Independent numerical cross-check of PR4100's exact-root and paid-power ledger (finite layer; no new kappa)

[petalas]

What this is

An independent, second-implementation numerical cross-check of the exact-root certificate and paid-power ledger behind PR #4100's claimed exponent κ = 0.00173097911567079247276057 (retained head 337d04939571a637aeab54a0f58f54fde811663a). The package re-implements the moment certificate from scratch — its own rational atanh/Taylor interval enclosures, its own bill evaluator and its own root bisection — and imports no code from the audited package or from staged544-mixed-fourth-entry/vendor. Constants are transcribed from PR4100's PROOF.md; the profile and benchmark come from the pinned PR #4088 RESULT.json.

What was established (finite facts, second implementation)

For the pinned profile {1:8060060, 2:2310120, 3:751813, 4:327210, 5:66425, 8:25920, 48:138492}, m = 73, W = 322,560, b = .001730979115670792472760579929302792893760378764:

• bill(b) < 1 with certified margin ≤ -1.800e-48 and bill(b + 1e-48) > 1 with margin ≥ +1.030e-48; the model root therefore lies strictly inside (b, b + 1e-48), localized at b + 6.363e-49 by an independent 90-digit bisection.
• bill(κ) < 1 with margin ≥ 2.815e-26; also bill(a_eff) < 1, bill(q) < 1.
• All twelve paid-power slacks and fifteen side-condition slacks are strictly positive as exact rationals; all five obsolete-route rejects remain strictly negative.
• Bootstrap constants reproduce exactly (a1 = .00173039532444, a2 = .001733394791695384296 from c = 8667/5,000,000); 7^16 < 2^45.
• Exact gap over PR Improve conditional κ with a mixed fourth staged entry (+8.696e-9) #4088's published conditional exponent: κ − conditional_kappa = 5.700000000000000000001e-25 > 5.7e-25; profile agrees with the pinned RESULT.json.
• Four controls rejected: inflated rank-1 count, bracket shifted up by 1e-12, root claimed at/above b + 1e-48, unrounded κ = b.

Scope — exactly

Finite numerical confirmation of stated arithmetic only. This is not validation of the all-size construction: the CRT total repair (audited separately in #4169), the free-algebra lift (COMPLEX.md), precision/conditioning and fixed-tape obligations, and physical realization remain open. No new κ, no supplier admission, no selected-frontier change. Selected κ stays 0.001672660197076918.

Related open work

• Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 — the audited predecessor; every number confirmed here is its own stated value at head 337d049.
• Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169 — audits the same Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 head's compact CRT repair inverse; complementary proposition, no overlapping number, agrees that all-size premises remain open.
• Prove the sharp pooled optimum for fourth staging #4091 / Independent transfer audit (no new κ): PR4091's pooled ceiling applies to the retained PR4088 head; remaining headroom 1.140318244487202e-9 #4170 — Independent transfer audit (no new κ): PR4091's pooled ceiling applies to the retained PR4088 head; remaining headroom 1.140318244487202e-9 #4170 measures the Prove the sharp pooled optimum for fourth staging #4091 pooled ceiling's remaining headroom over the retained Improve conditional κ with a mixed fourth staged entry (+8.696e-9) #4088 fill as 1.140318244487202e-9. Numerical agreement: 0.00173097911567079247276057 + 1.140318244487202e-9 = 0.00173098025598903696000259… ≈ 0.001730980256, matching the Prove the sharp pooled optimum for fourth staging #4091 root quoted in the family record. Different objects (pooled ceiling transfer vs this moment ledger); no contradiction.
• Improve conditional κ with a mixed fourth staged entry (+8.696e-9) #4088 — source of the pinned profile and conditional_kappa; both reproduced byte-for-byte from its retained RESULT.json.

Reproduce

python3.12 -B scripts/research_package.py --package research/unconditional-moment-crosscheck
python3.12 -B research/unconditional-moment-crosscheck/verify.py

Registered check runtime is well under one second (0.5 s measured). Locally certified with CPython 3.12.15: PASS_REGISTERED_RESEARCH_PACKAGE … outcome=partial and PASS_RESEARCH_COMMIT_BYTES with --check-source-history. CI is asked to certify the registered replay under its pinned interpreter; base is canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244 (priority revision 7).

Only research/unconditional-moment-crosscheck/ is added; no other file is touched. Coordination: handoff thread claim.

Prepared by Nick Petalas with AI assistance. Apache-2.0; predecessor attribution retained.

[rcfc-jreed]

Independent scoped audit of the PR4100 exact-root / paid-power ledger cross-check at a3be6bec: no defect found in the finite arithmetic layer

Code-disjoint reimplementation (own Fraction ledger + high-prec Decimal bill/bisection); does not import package verify.py. Package + pinned PR4088 RESULT.json fetched via GitHub API at base 12fa8c5c…. CLAIM.

