# RUN-063 — BCODR LUC 01: the arrival-time field over root 6217 rebuilds from the 200 subroots, cell for cell, and the most balanced eight are the text's; but the boxes touch the prune seeds in 64 pairs, not 35, and for 178 of the 195 cells the "distance to the critical ray" is the distance to its end ρ = 0.003

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *BCODR-LUC-01 — Root-6217 參數空間的雙向圓核閉包 / Bidirectional Kernel Closure of the Root-6217 Shape
Space* (BCODR-LUC-01-v0.1, 2026-09-21), with its arrival information ledger (JSON and CSV), kernel-metric schema,
frontier graph (DOT and PNG), verifier, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run063_bcodr_luc01.py`](../code/run063_bcodr_luc01.py)

- Cycle 11's 200 subroots of root 6217, as RUN-048 reads them;
- the boxes on the exact dyadic grid of root 6217's box;
- Paper 09B's critical ray, by golden-section distance;
- the gate log of RUN-058.

**Logs:** [`run063-bcodr-luc01.json`](../data/gate-logs/run063-bcodr-luc01.json)

**Verdict: PASS for the domain, every arrival time, balance, entropy and score, the nexus, the finite-radius cells, the
most balanced eight, and the records. Not as stated: the active/prune boundary contacts are 64 pairs, not 35. Noted:
the frontier is set mostly by where the ray is cut.**

- **The domain (§1–2).** The 200 subroots' union gives the printed ℓ, u and s to the last digit, and the ledger's
  normalization exactly.
- **The arrival times (§4–13, §31).**
  - T₋ (normalized distance to the five A_ref-pruned boxes) is the ledger's to 3×10⁻¹⁶ on every one of the 195 cells.
  - T₊ (distance to Paper 09B's critical ray for 0 ≤ ρ ≤ 0.003) is the ledger's to 2×10⁻⁹, which is the ledger's own
    ray grid.
  - Every nearest prune root is the ledger's, or one of two at equal distance. Nine cells sit at equal distance from
    two prune boxes; the ledger names one of them.
  - The meeting radius, the balance and the entropy follow their definitions to 10⁻¹².
  - The scheduling score follows its formula with the median over the 140 finite-radius cells (0.766444), not over all
    meeting radii (0.696945) as §31's median(T∩) reads.
- **Overlaps and contacts (§14–16, §25).** On the exact grid of root 6217's box (2²⁰ units a side, every box edge an
  integer):
  - the 200 boxes have pairwise disjoint interiors, so they are atomic cells as §25 says;
  - no active box overlaps a prune box;
  - **64 active/prune pairs touch, not 35**: 40 across a 4-dimensional face, 14 across a 3-face, 8 across a 2-face and
    2 along an edge.
  - 55 active cells touch a prune box. That is the state's own 54 contact cells plus the nexus, which could not happen
    with 35 pairs.
  - In floating point most of these contacts are 10⁻¹⁸ gaps or overlaps. A test for an exact zero finds 30 of them;
    the ledger's 35 is such a count.
- **The nexus and the frontier (§17–24).**
  - Cell 131 is the only cell with T₊ = T₋ = 0. More than touching the ray, it holds all of it: every point of the ray
    for 0 ≤ ρ ≤ 0.003 lies in box 131, and the barrier itself sits on its corner (ρ = 0, y₃ = 0).
  - 140 cells have finite radius on both sides, and the most balanced eight are 166, 167, 164, 165, 127, 101, 104 and
    103, the text's order included.
  - **The ray is short, so for 178 of the 195 cells the nearest ray point is its end, ρ = 0.003**, the local gate's cut.
    T₊ there is the distance to one point, and the frontier moves with the cut:
    - at 0.001 and 0.002 the eight are the same cells in another order;
    - at 0.004, cells 160 and 168 replace 103 and 104;
    - at 0.006, cells 100 and 168 come in.
- **The ledger's domain (§0, §35).**
  - The cells are Paper 08's 195, split 10 local-only, 44 crossing and 141 tail-only (RUN-058).
  - The 200 boxes cover 55.9 % of root 6217's box. The 14.6 % that lies in the far branch and was closed for A₃⁺ only
    has no cell in this field (RUN-058).
- **The records.**
  - The state's counts are the ledger's, and the CSV is the ledger's 195 cells value for value. The CSV starts with a
    byte-order mark.
  - The graph draws the nexus and the eight, each with the T₊ and T₋ edges.
  - VALIDATION's "graph_sha256" is the PNG's digest; the DOT is not digested.
- **The verifier**, read after the rebuild and never run, has 32 lines.
  - It reads the ledger alone, asserts its counts as constants, and recomputes no distance from any box.
  - The "boundary contact pairs = 35" it prints is the ledger's, unchecked.
- **The package (P1).** Every digest in VALIDATION matches its file and the archive's manifest.

