# RUN-062 — BCODR: a closure architecture, as it says of itself; its two differences are metrics on discs, its arrangement's cells are counted right (O(N²), exactly N(N−1) + 2 when every pair crosses), its certificate weights and its UESFCM link are consistent; its root-6217 map would start from the 195-root ledger that leaves out 14.6 % of the box

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *雙向圓核重疊分解法：從切分、分離到資訊重組 — Bidirectional Circular Overlap Decomposition and
Recomposition* (BCODR-v0.1, 2026-09-21), one of the 42 canonical documents (RUN-001), with its VALIDATION record. Its
case studies (LUC 01–05 and 07) are RUN-063 onward.

**Tools:** [`run062_bcodr.py`](../code/run062_bcodr.py)

- exact disc–disc intersection areas (the lens formula);
- Euler's formula for circle arrangements, against sampled membership signatures;
- UESFCM's methodology and the LESR midpoint handoff, as texts;
- the gate logs of RUN-048, RUN-058, RUN-059 and RUN-060.

**Logs:** [`run062-bcodr.json`](../data/gate-logs/run062-bcodr.json)

**Verdict: PASS. BCODR is a methodology: it computes nothing, and it says so (§78). Its one status phrase in a judgment
position is a label. Its mathematical statements hold where they can be instantiated, and its links to UESFCM and to
the certificate grades are consistent. One thing is noted: the root-6217 map it proposes as the next step starts from
an incomplete ledger.**

- **The status registry.** BCODR has one status phrase in a judgment position, CONFLICT CELL (§38). It is a label:
  a cell with both strong proof support and strong counter support, which is marked rather than averaged. It is
  defined here, and its occurrence reviewed as n/a.
- **The differences (§10–11).**
  - D_μ(A, B) = μ(A △ B) and D_J(A, B) = μ(A △ B)/μ(A ∪ B), the Jaccard (Steinhaus) distance.
  - On 20,000 random triples of discs, with exact lens areas, the triangle inequality holds for both, with no excess
    above 10⁻¹². D_J stays in [0, 1].
  - Scaling a pair by 3 multiplies D_μ by 9 and leaves D_J unchanged. That is §11's reason for normalizing.
- **The arrangement (§18–21, §59, §79 B).**
  - For circles in general position (no tangency, no three through a point), Euler's formula gives F = V + 1 + C
    cells. Here V is the crossing points and C the components of the crossing graph; the unbounded cell is included.
  - On 36 random families of 3 to 5 circles, the membership signatures met by 200,000 sample points never exceed F
    (at most 22 cells). Each signature is at least one cell.
  - When every pair crosses, F = N(N − 1) + 2 exactly: 134 at N = 12. So §59's O(N²) is tight.
  - Three crossing circles form a Venn diagram: 8 cells, all 8 signatures met.
  - §79 B, "does the arrangement capture all membership transitions?", is immediate: membership changes only across
    a circle.
  - A grid count of the cells is not a sound check. Near-tangent circles make slivers thinner than any grid, and
    pixels split them; a first attempt here counted 10 cells for 3 circles, where 8 is the most there can be.
- **The links (§16, §45).**
  - §16's six certificate weights (0.1, 0.3, 0.5, 0.7, 0.9, 1.0) rise in the same order as the handoff's five grades.
    BCODR adds "numerical replay" between search evidence and a research certificate.
  - §45 maps UESFCM's Expand, LinkBack, FiniteClose and Recenter, all of which UESFCM's text has.
- **The Lebesgue link (§47–50, §84).**
  - The root-6217 parameter-space map §84 proposes starts from "195 shape-active parent roots". That is Paper 08's
    ledger, which leaves out the 14.6 % of root 6217's box closed for A₃⁺ only (RUN-058). A map built on it inherits
    the gap.
  - Its "positive-rotation candidate branch" has no lower branch found (RUN-048, RUN-059).
  - The Paper 09C it would not yet replace is not in the archive (RUN-060).
- **The honest boundary (§78).** BCODR is not a convergence theorem, and nothing here tests one. §79's other questions
  (overlap connectivity, packing, information monotonicity, spine convergence) stay open.
- **The package (P1).** VALIDATION's digest is the document's and the archive's. Its core cycle is §0's, and its
  overlap rules are §9.1's and §9.2's.

Drill: 11 / 11 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 87 / 87 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 document claims

1. A closure architecture: outward circles from inner seeds and inward circles from outer constraints, overlapped;
   overlap kept as information (§0–9).
2. Minimum distinguishability by D_μ or D_J; multiplicity m, balance b = m₊ − m₋, novelty, certificate strength and a
   default information field (§10–17).
3. The arrangement cuts the domain into atomic cells of constant membership. They are separated by geometry,
   topology, information, direction and certificate role, recomposed under rules that keep provenance, and quotiented
   to a surviving spine. Conflict cells come first (§18–38).
4. A kernel-deformation gate; calculus after localization; relations to topology, groups, the skew field, UESFCM and
   LESR; a root-6217 map as the next step (§39–50, §84).
5. Colours, stop conditions, schedules, thresholds, hierarchy, a data model and an algorithm, certificate outputs,
   counter-cells, six principles, an honest boundary and open questions (§51–83).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | D_μ and D_J on 20,000 disc triples | metrics; D_J in [0, 1] and scale-free |
| W2 | the arrangement's cells | Euler ≥ every sampled signature count; N(N − 1) + 2 tight; a Venn diagram |
| W3 | the weights and the UESFCM stages | consistent with the handoff's grades and UESFCM's text |
| W4 | the root-6217 map's inputs | the 195 roots miss 14.6 %; no lower branch; 09C not archived |

## P — the package

**P1.** The VALIDATION digest matches the document and the archive, and its fields are the text's.

## The document, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §0–9 | the bidirectional fields; connected overlap by default | n/a | definitions |
| §10–11 | D_μ, D_J | PASS | metrics on discs (W1) |
| §12–17 | multiplicity, balance, novelty, certificate strength | n/a | definitions |
| §18–22 | the arrangement's atomic cells | PASS | W2 |
| §23–38 | separation, recomposition, graph, spine; CONFLICT CELL | n/a | rules and a label |
| §39–46 | the deformation gate; relations to other layers | PASS / n/a | the UESFCM stages are UESFCM's (W3) |
| §47–50, §84 | the root-6217 map | PARTIAL | its 195-root input is incomplete (W4) |
| §51–58 | colours, stop rules, thresholds | n/a | |
| §59–60 | O(N²); hierarchy | PASS | tight (W2) |
| §61–77 | data model, algorithm, certificates, principles | n/a | |
| §78–83 | the honest boundary; open questions; extensions | PASS | stated as open |

## The status registry

BCODR's one judgment-position phrase is reviewed and defined: CONFLICT CELL, a label (n/a). Census v5 adds none.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's O(N²) | 1 | A0 |
| a triangle-inequality excess | 1 | W1 |
| a pairwise cell count; a sampled signature count | 2 | W2 |
| a UESFCM stage; a handoff grade | 2 | W3 |
| RUN-058's missing share; Paper 09C archived | 2 | W4 |
| a review dropped | 1 | R1 |
| the VALIDATION digest; its cycle | 2 | P1 |

Three controls:

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

**11 / 11 plants, 3 / 3 controls, 87 / 87 must-stay-green assertions; no check without a plant.**

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

## What this run does not establish

- **That BCODR works.** It is an architecture; its case studies are RUN-063 onward.
- **§79's open questions**, apart from B.