Conditional κ ≈ 0.0017224999064433: cap-attaining allocation of #2917's unchanged words (+9.11e-7)

[mdumitrean]

PR #2917 ships a literal T74 fill that is 9.11e-7 below the exact fixed-capacity packing optimum of its own inventory. This package packs PR2917's unchanged words at that optimum and passes the complete admission:

κ = 0.001722499906443274564357 − 10⁻⁴⁶ − 10⁻⁷⁰, exactly 113907405684085469/125000000000000000000000 ≈ +9.1126e-7 over #2917 at 224bd5e56041c40078ff6b5fa7393733a215de5c (about 0.053%).

What changes

Only the literal allocation: allocation/FILL.json.gz, PROFILE.json, SEARCH.json. The two provider words, ORDER, DISPATCH, per-helper widths, the 82 private buffers, the owner engine and its plan/template are #2917's, byte for byte (pinned in SOURCES.json). Physical stock stays 66,804; 271,728 helper roles.

	complete paid profile
#2917	{1:1625724, 2:426240, 3:191980, 4:73260, 5:24951, 6:167, 7:45, 8:5328, 24:1, 32:207, 48:26958}
this PR	{1:1626372, 2:426240, 3:192474, 4:73260, 5:24198, 8:5328, 48:27158}

Why this is the optimum for these words

The fractional relaxation (bins 672/448/448/448 per replica of capacity 8/16/32/48, all 1,163 count-vector patterns, one idle child per bin, the complete local profile paid) of the inventory {4:26,5:2,7:62,8:96,9:222} with local profile {1:2416,2:640,3:280,4:110,5:26,8:8} has cap [0.001722499906443274564357, …4357 1] and an integral optimal basis per replica, so an allocation with exactly that pattern mix has no rounding residue. check_cap.py verifies, with exact rationals only, the basis' primal feasibility, the dual sign of all 1,163 reduced costs at the upper end of the root bracket, that the literal complete profile equals the optimum's (666 × local + 74 × idle), and that its root bracket is the cap bracket. #2917's fill pays 207 empty cap-32 bins and 213 rank-6/7/24 idle children; every single relocation from it is uphill (a single-move fixed point), which is why exchange repair stopped there.

The allocation was built with #2857's chronological flow model (PR2843 lineage) on PR2833's T37 chronology with the integral mix as pattern targets (single-thread CP-SAT, edge-colouring realization), then doubled exactly onto #2917's T74 order (which is two interleaved copies of that T37 chronology).

Validation

• Registered (run_registered.py, stdlib): package pins; Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917's vendored owner engine on the new allocation with five rejection controls; Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917's exact dyadic moment auditor with its baseline re-pinned to the Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917 headline (strict gain); the Fractions-only cap certificate with two controls; binding of the stored receipts to the exact words, allocation and claim.
• Complete replay (reproduce.py): exact Git exports of Conditional κ ≈ 0.001720398337: two-sided helper extraction and charged T74 packing #2908/Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917 (and through them Conditional κ = 0.001720184528377469: two more staged quartets on #2863's shared-kernel words #2894/Conditional κ = 0.001718657437166855: dirt-entry rewrite of helpers 530 and 542 on #2802's construction #2833), fresh retained bootstrap, Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917's derived source (manifest 98d57c82…, 632 files), closed adapters with the Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917 headline as comparator, fresh five-stage native chains for both providers, complete admission (owners, parent lift, rebind, discovery, block norm, invoices, independent outer checks, release): PASS_REPLICA_ORDER_FILL_DERIVATIVE_COMPLETE_RELEASE, PASS_RELEASE, strict gate κ > Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917 passed. Recorded as one fresh invocation (macOS arm64, CPython 3.12.13): bootstrap 279 s, natives 25 s, admission 256 s, 563 s wall.
• Independently, our chain reproduced Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917's own claim exactly before admitting the new fill, and the T37 form of the same allocation (333 calls) passes the same admission at the same κ.

git fetch https://github.com/CrocSwap/integer-mult-bounds.git 224bd5e56041c40078ff6b5fa7393733a215de5c \
  51309592971e3b5be935599eed2a95404dc7781e 18d03c447886895ef9abf888398178e0428533ca e9307b48d7d1b690939935ecef0f8e524e6d92f4
python3.12 scripts/research_package.py --package research/paired-extension-cap-fill
python3.12 -B research/paired-extension-cap-fill/reproduce.py --repo . --work /tmp/cap-fill-complete

Scope and next

Conditional finite supplier/outer certificate with all inherited hypotheses; no selected-frontier change, no unconditional theorem, no independent human review. The optimality statement is about packing these fixed words under this relaxation. The fill layer of #2917's words is now exhausted: the next gain must change the words (further paired extractions on the final 82-endpoint word; one at #2917-step-1 cost would be worth about +3.6e-7 after a cap re-pack). Coordination: Discussion #2612.

Credit: Mason Hale (#2917) for the words, chronology and admission tooling reused unchanged; Jeremy Diviney (#2908); Jeremy Kothe (#2894/#2833); Yiyang Lu (#2857/#2848) for the flow model and realization; Dugongue (#2863/#2839); Tekalign for the owner checker. Marius-Paul Dumitrean, prepared with substantial AI assistance (Claude Code). Apache-2.0.