Certify the ideal combined-lever budget grid of the selected complex local word (screening; no new kappa)

[petalas]

Summary

Adds the full ideal combined-lever budget grid for the selected complex local word inside the retained research/complex-local-word-ceiling model: centre charge {64 selected, 32 shared, 0 free} x local shape {published word, all paid mass delivered as (fractional) children of rank 8/16/32} x unpaid coordinates per live family {1, 2} — 24 cells, each an exact rational bracket at grid 1e-12 from the parent ceiling() (outward 2^-256 enclosures). Produces the planning/screening answer the one-at-a-time lever table leaves open: which lever combinations can reach the ambitious κ ≥ 1/99 target inside the model, and which cannot.

Scope, up front: ideal favorable relaxations for planning — fractional child counts, ideal idle at rank 51, no realization/hosting/restoration/precision/all-size cost for any lever. No new κ (selected record unchanged at 0.001672660197076918), no supplier admission, no frontier change. Complex side only.

Package: research/ideal-lever-grid-budgets/. Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90 (canonical main, fetched this round). Claim: discussioncomment-19027939.

What is established (finite, exact, inside the stated model)

• 12 of 24 cells reach κ cap ≥ 1/99 (≈ 0.010101010). The reaching set is upward closed in the lever lattice (0 monotonicity violations across all 276 comparable pairs); its minimal elements are exactly:

minimal reaching cell	a cap	κ cap = a/(1+a)
selected centre (C=64) + all-mass rank-8 + unpaid 2	0.010354258	0.010248146 (+1.46 % over target)
shared centre (C=32) + all-mass rank-32 + unpaid 1	0.012887708	0.012723728
free centre (C=0) + all-mass rank-16 + unpaid 1	0.011008425	0.010888560 (the published pair row)

• Notable non-reaches: free centre + published word + u = 2 tops out at 0.009322242 (7.7 % short); free centre + rank-8 + u = 1 at 0.007655385. So within these lever families no {centre relief} x {coverage relief} combination reaches the target without reshaping the delivery rank, and coverage relief (u = 2) is the only single lever that paired with rank-8 delivery at the full centre charge crosses 1/99.
• The tightest reaching margin (1.46 %) is the same order as the ≈ 1.35 % worst-case joint-integrality loss reported by #3811 in the baseline centre model, so that cell is not target-clearing under integer counts until its integer twin is computed (Open next question 1 in the package).

Controls

verify.py (PASS_IDEAL_LEVER_GRID_BUDGETS) runs: byte-exact recomputation of all 24 brackets against RESULT.json; endpoint certification relax(a_lo) < 1 < relax(a_hi) on every row; profile invariants (idle mass 1,646 at the selected cell; child ranks < m and ≤ R; paid masses 5,586/5,554/5,522); live execution of the pinned parent levers.py with agreement on all seven published rows (×1.00/×2.80/×1.90/×2.31/×1.62/×3.77/×6.51, baseline cap 0.001690054109089); lattice monotonicity; fail-closed guards (R = 52, unpaid 0, over-weight profile all rejected); bracket nesting at 1e-8 around 1e-12 on three cells.

Reproduce

python3 -I -B research/ideal-lever-grid-budgets/verify.py   # ~11 s locally, PASS_IDEAL_LEVER_GRID_BUDGETS
python3 -I -B research/ideal-lever-grid-budgets/grid.py     # ~7 s, prints the 24-row table
python3 scripts/research_package.py --package research/ideal-lever-grid-budgets
python3 scripts/check_research_commit.py --package research/ideal-lever-grid-budgets --check-source-history

Locally run and passing under CPython 3.12 at head fb86b078a53042974d88b9758fe36622571349d0 (registered verify ~11 s, well under the CI budget; the interpreter used was a repository virtualenv 3.12). CI replay certifies the registered check under the canonical environment.

Related open work

Four open peers price budgets/ceilings for the same selected complex local word (none modified here). Cross-checks below were run numerically against the package RESULT.json; agreement is claimed only where stated.

• feat(research): exact integer idle-histogram ceiling for the selected complex word #3729 — same model as my baseline cell (sel|word|u1) with integer idle counts. Overlap: the shared inherited ceiling 0.001690054109089 lies inside my baseline bracket [0.001690054109, 0.001690054110] (checked), and its attained integer ceiling 0.001688603446527 sits strictly below it, as integer tightening implies; its 0.0858 % integrality-loss column recomputes to 0.0858 % against my grid cap. Agreement.
• Research: price the paid-mass elimination budget of the selected complex word (screening budget; no new kappa) #3799 — same inherited relaxation at φ = 0 (anchor identical to the quote above, checked inside my bracket), then prices free removal of paid mass re-packed as ideal idle. That removal axis is disjoint from my rank-reshaping grid; no shared cell beyond the baseline. Its full-elimination figure 0.013675160009708 reproduces the stated 8.0916× against my grid-level cap (checked).
• Certify the idle rank-cap sensitivity of the selected packing endpoint (no new kappa) #3805 — same word/coverage/centre but sweeps the idle rank cap with optimal integer idle packing (aggregate 144-call normalization, same w = 896/9). My grid fixes idle at rank 51, so no cap endpoints coincide; the quoted fractional ceiling κ < 0.001687202646 agrees with my baseline κ cap to the quoted 12 decimals, and its per-call integer bracket lower end 0.0016886034465262156 lies below my baseline bracket as integer tightening implies. Agreement on the shared fractional ceiling; otherwise disjoint.
• feat(research): certify the joint integer rank-cap ceiling of the complex local mass #3811 — the integer twin of my shape lever at baseline centre/unpaid (joint integer paid+idle packings, caps 1..51). Shared quantity: its fractional class-relaxation control at cap 16 is the parent rank-16 lever row, which my sel|16|u1 cell reproduces live ([0.003909456001, 0.003909456002]); its cap-8 integer-idle row 0.00273511 sits 0.160 % below my fractional sel|8|u1 cap — the consistent direction (integer idle tightens the fractional ceiling). Its cap-44 κ 0.01020939 is close in magnitude to my minimal reaching cell sel|8|u2 κ 0.010248146 but rests on different premises (integer caps at baseline centre/u1 vs fractional rank-8 delivery with u = 2): no shared quantity, no comparison claimed. Consistent where premises coincide.

No contradiction found with any of these; the shared (or transitively shared) quantities — the baseline relaxation ceiling, the parent rank-16 lever row, and #3729's integrality-loss column — all agree numerically.

Limits and unrun obligations

• Every reaching cell is a screening number, not feasibility: u = 2 has no known priced contract (the line-port coverage law forces one unpaid coordinate), shape delivery ranks have no producer here, and center sharing/free rows omit their realization bills. See the package README's Open next questions.
• Nothing here addresses the bit supplier, chronology, interfaces or all-size admission; κ > 1/99 needs both sides.
• No frozen input, parent package or another contributor's file is touched; no κ, catalogue or frontier change.

Credit: Nick Petalas. Produced by Nick Petalas with AI assistance. Independent human and formal proof-kernel review are unclaimed. Model, coverage identity and ceiling tool credit their authors in the parent package. Apache-2.0.