Independent verification: exact enclosure of the pivot-566+554 T74 complex root behind the top pending kappa claims (no new kappa)

[navinor]

Independent enclosure of the pivot-566+554 T74 complex root (no new κ)

Card: independent-moment (independent-verification track, scope key c6f2494b197df3bc3591, chooser OS nonce 9dd43e38694b6907874fe4e993da7bf9, priority revision 3). Scope changed, with the reason in TASK.json. Claim: #1166 (comment). Base ae95f60feec79897b153dba6459bb3f9fd4d7b8e. Adds only research/t74-k554-root-enclosure/.

Why this object

The selected #1094 pricing already has independent ledger, moment, invoice and outer checks (frontier1094-emitted-ledger-seed78, selected-suppliers-de-diagnostic, pr1407-complex-moment-audit-seed39, #1735). So I applied the card's enclosure method to the pending supplier behind the highest open claims.

Every claim from #1800 to #1928 prices the pivot-566+554 T74 supplier. At their "free ε → 1" point, κ approaches this supplier's complex moment root. #1928 sets b to its certified native endpoint minus 10^-10010. Until now, only the authors' own engines had certified that endpoint.

What is checked

The inputs are the global profile {1:1560432, 2:336336, 3:278724, 4:84912, 5:54612, 8:5328, 48:22748}, m = 73 and W = 66304. They are pinned by Git blob and SHA-256 from new/SPEC.json at #1672 (head b8ab040d) and #1928 (head 07e4e423). extract_inputs.py --check rebuilds INPUTS.json byte for byte from the fetched pull refs.

The moment definitions come from the written review rather than code: native moment from research/native-complex-moment552/PROOF.md, conservative fallback from notes/three-stage-cover-assembly.tex.

The method shares no algorithm with the authors' Mercator and binary-reduction engines or with earlier audits:

• ln 2, ln 3 and ln 5 are integer combinations of ln(16/15), ln(25/24) and ln(81/80). Each of these is 2 atanh(1/k), and the verifier checks the identities exactly.
• ln 73 = ln 72 + 2 atanh(1/145).
• exp uses argument halving, a Taylor polynomial with a tail bound, and repeated squaring.
• All values are outward-rounded integers. The final comparison is exact. Undecided signs fail closed.

Case	Grid	Bits	M(e)	M(e + grid)
1672 native	1e-70	360	below 1	above 1
1672 conservative	1e-70	360	below 1	above 1
1928 native	1e-10000	33347	below 1	above 1
1928 conservative	1e-10000	33347	below 1	above 1

Also checked:

• For both models, the 1e-70 endpoints equal the 1e-70 floors of the 1e-10000 endpoints.
• At both heads, 249 gauge44 calls and 417 extra calls with their per-call profiles, plus 22,748 idle rank-48 children, sum to the global profile. Rank mass is 4,816,512 and the deficit is 23,680.
• The headline κ of PR 1928 is 2.003e-10010 below its native endpoint, as its choice of b implies. PR 1672's is 2.817e-6 below.
• Ten semantic controls are rejected: next and previous grid point as endpoint, one child moved from rank 1 to rank 2, m = 72, W − 1, and the surcharge omitted or added in the wrong model. Three are repeated at 1e-10000.

The optional crosscheck_decimal.py repeats the 1e-10000 case with Python's decimal ln and exp at 10,060 digits. It agrees on all seven log enclosures and all four signs. It takes about 4 minutes and corroborates the result; it is not a certificate.

No discrepancy was found.

Reproduce

python3 scripts/research_package.py --package research/t74-k554-root-enclosure --check-only
python3.12 scripts/research_package.py --package research/t74-k554-root-enclosure

Local results on Windows, Python 3.12, stdlib only:

PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_T74_K554_ROOT_ENCLOSURE_INDEPENDENT | brackets 4 | nesting 2 | controls 10 rejected | 13.5 s
Ran 10 tests ... OK
PASS_REGISTERED_RESEARCH_PACKAGE research/t74-k554-root-enclosure outcome=completed
PASS_DECIMAL_CROSSCHECK | 7 log enclosures | 4 moment signs agree
PASS_INPUTS_MATCH_PINNED_PR_HEADS

tests.test_research_onboarding, tests.test_research_replay_selection and tests.test_research_assignment also pass (32 tests). make is not installed on this host, so no make target was run. Hosted CI has not run.

Scope

This is a finite check of recorded endpoints and of the profile arithmetic. The following are taken as given:

• the profiles themselves;
• the native, surcharge-free complex recurrence;
• the outer reduction from b to κ, the fine source-prime padding and the ε → 1 transfer (Audit: PR1786 fine-prime transfer lemmas checked exactly; outer headroom 1.68e-10 on the PR1699 supplier (no new kappa) #1797 audits the PR 1786 lemmas on a different supplier);
• all inherited all-size/compiler, realization, hosting, precision/tape, recovery, analytic and bit H1-H5 assumptions.

No new κ is claimed and the selected record is unchanged.

Open next questions

1. Does PR 1928's outer/LEDGER.json follow from this endpoint, checked by code that does not import outer/*.py?
2. Does the global profile equal the profile emitted by the T74 template and owner map? The seed78 emitted-ledger rebuild could be adapted to the two-provider layout.

Credit

• Supplier: mdumitrean (Conditional κ = 0.001676896669021299: pivot-566 helper-donor kernel on PR1319's T74 exact fill (+4.14e-8) #1625), Jeremy Kothe (Conditional κ = 0.001676938111022438: pivot 554 helper-donor kernel (k554-signed-001) stacked on the previous record #1672 through Conditional κ = 0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263056561328: recertified 1e-10000 supplier endpoint and 1e-10010 slacks (kappa +2.00336e-500 over PR1897): fine source-prime padding (PR1786) at the free epsilon -> 1 point, with gauge-free interiors and the radix-distributed union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1928), Rohan Arun (Conditional kappa 0.001676855229071321: verified exact fill on 74 replicas #1319) and their credited lineage.
• Outer: Vlado Vojdanovski (Conditional kappa = 0.001677160151714381 via capped phase-cell fusion #1673) and chafreaky (Conditional κ = 0.001679754788150048 from fine source-prime padding #1786).
• Audit pattern: Sagar Thakrar's seed78 package and the earlier independent moment packages.

Prepared by Aaron Wang, with help from opencode. Apache-2.0.

Workflow feedback: the overlap search finds claims by κ digits but not "who has independently checked this number". A per-claim "independent checks" field in the PR ledger would have saved most of the 30-minute survey.

[Mister-Meeseeks]

Batch review at b10c1a196cf5fbbdedc43c9fd26ff2e834849600.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Independent outward-rounded integer atanh/Taylor arithmetic encloses both native and conservative T74 roots on the70- and10000-digit grids, checks profile aggregation and ten semantic mutations. Provider counts and inherited recurrence/outer obligations are inputs; certifying this root does not admit pending suppliers or improve the selected frontier.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.