# Computational status of the monograph

This ledger records the maintained numerical foundations and computational coverage of the [canonical source](polyhedral_volumes_and_geodesic_identities.tex). The [final-chapters checkpoint](../docs/execution/final-chapters-review-2026-11-01/README.md) adds Sections 10–11, 17–22, 32 and Appendix B and their finite quantum/geodesic support. The [counting/mixing checkpoint](../docs/execution/counting-mixing-review-2026-11-01/README.md) adds independently reviewed Sections 23–31, finite formula checks and a new full torus diagnostic with its own provenance. The [Section 15 checkpoint](../docs/execution/section15-analytic-review-2026-11-01/README.md) supplies an independently reviewed regular-family quantum Airy proof and new exact-integrand numerical evidence, with separate source and error limits. The [prior quantum and surgery increment](../docs/execution/quantum-surgery-review-2026-11-01/README.md) covers finite period formulas and local peripheral/surgery identities in Sections 12–14 and 16. The [prior normalization checkpoint](../docs/execution/normalization-review-2026-11-01/README.md) covers Sections 1–8 and Appendix A with 41 claims and an exact finite companion; Section 9 retains its separate bounded mathematical review. Each checkpoint records its independent reviews, source identities and integrated build verification separately. The [prior Airy and torus checkpoint](../docs/execution/monograph-revision-2026-11-01/README.md) introduced finite checks and a bounded replacement experiment without designating either whole chapter mathematically reviewed. The original source snapshot and supplied reference PDF retain their original bytes. A reproduced finite example does not audit a general proof, and missing computational evidence does not by itself disprove a mathematical statement. Historical line references refer to the monolithic source at `26b3c134d9308d345cb76f1240db9de982b2e5a6`; stable labels identify the current text.

## H³ numerical foundations

The [numerical errata](numerical-errata.md) records the bounded software corrections and changed values. The [execution checkpoint](../docs/execution/numerical-correctness-2026-10-31/README.md) preserves author results, independent review findings and their separate resolution stages. Existing scientific tables and research outputs have not been comprehensively replayed or rewritten.

| Evidence type | Maintained contract | Limit |
| --- | --- | --- |
| Binary64 volume | [Canonical Bloch–Wigner kernel](../packages/volume-toolkit/docs/numerical-precision.md), explicit method in final angle/optimizer reports; finite large arguments handled without magnitude overflow. | Floating-point values, not certified enclosures. Legacy `fast` defaults remain finite Fourier approximations. |
| Multiprecision evaluation | mpf-returning kernels preserve supported inputs and scope working precision; legacy `exact`/`eval_only` names alias numerical mpmath evaluation. | High-level output and Qhull/projection geometry remain binary64; extra working digits cannot recover lost input information or validate topology. |
| Angle feasibility and stationarity | [Common-margin LP and full numerical KKT audit](../packages/volume-toolkit/docs/rivin-numerics.md), with feasible steps and active-constraint release. | A verified finite angle-program witness or numerical stationary point is not a global embedding, exact infeasibility or interval-optimality certificate. Exact flat-angle optima are outside the differentiable solver. |
| Optimizer endpoint | [Fresh canonical endpoint checks](../packages/volume-toolkit/docs/optimizer-result-contract.md), separate raw convergence and candidate-validity fields, strict schema-v2 JSON. | A valid unconverged candidate can be retained; no global maximum follows. Historical v1 files retain their original status. |
| Exact finite identities | Section 9 integer/rational matrices and forest enumeration; the strict LP fixture's rational primal and dual witnesses establish common margin `1/7`. | Exact statements about the named fixtures only. They do not certify general numerical routines. |

The free-corner-angle optimizer with dihedral inequalities and Section 9's prescribed-dihedral leading–trailing coordinates describe different feasible families. A KKT result for the former is not a proof of a coordinate-space assertion in the latter. The imported [historical companion](../research/ipoly/duffin_rivin_volume_laplacian_solvers.tex) remains unchanged; the [reconciliation note](companion-reconciliation.md) records the reviewed positivity, exterior-angle Schläfli sign, complexity and benchmark corrections.

