# RUN-068 — BCODR LUC 07: the midpoint-separation and mirror theorems hold, and Paper 07's barrier is exactly up/down symmetric, hence achiral; but the candidate vector that §6 says is not recorded is in Cycle 17's diagnostic — 7.7×10⁻⁴ from the ray, 0.109 from the IMBG point — and it is not 131's lowest point: 131's corner holds the barrier, 1.38×10⁻⁷ lower

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *BCODR-LUC-07 — Root 131 Candidate/Germ Alignment and Mirror-Covariance Reduction* (BCODR-LUC-07-v0.1,
2026-09-21). A single document, without a package. It is evidence, not a canonical document.

**Tools:** [`run068_bcodr_luc07.py`](../code/run068_bcodr_luc07.py)

- RUN-067's nearest ray–155 pair at 50 digits, and the ray at the printed ρ\*;
- Cycle 17's 68-cell min-search diagnostic (RUN-049) and this tree's exact hull;
- Paper 07's barrier pieces at 50 digits (RUN-057);
- LUC 03's ledger, and the gate logs of RUN-060 and RUN-067.

**Logs:** [`run068-bcodr-luc07.json`](../data/gate-logs/run068-bcodr-luc07.json)

**Verdict: PASS for the midpoint separation theorem, T\* and κ₁₃₁, the mirror algebra and the support-function covariance,
and the up/down symmetry of Paper 07's barrier (CERT-07.4, replayed body by body and at 50 digits). Not as stated: that
Cycle 17's candidate is recorded by value only (§6, §20), and the positions of the B5 slivers (§22). Noted: the printed
p\*, q\* and z\* are LUC 03's optimizer values, 5×10⁻⁷ from the exact pair; LUC 06, and the schema and verifier that §23
describes, are not in the archive.**

- **The data (§1).**
  - The printed p\*, q\* and z\* (normalized and original) are LUC 03's ledger values for 131, to the last digit. The one
    exception is −0.24999999999999983, printed as −0.25.
  - They belong to LUC 03's ray parameter, 0.0025406761, not to the printed ρ\* = 0.00254068891729192. That ρ\* is LUC
    05's, and it lies 4.2×10⁻¹¹ from the exact 0.0025406888749283. At the printed ρ\* the ray's point is 5×10⁻⁷ from the
    printed p\*.
  - Against the exact pair, p\* is off by 5×10⁻⁷, z\* by 2×10⁻⁷ and q\* by 6×10⁻⁸.
  - The printed q\* carries z\*'s y₃ and x₅. The exact projection carries p\*'s.
  - T\* = 0.0473333498919449 is right to all its digits, and κ₁₃₁ is RUN-067's.
- **The separation (§3–4).**
  - A midpoint at T\* > 0 lies T\* from both seeds; this is immediate.
  - The three printed distances are those of the printed vectors.
  - The printed |p\* − q\*| = 0.09466669978536 is 1.5×10⁻¹² longer than the exact pair, 2T\*. The exact midpoint is
    exactly T\* from both ends.
- **The candidate (§6, §18, §20).**
  - **Its vector is in the archive.** For cell 131, Cycle 17's 68-cell min-search diagnostic stores the minimizer behind
    A_cand. Its area is 0.8349862484449208, the printed value to every digit, and it is the lowest of the 68 cells
    (RUN-049). So the archive already answers CERT-07.1, and CERT-07.2 can be computed.
  - In LUC 01's normalization the point lies in 131, 6.6×10⁻⁶ from 131's lower y₃ face. It is:
    - 0.109 from the IMBG point;
    - 7.7×10⁻⁴ from the ray, nearest near ρ = 1.7×10⁻⁴;
    - 0.124 from 155, the nearest prune box.
  - **And it is not 131's lowest point.** 131's corner (y₃ = 0, ρ₅ = 0, y₅ = 0) holds the barrier itself, which is where
    the ray starts. Its area is m₀ = 0.83498611055922 (RUN-057), 1.379×10⁻⁷ below A_cand.
  - So the lowest known value in the cell lies on the ray, at its start. The stored minimizer is a search point near it.
  - Root 131 still cannot be closed by A > A_ref, as §6 says. The barrier alone already prevents it.
- **The symmetry (§7–9, §11, CERT-07.4).**
  - S_x = R_π S_y as matrices. So S_yK = K gives S_xK = R_πK, and K is achiral.
  - h_{S_xK}(θ) = h_K(π − θ) and h_{S_yK}(θ) = h_K(−θ) hold.
  - At Paper 07's barrier, all three bodies are centred on the x-axis: the disc, B3 with corners at 0, 2π/3 and 4π/3,
    and B5 with corners at 2πk/5. Each is carried onto itself by y ↦ −y. So S_yK = K holds exactly, not only within the
    reconstruction.
  - At 50 digits, h_K(θ) = h_K(−θ) on 1,440 angles, and h_K(π − θ) = h_K(θ + π) on 360, both to 10⁻⁴⁵. The half-turn is
    the witness, as CERT-07.5 expects.
  - This replays CERT-07.4 numerically. No interval certificate is attempted.
- **The slivers (§22).** §22 puts B5's two corner slivers at π ∓ θ\*, as Paper 09B did. Along the critical velocity
  they open at B5's arcs a3 and a2, at θ\* and 2π − θ\*, beside the disc's at π/2 (RUN-060).
- **The archive (§0, §23).**
  - The archive holds LUC 01 to 05 as packages, LUC 07 as this one document, and no LUC 06.
  - So no document here supports §0's narrowing of the spine to 131, which it attributes to LUC 06.
  - The "companion JSON schema and verifier" that §23 describes are not in the archive either.
- **The document (P1).** Its bytes match the archive's manifest. It uses only $ delimiters and ends with ===END===.

