# RUN-067 — BCODR LUC 05: root 131's interior-midpoint germ rebuilds at 50 digits, and there the straight-line stiffness is the right one; but the "same second-order mechanism" — κ₁₃₁ / κ₁₂₇ = 1.0008 — is a ratio of 1.92 for the overlaps' own constants, and the taxonomy's rule can neither reach the unstable class it defines nor certify isolation

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *BCODR-LUC-05 — Balanced Overlap Germ Taxonomy* (BCODR-LUC-05-v0.1, 2026-09-21), with its result record,
plot, verifier, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run067_bcodr_luc05.py`](../code/run067_bcodr_luc05.py)

- the nearest pair of the ray and box 155, from the stationarity of its squared distance, at 50 digits;
- the same second-order construction as RUN-066, on the four-dimensional critical space;
- RUN-065's dual, on slices of 131's box;
- RUN-066's germ for 127.

**Logs:** [`run067-bcodr-luc05.json`](../data/gate-logs/run067-bcodr-luc05.json)

**Verdict: PASS for 131's nearest pair, T\*, the midpoint and its place inside 131, the rank-1 first order and the
four-dimensional critical space, H±, κ₁₃₁, its C_R and C_H, and the isolation of both germs. Not as stated: the
comparison of the two germs (§13–14), and §22's rule as a test of isolation. Noted: the midpoint theorem needs the
nearest pair of all, and its proof the meeting point's own nearest points; §3 "reproduces" LUC 03 to 8×10⁻¹³; LUC 06 is
not in the archive.**

- **The midpoint (§2–5).**
  - The nearest pair of the ray and 155 is at ρ\* = 0.0025406888749283. The printed 0.00254068891729192 is 4.2×10⁻¹¹ off.
  - T\* is half the pair's length: 0.04733334989194496, which the printed 0.0473333498919449 matches to all its digits.
    LUC 03 printed 0.0473333498927312, 7.9×10⁻¹³ higher, so §3 "reproduces" LUC 03 only to that much. It agrees with
    RUN-065's value.
  - The midpoint lies inside 131, at least 0.0095 from every face.
  - **The theorem needs the nearest pair of all, not a local one, and its proof must go through the meeting point's own
    nearest points.**
    - §5 writes |p\* − z| + |z − q\*| ≤ 2T. But a common point at radius T is within T of some point of each seed set,
      not necessarily of p\* and q\*.
    - The bound follows from d(S₊, S₋) ≤ 2T, which needs p\* and q\* to realize d(S₊, S₋).
    - Here they do: 155 is the nearest box (0.0947), and 144 the next (0.2464).
- **The first order (§6–8, §10).**
  - g₋ = −g₊ exactly, so the rank is 1 and the critical space is four-dimensional.
  - The ray point is the nearest one (g₊·p′ = 0). It lies outside 155 only in x₃, ρ₅ and y₅, so H₋ = diag(1, 0, 1, 0, 1).
  - D₊ = 1709.2110247947.
- **The stiffness (§9, §11–12, §15–19).**
  - The straight-line κ₁₃₁ is 0.45464980777815, at t = 0.50003; the printed value is 1.4×10⁻¹³ higher.
  - C_R = 4.802637134452 and C_H = 0.4563101327193, as printed.
  - Here Q₊ is positive definite, with eigenvalues 0.0001098, 1, 1 and 1: the ray bends away from the meeting point.
- **The growth.**
  - At a midpoint the KKT weight is exactly 1/2, and the straight-line optimum sits at t = 0.50003, so here the two
    constants agree: the Lagrangian minimum is 0.4546497978, 10⁻⁸ lower.
  - Slices confirm it. With x₅ pinned the least radius grows like 0.5000 h², with y₃ pinned like 0.4918 h², both within
    3×10⁻⁷ of the prediction.
  - So 131's printed constants are the overlap's own.
- **The comparison (§13–14, §23, §27).**
  - The printed ratio 1.00078142539226 is the ratio of the two straight-line constants, right to 3×10⁻¹¹.
  - But 127's straight-line constant is not its overlap's own (RUN-066); 127's own is 0.2366. **The overlaps' constants
    are 0.2366 and 0.4546, a ratio of 1.92, not 1.0008.** Both are positive, so both germs are isolated, as stated.
  - The radius costs are 0.932 and 4.80, a ratio of 5.15, not 2.68. The lens coefficients are 1.036 and 0.456, a ratio
    of 0.44, not 0.61.
  - The printed κ's nearly agree for two reasons: H₊ and H₋ are much the same at both points (the same ray, the same
    box 155), and both straight-line optima sit near t = 1/2. The weights that matter, 0.24 and 0.5, differ.
- **The rule (§9, §20–22).**
  - §22 decides isolation by the straight-line κ, which does not depend on the KKT weight. The second-order condition is
    the Lagrangian at that weight.
  - The inward seed is a box, so Q₋ ⪰ 0, and the straight-line κ cannot be negative. **§21's "unstable germ",
    κ_germ < 0, cannot occur under its own definition.**
  - With 127's own forms, the Lagrangian would turn negative for any KKT weight above 0.9987, while κ_germ stayed at
    0.4543. 127's actual weight is 0.24, so 127 is isolated; but a positive κ_germ is necessary for isolation, not
    sufficient.
- **The records.**
  - The result record's ratios follow from its own numbers.
  - Its "container separation gap" for 131 is LUC 03's, 7.9×10⁻¹³ more than its own T_true − T_box.
  - The state classifies both germs as isolated.
  - The next node, LUC 06, is not in the archive: the LUC files there are 01 to 05 and 07.
- **The verifier**, read after the rebuild and never run, has 45 lines. It reads the record alone and checks:
  - the types, dimensions and birth dimensions, as constants;
  - that both κ's are above 0.4;
  - the interior flag, the midpoint error and the soft balance.

