# RUN-066 — BCODR LUC 04: 127's germ, its KKT relation, its critical tangent space and both reduced Hessians rebuild at 50 digits, and the germ is isolated; but the overlap's second-order law is the Lagrangian α H₊ + (1 − α) H₋, softest 0.2366, not the straight-line 0.4543 — the lens grows to 1.036 √ε, not 0.748 √ε

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *BCODR-LUC-04 — True Overlap Germ Tangent Decomposition* (BCODR-LUC-04-v0.1, 2026-09-21), with its result
record, plot, verifier, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run066_bcodr_luc04.py`](../code/run066_bcodr_luc04.py)

- RUN-065's reduction for 127 against 155, with r\* from its stationarity, at 50 digits;
- the ray's first and second derivatives in normalized coordinates, in closed form;
- RUN-065's dual, on slices of 127's box with one coordinate pinned.

**Logs:** [`run066-bcodr-luc04.json`](../data/gate-logs/run066-bcodr-luc04.json)

**Verdict: PASS for the meeting point, the KKT relation, the rank and the critical tangent space, D₊, H± and Q±, the
stiffness as §14 defines it, and the isolation of the birth germ. Not as stated: that stiffness as the overlap's
second-order law, and the growth law built on it (§16–19, §22). Noted: the printed z\*, g±, α\* and slope carry the
printed r\*'s error; the printed T\* is LUC 03's 5-D value.**

- **The point (§0–3).**
  - r\* = 0.0025452863403045615 comes from the reduction's stationarity. Then T\* = 0.12695366779092487911.
  - z\* lies on 127's lower x₃ face, inside 127. Its y₃ and x₅ are the ray's, and lie inside 155's ranges.
  - The printed z\* is 3×10⁻¹⁰ off, the error the printed r\* gives it. The printed T\* is LUC 03's 5-D value, 1.7×10⁻¹⁴
    high (RUN-065).
- **The KKT relation (§4–6).**
  - g₊ and g₋ match to 10⁻⁹ and 3×10⁻¹⁰.
  - α\* = 1 − λ\* = 0.23944475705941908, 7.1×10⁻¹² from the printed value. η\* = 0.12035442698726, 3×10⁻¹³ from it.
  - The relation holds in ρ₅ and y₅ to 10⁻⁴⁵, and the ray point is the nearest one (g₊·p′ = 0).
  - The rank is 2. The singular values of [g₊; g₋; e_x₃] are 1.0133, 0.07374 and 0.
  - The reason is the reduction itself. It puts z on the segment from the ray point to 155's corner in (ρ₅, y₅), so g₊
    and g₋ are collinear off x₃ for every r, not only at r\*.
- **The tangent space (§7–13).**
  - The dimension is 3.
  - The slope is −0.4041979698. The printed −0.404197989002394 is 1.9×10⁻⁸ off, again from the printed r\*.
  - D₊ = 1706.98000390142, 4×10⁻¹¹ from the printed value. Q₊ matches to 6×10⁻¹¹, and Q₋ = diag(0, 0, 1).
  - Q₊ has one small negative eigenvalue, −0.0011924, from the ray's curvature, as the record says.
- **The stiffness as defined (§14–15).**
  - The minimum over unit critical v of max(vᵀQ₊v, vᵀQ₋v) equals the largest value over t in [0, 1] of
    λ_min(tQ₊ + (1 − t)Q₋).
  - It is 0.45429481026314, at t = 0.4997, and the printed value is 1.5×10⁻¹¹ higher. At its direction the two forms
    are equal.
  - The printed direction's x₅ component, 1.7×10⁻⁷, is noise; it is 0.
- **What the overlap does (§16–19, §22–23).**
  - §16's expansion holds along straight lines: d±² = T\*² + h² vᵀH±v + o(h²).
  - **But the overlap at radius T\* + ε is not made of straight lines.** A second-order correction along the active
    normal shifts weight from one distance to the other, at the KKT rate α. So the overlap's own second-order law is
    the Lagrangian form αQ₊ + (1 − α)Q₋.
  - Its eigenvalues are 0.236577, 0.239445 and 0.763138. All are positive, so the birth germ is isolated, as §16–17 say.
    But the softest stiffness is 0.2366, not 0.4543.
  - **Measured directly, without expansion:**
    - With x₅ pinned at z\* + h and max(d₊, d₋) minimized over the rest of 127's box, the least radius grows like
      0.239445 h² (= α).
    - With y₃ pinned instead, it grows like 0.236854 h².
    - Both match the Lagrangian prediction to 3×10⁻⁷ at h = 10⁻⁴.
    - For these two axes the straight-line model gives 1 and 0.9918, and §14 says no direction costs less than 0.4543.
  - So at T\* + ε the lens reaches 1.036 √ε in its softest direction (1.030 √ε along x₅), not 0.7476 √ε (§18). The least
    radius cost is 0.932 h², not 1.789 h² (§19).
  - §22's certificate target κ > 0.4 would fail on the true constant. Isolation needs only a positive one.
- **The records.** The result record's coefficients follow from its κ, its meeting radius is LUC 03's, and the state
  repeats the rank, the dimensions and κ.
- **The verifier**, read after the rebuild and never run, has 44 lines.
  - It reads the result record alone and checks:
    - the rank and the dimension, as constants;
    - that Q± are symmetric;
    - that κ > 0.4;
    - that the stored direction reaches the stored κ;
    - the stored KKT residual.
  - It computes no geometry. Given the Lagrangian constant, its κ > 0.4 test would fail.
- **The package (P1).** Every digest in VALIDATION matches its file and the archive's manifest.

