Prove a complete terminal-band QQ16 cost exclusion

[Clear1nsight]

Result and construction decision

Frozen #4166 QQ16 D costs strictly more reads, writes and unit moves than original-arithmetic B* on a nonempty, explicitly admitted terminal-child domain. This closes a complete signed comparison beyond #4172; it is not another native-component ratio or ledger with unexplained E1/E8/Delta_N prices.

The conclusion is scoped. Keep actual RootPlan fast entry, post-remainder e=73 lambda with d<=e^1000, the genuine C8 terminal predicate 8 lambda<73 OR (8 lambda)^1000<d, physical H_b>=L^4, complete packet v>=2685, the certified positive-slope column threshold, and compatible admitted q/profile/program/capacity choices. Original source/numerical/hosting/stock/long-record premises remain. Forty exact production-descriptor certificates at p=2^8500 establish lambda=5..44 in this domain. Their enormous payloads were not executed.

Stop expanding the fixed recovery in this proved band. No verdict is claimed for nonterminal C8 calls, changed source-compatible demands/schedules, arbitrary compiled-control tariffs, total historical backing, all-node suppliers or kappa.

What resolves the remaining prices

The actual source occurrence partition removes32 centre-wrapper monomials,16 rank-eight children,8 private copies and6 unit subtracts. The six subtracts match new identical-literal calls at the same full geometry. Monomials split at literal source call sites into equal setup/phase/return bases and exact completed gate atoms. Inverse reverses the word; it does not change a gate selector. #4172 supplies successful-payload/closed-history equality only at matching full geometry.

The736 old wrapper gates inject into7150 fixed new gates except4 Gamma add(6,8) and Gamma add(8,6). A new quantified finite-group result closes those deficits: every fixed9x9 head has more than half its independent-bit mass in the complete O(73,2) cover, using64-coordinate tails and an explicit characteristic-vector exclusion. Leading-block events force each literal gate with frequency greater than Gamma/32. The736 paid owner words supply more than23 Gamma of each. Identity-owner witnesses are not extrapolated and no axis-cost invariance is assumed.

The complete terminal original C call, including descriptor/shape work, the paid capped-power cutoff, all coefficient axes, return, disposal and borrowed heads, has componentwise bound

E_w <= 2^32(w+1)[v+(G_s+1)L^4] <= 2^33(w+1)v,
G_s=v/(2H_b).

There are exactly zero descendants on this source branch. Added E1 calls have this same derived bound and are conservatively dropped as positive work in the exclusion. All removed16 E8 calls use an UPPER bound. Eight copies and sixteen B external shells have separate source-derived upper debits; A's embedded parking is never used as avoided baseline work.

After matching the original two core bank cycles, actual candidate-only gathers and disjoint scatter appends give

(R,W,M) >= (2,1,2)(281593/816) Gamma B_bank,
B_bank=14336 Gamma v; service_bank=B_bank/32.

This is a literal source-action lower bound, not a diagnostic volume subtraction. Every other acquisition/compiler/service/scalar/map/publication/disposal/fence/setup term is matched, debited on the correct side, or retained nonnegative. Exchange-internal negations are not added twice.

Complete fair-baseline certificate

Let a=(281593/816)*14336 Gamma and

C(lambda)=16*2^33(8lambda+1)+8*2^22+16*2^26.
s=(2a+5*510662-C, a+2*510662-C, 2a+4*510662-C).
h=(12*510646+2*510662, 4*510646, 12*510646+2*510662).

The entire root comparison satisfies, componentwise:

(T_D-T_B*)_i >= Gamma v s_i + Gamma h_i - 66496 > 0.

All510646 Gamma retained bodies cancel under #4172, including6972 changed-core and493097 later calls. Their14,298,088 Gamma header actions remain. B*'s established adoption saving is credited exactly once; frozen D gets no D* adjustment. The703-write original-backend potential cancels once globally. The66496 total B-controller startup/retirement upper is pessimistically subtracted in EACH coordinate.

The last positive column integer is derived explicitly, has2610 bits, and passes an endpoint/next-integer check. It is a sufficient bound threshold, not a claim every geometry below it is reachable. For arbitrary positive action weights, exclusion holds in this source currency. Full compiled-step exclusion additionally requires signed control difference at least minus the weighted lower bound. Fixed inventories remain B/B*:678+M(q)+1023, D:conservative678+3M(q)+2^20; historical per-role backing is separate and erasure does not reclaim it.

