# RUN-041 — DLMVC: the lagged line's mathematics holds and in two places goes further than this tree's first reading; its one "independent reproduction" re-ran the author's script and reproduced the script's error — the failure DLMVC exists to prevent

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** `DLMVC_Deep_Lag_Multi_Pass_Verification_Closure_Methodology_v0.1.md` (EML-DLMVC-v0.1, dated 2026-09-19) and the case studies in its folder, read from the archive:

- Round 00's state crystal;
- the lagged verification of Round 01 (`…Verification_Round_01_by_LaggedAI_v0.1.md`);
- the lagged verification of Round 04 and its package (the document, `sharp_radius.py`, `VALIDATION.json`).

The folder's four deep-audit passes of Round 04 are RUN-042's. Rounds 00–37 are searched for what the
frontier took back.

**Tools:** [`run041_dlmvc.py`](../code/run041_dlmvc.py) — exact rationals for E_M; Fourier series of three curvature densities; random placements for JPA-01; the hull's support function and mpmath at 40 digits for R_sharp; RUN-007's outer polygon for Round 04's sanity example
**Logs:** [`run041-dlmvc.json`](../data/gate-logs/run041-dlmvc.json)

**Verdict: PASS for DLMVC's nine judgments; the other nineteen are labels. Its "Empirical validation: PARTIAL" is right — for a reason it does not give.**

- **What DLMVC is.** A governance text: nothing in it is computed, its six DEFINED parts are defined, and its
  skeleton is closed as a text. The claim that can be tested is §26's "initial successful mathematical case
  studies", and they are in its folder.
- **Round 01's lagged verification.**
  - The Fejér compiler P_M keeps a density legal. On the Reuleaux triangle and on the case study's own sign
    family r₉, at every M of its table, the image stays in [0, 1], antipodal, with no first moment.
  - E_M's closed form equals the bound summed mode by mode, exactly, for M = 3 … 129. The table the case
    study deferred is computed here: 0.2188, 0.1806, 0.1342, 0.09093, 0.0575, 0.03465, 0.0202.
  - The bound can be tightened. On the domain |r̂ₙ| ≤ 1/π, not 1/2, and the Reuleaux triangle attains it at
    n = 3, so (2/π)·E_M is a bound too. The measured errors lie below even that: the Reuleaux triangle at
    M = 129 is off by 0.0017, against E_M = 0.0202.
  - JPA-01's coercivity, J(Q, t) ≥ ‖t‖ − R_K − R_U, holds on 4,000 random placements, reflections
    included. 2,236 of them lie beyond the radii, with a least slack of 1.27.
  - The one-active branch is in Round 01 and is corrected in Round 03 §8 (R01-CORRECTION-001). The case study
    says it had seen Round 03's crystal by accident, so its finding is not blind, and it says so. This tree's
    RUN-004 found the same thing independently.
  - JPA-01 goes further than RUN-004, which did not flag the joint-attainment step.
- **Round 04's lagged verification.**
  - Φ_r(d) = πr² + r√(d² − r²) − r² arccos(r/d) is the area of conv(B_r ∪ {x}). It is re-derived here from
    the hull's support function max(r, d cos t), and from a polygon of 2²⁰ sides. Φ'_r is the closed form.
  - R_sharp = Φ_{1/2}⁻¹(Ā) = 0.721238556039142255 (the case study: 0.721238556039143). The bound is sharp,
    since the hull at R_sharp attains it.
  - The constants at M = 128, q = 5×10⁻⁴ — R_P, ε and β for both radii — recompute to every digit printed,
    and so do the level counts 2,379 and 445 (a ratio of 10^−93.19).
  - This goes further than RUN-007, which kept R = 2Ā and measured the excess with the test cover's true
    radius instead.
  - Two formulas lost their braces in a substitution: `d\le\frac0.84409359440.5` (meaning A/r) and
    `\Delta=\frac{\pi}128` (meaning π/M). VALIDATION passes both; its checks are syntactic.
- **The sanity example.**
  - §I.2 says it re-ran Round 04's `sanity.py`, got the package's numbers, and marks this `SANITY:
    INDEPENDENTLY REPRODUCED`.
  - Those numbers — polygon area 0.8337053574119659, excess 6.4598497001×10⁻⁴ — are Round 04's recorded
    ones. RUN-007 showed that these are not the polygon Round 04 defines: the script's quantization guard
    sends six of the eight exact-level heights one level up. The polygon of the definition has area
    0.833639164382 and excess 0.000622493675876.
  - So re-running the script reproduced the script. This is the shared-path dependence DLMVC §1 warns of, in
    DLMVC's own case study, labelled independent.
  - RUN-007 wrote, before reading this, that "re-running the package's own script would never have shown it".
- **What the frontier took back.**
  - Round 03 records the one-active correction.
  - Rounds 00, 11, 12, 13 and 17 give a lagged line a place in their audit protocols.
  - JPA-01, the Fejér compiler FLC-01 and the sharp radius AR-04 appear in no round. Round 02 built its own
    compiler, squared Fejér on the support function (RUN-005).
- **Round 00's crystal.** It is the only copy in the archive, its parent Round 00 v0.2 is there, and its
  baseline bounds are RUN-002's.
- **The script,** read after the rebuild and never run (W6).
  - It uses the same closed form, a float bisection of 200 steps, and Round 04's constant formulas.
  - The keys it prints are not the keys VALIDATION records, so VALIDATION was not written by it as shipped.