Drill: 15 / 15 plants caught by exactly the named checks, 4 / 4 controls undisturbed, 155 / 155 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. The 200 subroots fix a normalized 5-D space; the first kernel is the Euclidean ball there (§1–3).
2. Outward: Paper 09B's critical ray for 0 ≤ ρ ≤ 0.003. Inward: the five A_ref-pruned boxes. Arrival times T₊ and T₋,
   the meeting radius, the balance and the balance entropy, with no fixed radius (§4–13).
3. No positive-measure active/prune overlap, but 35 boundary-contact pairs; contact is not overlap (§14–16).
4. Root 131 is a zero-radius contact nexus, not a prune; the most balanced finite-radius cells are 166, 167, 164, 165,
   127, 101, 104 and 103 (§17–24).
5. The 200 boxes are measure-atomic cells, cut on kernel events, merged under rules; a scheduling score; colours;
   Φ₂ only on the surviving responsibility; next, LUC 02 (§25–43).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the normalization | to the last digit |
| W2 | T₊, T₋ and what follows from them | cell for cell; the score's median is the finite cells' |
| W3 | overlaps and contacts on the exact grid | atomic; 64 contact pairs, not 35 |
| W4 | the nexus and the frontier | 131 holds the ray; the eight; they move with the cut |
| W5 | the ledger's domain | Paper 08's 195, covering 55.9 % of the box |
| W6 | the state, CSV, schema and graph | consistent; the CSV has a BOM |
| W7 | the verifier, read | reads the ledger alone |

## P — the package

**P1.** The paper's, the ledger's, the CSV's, the schema's, the PNG's, the verifier's and the state's digests match
the bytes and the archive's manifest.

## The document, read

| section | statement | verdict | note |
| --- | --- | --- | --- |
| §1–3 | the normalized space and its metric | PASS | W1 |
| §4–13 | the two fields and the arrival times | PASS | W2 |
| §14 | no positive-measure active/prune overlap | PASS | W3 |
| §15 | 35 boundary-contact pairs | FAIL | 64 on the exact grid (W3) |
| §16 | contact is not overlap | PASS | |
| §17–19 | 131 the zero-radius nexus; not a prune | PASS | it holds the whole ray (W4) |
| §20–24 | the finite-radius frontier | PASS | the eight are the text's; they depend on the cut at 0.003 (W4) |
| §25 | the 200 boxes are measure-atomic | PASS | W3 |
| §26–34 | cuts, merges, score, colours, Φ₂ later | PASS / n/a | the score uses the finite cells' median (W2) |
| §35 | 195 = 10 + 44 + 141 | PASS | the 195 miss 14.6 % of the box (W5) |
| §36–43 | the first graph; what it does not claim; LUC 02 | PASS / n/a | W6 |

## The status registry

Not touched: LUC 01 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's contact pairs | 1 | A0 |
| the ledger's domain | 1 | W1 |
| a T₊; a tie | 2 | W2 |
| the contact pairs; the contact cells | 2 | W3 (the second also W6) |
| a ray holder; a cut's eight | 2 | W4 |
| the box coverage; a ρ class | 2 | W5 |
| a graph node; a CSV value | 2 | W6 |
| the verifier reading boxes | 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 ledger with the keys sorted.

**15 / 15 plants, 4 / 4 controls, 155 / 155 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 |
| W5 | 2 | 2 |
| W6 | 3 | 3 |
| W7 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Anything about areas.** The field is geometric bookkeeping in a parameter space; no cell's area is computed here.
- **That the normalized Euclidean metric means anything for shapes.** The document does not claim it does (§3, §38).