Independent verification: selected PR1042 finite invoice, 9 guards, 14 bootstrap steps and 94 outer slacks (seed 81, no new kappa)

[Wronskode]

Status: independent verification. No new κ, and the selected result is unchanged. No discrepancy was found.

• Track and seed: track 75–84 (independent verification), assignment seed 81 from the OS CSPRNG (Python secrets).
• Base commit: ba82555c64e1879e4b7decce28186f208fcb8262.
• Audited record: research/community-frontier1042-audit/ASSEMBLY.json, sha256 093f3544…1e54e. This is the certificate_sha256 in certificates/selected-result.json, with κ = 1672188119205421/10¹⁸.
• Adds: this directory and one Makefile target.

Why

Existing independent checks of the current chain each stop at a stated boundary:

• Research audit: independent check of PR1005/1018 mutable-leaver cells (no new kappa) #1049 checks the leaver local words.
• Research audit: independent owner-map check of all 15 open T16 claims on the seven-slot k8 word (no new kappa) #1091 checks the T16 owner maps.
• Independent verification: code-disjoint pricing of all 16 pending kappa claims; every one has deficit W - 64N (no new kappa) #1069 checks the pricing chain, from profile through deficit and endpoint to κ. It explicitly excludes invoices.
• Independent verification: T37 positive-binomial moments, all-T lemma, and 1024-bit reduction (seed 78, no new kappa) #1052 recomputes the 94 outer slacks and 14 bootstrap steps, but only for the historical Conditional κ = 0.001661960547713947: T37 whole-owner packing and all-T bound #942 T37 record.
• Research audit: binary cover order and exact finite invoice guards #771 (theon199, open) independently reproduces the nine finite invoice guards and the 2,628-bit group order, but for the earlier Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 invoice (row coefficient 16,467), not the selected Conditional κ = 0.001672188119205421: improved T16 owner packing on the seven-slot k8 word #1042 invoice (16,469).

That left the finite invoice of the selected record without a check outside the retained assemble.py. The same was true of its seven-plus-seven bootstrap levels and its 47 + 47 outer slacks. The finite invoice covers the per-vertex envelope and the nine strict size guards (router, literal, precision, tape, row and the others listed below).

What verify.py checks

It uses only the standard library. It imports and executes nothing from the repository. All inputs are pinned and read as data:

• the committed assembly;
• the members t16/global/audit/OWNERS-AND-PROFILE.json and PREFIX-AND-INVOICE.json from the committed selected-replay.tar.gz. Each is checked against the assembly's own input_sha256, and the archive's t16/ASSEMBLY.json must be byte-equal to the committed file;
• PR589's EXPECTED.json, checked against git blob e293ddc5….

The verifier recomputes every quantity and compares it exactly with the record.

1. Profile composition.

   • There are exactly 144 calls, one per (stage, replica) in 9 × 16, with forward and backward orientations alternating.
   • The per-call paid profiles sum to the selected local profile.
   • The idle remainders {1:2249, 2:1, 3:400, 7:3, 16:47, 48:4850} are nonnegative and number 7550.
   • Rank mass is 1,041,408 and the deficit is 73 · 14336 − mass = 5120.
   • The halving guard holds: 73² = 5329 > 2 · 48² = 4608.

2. Per-vertex envelope.

   • Actual selected primitives: 28,625,376. Wrapper factors: 9,953,280.
   • Adapter formula, compiler 64 · 74³, nonlocal item count, unhosted per-vertex 3,588,959,010,714,576 and hosted rebased per-vertex value.
   • The retained prior overcharge gives a used allowance of 6,466,906,000,439,144.
   • The physical stock identity sums to 14,826.