Drill: 16 / 16 plants caught by exactly the named checks, 4 / 4 controls undisturbed, 163 / 163 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 the document claims

1. At 127's earliest common point, g₊ and g₋ satisfy a KKT relation with the active x₃ face. The active normals have
   rank 2, which leaves a three-dimensional critical tangent space (§0–8).
2. First-order tangency is not enough. The reduced Hessians H₊ (a Schur complement over the ray parameter) and
   H₋ = diag(1, 0, 1, 0, 1), restricted to that space, give Q₊ and Q₋ (§9–13).
3. The combined stiffness min max(vᵀH₊v, vᵀH₋v) is 0.45429481027797 > 0, so every critical direction raises the
   overlap radius at second order, and the birth germ is isolated (§14–17).
4. After birth the overlap grows as a lens: |h| ≲ 0.7476 √ε, and ε ≳ 1.789 h² (§18–19, §23).
5. No hidden thick corridor; no Φ₂ needed for flatness; interval certificates (κ > 0.4, "generous"); 131 next
   (§20–27).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | z\*, T\*, the active pattern | the printed z\* to 3×10⁻¹⁰ |
| W2 | g±, α\*, η\*, the rank | rank 2 for every r, by the reduction |
| W3 | the tangent space, D₊, H±, Q± | as printed, to the printed r\*'s rounding |
| W4 | the straight-line stiffness | 0.45429481026314, as defined |
| W5 | the overlap's growth, on slices | Lagrangian: 0.2366; lens 1.036 √ε |
| W6 | the result record and the state | consistent |
| W7 | the verifier, read | checks the record against itself |

## 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 |
| --- | --- | --- | --- |
| §0–3 | z\*, T\*, the active face | PASS | the printed z\* to 3×10⁻¹⁰; T\* is LUC 03's 5-D value (W1) |
| §4–5 | g±, α\*, η\* | PASS | to the printed r\*'s rounding (W2) |
| §6 | rank 2, for a structural reason | PASS | the reduction makes it so for every r (W2) |
| §7–8 | the critical tangent space and its basis | PASS | W3 |
| §9 | first-order tangency is not enough | PASS | |
| §10–12 | D₊, H±, Q± | PASS | W3 |
| §13 | Q₊ alone indefinite; Q₋ flat | PASS | W3 |
| §14–15 | κ_germ = 0.45429481027797, its direction | PASS | as defined: the straight-line stiffness (W4) |
| §16 | the expansion; isolation to second order | PASS / not as stated | isolated, yes; the bound holds on straight lines only (W5) |
| §17 | from dimension 3 to 0 | PASS | W5 |
| §18–19 | 0.7476 √ε; 1.789 h² | FAIL | 1.036 √ε; 0.932 h² (W5) |
| §20–21 | no hidden thick corridor; no Φ₂ for flatness | PASS | the germ is isolated (W5) |
| §22 | CERT-04A–C; κ > 0.4 "generous" | not as stated | the true constant is 0.2366 (W5) |
| §23 | a lens growing as O(√(T − T\*)) | PASS | the rate, not the coefficient (W5) |
| §24–27 | information, spine, 131 next, conclusion | PASS / n/a | |

## The status registry

Not touched: LUC 04 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the stiffness in the text | 1 | A0 |
| the meeting point; LUC 03's T\* | 2 | W1 |
| α\*; the collinearity | 2 | W2 |
| D₊; an entry of Q₊ | 2 | W3 |
| the straight-line κ; the recorded κ | 2 | W4 (the second also W6) |
| a slice; the Lagrangian | 2 | W5 |
| a coefficient; the state | 2 | W6 |
| the verifier touching geometry | 1 | W7 |
| 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.

**16 / 16 plants, 4 / 4 controls, 163 / 163 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 | 2 | 2 |
| W6 | 3 | 3 |
| W7 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Anything about areas.** The germ lives in a parameter space.
- **Any interval certificate.** Everything here is 50-digit numerics, like the document's, and CERT-04A–C stay open.
  The slices measure the growth; they bound nothing rigorously.