Evidence and reproducibility

Scientific main12fa8c5c844be21bfefb38a19503fbd867a11244, priority7. Retained #4172 head59b46625a8b2f0dfff3e38ba0949b4247423938a; #4167 head6749e458800f461535d64cc91c017552d9f55e5f; frozen #4166 head0b082d22a0e9a442d264ab0633e35ef26c790630. All108 nested-loader files authenticate before inherited execution. No cached foreign Git object is needed.

Reproduce:

python3.12 -B scripts/research_package.py --package research/qq16-complete-cost

Default fresh evidence includes17 exact countdown calibrations,7 complete selected C leaf executions,1 explicitly reduced-binding complete cutoff execution,3 encoded cutoff kernels,12 phase cases,3 complete bank cycles, literal O73 gate witnesses, full occurrence/later/service census,40 exact RootPlan geometries and independent signed/counting/schema controls. The optional6190-bit cutoff kernel was executed once (209.934s), not repeated by default and not represented as a width73 payload run.

Fresh ordinary/optimised primary and six source/export/typed-receipt boundary groups passed. The full registered launcher passed, with six boundary groups in95.511seconds. Committed-byte/source-history preflight passed108files at the exact publication head; the official108-payload-file export/check passed. Actual committed read-only Git-free scientific replay results are recorded below. Archive SHA256 36ff7e1e6613a0669c1915428c0e7472f76babe1ad24740d3653242fe5a1f76c. Frozen16-file read-only adversarial review found no unresolved blocker; it independently reconstructed cover arithmetic and representative signs, inspected source interfaces, and ran fresh/check-only verification. Shared sources/checkers are disclosed; this is assisted review, not human/formal-kernel acceptance.

Publication head: 4cdda81.

Workflow, attribution and remaining work

Assigned joint-query-controller scope73442cf25caf1ed45f11. Main/priority, exact predecessor heads/reviews and overlap were refreshed at entry, coupling checkpoint and submission. Top30/top20 connector searches were capped; helper remote coverage was unavailable without gh; bounded cached Discussion pages were checked. No novelty guarantee is asserted. The canonical #3985 handoff will preserve prior publications and point here.

Credit Jose Manuel/#3385/#2899 and retained physical/Compact authors; ywz65 parking/monomial/countdown lineage; Sagar Thakrar/#3464; Tekalign/#3380 and already credited adaptations; Ali Romman/Arham Tahir and #3985/#4015/#4106/#4109/#4129/#4166/#4167/#4172. Nick Petalas/#2982/#2951, commdata2338/#3233, Ross Richardson/#2831, Nick/#2923 and Wentao Li/#3993 retain their scoped method/backing credit. Arham/#2877 and ywz65/#2473 retain predecessor countdown credit. No BIT/ADD/wrapper coefficients are transplanted. Chris R/Clear1nsight with substantial coordinated OpenAI Codex assistance. Original notices retained.

The next discriminating domain is a genuinely nonterminal C8 call after the actual global height cutoff. It needs complete embedded descendant parking and candidate local/native prices. Width>=73 alone does not certify recursion. No additional decoder search is justified by this scoped result. Selected kappa is unchanged.

Committed scientific export replays both PASS: ordinary50.928s, optimised58.403s, empty stderr, Git absent, executable search path unavailable, and unchanged bytes/modes/mtimes/inodes. These reran all default physical calibrations from the exact committed108-file export plus manifest; check-only transport is separate. No scientific source changed after the frozen review and committed verification.

[rcfc-jreed]

Independent scoped audit of the QQ16 terminal-band complete cost exclusion at 4cdda81c: no defect found in the finite receipt / seal / RWM-exclusion layer

Code-disjoint reimplementation; does not import package verify.py / checker.py / cost.py. CLAIM.

What passes (24 of 24):

1. Pins. SOURCES.base / RESULT.scientific_main = 12fa8c5c…. Status PASS_QQ16_COMPLETE_TERMINAL_COST, no_kappa_change, not_a_universal_QQ16_obstruction. Frozen predecessor 59b46625… (Prove native-history action coupling and reduce the QQ16 comparison #4172). Selected κ untouched.
2. Seals. All seven package scripts match SEALS.json SHA-256 on HEAD bytes.
3. Root census. A / B / B_shared identical at 510662 calls / rank mass 1041408. C_D: +16 calls, −96 rank mass (rank-8 1152→1136, rank-1 +32). Retained mass consistent with ranks.
4. Cost cases. Four columns {5,9,10,44}: strict_all_RWM_exclusion, positive Γ-linear and Γ-constant coefficients, no unknown prices / production payload, bit lengths [5290,5289,5290], terminal predicate 8 λ < 73 OR (8 λ)^1000 < d.
5. Gate cover. Identity negative control; source group orders match; 9×9 heads supported; formal_kernel_verification=false; 736 owners/Γ; strict inequalities in count certificate.
6. Scope. LARGE-CUTOFF and EXECUTION C calibrations both mark complete_C_payload_executed=false; default registered replay false.