## Sections 1–8 and Appendix A: normalization and exact finite checks

The [Sections 1–6 and Appendix A ledger](claims/sections1-6.json) contains 28 claims; the [Sections 7–8 ledger](claims/sections7-8.json) contains 13. They separate local derivations, checked source statements, hypotheses, finite computations and remaining work. The intrinsic realization statement is checked against Fillastre's version-pinned restatement, Theorem 1.1 of arXiv:0801.0538v1; the original Schlenker proof has not been inspected. This increment does not designate every external proof or the whole monograph reviewed.

The [fixed companion](../experiments/normalizations/README.md) uses exact rational arithmetic with explicit strings as inputs. It checks the tetrahedral perimeter determinant and two oriented sheets; torus stack/effective factors; cone-wall corrections and strict final chambers; n=4/5 compactified intersections against an independent set-partition formula; a symmetric hyperideal Gram/density fixture; a Pfaffian; and SL₂ Fricke/twist identities. Six flat-weight fixtures include balanced perturbations and a light triple represented by its complementary pair boundary. Coordinates use `a=A/pi`, `V_WP/pi^(2d)` and curvature −1 `V_CH/pi^d`, where `d=n−3`.

Independent computational review passed 497 rational weight vectors and 1,491 direct class evaluations, using a separate five-coordinate intersection model; 512 SL₂ triples; 195 n=6 F-curve probes; and all 11 regression tests. The larger probes remain finite checks conditional on the mathematical intersection rules. They do not prove a general divisor identity, analytic metric convergence, global realization of the SL₂ fixtures, or a full hyperideal cell integral. The three n=6 probes in the maintained driver are illustrative rather than a general volume implementation.

The initial result records a dirty development tree. The checkpoint's output paths are `artifacts/normalizations/result.json` and `artifacts/normalization-review-2026-11-01/monograph-working-draft.pdf`, with generation, source identities and verification recorded separately. Source hashes, exact inputs, units, runtime and evidence limits accompany the result. Earlier Section 15/30 results, PDFs, Site exports and downloads remain frozen and retain their own source records.

## Section 9: reviewed mathematics and reproducible finite examples

The [modular chapter](chapters/09-volume-hessians.tex) and [claim ledger](claims/section9.json) now separate general derivations, checked external theorem statements, conditional algorithmic consequences, exact finite checks, numerical comparisons and open questions. The [core audit](../docs/execution/section9-chapter-2026-10-31/math-audit.md) supplies coordinate completeness and grounded strictness, including zero Delaunay weights. The [companion audit](../docs/execution/section9-chapter-2026-10-31/companion-audit.md) supplies the finite-centering proof and source-scope checks. Neither report claims a complete re-refereeing of the cited literature or the rest of the monograph.