3. The nine strict finite guards. The table below gives the headroom in bits; all nine recorded bit lengths match.

   • Group order. The charged group order is computed in two independent ways: 2^(n²)∏(2^(2j)−1), and the orbit–stabilizer count of hyperbolic pairs. Both give n = 36, i.e. |Sp(72,F₂)|.
   • Interpretation. In characteristic two, the orthogonal group of a nondegenerate quadratic form in odd dimension 2n+1 is isomorphic to Sp(2n,q). So this number is |O(73,F₂)| for the parabolic quadratic form, for m = 73. That is one reading of the "full finite orthogonal group" named in research/native-complex-moment552/PROOF.md.
   • Relation to Research audit: binary cover order and exact finite invoice guards #771. Research audit: binary cover order and exact finite invoice guards #771 reads the same group as the isometry group of the standard dot product on F₂⁷³. It checks that reading by enumerating orthonormal bases in dimensions 1–7. That group has the same order, so the invoice is identical under either reading.
   • Brute-force check. The verifier enumerates every invertible matrix preserving the parabolic form x₀² + Σ x₂ᵢ₋₁x₂ᵢ. It finds |O(3,2)| = 6 and |O(5,2)| = 720, matching the formula. This is a different object from Research audit: binary cover order and exact finite invoice guards #771's, so the two checks complement each other.

   Guard	Bound	Headroom
   local primitive bound	2⁴⁸	4 bits (10,038,040,553,630 ≈ 2^43.19, about 28×)
   logical (group · used + 64)	2⁶⁰⁵⁰	3,370 bits
   router	2³⁰⁰⁰⁰	22,012 bits
   literal	2¹⁰⁰⁰⁰⁰	64,716 bits
   mass (streamed mass vs 2¹⁰⁰⁰⁰⁰)	2¹⁰⁰⁰⁰¹ total	97,352 bits
   precision 32·73·B²	2²¹⁰⁰⁰⁰	9,988 bits
   tape	2²¹⁰⁰⁰⁰	109,998 bits
   row coefficient 2·2642+9909+252+1024	20,161	16,469 < 20,161
   halving	—	5329 > 4608

   Prefix: the local maximum is 68 bits. The global prefix is 68+37+144+144+4 = 397, charged at 512; the row reserve is 2 · 512 = 1024.

4. Bootstrap and outer assembly, for both the native and the conservative endpoint.

   • Bootstrap. It has seven affine levels from 384599/10¹⁰, with target s = min(c, (1−β)b) − 10⁻²⁷ and η = β = 10⁻²⁴. All 14 (exponent, gap, cutoff) records match exactly.
   • Outer slacks. All 47 + 47 slacks equal the recorded rationals exactly and are positive.
   • κ. Each κ is the largest 10⁻¹⁸ grid point strictly below the absorption margin εq.
   • Controls. The four recorded outer controls (next_grid, zero_eta, zero_beta, unit_beta) are re-derived as failures.

5. PR589 bit side.

   • The bit endpoint equals PR589's recorded bit_lower.
   • The bit deficit is 32(2·4·111 − 9·63) = 10,272, and it equals 64·live − mass for PR589's profile.
   • The endpoints are ordered conservative < native complex < bit.

Additional observations

• The bootstrap cutoffs do not depend on the prefix reserve. At every one of the 14 steps the quadratic term ⌈36/gap²⌉ decides the cutoff. It keeps deciding it for any reserve P ≤ 6,566,539 in ⌈2(4+P)/gap⌉. So this lineage's 512-bit linear term, where T37 used 1024, does not affect the recorded cutoffs.
• The tightest finite guard is the local primitive bound, not precision. Its value is dominated by local adapters × compiler (387,056 × 25,934,336). Future provider words with more macros or children consume this 28× margin first.
• The smallest outer slacks are η-scale. For example, λ − τ = aη ≈ 1.7 × 10⁻²⁷, followed by β = 10⁻²⁴. They are positive by construction, not by numerical margin.

Negative controls

There are 15 controls, and each must be rejected. Every one is rejected at its intended check:

• one extra idle child;
• one call's primitive count changed;
• one call dropped;
• one orientation swapped;
• one extra physical register;
• the retained prior overcharge dropped;
• recorded router bits off by one;
• row coefficient set to the bound;
• κ + 10⁻¹⁸;
• one outer slack edited by 10⁻⁴⁰;
• one bootstrap cutoff off by one;
• bit endpoint − 10⁻²⁴;
• complex endpoint + 10⁻²⁴;
• a count- and mass-preserving idle edit introducing a rank-52 child, which fails only the halving guard;
• group |Sp(70,2)| in place of |Sp(72,2)|.

