Conditional bound: halving degree of the selected complex invoice under the inherited degree-t ledger (t_max 3 retained-binding / 181 bridge-only; R<=57/72; ideal +0.875558%/+2.76469%; no κ)

[thakrarsagar]

Statement

Scoped finite statement, conditional on inherited hypotheses H1–H4; an optimistic opening only. No new κ, no supplier, no frontier change.

Every complex ceiling screen caps child ranks at 51 because of the halving guard 73^t > 2R^t at halving degree t = 2. This package asks whether the selected #1094 finite invoice forces t = 2.

• Bound that holds for any invoice. R = 73 never meets the guard, so every child rank, paid or idle, is ≤ 72 for every t. The idle-cap component (the selected word's paid children held fixed, only idle mass repacked) is worth at most +2.76469% in Colkitt's ideal relaxation, under any row contract.
• Under the inherited degree-t ledger Row(t, S) = t·S + 9909 + 252 + prefix(t), with S = 2642 (t = 2 gives the invoice's 16469):

Row contract	prefix(t)	t_max	Largest admissible R
B, bridge-only: Row < 490196 = ⌊(25·10^6 − 1)/51⌋	1024	181 (row 489387)	72 (needs t ≥ 51)
B	512·t	152 (row 489569)	72
A, retained-binding: Row < 20161	1024	3 (row 19111)	57
A	512·t	3 (row 19623)	57

• Ideal ceilings (Colkitt's relaxation with idle ranks up to R, each certified to a 10^−15 grid cell): R = 51: 0.001690054109089 (the published value, reproduced); R = 57: 0.001704851516268 (+0.875558%); R = 72: 0.001736778936705 (+2.76469%). The ceiling increases strictly over R = 51..72. R = 71 (t_min = 25) is the least rank whose ceiling lies above the selected bit endpoint 0.001733427039589844; for R ≥ 71, κ ≤ a/(1+a) is still an upper bound but no longer the binding one (bit cap b/(1+b) = 0.001730427469823603).
• Conditional what-ifs under reading A (asserted in verify.py, not adopted):
  • Pricing t·S at the sharper coefficient 1/log2(73/R) from selected-coarse-depth-stock gives R ≤ 59 (+1.1501%) with the fixed prefix, or R ≤ 58 (+1.0138%) with the scaled prefix.
  • Research: selected m=73 row-ledger terms do not bind rho (no new kappa) #3667's lean row gives R ≤ 66 or R ≤ 65.
  • All of these stay below the R = 71 crossing.

Milestone and decision informed

Declared milestone: a local module, namely a scoped finite invoice theorem giving the exact admissible (t, R, S) region and exact ideal ceilings. It is conditional on the inherited transfer and bridge hypotheses.

Decision informed: R = 51 is not forced by the invoice under either reading, given H1–H4. How much it is worth depends on one contract question: does retained_row_coefficient 20161 bind?

• Reading B. Favoured by the inherited bridge text, which derives 20161 = 10000 + 9909 + 252 as a halving-degree-1 overcharge: "Increasing these finite overcharges changes constants…" (FINITE_BRIDGE.txt §4, lineage evidence outside the selected manifest). Under B, R = 72 is admissible, and from R = 71 this relaxation no longer forces the complex side to bind.
• Reading A. The fallback, which keeps the replay's pinned constant: R ≤ 57.

The guard caps every child rank. The paid-rank cap s = 51 of the rank-shape screens (open #3701/#3811) comes from the same t = 2 guard, and under H1–H4 it lifts to 57 (A) or 72 (B). That paid-rank lever is not priced or claimed here (open question 5).

Base, pins, reproduce

• Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90 (priority revision 6). origin/main was unchanged at submission.
• Every input is pinned by git blob and SHA256 in SOURCES.json. The selected assembler research/complex-sw4-frame8/code/{assemble,invoice_allcold}.py is hash-checked against SELECTED-SOURCE.json, as resolved from certificates/selected-result.json.
• The selected invoice is tar member PREFIX-AND-INVOICE.json, sha256 212cf4ee…0d12. verify.py checks 41 source bindings.

python3 -I -B research/complex-halving-degree-invoice/verify.py

Expected output (~3.7 s, ~22 MB; byte-equal to RESULT.json):

PASS_COMPLEX_HALVING_DEGREE_INVOICE_SCOPED {'A_retained/fixed': 3, 'A_retained/per_level': 3, 'B_bridge/fixed': 181, 'B_bridge/per_level': 152} {'51': '0.001690054109089', '57': '0.001704851516268', '72': '0.001736778936705'} 11 controls

scripts/research_package.py --check-only gives PASS_PACKAGE_METADATA_AND_PINS. The verifier was tested with Python 3.14 only. The runner's Python 3.12 replay was not run locally because no 3.12 interpreter is available here.