| Source claim or convention | Current implementation and evidence | Scope and next review |
| --- | --- | --- |
| Leading–trailing angle coordinates, `eq:ipolyAngleCoordinates` | [Maintained operators](../packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py) and [fixture](../experiments/section9/fixture.json) use the directed opposite-edge convention. | Completeness is proved for a disk; an exact annulus counterexample records the topology restriction. Fixed boundary angle coordinates and electrical boundary voltages are separate variables. |
| Negative volume Hessian equals twice the grounded FEM matrix, `eq:ipolyHessian` | Integer grid gives `L_II=[[4,-1],[-1,4]]` and `K=[[8,-2],[-2,8]]`. Independent rational FEM assembly and high-precision numerical differentiation check this fixture. | The general identity, CCW gradient convention and grounded kernel are derived in the chapter and independently audited. A planar FEM realization is additional geometry; arbitrary deformed angles remain abstract. |
| Dirichlet-to-Neumann and Schur operations, `eq:ipolyDtN`; `eq:ipolySchur` | Eliminating the first variable of `K` gives `15/2`, with determinant `8*(15/2)=60`. Boundary voltage `x/3` gives interior values `1/3,2/3` and energy `2/3`. | General energy-minimization and block identities are proved; computations use dense small matrices. No nearly-linear solver or complexity benchmark is supplied. |
| Forest determinant formula, `eq:ipolyForestDeterminant` | Exhaustive boundary-rooted forest enumeration gives fifteen unit-weight terms; multiplying by `2^2` gives `det(K)=60`. | The general incidence-minor proof is separate from exact finite enumeration. Enumeration is an oracle for this fixture, not an efficient general algorithm. |
| Barrier and short-step bounds, `prop:ipolyCompatibility`; `eq:ipolyStepBound` | The [dense affine-angle solver](../packages/volume-toolkit/ideal_poly_volume_toolkit/angle_barrier.py) implements the log-sine barrier, separately budgeted centering and the specified path schedule. Its [fixed-fixture driver](../experiments/section9/barrier_run.py) compares against independent multiprecision references and records all accepted iterates. | The mathematical finite-center bound is `17m/(12t)` at decrement at most `1/4`; its binary64 evaluation is a numerical estimate. No certified energy-error guarantee or nearly-linear backend is supplied. Feasibility construction, initial centering, precision and geometric error have separate costs. |
| Full three-manifold criticality, `subsec:ipolyThreeManifolds` | Checked scope of the Casson–Rivin theorem and an exact local non-SDD Hessian example. | Requires an orientable ideal-triangulated compact manifold with torus boundary and full angle-space criticality, including peripheral directions. No complete peripheral basis, solver or metric certificate is supplied. |

The [worked companion](../site/section9/index.html), [reproduction guide](../experiments/section9/README.md), [runnable archive](../site/section9/downloads/section9-reproduction.zip), and [result](../artifacts/section9/result.json) connect the text, implementation and evidence. The [initial mathematical review](../docs/execution/2026-10-31/section9-review.md) is preserved, with its three implementation findings addressed in the [independent re-review](../docs/execution/2026-10-31/section9-rereview.md). A separate [installed-package verification](../docs/execution/2026-10-31/package-verification.md) ran both the corrected wheel and extracted archive outside the source tree in a minimal NumPy/SciPy/mpmath environment. The result records the actual source and input hashes, methods, versions and numerical tolerances. Its source commit and dirty-state fields identify the run precisely.

Zero-weight square diagonals make this example non-strict Delaunay. Deformed off-origin angles are abstract angle structures; geometric closure is not asserted. The example supplies neither a strict convex-polyhedron certificate nor an interval enclosure. Its authored HTML companion remains distinct from the complete chapter conversion produced by the [chapter build](../site/README.md). Build, PDF rendering and browser verification have separate status in the [chapter execution evidence](../docs/execution/section9-chapter-2026-10-31/README.md).

The separate [barrier examples](../site/section9/barrier.html),
[result](../artifacts/section9/barrier-result.json), and
[reproduction archive](../site/section9/downloads/barrier-reproduction.zip)
cover the new dense solver. Its contract requires a supplied positive affine
slice; it does not infer that a chosen basis spans every geometric deformation.
The result distinguishes a finite gap estimate, a centered point, a reached
target parameter, and numerical convergence. It returns unsuccessful states
for exhausted budgets or failed numerical safeguards, and never marks a result
as formally certified. The [barrier checkpoint](../docs/execution/section9-barrier-2026-10-31/README.md)
owns the new numerical and export review. The earlier mathematical review is
still tied to its frozen source hashes, with proof mathematics retained and
computational prose updated separately. Original chapter-generation reports
remain historical records rather than checks of these changed bytes.