Reproduce

make frontier1042-invoice-audit-verify

Expected output:

PASS_FRONTIER1042_INVOICE_OUTER_INDEPENDENT | guards 9 | slacks 94 | bootstrap 14 | controls 15

Six unit tests should then pass. The run takes about 1.5 s on Python 3.10+, needs no network, and reads the committed archive without extracting it. RESULT.json is regenerated and compared byte for byte; --write rewrites it.

Scope and independence

• How the formulas were obtained. The definitions of the envelope, the guards, the bootstrap and the 47 constraints exist only as code. They live in the retained code/assemble.py and the engines/outer.py engine (sha c1ebb521…). Their conceptual source is notes/three-stage-cover-assembly.tex, § "Full finite router and completed-child guard". I read those definitions and re-implemented them separately; nothing from them is imported or executed.
• What this does and does not catch. It is an independent re-computation. It catches transcription, data-binding and arithmetic errors in the selected record. It does not independently justify why these particular formulas are the right charges.
• What is taken as data. Taken as given are:
  • the semantic content of the receipts: local words, Clifford phases, owner windows and hosting. Those are covered by Research audit: independent check of PR1005/1018 mutable-leaver cells (no new kappa) #1049, Research audit: independent owner-map check of all 15 open T16 claims on the seven-slot k8 word (no new kappa) #1091 and the maintainer replay;
  • the two moment endpoints complex_lower and conservative_complex_lower. Those are certified by the two retained engines and independently by Independent verification: code-disjoint pricing of all 16 pending kappa claims; every one has deficit W - 64N (no new kappa) #1069.
• Inherited assumptions. All the record's conditions remain: all-size/compiler, common-label/mutable-source realization, whole-call hosting, precision/tape, exact recovery, analytic transfer and bit H1–H5.
• What this is not. It is a finite check of one certificate. It is not a formal proof, and it is not a verification of the multiplication theorem.

Credit

• Construction: Jeremy Kothe (Conditional κ = 0.001672188119205421: improved T16 owner packing on the seven-slot k8 word #1042 and its chain), with Rohan Arun's Conditional κ = 0.001662039077758742 — U10 (two new triangle-hybrid slots) on PR942's T37 owners #987 slot search, Hao Yan's Research: T16 owner concentration raises conditional κ to 0.001661642255531641 #858 T16 controller and the full lineage credited in research/gauge44-slots-t16. The bit supplier is DreamingOfClouds' PR589 on eumemic's PR555.
• Assembly and guard engines: the retained engines and their authors as named in their headers: icekylinx (PR144), Zhihao Chen (PR23), eumemic and hipotures' balanced transform.
• Predecessor guard audit: theon199's Research audit: binary cover order and exact finite invoice guards #771 established the group order and the nine-guard recomputation for the Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 invoice, including the row-guard sensitivity to replacing the group with GL(73,2). This package extends that check to the selected Conditional κ = 0.001672188119205421: improved T16 owner packing on the seven-slot k8 word #1042 invoice, the per-vertex envelope, the bootstrap chain and the outer slacks. It does not claim priority for the group-order identity.
• Neighbouring audits whose boundaries this complements: Independent verification: code-disjoint pricing of all 16 pending kappa claims; every one has deficit W - 64N (no new kappa) #1069 and Independent verification: code-disjoint T37 moment endpoints and kappa reduction (no new kappa) #1051 (Sagar Thakrar), Independent verification: T37 positive-binomial moments, all-T lemma, and 1024-bit reduction (seed 78, no new kappa) #1052 (steph4n-gh), Research audit: independent owner-map check of all 15 open T16 claims on the seven-slot k8 word (no new kappa) #1091 (JosieElliston), Research audit: independent check of PR1005/1018 mutable-leaver cells (no new kappa) #1049 and Audit selected T37 bootstrap depth: six levels suffice and five cannot #1009.

Prepared by Wronskode (@Wronskode) with substantial AI assistance from Claude (Anthropic) via Claude Code. New material is Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 3a6a13cf4fbda16e1d191553f8b08cb6a3d5ed27.

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

Reviewed scope: Independent historical PR1042 invoice, nine guards, fourteen bootstrap levels and 94 strict outer slacks; inputs are authenticated data, not a fresh supplier rebuild.

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.