Not covered: production fullRootLayer payload execution; formal-kernel acceptance; universal QQ16 obstruction; nonterminal C8 domain; new κ.

This is a finite replay of committed receipts, not a formal proof.

check_4177.py

#!/usr/bin/env python3
"""Independent finite controls for PR4177 (QQ16 terminal-band cost exclusion).

Recomputes root-census rank masses, seal hashes, and cost-case exclusion flags
from pinned receipts. Does not import package verify.py / checker.py / cost.py.

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).
Apache-2.0.
"""
import argparse, hashlib, json, subprocess, sys

HEAD = '4cdda81c2ad99ffe5add0ac5d894094c3cca101a'
BASE = '12fa8c5c844be21bfefb38a19503fbd867a11244'
PKG = 'research/qq16-complete-cost/'
results = []


def check(name, ok, detail=''):
    results.append((name, bool(ok), detail))
    print(('PASS ' if ok else 'FAIL ') + name + ((': ' + str(detail)) if detail else ''), flush=True)
    return bool(ok)


def show(repo, path, rev=HEAD):
    return subprocess.run(['git', '-C', repo, 'show', f'{rev}:{path}'],
                          capture_output=True, check=True).stdout


def js(b):
    return json.loads(b)


def mass(ranks):
    return sum(int(r) * int(c) for r, c in ranks.items())


def load(repo):
    g = lambda p: show(repo, PKG + p)
    return dict(
        sources=js(g('SOURCES.json')),
        result=js(g('RESULT.json')),
        bill=js(g('BILL.json')),
        admission=js(g('ADMISSION.json')),
        seals=js(g('SEALS.json')),
        review=js(g('REVIEW.json')),
        execution=js(g('EXECUTION.json')),
        large=js(g('LARGE-CUTOFF.json')),
    )


def run(d, repo):
    r, b, a = d['result'], d['bill'], d['admission']

    check('I1 SOURCES base / RESULT scientific_main == 12fa8c5c…',
          d['sources']['base_commit'] == BASE == r['scientific_main'])
    check('I2 RESULT status PASS_QQ16_COMPLETE_TERMINAL_COST; no_kappa_change',
          r['status'] == 'PASS_QQ16_COMPLETE_TERMINAL_COST'
          and r['no_kappa_change'] is True)
    check('I3 RESULT not_a_universal_QQ16_obstruction',
          r['not_a_universal_QQ16_obstruction'] is True)
    check('I4 ADMISSION excludes newkappa / production payload / formal-kernel',
          'newkappa' in a['excluded_claims']
          and 'production payload/fullRootLayer execution' in a['excluded_claims'])
    check('I5 REVIEW PASS with zero blocking defects',
          d['review']['status'] == 'PASS_ASSISTED_ADVERSARIAL_REVIEW'
          and d['review']['blocking_defects'] == [])
    check('I6 predecessor head matches RESULT frozen_predecessor',
          d['sources']['retained_predecessor']['head'] == r['frozen_predecessor']
          == '59b46625a8b2f0dfff3e38ba0949b4247423938a')

    # Seals
    bad = []
    for rel, digest in d['seals'].items():
        got = hashlib.sha256(show(repo, PKG + rel)).hexdigest()
        if got != digest:
            bad.append(rel)
    check('S1 SEALS.json matches HEAD bytes for all seven scripts', not bad, bad)