Drill: 14 / 14 plants caught by exactly the named checks, 3 / 3 controls undisturbed,
137 / 137 must-stay-green assertions held. Each of the nine checks was blinded in turn, and each was
reported blind in exactly as many cases as plants name it.

---

## What DLMVC claims

1. **Two lines** (§1–4). A second line lagging the frontier in rounds, time and structure keeps
   verification independent. Synchronized agreement is not independence.
2. **Blindness** (§5). Blindness is recorded for every target. Contamination downgrades the independence
   label but does not invalidate the work.
3. **Deep passes** (§6–9). A target is worked through a ladder of passes, each judged by the freedom it
   removes, with rules for when to stop.
4. **Resolution** (§10–11). Closure within one resolution is not saturation across resolutions; what is not
   known is pressed into ever smaller domains.
5. **Returns** (§12–14). Results go back to the frontier as typed returns, never overwriting canonical
   history, and evidence is ordered rather than votes counted.
6. **Status** (§26). The skeleton is closed, six parts are defined, empirical validation is PARTIAL, and no
   universal claim is made: "initial successful mathematical case studies".

## A — sources

**A0.** 43 anchors in the four documents. DLMVC has no reference list. The case studies' external
references (Papini–Wu, Hynd, Gibbs, Mishra, Zeng, Gonzalez) are not fetched here; RUN-002 and RUN-007 read
the bounds they quote.

## W — rebuilt

| check | source | rebuilt | result |
| --- | --- | --- | --- |
| W1 | Round 01's case study, J | the compiler's legality; E_M exactly; errors on three densities; the coefficient bound | PASS; 1/2 improvable to 1/π |
| W2 | the same, D.7 and D.12 | JPA-01 on 4,000 placements; the one-active branch in Rounds 01 and 03; what rounds took back | PASS |
| W3 | Round 04's case study, J | Φ_r from the support function; Φ'_r; R_sharp; the constants | PASS |
| W4 | the same, I.2 | the "independently reproduced" numbers against Round 04's definition | the script's numbers, not the definition's |
| W5 | DLMVC §26 | the case studies, their blindness records, the frontier's lagged role; Round 00's crystal | PASS |
| W6 | — | the case study's script | read; not the source of its VALIDATION |

## P — the package

**P1.**

- VALIDATION describes the case study's document correctly: its digest, size, `$$` count, control
  characters and delimiters.
- Its constants are this tree's to 10⁻¹⁵: R_sharp, the hull area there, R_P, ε, β, K_q and the level counts
  for both radii, and the ratio's logarithm.
- It passes the two formulas broken by substitution.

## DLMVC, read

| DLMVC | statement | verdict | note |
| --- | --- | --- | --- |
| §26 | methodology skeleton: CLOSED | PASS | as a text; nothing in it is computed |
| §26 | six parts DEFINED | PASS | §3–5, §6–10, §18–19 |
| §26 | universal validity claim: NO | PASS | none is made |
| §26 | empirical validation: PARTIAL | PASS | the case studies' mathematics holds (W1–W3); one calls a re-run of the author's script independent (W4) |

## The status registry

DLMVC's 28 judgments are reviewed: nine are judgments, and nineteen are labels of the ladder and of the
blindness states, reviewed n/a. It uses seventeen phrases for the first time, each defined where it is
introduced: DEFINED in §26, the four blindness states in §5, the pass outputs in §7, and
FORMALIZATION-DEFERRED in §15.

| verdict | judgments |
| --- | ---: |
| PASS | 9 |
| n/a | 19 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's L_r | 1 | A0 |
| the case study's R_sharp | 1 | A0, W3 |
| a Fejér image out of [0, 1]; an error above the bound | 2 | W1 |
| a JPA-01 violation; Round 01's taxonomy | 2 | W2 |
| a level count | 1 | W3 (also P1) |
| Round 04's recorded polygon | 1 | W4 |
| a second copy of the crystal | 1 | W5 |
| the script's facts | 1 | W6 |
| a VALIDATION digest; a VALIDATION constant | 2 | P1 |
| a review dropped; a definition dropped | 2 | R1 |

Three controls:

- the unmodified inputs;
- the Round 04 case study with CRLF line endings;
- its VALIDATION with the keys sorted.

**14 / 14 plants, 3 / 3 controls, 137 / 137 must-stay-green assertions; no check without a
plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 2 | 2 |
| W1 | 2 | 2 |
| W2 | 2 | 2 |
| W3 | 2 | 2 |
| W4 | 1 | 1 |
| W5 | 1 | 1 |
| W6 | 1 | 1 |
| P1 | 3 | 3 |
| R1 | 2 | 2 |

## What this run does not establish

- **The four deep-audit passes of Round 04.** They are RUN-042's.
- **The case studies' blindness.** Whether the Round 04 case study really read nothing of Rounds 05–06
  cannot be told from here.
- **DLMVC as a method.** There is one corpus, one frontier and two case studies here, so PARTIAL is the
  honest word. This tree is itself such a lagged line — content-exposed for every round it read, in order —
  and RUN-007 is the independent reconstruction whose absence §I.2 shows.