Independent verification (no κ): PR3620's 425/426 P104 deletion threshold holds on PR3609's dual (conditional); slack valid as scoped, sharp ceilings at κ/(1-κ)

[thakrarsagar]

PR3620's 425/426 P104 deletion threshold holds on PR3609's dual, under the optimistic named law. The rejection is proved by two exact methods at a = κ, so it holds at every a ≥ κ. The coverage grant is favourable. 426 is the least uniform per-call deletion; uneven calls need a total of 7·425+3 = 2978 at p = 7. PR3620's 4.557071 seven-pair slack at a = 1/596 is correct as scoped. The sharp ceilings at a_req = κ/(1-κ) are 0.710094 per dual call (best at p = 7) and 5.524500 for a purely fixed cost (best at p = 8). A mixed cost passes iff c0 + c1·p < -E426(p,a) for some p.

No κ claim, no supplier, no frontier change. This is a conditional independent verification of one relaxation, not a barrier or multiplication theorem.

Statement

Object: PR #3620 (Clear1nsight), head 5a4354c669dc05a9368f58321b1951ebac36d80d, research/p104-caller-feasibility/consumer/PROOF.md sections 1 and 3. It is evaluated on PR #3609's changed full dual D (Clear1nsight, head 13f974dae4af66027afb01f3ab42df9e830c2e90, RECEIPT.json: D = (1316,61,74,86,33,25,46,8,93), 1742 calls, mass 3542) and PR2744's forward profile F = (1216,74,86,104,48,42,68,8) on main.

Model, as in PR3620 section 1: m = 16p+9, W = 208p, P = 3014, p+1 forward calls, p dual calls, fully paid banks B. Deleting d rank-one children from every dual call passes iff

E_d(p,a) = (16p+9)^a (p·U_d + V) - 6236p - 3014 < 0.

• Rejection, confirmed. Deleting at most 425 rank units per dual call fails for every p ≥ 1 and every finite number of fully paid banks. The minimum is E425(7,κ) = 1.600686410509596…. This is proved twice:
  • (A) Fraction intervals plus convexity in p.
  • (B) Decimal values for p = 1..23 plus a linear tail bound from p = 24, without convexity.
• Pass. Deleting 426 passes: E426(7,1/596) = -4.557071360876949…. The passing set is p = 6..10 at 1/596 and p = 6..11 at a_req.
• Uneven deletion. The excess depends only on the total deletion, so 426 is the least uniform per-call deletion. With uneven calls a total of 2978 passes at p = 7, at both a_req and 1/596, and 2977 fails. Some call must still delete more than 425.
• Reductions, all valid. The general deletion lemma, the clamp, bank cancellation, monotonicity in a and d, and strict convexity in p.
• Coverage grant, favourable. P = 3014 = 104·8+104·8+150·9 is fixed by the boundary frame dimensions. It is exact for PR3609's dual, given PR3609's claim that it keeps all 358 final frames; that check was not re-run here. W = 208p is the inherited law. The 145 completion calls (52 H1 + 93 H9) are charged once, inside D.
• Slack: PR3620's sentences are valid as scoped (p = 7, a = 1/596), with a clarification. Under the retained conversion κ = a/(1+a), an improvement needs a > a_req = κ/(1-κ). E increases in a, so the sharp ceilings are the values at a_req (suprema, not attained). All figures below are optimistic ceilings for 426 rank-one deletions in every dual call, truncated toward zero.
  • A cost proportional to the number of dual calls is best at p = 7: 0.710094 per dual call at a_req and 0.651010 at 1/596. The seven-pair totals are 4.970660 and 4.557071. The marker label best_p_for_per_call_costs refers to these per-call values.
  • A purely fixed cost is best at p = 8: 5.524500 at a_req and 5.040108 at 1/596.
  • A mixed cost passes iff c0 + c1·p < -E426(p,a) for some p. For example, at a_req, 5 + 0.05p passes only at p = 8.

Milestone and decision informed

Milestone: independent conditional audit. Decision informed: the P104 expansion gate in research-family-status ("complete cost allowance"). A realised correction that acts by rank-unit deletion from D must reach the 426 / total threshold, and its extra cost must fit inside the ceilings above, before the stopped P104 supplier can reopen. The threshold and the stop decision are unchanged. Other replacement profiles must pass PR3620 section 2's general criterion.