  It computes nothing.
- **The package (P1).** Every digest in VALIDATION matches its file and the archive's manifest.

Drill: 17 / 17 plants caught by exactly the named checks, 4 / 4 controls undisturbed, 190 / 190 must-stay-green
assertions held. Each of the ten checks was blinded in turn, and each was reported blind in exactly as many cases as
plants name it.

---

## What the document claims

1. 127 is a face-constrained balanced germ (FCBG) and 131 an interior-midpoint one (IMBG). By a midpoint theorem,
   131's meeting point is the midpoint of the nearest ray–155 pair, at T\* = 0.0473333498919449 (§0–5).
2. At 131 no face is active, g₋ = −g₊, the rank is 1 and the critical space four-dimensional (§6–8).
3. κ₁₃₁ = 0.454649807778291 and κ₁₂₇ = 0.45429481027797, a ratio of 1.00078: less than a tenth of a percent apart.
   Different first-order classes, the same second-order isolation (§9–14).
4. C_R = 1.789 and 4.803, C_H = 0.748 and 0.456 (§15–19).
5. A taxonomy (IMBG, FCBG, edge, corner, provenance-degenerate, flat, unstable) and a rule: κ_germ > 0 isolated,
   = 0 flat, < 0 unstable. LUC 06 is next (§20–27).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the nearest pair, T\*, the midpoint | as printed; the pair is the nearest of all |
| W2 | the first order at 131 | rank 1, dimension 4, H₋ as printed |
| W3 | κ₁₃₁, its direction, C_R, C_H | as printed |
| W4 | the Lagrangian at 1/2, and slices | agrees with the straight-line constant |
| W5 | the comparison with 127 | overlap constants 0.2366 and 0.4546: ratio 1.92 |
| W6 | what the rule can decide | κ_germ ≥ 0 always; necessary, not sufficient |
| W7 | the record and the state | consistent; the gap is LUC 03's |
| W8 | the verifier, read | asserts constants and flags |

## P — the package

**P1.** The paper's, the result record's, the plot's, the verifier's and the state's digests match the bytes and the
archive's manifest.

## The document, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §0 | different first-order class, same second-order outcome | PASS | both isolated (W5) |
| §1–2 | FCBG and IMBG | PASS | |
| §3 | ρ\*, T\*; reproduces LUC 03 | PASS | ρ\* 4×10⁻¹¹ off; LUC 03's printed value is 8×10⁻¹³ higher (W1) |
| §4–5 | the midpoint theorem and its proof | not as stated | needs the nearest pair of all, and z's own nearest points; true here (W1) |
| §6–8 | no face; g₋ = −g₊; rank 1; dimension 4 | PASS | W2 |
| §9 | κ_germ decides | not as stated | the Lagrangian at the KKT weight decides (W6) |
| §10–12 | H₋, H₊, κ₁₃₁, birth dimension 0 | PASS | here the straight-line constant is the overlap's (W3, W4) |
| §13 | ratio 1.0008, under a tenth of a percent | not as stated | the overlaps' constants: ratio 1.92 (W5) |
| §14 | the same second-order mechanism | PASS / not as stated | both isolated; not alike in stiffness (W5) |
| §15–19 | C_R, C_H | PASS for 131 | 127's are 0.932 and 1.036 (RUN-066, W5) |
| §20 | the taxonomy | PASS | |
| §21–22 | the future types; the rule | not as stated | κ_germ < 0 cannot occur; κ_germ > 0 does not suffice (W6) |
| §23–24 | isolated germs; no Φ₂ for flatness | PASS | |
| §25–26 | LUC 06 next | n/a | not in the archive (W7) |
| §27 | the conclusion | PASS / not as stated | "κ ≈ 0.454 for both" holds for the straight-line constants only |

## The status registry

Not touched: LUC 05 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the ratio in the text | 1 | A0 |
| the nearest box; LUC 03's T | 2 | W1 |
| the rank; the outside dimensions | 2 | W2 |
| κ₁₃₁; the recorded direction | 2 | W3 (the first also W4 and W5) |
| a slice | 1 | W4 |
| 127's Lagrangian; the record's cost ratio | 2 | W5 (the second also W7) |
| the critical weight; 127's Q₋ | 2 | W6 |
| the state; the record's gap | 2 | W7 |
| the verifier computing | 1 | W8 |
| a VALIDATION digest; the paper's bytes | 2 | P1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the VALIDATION record with the keys sorted;
- the result record with the keys sorted.

**17 / 17 plants, 4 / 4 controls, 190 / 190 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 | 3 | 3 |
| W6 | 2 | 2 |
| W7 | 3 | 3 |
| W8 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Anything about areas.**
- **Any interval certificate.** As in RUN-066, the numerics are at 50 digits, and the slices measure the growth but
  bound nothing.
- **The taxonomy beyond these two germs.** Two examples name two classes. Whether the other classes occur is not
  examined.