At the numerical-correctness checkpoint, the result and reproduction archive were regenerated from clean source `9188e0624cdf6d9949af1f620336eff67c8b59d7` and committed as `29440cfbfcdaa23711cfdac4e09bf5f939404273`. That run passed all twelve numerical checks and exact fixture identities, with all seven recorded source/book hashes matching. Its archive was extracted outside the checkout and replayed in the minimal environment, verifying all nine payload hashes and its clean originating commit. These are results for that source state, not hash verification of the subsequently revised chapter; see the [numerical closeout](../docs/execution/numerical-correctness-2026-10-31/closeout.md). A later generated result must identify its own source and artifact provenance. Earlier source `c02b358de5a6f724faf3368a641ba64f5841d859` and artifacts `74e2abc67e434e0d7eadf70d7becaa6de2f77785` remain preserved in Git history and the first checkpoint. The [preliminary installed-wheel and detached-archive replays](../docs/execution/numerical-correctness-2026-10-31/integration-section9-replays.json) remain separate records of the modified checkout examined before the clean-source run.

## Sections 10–11: quantum conventions and classical degeneration

The [61-claim ledger](claims/sections10-11.json) records local derivations, exact dependency snapshots and selected external statements. The refined Chen–Murakami prefactor requires the exact arithmetic label sequence stated in the chapter, beyond the broader leading-volume theorem. Costantino's pole-removed evaluation and doubled geometric block retain their separate convention. The local phase/symbol assumptions, regular-family Airy scope, convex Klein path and differentiated curvature remainder are explicit. Selected source statements were inspected with the access limits in the checkpoint; no complete external-proof audit is claimed.

The finite companion's quantum groups connect these formulas to the following checks.

| Group | Formula checks | Limit |
| --- | --- | --- |
| Q1 | Figure-eight positive products, the finite complex root evaluation, terminal sine product and maximizing indices for the declared N list. | Finite knot evaluations; no general colored-Jones/6j solver or numerical asymptotic proof. |
| Q2 | Exact Klein radial moments, independent simplex/box moment formulas, vertex-derived Lorentzian face coefficients and the curvature sum for three polyhedra. | Named homothetic fixtures; the positive series tail controls truncation only. No general volume quadrature or certified geometric realization. |
| Q3 | Multiplier/half-log conventions, the Weyl shift, determinant-one filling transforms and formal kernel/Hessian identities. | Guarded reuse of exact filling algebra; no old result is imported and no actual filling, torsion invariant or global surgery contribution is computed. |
| Q4 | Rational charge weights, first-moment inequalities, the screening exponent, curvature factor and top exterior powers. | Finite specialization; no GMC construction, spectral-tail estimate or moduli integral. |

The independently reviewed author result records 1,342 exact checks, 4,266 floating comparisons and 509 cross-precision observations. The [implementation acceptance](../docs/execution/final-chapters-review-2026-11-01/computation-review/revision-v2/final-acceptance.json) also records a fresh reviewer run with identical scientific fields and all 50 focused tests passing normally and under `-O`. Both runs retain their recorded dirty source state; the later clean-source generation has its own receipt. The earlier 400 exact and 1,322 alternate numerical oracle checks are reused by unchanged mathematical/fixture identity, not claimed as newly rerun.

Requested decimal precisions are 70 and 110. Ordinary contexts use 90/130 actual digits, G3 uses 170/210, and G5 uses 150/190; these are working settings, not certified achieved digits. Positive numerical signals use relative or explicitly scaled comparisons; a tiny positive value cannot pass merely by being rounded to zero. Exact algebra, numerical agreement, cross-precision observations and analytic truncation bounds remain distinct. Reproduction instructions are in the [companion README](../experiments/quantum_geodesics/README.md); the final result path is `artifacts/quantum_geodesics/result.json`. The checkpoint binds implementation review and author evidence separately from the later clean-source generation.

## Sections 12–14 and 16: finite periods, local geometry and formal surgery

The modular chapters and [Sections 12–14 ledger](claims/sections12-14.json) separate the elementary first-moment and period derivation from an additional spectral-continuation assumption, and the rank-two screened Gram data from the rank-three compact classical fold. The [Section 16 ledger](claims/section16.json) records multiplier-log conventions, local analytic-germ and constrained-response hypotheses, and the extra phase, amplitude, cycle and dominance assumptions needed for quantum transfer. These source and hypothesis reviews do not turn the corresponding global quantum claims into finite computations.

