Negative result: +10% centre sharing at c* ≈ 0.0018433 needs 511 net complex rank-1 units (3.55/call, not 1.33); no admitted bit lever nets any of the 649 needed (no κ)

[thakrarsagar]

Statement

Planning bound and scoped proof. No construction, no κ claim, and not a multiplication theorem. The selected record is unchanged: κ = 0.001672660197076918 (#1094), complex-bound.

This prices the +10% centre-sharing route at the bit endpoint c* = 921658892752012102303/5e23 ≈ 0.0018433. That endpoint is 1.0634x the current value, and PR1791 (centre-sharing-overhead-budget) derives it for +10% whole-bound κ. It also answers PR1791's open question 1. The bit side uses the actual PR589 histogram: deficit 10,272, m = 64, W = 28,416, T = 32, 288 calls, nine rank-7 copies per call, cst = 63. Every lever changes exactly one premise, and the surcharge is recomputed each time.

Complex side (correcting PR1791 OQ1). κ ≥ 1.1x selected requires the complex endpoint to reach c* itself. PR1791's figure of ≥ 1.33 units per call is not enough. The requirement is a sharp 511 net rank-1 children (510 fails), about 3.55 per call over 144 calls. Necessity is asserted: κ(c*−1e-24, c*) < target ≤ κ(c*, c*). Under two-call sharing, the complex overhead budget at c* is 4,079 rank-1 or 511 rank-8 children, about 28.3–28.4 units per call. That is an additive rank-1/rank-8 budget under PR1791's treatment; mixed ranks allow slightly more (PR1888).

Bit side. The requirement is 649 net rank-1 units; 648 fails.

• Every admitted lever nets ≤ 0:
  • (a) Halving depth 3/4 (idle cap 45 → 50/53) nets exactly 0. The recipe's idle windows are {7, 14, 28, 42}, so no child exceeds 42 < 45. Bridges delimit the windows, so no child can span two of them. PR1334's cap-50/53 columns and its HD-4 escape therefore cannot be reached by any recipe in PR1487's nine-stage class.
  • (a') Width-8 re-hosting nets exactly 0. This is certified by a concavity argument and re-certifies Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023 at c*.
  • (b1) Rank-8 copy carriage nets −2,577.6.
• Clearing 649 needs a new word, decoder or coverage law. Sharp sizes:
  • (b2) Remove 93 of the 2,592 copies record-wide (92 fails). A uniform decoder change removes 288, one per call.
  • (b3) Bit-side two-call centre sharing (cst 31.5) gives +9,039.5, with overhead ≤ 8,391 rank-1, 1,203 rank-7 or 201 rank-42 children (29.1–29.3 per call).
  • (c) A second unpaid coordinate on ≥ 649 families. This is necessary for pure one-unit removal.
  • (d) ≥ 40 pure helper deletions per call (RANK1 model, stage-relaxed pass). 39 fails for every hosting. PR1043's TEMPLATE delta is exhausted at d = 19 with ≤ +170.4.
  • (e) Ideal in-window shapes in ≥ 52 of 288 calls (51 fails).
  • (f) Partial width-7 conversion clears only in Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023's window-free relaxation, at k = 71 per call. Under replica-local windows no k ≤ 74 clears (PR1736). The full all-439 conversion clears, but it needs a changed prefix (Research: two-sided boundary obstruction for R439 width-seven helpers #1128).

Milestone and decision informed

The milestone is a planning bound for centre-path-sharing, which is recursive-interfaces work. The package is filed under bit-helper-contract with a recorded scope change. The decision it informs:

• No admitted bit parameter lever makes two-call centre sharing part of a +10% route.
• On the complex side, the route needs ≥ 511 net rank-1 units, about 3.55 per call.
• On the bit side, it needs one of the new mechanisms above, at its stated size.
• Halving-depth 3/4 admission is worthless as a bit lever on nine-stage windows.

Base, pins, reproduce

• Base: canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. Pins are in research/bit-endpoint-lever-budgets/SOURCES.json, with sha1 and sha256 recorded.

python3 -I -B research/bit-endpoint-lever-budgets/verify.py && python3 -I -B research/bit-endpoint-lever-budgets/independent.py

Expected output (under a minute, standard library only):

PASS_BIT_ENDPOINT_LEVER_BUDGETS need_rank1=649 admitted_levers_max_net=0 copies_sharp=93 deletions_per_call=RANK1:40/TEMPLATE:None width7_windowfree_per_call=71 ideal_shape_calls=52 complex_need_rank1=511 no_new_kappa
PASS_BIT_ENDPOINT_LEVER_BUDGETS_INDEPENDENT signs=75 at c*-1e-24, c* and c*+1e-24 (levers.py interval arithmetic)

Other checks:

• python3 -I -B -m unittest discover -s research/bit-endpoint-lever-budgets -p test_package.py passes; 11 tamper cases are rejected, including a consistent-shift tamper that only the independent sign arithmetic catches.
• scripts/research_package.py --check-only passes, and so does the registered python3.12 replay (round 1).
• verify.py regenerates RESULT.json byte-identically.