    # Root census masses
    census = b['complete_root_census']
    for key in ('A_original', 'B_protocol_only', 'B_shared_protocol_only'):
        c = census[key]
        check(f'R1 {key} calls=510662 rank_mass=1041408 matches ranks',
              c['calls'] == 510662
              and c['rank_mass'] == 1041408 == mass(c['ranks']))
    cd = census['C_D_root']
    check('R2 C_D_root calls=510678 rank_mass=1041312 matches ranks',
          cd['calls'] == 510678
          and cd['rank_mass'] == 1041312 == mass(cd['ranks']))
    ret = census['retained']
    check('R3 retained mass/calls consistent with ranks',
          ret['rank_mass'] == mass(ret['ranks']) and ret['calls'] == sum(ret['ranks'].values()))
    check('R4 A/B/B_shared profiles identical (protocol-only baseline)',
          census['A_original']['ranks'] == census['B_protocol_only']['ranks']
          == census['B_shared_protocol_only']['ranks'])
    check('R5 C_D vs A: +16 calls, −96 rank mass (8s 1152→1136, 1s +32)',
          cd['calls'] - census['A_original']['calls'] == 16
          and census['A_original']['rank_mass'] - cd['rank_mass'] == 96
          and cd['ranks']['8'] == 1136
          and census['A_original']['ranks']['8'] == 1152
          and cd['ranks']['1'] - census['A_original']['ranks']['1'] == 32)

    # Cost cases
    cases = b['derived_cost_cases']
    check('X1 four derived cost cases (columns 5,9,10,44)',
          len(cases) == 4
          and sorted(c['columns'] for c in cases) == [5, 9, 10, 44])
    bad_x = []
    for c in cases:
        if c['variant'] != 'frozen D minus Bstar':
            bad_x.append(('variant', c['columns']))
        if not c['strict_all_RWM_exclusion']:
            bad_x.append(('strict', c['columns']))
        if c['unknown_prices_used']:
            bad_x.append(('unknown', c['columns']))
        if c['production_payload_executed']:
            bad_x.append(('payload', c['columns']))
        lb = c['lower_bound_RWM']
        if not (all(x > 0 for x in lb['Gamma_times_v_coefficients'])
                and all(x > 0 for x in lb['Gamma_constant_coefficients'])):
            bad_x.append(('pos', c['columns']))
        if c['minimal_component_lower_bit_lengths'] != [5290, 5289, 5290]:
            bad_x.append(('bits', c['columns']))
        if '8 lambda < 73 OR (8 lambda)^1000 < d' not in c['terminal_predicate']:
            bad_x.append(('term', c['columns']))
    check('X2 every case: strict RWM exclusion, positive Γ-linear lower bounds, no unknown prices',
          not bad_x, bad_x)

    # Gate cover
    check('G1 gate_cover identity negative control + source group orders match',
          b['gate_cover']['identity_negative_control'] is True
          and b['gate_cover']['source_group_orders_match'] is True)
    check('G2 independent_gate_check: 9×9 heads supported; formal_kernel=false; 736 owners/Γ',
          b['independent_gate_check']['all_9_by_9_heads_supported'] is True
          and b['independent_gate_check']['formal_kernel_verification'] is False
          and b['independent_gate_check']['owner_calls_per_Gamma'] == 736)
    check('G3 count_certificate strict_inequalities',
          b['gate_cover']['count_certificate']['strict_inequalities'] is True)

    # Execution / large cutoff scope
    check('E1 LARGE-CUTOFF production_payload_executed False; default replay False',
          d['large']['production_payload_executed'] is False
          and d['large']['default_registered_replay'] is False)
    check('E2 EXECUTION C calibrations mark complete_C_payload_executed False',
          all(not x['complete_C_payload_executed']
              for x in d['execution']['C']['cutoff_kernel_calibrations']))

    # Negatives
    check('NEG1 claiming universal QQ16 obstruction contradicts RESULT flag',
          r['not_a_universal_QQ16_obstruction'] is True)
    check('NEG2 zero Γ-linear coefficient would break positivity (spot)',
          all(c['lower_bound_RWM']['Gamma_times_v_coefficients'][0] > 0 for c in cases))

    failed = [name for name, ok, _ in results if not ok]
    print(f'\n{len(results) - len(failed)}/{len(results)} PASS', flush=True)
    if failed:
        print('FAILED:', ', '.join(failed), flush=True)
        return 1
    return 0


def main():
    ap = argparse.ArgumentParser()
    ap.add_argument('--repo', required=True)
    args = ap.parse_args()
    head = subprocess.run(['git', '-C', args.repo, 'rev-parse', 'HEAD'],
                          capture_output=True, text=True, check=True).stdout.strip()
    check('HEAD is PR4177 tip 4cdda81c…', head == HEAD, head)
    return run(load(args.repo), args.repo)


if __name__ == '__main__':
    sys.exit(main())

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).