What passes (20 of 20):

1. Pins. Base 12fa8c5c…. TASK partial / VERIFY ready. Profile matches pinned PR4088; W=322560.
2. Ledger. All 12 paid-power slacks > 0; all 15 side slacks > 0; all 5 obsolete rejects < 0; a1/a2 and 7^16<2^45.
3. Gap / truncation. kappa is the 26-place truncation of b; kappa−conditional_κ > 5.7e−25.
4. Bill / root. bill(b)<0<bill(b+1e−48) (margins ≈ −1.80e−48 / +1.03e−48); bill(kappa), bill(a_eff), bill(q) < 0; root localized inside (b,b+1e−48).
5. Controls / cross-link. Inflated rank-1 and shifted bracket rejected; kappa+#4170 headroom ≈ 0.001730980256. Selected κ untouched.

Not covered: CRT total repair; free-algebra lift; precision/tape; supplier admission; package interval-enclosure code path (Decimal used here).

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

check_4171.py

#!/usr/bin/env python3
"""Independent finite controls for PR4171 (PR4100 moment ledger cross-check).

Recomputes paid-power / side / reject slacks as exact rationals and a high-prec
Decimal bill at the stated bracket. Does not import package verify.py.

Pinned PR4088 RESULT.json fetched separately (API) at base 12fa8c5c….
Package head a3be6becf547c5c7657446843b59fdb50e655eac.

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).
Apache-2.0.
"""
from __future__ import annotations

import argparse, json, sys
from decimal import Decimal as D, localcontext
from fractions import Fraction as Q
from pathlib import Path

HEAD = "a3be6becf547c5c7657446843b59fdb50e655eac"
BASE = "12fa8c5c844be21bfefb38a19503fbd867a11244"
PROFILE = {1: 8060060, 2: 2310120, 3: 751813, 4: 327210, 5: 66425, 8: 25920, 48: 138492}
W = 896 * 360
B = Q(".001730979115670792472760579929302792893760378764")
KAPPA = Q(".00173097911567079247276057")
QW = B - Q(1, 10**28)
A_EFF = (B + QW) / 2
EPS = 1 - Q(1, 10**30)
BETA = 1 - Q(3, 10**30)
XI = EPS * (1 + QW) - Q(3, 32) + Q(1, 10**30)
STOP = Q(1, 10**32)
C = Q(8667, 5000000)
A1, A2 = C * (1 - C), C * (1 - C * C)

POWERS = {
    "main": 1 - EPS * QW,
    "global_router": 1 - A_EFF,
    "gaussian": EPS - QW * BETA,
    "group_router": EPS - A_EFF * BETA,
    "group_scans": EPS - BETA,
    "boundary": EPS - XI + Q(29, 32),
    "union": EPS + Q(1, 2) - XI,
    "products": 1 - EPS,
    "crt": 1 - A2 + 2 * (1 - EPS),
    "halo_arithmetic": EPS + BETA + 144 - 400,
    "halo_metadata": EPS + 50 - 400,
    "cached_phases_and_recovery": Q(0),
}
SIDE = {
    "dimension_below_one": 1 - EPS,
    "group_below_dimension": EPS - BETA,
    "group_positive": BETA,
    "group_headers": 1 - BETA,
    "membership": 1 - BETA - 2 * (1 - EPS),
    "shifted_guard": 1 - BETA - 2 * XI + 2,
    "cap_above_band": XI - Q(1, 2),
    "cap_below_phase": 2 - XI,
    "bounded_individual_chirp": 1 - XI,
    "core_above_faces": 400 - 48 - BETA,
    "leaf_saving": (1 - STOP) * B - QW,
    "bit_above_internal": A2 - A_EFF,
    "internal_above_q": A_EFF - QW,
    "root_above_q": B - QW,
    "crt_table_guard": Q(64) - 4,
}
REJECTED = {
    "old_Vd_crt": 1 - KAPPA - EPS,
    "old_g_union_scans": 1 - KAPPA - (EPS + BETA + Q(1, 2) - XI),
    "old_per_record_d_chirp": 1 - KAPPA - EPS,
    "old_theta_chirp_guard": 1 - BETA - 2 * XI + EPS,
    "no_group_gap_membership": 1 - EPS - 2 * (1 - EPS),
}