Certificates and negative controls

• Exact arithmetic. Sharp integers are computed exactly at c*.
• Independent second method. independent.py rebuilds every boundary histogram with separate code and recomputes all 75 pass/fail signs with bit-revision3-levers-seed03/levers.py interval arithmetic (PR1334: atanh prime-chain ln, Taylor exp with remainder, outward rounding). It checks c*−1e-24, c* and c*+1e-24. The lever definitions and hosting model are shared with verify.py; only the histogram code and the arithmetic are independent.
• Negative controls. Each of these fails:
  • 648 rank-1 children;
  • 92 copies;
  • 39 deletions per call;
  • 51 ideal-shape calls;
  • every budget + 1 child;
  • complex 510 rank-1, and complex budget + 1;
  • TEMPLATE at exhaustion;
  • width-7 at k = 70 (concavity bound);
  • the selected endpoints' next grid points;
  • κ at c*−1e-24.

Scope, assumptions, unresolved obligations

• Every passing row (b2, b3, c, d, e, f) is a pricing pass for a hypothetical word or decoder. None of these is paid:
  • the word or decoder itself;
  • the dirty-input endpoint and restoration;
  • hosting under the real stage windows;
  • the row-coefficient invoice and native precision/tape;
  • admission.
• (b3) also needs the bit analogue of PR1748's ownership/sink blocker.
• Rows (c) and (d) hold only within their stated models.
• Inherited: PR1791's endpoint model, the fixed-histogram bill and surcharge law, the coverage identity at fixed m and W, PR589's H1–H5, and the all-size, compiler, precision and analytic obligations.
• The PR1736 window bound (0.0018099 at k = 74) is cited, not re-asserted.
• PR1334's rank-8 (h = 8) and rank-16 rows are noted as uncertified estimates of about 50 and 34 calls.

Search record

There was no search. Each lever is a one-parameter family, monotone because the bill is affine in the counts. Boundaries come from bisection or an exhaustive scan. Re-hosting optimality uses a concavity certificate.

Credit, concurrent work and what this adds

Predecessors:

• PR1791 (rtyu193-rgb, centre-sharing-overhead-budget): the 649 requirement, OQ1 and the sharp-removal method.
• PR1334 (bit-revision3-levers-seed03): lever premises and interval arithmetic.
• Navinor's selected-suppliers-de-diagnostic: moment enclosures.
• Wronskode's frontier1042-invoice-audit-seed81 (Independent verification: selected PR1042 finite invoice, 9 guards, 14 bootstrap steps and 94 outer slacks (seed 81, no new kappa) #1112): bootstrap.
• PR589 and PR555: bit supplier and coverage law.
• PR1487 (bit589-recipe-floor).
• Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023 and PR1736: width-7 pricing and windows.
• Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281 (mightyOil): partial width-7 slot optima.
• Research: two-sided boundary obstruction for R439 width-seven helpers #1128 (Ali Romman): bit-width-seven-boundary.
• PR2465 / PR1043 / PR1250 (bbMisk): deletions.
• PR1808 (JosieElliston): plane-port copy budgets.
• PR1888 (RossRichardson): mixed-rank overhead.
• PR3075 (navinor): the k = 2 charged costs, 4 ADD + 4 scratch CLEAR + 4 precision per call. Its charged costs fit inside the budget; eligibility and demand coupling remain the blockers.
• PR1708 (1337isnot1337) and PR1748 (ToikY): the complex two-call phase.
• r740-hosted-eleven row_bound: HD admissibility.
• r439-copy-path-sharing-barrier.

The README gives full heads.

Concurrent work: #3873 (Clear1nsight) bounds causal scalar-oracle centre sharing and audits the coverage ledger. That is a different object; neither prices bit levers at c*.

New here:

• pricing at c*;
• the slack-cap observation;
• the bit-side sharing budget;
• the sharp sizes;
• the complex-need correction (511 rank-1, 3.55 per call, against PR1791's 1.33).

Claim: #1167 (comment)

Review provenance

Reviewed by 7 independent AI review passes in the same loop, across two rounds: reproducibility and negative controls; adversarial math and accounting, including a python-flint arb recheck at 320 bits; novelty, provenance and compliance; and a referee. Two revise rounds followed. No human has reviewed this.

Attribution

Sagar Thakrar (@thakrarsagar), with substantial AI assistance from Claude Opus 5.5 via Claude Code in an automated research loop.

🤖 Generated with Claude Code