Base and pins

• Base: canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. Re-checked at submission: main unchanged; PR3620 and PR3609 still open at the audited heads.
• F: research/p104-consumer-budget-audit/RESULT.json on main, sha256 08705a91….
• D: PR3609's RECEIPT.json, vendored byte for byte as inputs/PR3609_RECEIPT.json (git blob 6faa5157…, sha256 6d06db04…). verify.py checks both digests.

Reproduce

python3 -I -B research/p104-consumer-threshold-audit/verify.py
python3 -I -B research/p104-consumer-threshold-audit/controls.py
python3 scripts/research_package.py --package research/p104-consumer-threshold-audit

Expected:

PASS_P104_CONSUMER_THRESHOLD_AUDIT d425_rejects_every_p>=1 d426_passes(a596:p6..10,a_req:p6..11) total_slack_p7(a596=4.557071,a_req=4.970660)(best_p_for_per_call_costs) fixed_cost_slack_best_p8(a596=5.040108,a_req=5.524500) a_req=kappa/(1-kappa) conditional_on_PR3609_D_and_named_law
PASS_P104_CONSUMER_THRESHOLD_CONTROLS rejected=14 positive=1

Standard library only. The verifier takes about 2 s and the controls 10-25 s. The registered hook passed under CPython 3.12.15, and --check-only passes.

Certificates and negative controls

verify.py recomputes 31 exact claims and compares them with RESULT.json. Each check is an explicit raise, so it also runs under -O.

controls.py must reject 14 mutations, each of which breaks a named claim:

• P = 3015 and P = 3013;
• D9 = 92, D1 = 1317, the rank-9 stratum dropped, and the old PR2712 dual;
• F7 = 76, m = 16p+8 (value binding), and W = 209p;
• the threshold swaps at 425/426 and at p = 4 (427/428);
• one byte of the vendored receipt changed.

The unmutated positive control passes. The code was frozen before the sweep.

Known limit: removing the clamp is not detected. This is harmless, because the clamp is active only at p = 1, and method B proves p = 1 without it.

Scope, assumptions and unresolved obligations

• Inherited: the named coverage law, which is optimistic and unrealised; per-role progress caps; full credit after deletion; PR2744's forward recount.
• D is bound to PR3609's receipt only. It was not recounted from PROGRAM.json/NATIVE.json, and PR3609 is open and unreviewed. PR3609's endpoint, 358-frame and native proofs were not re-run.
• Unpriced costs on the passing side: Y-source writability, ownership, routing, scalar work, the 102 missing forward corrections, and any physical deletion network.
• Out of scope: PR3620's physical/ format half and section 4 (the PR3605 T740 bracket), PR3623, and the PR3259 support cut. PR3605's staged words are audited in open PR Independent verification: PR3605's staged 644/660 words pass an exact audit against the PR3133 control (no κ claim) #3687.
• Not run: the PR3620 and PR3609 verifiers (heavy, out of scope).
• Open next questions are in the README and TASK.json:
  • which ceiling form gates should quote;
  • a realised network meeting 426 per call or the total threshold;
  • a recount of D from PR3609's artifacts.

Search record

No search was run. Every evaluation is a finite, deterministic exact computation. The all-p claims rest on proofs (convexity, and separately a linear tail bound), not on sampling.

Credit

• Clear1nsight: PR3620 and PR3609.
• Vasek (flyingviper): PR2744.
• Ross Richardson: PR2458.
• Nick Petalas: the all-pair consumer model.
• Jose Manuel (jmbarrancoml): the four-pair screen and the deletion discount.
• ywz65: PR2712, used in a control.
• Hiroki Kumazaki (kumagi): the PR3446 projection.
• Dominik Scholz and PR1012: as credited in the all-pair package.
• PR3609's own notices remain in force for the vendored receipt.

Concurrent: jwr404's open PR #3654 (head 4068c58d) proves the rank-one deletion order and a convexity bracket. REVIEW.md 3.1, 3.2 and 3.5 were written independently and do not rely on it.

What this adds:

• binding to PR3609's changed dual;
• the grant and double-count check;
• a second all-p proof without convexity;
• the uniform-versus-total deletion distinction;
• the slack-ceiling clarification at a_req.

Claim: #1166 (reply in thread)

Review provenance

This package was reviewed by 10 independent AI review passes over three rounds in the same loop (reproducibility, mathematics, provenance, referee), with a revision after each round. It has not been reviewed by a human and is not a proof-assistant check.

Attribution

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

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