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 decimal_bill(a: Q, profile, w, prec=90) -> D:
    with localcontext() as ctx:
        ctx.prec = prec
        total = D(0)
        for rank, n in profile.items():
            r = D(rank) / D(73)
            total += n * (r ** (D(1) - D(a.numerator) / D(a.denominator)))
        return total / D(w) - D(1)


def run(pkg: Path, pinned: Path):
    sources = json.loads((pkg / "SOURCES.json").read_text())
    task = json.loads((pkg / "TASK.json").read_text())
    verify = json.loads((pkg / "VERIFY.json").read_text())
    result = json.loads(pinned.read_text())

    check("I1 SOURCES/TASK base_commit == 12fa8c5c…",
          sources["base_commit"] == BASE == task["base_commit"])
    check("I2 VERIFY ready; TASK outcome partial; task_id uniform-transfer-audit",
          verify["status"] == "ready"
          and task["outcome"] == "partial"
          and task["task_id"] == "uniform-transfer-audit")
    check("I3 VERIFY inputs pin staged544-mixed-fourth-entry/RESULT.json",
          verify["inputs"] == ["research/staged544-mixed-fourth-entry/RESULT.json"])

    pinned_profile = {int(k): v for k, v in result["profile"].items()}
    cond = Q(result["conditional_kappa"])
    check("P1 pinned PR4088 profile matches transcribed PROFILE",
          pinned_profile == PROFILE)
    check("P2 W = 896×360 = 322560", W == 322560)

    power_slacks = {n: 1 - KAPPA - p for n, p in POWERS.items()}
    check("L1 twelve paid-power slacks all strictly positive",
          len(POWERS) == 12 and all(x > 0 for x in power_slacks.values()))
    check("L2 fifteen side-condition slacks all strictly positive",
          len(SIDE) == 15 and all(x > 0 for x in SIDE.values()))
    check("L3 five obsolete-route rejects all strictly negative",
          len(REJECTED) == 5 and all(x < 0 for x in REJECTED.values()))
    check("L4 bootstrap a1/a2 decimals and 7^16 < 2^45",
          A1 == Q(".00173039532444")
          and A2 == Q(".001733394791695384296")
          and 7**16 < 2**45)

    gap = KAPPA - cond
    trunc = Q((B * 10**26).__floor__(), 10**26)
    check("G1 kappa is 26-place truncation of b; kappa < b",
          trunc == KAPPA < B)
    check("G2 gap over PR4088 conditional_kappa > 5.7e-25",
          gap > Q(57, 10**26), str(gap))

    # Decimal bill signs at bracket (independent of package interval code)
    f_b = decimal_bill(B, PROFILE, W)
    f_up = decimal_bill(B + Q(1, 10**48), PROFILE, W)
    f_k = decimal_bill(KAPPA, PROFILE, W)
    f_e = decimal_bill(A_EFF, PROFILE, W)
    f_q = decimal_bill(QW, PROFILE, W)
    check("B1 decimal bill(b) < 0 < bill(b+1e-48)", f_b < 0 < f_up, (str(f_b), str(f_up)))
    check("B2 decimal bill(kappa), bill(a_eff), bill(q) all < 0",
          f_k < 0 and f_e < 0 and f_q < 0)

    # Bisection localization
    a1_, a2_ = B, B + Q(1, 10**48)
    for _ in range(140):
        mid = (a1_ + a2_) / 2
        if decimal_bill(mid, PROFILE, W) < 0:
            a1_ = mid
        else:
            a2_ = mid
    check("B3 localized root strictly inside (b, b+1e-48)",
          a1_ > B and a2_ < B + Q(1, 10**48))

    # Controls
    wrong = dict(PROFILE)
    wrong[1] += 1
    check("N1 inflated rank-1 profile makes bill(b) no longer < 0 (control)",
          decimal_bill(B, wrong, W) >= 0)
    check("N2 bill(b+1e-12) is not < 0 (shifted bracket excludes root)",
          decimal_bill(B + Q(1, 10**12), PROFILE, W) >= 0)
    check("N3 claiming kappa >= b fails (kappa < b)", KAPPA < B)

    # Cross-link to #4170 headroom arithmetic quoted in README
    check("NEG1 selected κ 0.001672660197076918 untouched (TASK partial / no new κ)",
          task["outcome"] == "partial")
    # README: kappa + 1.140318244487202e-9 ≈ 0.001730980256 (matches #4091/#4181 ceiling)
    headroom = Q("1.140318244487202e-9")
    ceil_approx = KAPPA + headroom
    check("N4 kappa + #4170 headroom rounds to 0.001730980256 at 1e-12 (#4091 cross-link)",
          abs(ceil_approx - Q("0.001730980256")) < Q("1e-12"), str(float(ceil_approx)))

    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("--pkg", required=True)
    ap.add_argument("--pinned-result", required=True)
    ap.add_argument("--expect-head", default=HEAD)
    args = ap.parse_args()
    check("HEAD pin recorded (API fetch)", args.expect_head == HEAD, args.expect_head)
    return run(Path(args.pkg), Path(args.pinned_result))


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

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