The [bounded companion](../experiments/quantum_surgery/README.md) has the following coverage:

| Source formulas | Computation | Evidence limit |
| --- | --- | --- |
| Section 12 Fricke/Heron expansion and charge conventions | Exact rational series and three Heron-chart points check coefficients, orientation, first density correction and the distinct heavy-charge/dictionary scalings. | Fixed algebra, not a metric-limit or global measure-convergence proof. |
| Section 13 finite fusion, Hermitian area, Wronskian and curvature −1 density | Two rational weight tuples with both signs of δ₀, real and conjugate complex points, hypergeometric formulas and independently derived Euler-arc quadrature at 60/90 digits. A separate balanced logarithmic/elliptic basis checks its series, fusion and Poincaré density. | Shared mpmath arithmetic backend; no interval enclosure, certified series tail, numerical monodromy continuation or spectral-tail estimate. |
| Section 14 screening and compact fold | Numerical planar Gram/cofactor/determinant identities; exact regular-area/dihedral Taylor coefficients, saddle factorization/discriminant and non-branch double root. | No tetrahedron realization from the screened matrix and no uniform quantum integral asymptotic. The existing Section 15 companion is not rerun or imported. |
| Section 16 local filling and response | Exact θ⁵ inverse and θ⁶ stationary action, peripheral covariance/canonical differential, figure-eight local-curve coefficients, constrained K/P/E cubic/quartic tensors, one-cusp Hessian and one conditional two-cusp block example. | Formal finite coefficients and matrix identities, not a filled-manifold computation, geometric volume verification, torsion invariant or global quantum result. |

The initial dirty-tree author run passes 192 exact identities, 300 numerical checks across 60/90 digits and 72 cross-precision comparisons, with 15 focused tests and Ruff passing. The largest observed scaled numerical residuals are about 2.00×10⁻⁶⁰ and 2.34×10⁻⁹⁰; the observed cross-precision difference is at most 4.32×10⁻⁶⁰ on these fixtures. These are measured agreements, not certified accuracy guarantees. Endpoint-denominator substitutions remove the Euler integrals' singular powers before quadrature, including the weight 9/10. Scientific inputs remain exact rational strings, and each numerical run uses a local precision context.

The prior quantum/surgery increment records its clean-source result at `artifacts/quantum_surgery/result.json` and its reviewed PDF at `artifacts/quantum-surgery-review-2026-11-01/monograph-working-draft.pdf`. Its [checkpoint](../docs/execution/quantum-surgery-review-2026-11-01/README.md) records independent review, final generation, source hashes and PDF verification separately from the initial author result. The prior normalization and Section 15/30 outputs remain frozen. Neither the finite fold checks nor the local surgery algebra supplies the quantum theorem or its exact-integrand evaluation; those have separate evidence in the current Section 15 checkpoint. The original quantum evaluator and raw runs remain unrecovered.

## Section 15: reviewed regular-family theorem and exact-integrand numerics

The [Airy chapter](chapters/15-quantum-airy.tex), `sec:quantumAiry`, now has an independent analytic review of its specified regular-family transition for fixed bounded real zeta. The proof explicitly fixes the convergent double-Gamma/double-sine normalization, including the factor-two subtraction typo in TV v3 (A.1); establishes the phase constant through the real Murakami–Yano formula; relocates arbitrary separating contours at fixed b; and proves uniform global tails, a displaced Gaussian bound, the integrated Taylor remainder and the oriented first correction. The [claim ledger](claims/section15.json) records these analytic dependencies separately from finite computations. The theorem does not cover arbitrary tetrahedral shapes, unbounded zeta, fixed nonzero volume or a global quantum invariant.

