Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa)

[Mister-Meeseeks]

Scope: a scoped obstruction plus a screening tool. No new κ, and the selected record is unchanged.

Task card

• Card: complex-new-helper-inventory (complex-mechanisms track, chooser seed f10085bdeb241ba1cd1390dfec98a448, base e3bb23b4). Scope is refined to the card's first experiment: an optimistic exact relaxation that rejects routes without headroom. It is applied to packing of the incumbent word and turned into a calculator for proposed inventories.
• Question: with the selected Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 local word fixed, how much can any hosting, owner or idle-remainder repacking gain? What would a changed helper inventory have to pay to reach a = 1/99?

Result

• The selected local word pays {1:2412, 2:418, 3:432, 4:142, 5:82, 8:8} per call (mass 5,586). Live stock is 896 data families per nine-stage replica, with all helpers hosted.
• PR1012's coverage identity (mass = 72W + 64N, re-checked on the selected profile) fixes the idle mass at 1,646 per call. Cost per unit mass r^(-a)·73^(a-1) does not increase with r, so the best idle histogram puts all idle mass at the halving-guard maximum rank, 51, with fractional counts allowed.
• Packing ceiling: a < 0.001690054109089, so κ < 0.0016872026 (+0.869%). This is certified with exact rational intervals.
• Previous record Conditional κ = 0.001672188119205421: improved T16 owner packing on the seven-slot k8 word #1042: it splits as the same local word plus an admissible idle histogram. The derived idle histogram {1:2249, 2:1, 3:400, 7:3, 16:47, 48:4850} matches its README exactly, and its endpoint lies below the same ceiling. So all of Conditional κ = 0.001672188119205421: improved T16 owner packing on the seven-slot k8 word #1042→Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 was packing, and less than 0.87% of that headroom remains.
• Planning target a = 1/99: written as Σ n_r p_r(a) ≤ B(a), the word's excess spend is 20.4 times the per-call budget (2.597 against 0.1275). Deleting all rank-1 children cannot reach it. A target-scale gain needs about 95% lower fully priced local excess spend, or a changed coverage law: unpaid coordinates per family, the 64-per-call centre charge, or families per call.
• Tool: ceiling.ceiling(local, 73, 896, 9, 64) screens any proposed per-call local profile in about a second.

Not covered

• This is one local word, port count, template, coverage law and outer recipe.
• Open claims that change the word (Conditional κ = 0.001675560947524449: joint k8 entrances, rank-seven ends, and T16 hosting #1148's joint-entrance word and its descendants, all ≤ 0.0016759) are not bounded by it. Their ceilings are an open question in the README.
• Fractional counts and every repacking are allowed, and realizability is not checked.
• κ < a/(1+a) is the retained planning ceiling.

Controls

• An idle-rank cap of 52 is rejected by the halving guard.
• A local mass beyond the coverage law is rejected.
• One extra rank-1 child lowers the ceiling.
• 25 sampled integer idle histograms all have exact moment > 1 at the ceiling.

Reproduce

python3 -I -B research/complex-local-word-ceiling/verify.py   # ~10 s
python3.12 scripts/research_package.py --package research/complex-local-word-ceiling
python3 -B -m unittest discover -s tests -p 'test_research_*.py'

Credit: the coverage identity is PR1012's. The fixed-word packing-ceiling lineage is static44, PR525 and PR643; the catalogue mixture bound is Ali Romman's heterogeneous-owner-assembly. The construction credits are in the selected review. Prepared by Douglas Colkitt with substantial Claude (Anthropic) assistance. Apache-2.0.

🤖 Generated with Claude Code