Research: certified bit-side lever budgets for PR589 at a = 1/99 under the proposed revision 3 (no new kappa)

[thakrarsagar]

This PR has scoped obstructions and a screening table. There is no new κ, and the selected record is unchanged.

• Track: bit supplier, seed 3.
• Card: bit-helper-contract, with the scope changed. The card question is saturated, and the register-model card is done in Research theorem: exact paid cost for binary affine register words #1333.
• Base: bce54fb3.

Why

The proposed priority revision 3 (open #1272, stacked on open #1240) prices the centre-path, rank-shape and coverage-law levers for the complex supplier only. This PR adds the bit-side numbers for the selected PR589 supplier.

Model and premises

The arithmetic uses exact rational intervals, standard library only, and imports no repository engine.

It is #1240's coverage-law relaxation, applied to the bit side:

• Per call, the paid mass is (m−u)w + cst, with m = 64, w = 888/9 and cst = 63. This gives D = W − 63N, as in format-limits (Dfc).
• Local children are either the replayed word or the most favorable in-window shapes: each data register is one rank-7 climb, and each helper is one climb whose rank equals its width.
• Copies are carried at rank 7. This is a premise; at rank 8 nothing changes.
• Idle mass goes to the halving-depth cap, with fractional counts and no surcharge. Each of these steps can only lower the bill, so a certified fail rules out every construction that meets the premises.

The idle cap is a premise tied to the halving depth.

• global_k.py raises the halving depth until 2·maxchild^HD < m^HD.
• The row bound limits HD:
  • The record uses HD 2 under the PR510 bound (row 18355), so the cap is 45.
  • The merged elementary-cover bound (row_bound.py) admits HD ≤ 4, which gives caps 50 and 53.
• Results are reported at caps 45, 50 and 53. Cap 63 is a robustness bound only.

Results (certified at a = 1/99)

Bit lever (cap 45)	root a	×	1/99
Packing ceiling (reproduces merged #1023's free concentration)	0.0019559	1	fail
Centre path free	0.0054196	2.771	fail
Centre path shared by two calls	0.0036864	1.885	fail
Two unpaid coordinates	0.0073120	3.738	fail
Rank-8 children / rank-16 children	0.0032114 / 0.0047309	1.642 / 2.419	fail
Centre free + rank-8 (shipped helper mass)	0.0089383	4.570	fail
Centre free + rank-16 / + two unpaid	0.0131671 / 0.0107587	6.732 / 5.501	pass
Centre free + ideal data + ≤ 332 width-8 helpers (in window)	0.0101021	5.165	pass
Two unpaid + ideal in-window shapes (copies kept)	0.0117941	6.030	pass

The fail rows also fail at caps 50, 53 and 63.

O1 (copy charge 63).

• Even with no helpers and ideal data, the bill exceeds 1 at a = 1/99 at every cap from 42 to 63.
• The ceiling is 0.006794 at cap 45 and 0.009665 at cap 63.

O2 (centre path shared, copy charge 31.5, full-frame width-8 helpers).

• The maximum helper count is 104, 131 and 145 at caps 45, 50 and 53. Each count k is certified: k passes and k+1 fails.
• The strict design-free floor of format-limits section D is 138. It applies to full-frame helpers and is cited as an argued premise.
• The obstruction holds at halving depth ≤ 3 and flips at depth 4.

Joint surface. The README has the table of copy charge × full-frame helper count at caps 45, 50 and 53, for ideal and shipped data shapes.

What it does not say.

• With the shipped helper mass, centre free needs rank-16 children or a second unpaid coordinate.
• But in this relaxation, fewer helpers (≤ 332) or a second unpaid coordinate reach 1/99 inside today's windows.

Complex side.

• Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240's ceiling is reproduced.
• Director: priority revision 3 — aim effort at centre path, rank shape and coverage law (stacked on #1240) #1272's ×6.5 combination (0.011008) needs rank-16 children beyond the window h = 9.
• Inside the window, centre free + rank-9 gives 0.008128 (fail), and two unpaid + rank-9 gives 0.010911 (pass).

Checks

• Bindings. Selected record → frontier1094 certificate → PR589-EXPECTED.json → bit word, endpoint, halving depth and row coefficient. The row-bound and halving-depth source lines are pinned.
• Positive controls. The selected bit endpoint is accepted and the next 1e-24 point is rejected, with the surcharge. Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240's complex ceiling is reproduced.
• Replay. The local histogram equals the record: mass 5,129 and the class split. The profile equals 288 local words plus idle.
• Tests (15). They cover:
  • the interval engine and the endpoint;
  • the relaxation steps;
  • the guards and the halving-depth cap table;
  • the O1 and O2 boundaries at each cap;
  • the in-window centre-free boundary (332 passes, 333 fails);
  • tampering.

Reproduce

python3 -I -B research/bit-revision3-levers-seed03/verify.py           # ~13 s; PASS_BIT_REVISION3_LEVERS
python3 -I -B -m unittest discover -s research/bit-revision3-levers-seed03 -p test_package.py
python3.12 scripts/research_package.py --package research/bit-revision3-levers-seed03

Overlap

• Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240 and Director: priority revision 3 — aim effort at centre path, rank shape and coverage law (stacked on #1240) #1272 (Douglas Colkitt / Mister-Meeseeks): this is the bit analogue, in independent code.
• Independent verification: prove and certify the a < D/E diagnostic for the selected complex and bit ledgers (no new kappa) #1293 (navinor): reconciled in the README.
• Merged format-limits sections A, A.4, A.7 and D. A.7's ×2.79 cross-checks the centre-free row.
• Research: T21/T37 interface budget and bit-binding headroom (no new kappa) #974 (avirazza).
• Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023 (ours, merged): the packing ceiling is reproduced, not claimed as new.
• Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281 (mightyOil).
• Prove a semantic zero-frame D contract for a fixed R439 word #1194, Research: independently check semantic-D bridge for composed R439 word #1311, Finite bit construction: sparse R439 kernel entrance gauge (no new kappa) #1179, Scoped negative result: all-support R439 common-frame kernel entrances #1253 and Research: two-sided boundary obstruction for R439 width-seven helpers #1128.
• Research theorem: exact paid cost for binary affine register words #1333 / Binary affine addresses: exact paid-call floor with arbitrary dirty registers #1317 (bitplane).

Disclosure

Prepared by Sagar Thakrar (thakrarsagar) with substantial AI assistance from Claude (Anthropic) via Claude Code. Parallel read-only scout and judge subagents ranked the question. A post-PR adversarial review found the halving-depth error and two overclaims; all are corrected in the second commit. No claim was posted to Discussions. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 5ad74a4a04d1c9264768c43762e130690918b043.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Certified bit-side lever budgets at a=1/99, rebuilding the retained histogram and comparing explicit coverage, copy and idle-cap relaxations. Infeasible relaxed rows are scoped obstructions; favorable rows are unconstructed targets, not new κ witnesses.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.