The [new companion](../experiments/section15_quantum/README.md) evaluates the exact cleared Barnes integrand using the centered identity for log S_b. A finite cotangent-derivative expansion of D_b has an exact rational truncation bound on the declared centered strip. The five logarithms are combined before exponentiation; the Airy approximation is computed only for comparison. The seven inputs are the six printed pairs and `b=1/160,zeta=-1`. Each has 60-digit `C=1,T=12`, 90-digit `C=1,T=14`, and 90-digit `C=5/4,T=14` evaluations, with two extra D terms in the last setting and separate shorter-cutoff extensions. Direct D and full-log-S integrals, reflection and shift identities, and an independent combined-Barnes QUADPACK evaluation provide additional finite checks.

The source-recorded 21-quadrature matrix has observed 60/90-digit differences below `4e-57` and contour/order differences below `1.3e-91`; the separate whole-Barnes reference agrees on all seven inputs within `2e-12`. These are measured agreements, not certified achieved digits. The exact rational bound controls D-series truncation conditional on the centered-strip hypothesis, not floating-point evaluation. Propagated correction errors use numerical absolute-integral estimates; outer quadrature errors, sign checks and tail-majorant formulas lack outward rounding. No total interval quadrature enclosure is claimed.

The new first quantum value is `0.459086488736078952...`, which rounds to `0.459086` rather than the supplied `0.459087`; the other five rounded values agree. The three new zeta=1 scaled residuals are approximately `1.208081684`, `1.211366819`, and `1.208878763`. The negative-parameter check gives `Q_(1/160)(-1)=0.517966671856531...`. The original evaluator, raw values and rounding convention remain unrecovered. The [historical JSON](../experiments/section15/historical-table.json) is unchanged; new evidence does not reconstruct its missing provenance.

The earlier [finite companion](../experiments/section15/README.md) and [result](../artifacts/section15/checks.json) remain frozen. They supply 31 exact checks over the rationals with sqrt(2), plus the six Airy approximation values at 80/120 digits and a separately coded ODE series. They neither evaluated the exact quantum integral nor established the global analytic proof at that earlier checkpoint.

The new result path is `artifacts/section15_quantum/result.json`. Its [checkpoint](../docs/execution/section15-analytic-review-2026-11-01/README.md) records the source snapshots, independent analytic and computational reviews, final clean generation and integrated PDF verification separately. Finite numerical agreement supports the seven evaluations; the chapter's uniform theorem rests on the analytic argument.

## Sections 17–22: finite geodesic identities

The [59-claim ledger](claims/sections17-22.json) distinguishes local proofs from external theorem inputs and finite formula checks. Unfolding retains nonnegative or absolute-integrability hypotheses and stabilizer/orbifold factors. Twist convexity uses fixed lifts, itinerary and winding, with disjoint curves as the constant-length exception. Geometric trace signs, the two N-type and one H-type torus labels, signed cusp truncation and the oriented/unoriented Basmajian coefficient remain explicit.

The companion's geodesic groups have the following bounded scope.

| Groups | Formula checks | Limit |
| --- | --- | --- |
| G1–G3 | Torus normalization/Fermi integrals, exact trace and multiplicity algebra, and finite universal-kernel coefficients and tails. | No moduli quotient proof or full surface-curve sum is computed. Cancellation-limited residuals are distinguished from positive tail evaluations. |
| G4–G5 | Lorentz distance, endpoint derivatives/Hessian, seam matrices and one-crossing torus twist holonomy. | Fixed local fixtures; no arbitrary multicurve minimizer, global strict-convexity proof by sampling or discrete-holonomy certificate. |
| G6–G7 | Coarea/beta integrals, sublevel formulas, H/K lattice partitions and stable finite tails. | Analytic truncation bounds exclude numerical quadrature and roundoff; no general infinite-spectrum tail is supplied. |
| G8 | Position/winding matching and a directional stationarity example. | Endpoint position alone does not establish smooth geodesic matching; no general gluing solver. |
| G9–G10 | Cusp parabolic/decoration and ideal-triangle identities; local Basmajian shadows and orientation factors. | Signed truncations and local shadows do not enumerate a complete orthospectrum or certify a surface. |

These groups use exact inputs and selected 70/110-digit comparisons with declared guards and scales. The [companion](../experiments/quantum_geodesics/README.md) and `artifacts/quantum_geodesics/result.json` map each record to its formula. Implementation review, actual generation and source identities belong to the [checkpoint](../docs/execution/final-chapters-review-2026-11-01/README.md). Earlier numerical campaigns are preserved, not replayed by this increment.