Certificates and negative controls

• Exact integer ledger: t_max is checked both from closed-form boundary rows and by a direct scan of t = 2..200, which confirms the admissible degrees form an initial segment.
• Guard thresholds t_min(R) for R = 1..72, cross-checked against an exact coarse recurrence. This recurrence is an implementation cross-check; it is algebraically the same comparison.
• Ceilings come from my own decimal interval bisection (correctly rounded ln/exp with 1-ulp widening). For every R = 51..72 they are cross-checked at both bracket endpoints against Colkitt's unmodified rational child_cost enclosures. At R = 51 the result equals the unmodified ceiling.ceiling() and the published RESULT.json.
• 11 internal negative controls run through the main check. Each must fail with its stated assertion message:
  • reading A at t = 4, with each prefix;
  • the guard at (t = 3, R = 58), (50, 72) and (200, 73);
  • reading B at t = 182 and, per level, at t = 153;
  • bridge cap + 1 (degree gap not positive);
  • prefix mutation, stock mutation and physical mutation.
• External checks: running with -O is refused, an unknown argument is rejected, and a one-digit edit of RESULT.json is caught.

Scope, assumptions, unresolved obligations

• H1. Degree-t instantiation of the finite-cover transfer. Lineage notes do this at t = 4 and t = 24, closed PR375 states the degree-t ledger, and Research: selected m=73 complex stock shape A=0 under finite-stock-as-C (no new kappa) #3421 gives S as the per-level allowance. The degree-genericity of the written proof is not re-proved.
• H2. prefix(t) ∈ {1024, 512·t}. Unsettled: this decides 181 versus 152, not R = 72.
• H3. Reading A or B. Unsettled: a contract decision. Both are shipped.
• H4. The router, literal, C0, precision, tape, mass and primitive checks do not depend on t.
• Unknown costs, not zero:
  • physical hosting of idle children above rank 48 (Certify the idle rank-cap sensitivity of the selected packing endpoint (no new kappa) #3805);
  • ρ and the per-level divisor for higher-rank children (Selected stock envelope at m=73: bound quantities and the exact missing shape lemma #3113/Research: selected m=73 complex stock shape A=0 under finite-stock-as-C (no new kappa) #3421);
  • native admission (hard-coded t = 2 guards at assemble.py:175,191, ceiling.py:111);
  • any staged544 template change.
• The ceilings are relaxations, not achievable values. This package admits nothing.

Search record

No search. Exact enumeration:

• t = 2..200 for every reading and prefix pair;
• guard thresholds for R = 1..72;
• ceilings on a 10^−15 grid over [0, 1/20] for R = 51..72;
• region rows for R ∈ {48, 51, 52, 56, 57, 58, 64, 68, 71, 72}.

Credit, concurrent work, and what this adds

• philippeVV, #2278 note (local-only), stated t ≤ 181 and ≤ +2.8%. This package confirms ≤ +2.8% exactly (+2.76469%) under every reading, and t ≤ 181 under reading B with a fixed prefix (152 with a 512·t prefix).
• New here: the t-dependence bound to the selected assembler and invoice bytes, reading A, the source evidence for B, the exact (R, S) region, certified ceilings for R = 51..72, and the bit crossing at R = 71.
• Gonzalo-Mier: kappa010-rank-cap-floor OQ3, now answered for the complex invoice. Depth 6 gives row 27037, which fits under B and fails under A.
• Package and code sources:
  • Sricharan Vagvala (@theon199): selected-coarse-depth-stock.
  • VictorLiwentao: Research: selected m=73 complex stock shape A=0 under finite-stock-as-C (no new kappa) #3421.
  • Douglas Colkitt: complex-local-word-ceiling, reused unmodified.
  • Dugongue: contextual-provider-pooling, PR563 (integrated by Colkitt).
  • rohanarun: the Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 invoice.
  • chafreaky: complex-sw4-frame8, PR741.
  • jeremykothe2: PR1087.
  • icekylinx: four precedent notes.
  • DaysSky: closed PR375.
  • navinor: Selected stock envelope at m=73: bound quantities and the exact missing shape lemma #3113.
  • petalas: open Exact rank-cap ceiling curve for the selected complex local word (no new kappa) #3701/feat(research): exact integer idle-histogram ceiling for the selected complex word #3729/research: pool the replica idle block; the two-unpaid premise is per-call-forbidden only #3754/Certify the idle rank-cap sensitivity of the selected packing endpoint (no new kappa) #3805/feat(research): certify the joint integer rank-cap ceiling of the complex local mass #3811, ceilings at caps ≤ 51.
• Concurrent open PRs: jwr404 Record the three degree-gap clearances #3768 (the same degree-gap arithmetic), Multiply the published row-ledger terms #3670 and Record the two selected row readings #3766 (t = 2 rows); Mainzu Research: selected m=73 row-ledger terms do not bind rho (no new kappa) #3667 (w-free reserve degrees; conditional lean row); Matarener Research: nine-stage bridge schedules force the 16/32/48 idle alphabet (negative, no new kappa) #3836 (16/32/48 alphabet, which qualifies the rank-56 bin).

Claim: #2278 (reply in thread)

Review provenance

Reviewed by seven independent AI review passes in the same loop, over two rounds. The lenses were reproducibility and negative controls, an adversarial math and accounting audit with independent recomputation scripts, novelty and provenance, and a referee. All findings rated blocking or major were addressed. No human has reviewed this.

Attribution

Prepared by Sagar Thakrar (@thakrarsagar) with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5), via Claude Code in an automated research loop. Claude bound the sources, chose the method, wrote verify.py and drafted the documents.

🤖 Generated with Claude Code