kappa #3734 lost 8.65 bill units: its clock, at the start. An exact per-tick LP (Hall conditions per width on every tick interval, all patterns priced) caps Stage helper544 after a returning producer excursion; improve conditional kappa #3734's clock at 0.0017304575193851, 9.2e-9 above its fill. The binding cut: each stage-1 call closes its owner's rank-16 pair (−1, τ1], whose ideal content is 224 × (7, 9), while a call supplies 53 width-7 helpers, and 8 replicas reach τ1 between ticks 40 and 47.
3. The ceiling on Stage helper544 after a returning producer excursion; improve conditional kappa #3734's own clock and stock. clock.py applies one rule to Stage helper544 after a returning producer excursion; improve conditional kappa #3734's ORDER.json (a stage-1 call waits while the placed stage-1 count would exceed 53/224 · (t + 1)); 727 of 3240 calls move. T360, 323,069 physical families (Improve conditional kappa beyond T488 with a T385 combined-clock allocation #3802: 345,469; both 19 stock bits, so no margin changes). Idle histogram {1:84240, 3:23328, 5:4320, 48:137952}, root 0.001730578437266246818740, which is the ideal root Stage helper544 after a returning producer excursion; improve conditional kappa #3734's PROOF.md calls infeasible on its fixed clock.
4. Stage helper544 after a returning producer excursion; improve conditional kappa #3734's complete admission, run end to end through its full_replay.py --reuse-current-native wrapper on the changed inputs: PASS_STAGED544_COMPLETE_CONDITIONAL_RELEASE, conditional κ 0.0017305784372662468187399999999999999999999998999999999999999999999999 (recorded in evidence/RELEASE.json, bound to the inputs by verify.py). Stage helper544 after a returning producer excursion; improve conditional kappa #3734 and Improve conditional kappa beyond T488 with a T385 combined-clock allocation #3802 each state their wrapper was not run end to end.

Verification

python3 scripts/research_package.py --package research/staged544-clock-ceiling --check-only
python3.12 scripts/research_package.py --package research/staged544-clock-ceiling

Both pass locally (CPython 3.12.14, about 15 s), as does check_research_commit.py --check-source-history. verify.py is standard library. It regenerates the clock from its rule and runs #3734's byte-pinned owner_audit.py on the changed ORDER/FILL/PROFILE beside #3734's unchanged DISPATCH/NATIVE/PLAN/TEMPLATE (PASS, three mutants rejected). It also checks the block-norm helper/data alias premise on every role, encloses the root with #3734's rational_price.py and checks the ceiling certificate. Six semantic controls are rejected: #3734's clock with the new fill, a same-replica stage swap, a bracket shifted by 10^-24, one idle rank-48 child split 47 + 1, the certificate at root_lower (bound not positive), and a dual perturbed by 10^-30 (infeasible).

The LP and the reverse-time fill builder (numpy, scipy, highspy) are in construction/ as provenance and are not run by the verifier.

Scope

Conditional finite construction in exactly #3734's sense: every inherited all-size compiler, realization, hosting, precision/recovery, analytic and BIT assumption remains. The certificate is scoped to this supplier, idle-child template and paid-remainder rule. No unconditional theorem, formal verification or selected-frontier change. If #3734 changes head, this needs a rebase and rerun.

Open next questions

Which changed T360 inventory or word raises this ceiling while keeping the complete admission (first step: price inventories with the fixed-inventory LP or the certificate's dual); and whether other owner fills of this model fall short of their own ceiling because of the clock (first step: the per-tick LP on their ORDER.json). Details in the package README and TASK.json.

Credit

Builds on @dataisfire's #3734 (owner audit, rational price, fixed-inventory ideal and the question answered here) and, through it, #3682, #3638, #3605, #3687/#3693 and the authors in its SOURCE-CREDITS.md. #3802 and #3795 (@hexapode) are concurrent; #3802 reached the same κ first. Prepared with substantial assistance from Claude (Anthropic, Claude Code). Apache-2.0.

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