## Section 32 and Appendix B: questions and convention dependencies

The [66-record ledger](claims/section32-appendixb.json) covers fifteen open questions, 43 convention rows (including all 28 original rows) and eight narrative/source records. It keeps exact reviewed dependency snapshots separate from integrated current files. The positive-boundary tree comparison now names Zonneveld's selected unequal pair and equal-length volume values by continuity, rather than using the origin-cusp formula as the all-positive comparison. Existing finite result identities are checked without claiming a fresh replay. New quantum/geodesic support is attached only at its reviewed finite scope. None of the fifteen questions is a requirement to solve before publication.

## Sections 23–31: counting, mixing and arithmetic

The four ledgers cover 168 independently reviewed claims: [52 in Sections 23–25](claims/sections23-25.json), [59 in Sections 26–29](claims/sections26-29.json), [32 in Section 30](claims/section30.json), and [25 in Section 31](claims/section31.json). They retain the distinction between local derivations, external theorem statements, conditional implications and open comparisons. The nonsquare genus-two restriction, endpoint weights and observable norms remain explicit. The all-point `O(L/log L)` target requires the compatible local row-width condition; infinite flatness alone does not provide it. The pinned recent OpenAI and Doan–Li–Nguyen statements were checked, but their complete proofs and formalizations were not audited.

The [finite companion](../experiments/counting_mixing/README.md) supplies these bounded checks:

| Source formulas | Implemented evidence | Limit |
| --- | --- | --- |
| Sections 23–25 trace and coefficient filters | Exact recurrences through power 12, four rational SL₂ word fixtures, global weights, support selectors and finite length-valued Euler convolution. | Formal lengths need not come from one hyperbolic metric; no infinite-product or general first-atom proof follows from the fixture. |
| Sections 26–28 counting and endpoint models | Exact periodic/aperiodic matrix examples, terminal exponential identities, inverse-residue characters and finite Fourier checks. | No general mixing theorem, spectrum, geometric classification or square-tiling/bisector comparison is computed. |
| Section 29 observables | Exact rational primitive/parity counts and Gaussian algebra; one- and two-mark Gaussian formulas, divisor/logarithm series and determinant-orbit quadrature at 60/90 working digits. | Analytic series/integral truncation bounds are separate from uncertified quadrature and roundoff. Finite collinear growth does not prove divergence. Optional rectangle clipping and geometric length-derivative illustrations were not implemented. |
| Sections 30–31 arithmetic | Exact square/diamond Möbius inversion, finite dyadic telescopes and event sweeps, signed multiples, closed row selectors and conditional balance exponents. | No geometric modulus, arithmetic hypothesis or uniform infinite-range estimate is established. |

The result path is `artifacts/counting_mixing/result.json`. Independent review found and required repair of the initial rank-two radial quadrature acceptance, where a tiny signal could pass an absolute tolerance despite poor relative accuracy. The corrected v2 source has independent approval with no open findings: the reviewer's fresh replay passes 4054 exact checks, 354 numerical comparisons and 297 cross-precision observations, and all 38 focused tests pass. The [targeted review](../docs/execution/counting-mixing-review-2026-11-01/computation-review/revision-v2/review-v2.md) and its [approved manifest](../docs/execution/counting-mixing-review-2026-11-01/computation-review/revision-v2/approved-manifest-v2.json) preserve that source identity. This approval does not extend to the rejected v1 numerical acceptance. The final canonical replay and its generation identity have a separate checkpoint record. Neither the earlier nominal pass count nor the repaired comparisons is a total error certificate.

