# RUN-042 — DLMVC's four deep passes on Round 04: every constant recomputes and the error bound holds on random bodies; two of Pass 04's quantities are grid samples on the unsafe side, and its figure for Pass 03's cell radius is a different quantity

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** the four deep lagged-audit passes of Round 04 in DLMVC's folder, each with its package (the document, a script, `VALIDATION.json`), read from the archive:

- Pass 02 (2026-09-18) — error-budget compression, admissible support dictionary, threshold-adaptive compiler;
- Pass 03 (2026-09-18) — exact support-cell geometry and branch-and-bound primitives;
- Pass 04 (2026-09-18) — phase-Lipschitz lower envelope, guaranteed inner core, bi-critical cells;
- Pass 05 (2026-09-19) — finite polyhedral verifier, quadratic branch closure, contact skeleton.

They are not among the 42 canonical documents; RUN-041 read DLMVC and the passes' predecessors.

**Tools:** [`run042_dlmvc_deep_passes.py`](../code/run042_dlmvc_deep_passes.py) — mpmath at 30 digits for the constants; RUN-007's outer polygon (from the definition) for the error bound on random bodies; the spike cell rebuilt from its support function, extremes found by zooming; the area forms' spectra
**Logs:** [`run042-dlmvc-deep-passes.json`](../data/gate-logs/run042-dlmvc-deep-passes.json)

**Verdict: PASS for the passes' theorems and every constant and sanity number but three. Pass 04's inner-core factor and sharp cell radius are grid samples on the unsafe side, so what it boxes with them is not established by its own argument; and its figure for Pass 03's cell radius is another quantity.**

- **Pass 02.**
  - All twenty constants its VALIDATION records, at the cap Ā and at the threshold 0.8344, recompute to
    10⁻¹².
  - The adjacent-normal error ε_deep = 0.013327 holds against the actual outer error of Round 04's
    compiler, on 60 random convex bodies between B_{1/2} and B_{R_sharp}: at most 63 % of it, median 35 %.
    It is 84.02 % below Round 04's.
  - The two identities of G.1 hold, the second with its maximum at t = Δ.
  - The adjacency step 51 (48 at 0.8344) is safe, but 50 (47) suffices, since the quantization error is
    strictly below q.
- **Pass 03.**
  - The spike cell U* = conv(B_{1/2} ∪ {R_sharp u_{0.013}}), rebuilt, has 103 active facets, by the
    redundancy criterion and by clipping alike.
  - Its outer area 0.846661292560721 agrees between the exact formula and the clipped polygon to 10⁻¹³.
    Perimeter 3.3771690420862; perimeter lower bound 3.36584089305767; area lower bound 0.841460223264417.
  - The cell radius is 0.013317, at j = 0, t = 0.02503 — the package's number, on the package's 4,001-point
    grid.
  - The uniform formulas hold on 200 random codewords, and the regular polygon's two areas agree to
    8×10⁻¹⁵.
- **Pass 04.**
  - The phase floor, the phase-integrated bound 0.842616436112935 and the radial bound 0.844007644444885
    recompute to fifteen digits.
  - **The inner-core factor λ_b** is a minimum. The recorded 0.9882053662605436 is its value on a grid of
    240,000 angles; zoomed, the minimum is 0.9882022551, at θ ≈ 0.81765. There the recorded core λ·h_P lies
    1.6×10⁻⁶ above the guaranteed envelope L♯. So the boxed 0.9882…·P(b) ⊆ U is not established for the cell
    by the pass's own bound. U* itself still clears it there, by 8×10⁻⁵.
  - With the true λ, L♯ ≤ h_U* ≤ h_P and λP ⊆ U* hold everywhere checked. The homothetic area bound becomes
    0.826801748731, against the recorded 0.8268069548.
  - **The sharp cell radius** is a maximum. The recorded 0.0059677037 comes from the same grid; the maximum
    is 0.0059692967, 1.6×10⁻⁶ higher, so the recorded radius is not an upper bound as stated.
  - **§S.1's "Pass 03 cell radius" 0.013060516** is not Pass 03's radius, which is 0.013316971 — Pass 03's
    own figure and this tree's. It is the same construction with the polygon's exact support h_P in place of
    Pass 03's H_j, sampled. Its true maximum is 0.0130615798.
- **Pass 05.**
  - The uniform area form at M = 128 has signature (1, 2, 125), numerically and in closed form: the zero
    modes are k = 1 and 127, and the extremes −81.470968 and 0.049097244.
  - On random fans of 5 to 32 normals the signature is always (1, 2, m − 3).
  - On 12 directions the translation dual has 6 antipodal pairs and 40 origin-enclosing triples, and their
    supports reach 6 candidate heights on a half-shifted grid.
  - The M = 8 contact basis is exact: b₃ = b₇ = (p − 3)/2, p on [4, 3 + √2], and
    A(p) = −p²/2 + (3 + √2)p − (6 + 2√2). The minimum is 2√2 − 2 at p = 4, the octagon around the disk.
    Its "perimeter mode" p is Σb, not the perimeter.
- **The scripts,** read after the rebuild and never run (W5).
  - Each prints exactly its VALIDATION's replay block. Pass 02 uses the formulas rebuilt here, with the
    step by a ceiling; Pass 03 quantizes without a guard; Pass 04 takes its extremes on the 240,000-angle
    grid with no refinement; Pass 05 gives its spectrum in closed form.
  - The rest of each VALIDATION comes from no shipped script: Pass 03's cell radius, Pass 04's phase and
    homothetic bounds and its figure for Pass 03, and Pass 05's eigenvalue extremes, sparsity example and
    toy basis.