Drill: 11 / 11 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 100 / 100 must-stay-green
assertions held. Each of the eight checks was blinded in turn, and each was reported blind in exactly as many cases as
plants name it.

---

## What the document claims

1. LUC 06 narrowed the selected spine to 131. 131's IMBG point is the midpoint of the nearest ray–155 pair, at
   T\* = 0.0473333498919449, and it is not the ray's point (§0–4).
2. Alignment comes in three strengths: point, seed and structural. The provenance link to Paper 09B is established
   (§2, §5).
3. Cycle 17's candidate in 131, A_cand = 0.8349862484449208, keeps 131 open. Its parameter vector is not recorded, so
   its distances to the germ are a local obligation (§6, §18, §20).
4. Exact up/down symmetry makes a planar body achiral. Paper 07's zero-rotation barrier has that symmetry, so it is
   achiral (§7–10).
5. The problem, the support function, the dual pressures and the germ stiffness are mirror-covariant. A decision tree
   follows, with eight certificates (07.1–07.8), a return to Paper 09C and its three slivers at π/2 and π ∓ θ\*, and a
   companion schema and verifier (§11–25).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the printed pair, meeting point, T\*, ρ\*, κ | T\* right; the vectors are LUC 03's, 5×10⁻⁷ off |
| W2 | the separation theorem and the distances | holds; the printed pair 1.5×10⁻¹² long |
| W3 | Cycle 17's stored minimizer for 131 | in the archive; 7.7×10⁻⁴ from the ray; the barrier lower |
| W4 | the mirror algebra and the barrier's symmetry | exact, body by body; even to 10⁻⁴⁵ |
| W5 | the secondary slivers | at θ\* and 2π − θ\*, not π ∓ θ\* (RUN-060) |
| W6 | what the document builds on and ships | no LUC 06; no schema or verifier |

## P — the document

**P1.** The document's digest and size match the archive's manifest.

## The document, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §0 | LUC 06 narrowed the spine to 131; 131's six properties | n/a / PASS | no LUC 06 in the archive; the properties hold (RUN-063, 065, 067) |
| §1 | the pair, the meeting point, T\*, ρ\*, κ | PASS | T\* right; the vectors are LUC 03's, 5×10⁻⁷ off (W1) |
| §2 | three notions of alignment | PASS | |
| §3–4 | midpoint separation; the distances | PASS | W2 |
| §5 | the provenance: the ray and 155 | PASS | RUN-065 |
| §6 | A_cand in 131; recorded by value only | PASS / not as stated | the vector is in Cycle 17's diagnostic, and the barrier is lower (W3) |
| §7–8 | S_x = R_π S_y; single-axis symmetry ⇒ achiral | PASS | W4 |
| §9 | the zero-rotation barrier is symmetric, so achiral | PASS | exact, body by body (W4) |
| §10 | marked and branch chirality | n/a | |
| §11 | support-function covariance | PASS | W4 |
| §12–15 | signature reversal; global, chart and dual covariance | PASS / n/a | |
| §16 | κ invariant under an isometric mirror | PASS | so is the Lagrangian constant |
| §17–18 | the decision tree; CERT-07.1–07.8 | n/a | 07.1 and 07.2 answerable now; 07.4 replayed (W3, W4) |
| §19 | closed A–E | PASS | |
| §20 | the candidate parameter needs export | not as stated | it is in the archive (W3) |
| §21–22 | Paper 09C; the three slivers | PASS / not as stated | the B5 slivers open at θ\* and 2π − θ\* (W5) |
| §23 | a companion schema and verifier | not as stated | not in the archive (W6) |
| §24–25 | the verdict; the conclusion | PASS / n/a | |

## The status registry

Not touched: LUC 07 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the area in the text | 1 | A0 |
| a printed coordinate; LUC 03's ray parameter | 2 | W1 (the first also W2) |
| a printed distance | 1 | W2 |
| the stored value; the barrier's place | 2 | W3 |
| the evenness; B5's corners | 2 | W4 |
| a sliver | 1 | W5 |
| a LUC 06 file | 1 | W6 |
| the document's bytes | 1 | P1 |

Three controls:

- the unmodified inputs;
- the document with CRLF line endings;
- LUC 03's ledger with the keys sorted.

**11 / 11 plants, 3 / 3 controls, 100 / 100 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 | 2 | 2 |
| W4 | 2 | 2 |
| W5 | 1 | 1 |
| W6 | 1 | 1 |
| P1 | 1 | 1 |

## What this run does not establish

- **Structural alignment (CERT-07.3).** The candidate's support topology is not compared with the Paper 09B branch.
  Its nearness to the ray's start is a distance, not a mechanism.
- **Any interval certificate.** The symmetry is exact by construction and replayed at 50 digits; the distances are
  numerical.
- **Marked or nonzero-rotation chirality, and the root atlas's mirror map.** These stay open, as §20 lists them.