## Section 30: new full torus campaign and frozen predecessor

The [new full campaign](../experiments/torus_campaign/README.md) uses the same two exact trace triples as the [sharp-diagnostics chapter](chapters/30-sharp-diagnostics.tex). It obtains 18,191,508 modular and 18,486,116 asymmetric curves through hyperbolic length 8192 and matches every historically printed M/Q entry at four decimals. The result path `artifacts/torus_campaign/result.json` records all twenty block rows, dyadic primitive counts, signed all-point `A(L)=2 sum floor(L/ell)` and `A(2L)-4A(L)`, selected precision checks and exact rational conditional sensitivity envelopes. Slope identity streams and sorted binary64 length streams have separate hashes and are not index-aligned; the recipe and checkpoint record their retention and regeneration.

The campaign compares the complete length-64 list with an exact-trace oracle at 100/140 digits, validates a moderate length-512 traversal independently, and retains 151/126 selected long samples, including 36 near-8192 curves per surface. The worst selected binary64 residual is below `1.2e-9`. Independent review additionally used 100/160-digit matrix-word samples and recomputed every dyadic primitive and all-point count from all 36,677,624 stored lengths. These are finite sampled comparisons and exact operations on stored binary64 values; they do not certify all true lengths, threshold membership, event order or numerical pruning.

The new rational sensitivity envelopes assume length error at most `1e-6` for each listed curve. Their maximum widths across all ten blocks per surface are below `1.232e-6` and `5.613e-7`. Applying them to the true spectrum also requires completeness and no entry by omitted curves under perturbation. The historical `7.5e-13` long-path precision and `7.1e-7` sensitivity claims retain their historical status; their implementation, sample choice and bound construction remain unrecovered. Reproducing the counts and rounded rows does not reconstruct that missing archive.

The [length-64 predecessor](../experiments/section30/README.md) and its [result](../artifacts/section30/result.json) retain their earlier source and artifact bytes. They record 1104/1127 curves and their own oracle, metric and sensitivity checks. The [historical torus-bundle script](../research/ipoly/v8_torus_bundles.py) concerns three-dimensional bundle volumes and is not the missing counting experiment.

The Section 31 signed-multiple extension uses hyperbolic length units. For systole `s_X`, the unit-shortest gauge is `lambda=ell/s_X`, so `A_B(L)=A_hyp(s_X L)`. The campaign's dyadic cutoffs must be interpreted with that scaling. No finite diagnostic proves an asymptotic linear, sublinear or logarithmic remainder, identifies the required flat rational-normal point, or validates a recent preprint proof.

The [checkpoint](../docs/execution/counting-mixing-review-2026-11-01/README.md) records the analytic and computational approvals, required numerical/parser repairs, final source generation, raw-stream identities and integrated PDF verification separately. The working PDF path is `artifacts/counting-mixing-review-2026-11-01/monograph-working-draft.pdf`. Reviewed author runs retain their actual source states and are not relabeled as final clean replays.

## Further editorial and release work

The claim and dependency records are tracked in the [full audit](../audit/2026-10-31/monograph-audit.md), the chapter ledgers and the [plan](../INTEGRATION_PLAN.md). The bounded chapter-by-chapter pass now covers Sections 1–32 and both appendices. This coverage does not mean every cited proof was re-refereed or every research computation implemented. The original Airy evaluator and torus precision archive remain unrecovered; replacement evidence keeps its own identity. Cross-chapter editorial consistency, the complete local edition and release checks remain. Revisit earlier chapters only where those checks reveal a concrete dependency or presentation issue.

Existing Section 9 site files, its broader monograph-source ZIP and the reading-preview bundle remain frozen earlier checkpoints. They have not been refreshed for this manuscript revision. The supplied reference PDF and its earlier visual sample do not verify a new build. The earlier full-suite failures and collection errors associated with unavailable optional dependencies also remain unresolved; focused companion tests are not a replacement for that suite. Manuscript revision, computational evidence, local edition checks, authorship/credits and text/companion licensing precede the final upload/deployment step. No Site has been published by this revision.