Drill: 11 / 11 plants caught by exactly the named checks, 2 / 2 controls undisturbed,
78 / 78 must-stay-green assertions held. Each of the seven checks was blinded in turn, and each was reported
blind in exactly as many cases as plants name it.

---

## What the passes claim

1. **Pass 02.**
   - The support function's angular Lipschitz constant is √(R² − r²), its exact variation V_{R,r}(t), and
     from adjacent normals an outer error ε_deep.
   - Perimeter ≤ 2A/r, and an area error β_deep.
   - Admissible codewords C1–C3 and a dihedral quotient.
   - A lower-bound search can use the threshold's own radius.
2. **Pass 03.**
   - A redundancy criterion, exact active-fan edge lengths and area, a cell's lower and upper envelopes and
     radius.
   - Closed-form perimeter and area lower bounds.
   - Refinement only for low-area, zero-margin cells.
3. **Pass 04.**
   - arcsin(h/R) is 1-Lipschitz, which gives a sharper envelope L♯, a guaranteed Wulff core and a homothetic
     core λ_b P(b) ⊆ U.
   - Three area lower bounds, and the bi-critical cells.
4. **Pass 05.**
   - The fixed-resolution verifier's certified region is a finite union of polyhedra, and each extreme
     witness is at most 6-sparse.
   - The fixed-fan area form has signature (+, 0, 0, −, …), so on a perimeter slice the minimum sits at a
     vertex and each contact basis reduces to one quadratic in p.
   - Cross-resolution saturation stays open.

## A — sources

**A0.** 26 anchors in the four documents. The passes cite standard convex geometry and Fillastre–Izmestiev
for the signature, which is checked here numerically rather than fetched.

## W — rebuilt

| check | pass | rebuilt | result |
| --- | --- | --- | --- |
| W1 | 02 | 20 constants at two caps; ε_deep on 60 random bodies; the identities; the step | PASS; 50 / 47 would do |
| W2 | 03 | the spike cell: facets, areas, bounds, cell radius; the formulas on random codewords | PASS |
| W3 | 04 | phase, radial and homothetic bounds; λ_b and ε♯ zoomed; the sandwich on U* | PASS for the bounds; λ_b and ε♯ recorded on the unsafe side |
| W4 | 05 | signatures (uniform, random fans); the dual's extremes and sparsity; the M = 8 basis | PASS |
| W5 | — | the four scripts | read; much of each VALIDATION has no shipped source |

## P — the package

**P1.** Each VALIDATION describes its document correctly: digest, size, lines and control characters. Its
exactly recomputable numbers are this tree's to 10⁻¹²: Pass 03's facets, areas, perimeters and bounds;
Pass 04's facets, outer area and phase and radial bounds; Pass 05's signature and dual counts. The two
passes' records of Pass 03's cell radius disagree (0.013316971 and 0.013060516), for the reason W2 and W3 give.

## The passes, read

| pass | statement | verdict | note |
| --- | --- | --- | --- |
| 02 | TSL, AV, AN: ε_deep; IP, BA: β_deep; ADC; TA | PASS | constants to 10⁻¹²; ε_deep holds on random bodies (W1) |
| 03 | exact cell geometry; area interval; cell radius | PASS | W2 |
| 04 | phase law, L♯, cores; area bounds | PASS | W3 |
| 04 | λ_b ≈ 0.988205366; 0.9882…·P(b) ⊆ U | PARTIAL | a sampled minimum 3.1×10⁻⁶ too large; the inclusion follows for λ ≤ 0.9882022551 |
| 04 | ε♯_cell ≈ 0.0059677 | PARTIAL | a sampled maximum 1.6×10⁻⁶ too small |
| 04 | Pass 03's ε_cell ≈ 0.0130605 | FAIL as labelled | another quantity (h_P for H_j); Pass 03's is 0.0133170 |
| 05 | signature; sparse witnesses; contact basis; quadratic closure | PASS | W4 |

## The status registry

Not touched: the passes are not canonical documents, and the census (RUN-003) does not read them.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| Pass 02's ε_deep in the text | 1 | A0 |
| a constant; the error bound exceeded | 2 | W1 |
| the facet count | 1 | W2 (also P1) |
| a uniform formula | 1 | W2 |
| the sandwich | 1 | W3 |
| the signature | 1 | W4 (also P1) |
| the toy quadratic | 1 | W4 |
| the scripts' facts | 1 | W5 |
| a VALIDATION digest; a VALIDATION number | 2 | P1 |

Two controls: the unmodified inputs, and the four documents with CRLF line endings.

**11 / 11 plants, 2 / 2 controls, 78 / 78 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 2 | 2 |
| W2 | 2 | 2 |
| W3 | 1 | 1 |
| W4 | 2 | 2 |
| W5 | 1 | 1 |
| P1 | 4 | 4 |

## What this run does not establish

- **A legal body that breaks the recorded core.** At the extreme angle the recorded λ·h_P exceeds the
  envelope L♯ by 1.6×10⁻⁶. Whether some legal U in the cell comes that close to L♯ there is not
  constructed here.
- **Pass 05's continuum lift.** It is marked DERIVED-CANDIDATE by the pass itself, and nothing here tests it.
- **The finite polyhedral verifier at scale.** Its structure is checked (the signature, sparsity and one
  basis), not a verifier run.