# RUN-064 — BCODR LUC 02: the 288 children, their signatures and the 17 classes rebuild exactly; but 127's provenance "flip" is a tie — it is 1/8 from 144 and from 155 alike — and a BOTH child is one that holds the point where the later field first touches its parent, which is all 127's "thick corridor" is

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *BCODR-LUC-02 — Atomic Cut and Structural Recomposition* (BCODR-LUC-02-v0.1, 2026-09-21), with its atomic
cut ledger, children CSV, structural recomposition, overlap persistence, spine graph (DOT and PNG), verifier, state
record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run064_bcodr_luc02.py`](../code/run064_bcodr_luc02.py)

- Cycle 11's 200 subroots of root 6217 on the exact dyadic grid of root 6217's box, 2²² units a side;
- T₋ as an exact fraction, and T₊ to 50 digits, so that a tie is a tie;
- the LUC 01 ledger and the gate log of RUN-063.

**Logs:** [`run064-bcodr-luc02.json`](../data/gate-logs/run064-bcodr-luc02.json)

**Verdict: PASS for the cut, all 288 signatures, the 17 classes and the seven BOTH classes, the second cut's counts, and
the records. Not as stated: the provenance flip at 127, and 127 as a thick, provenance-switching corridor. Noted: at its
parent's radius a BOTH child is one that holds a point where the later field first touches the parent; the 17 classes
need corner contacts and the parent's ρ class.**

- **The cut (§1–3).**
  - The nine parents are LUC 01's nexus and its most balanced eight (RUN-063). Each carries LUC 01's T₊, T₋ and meeting
    radius to the last digit.
  - On the grid every parent's width is even, so each child is exactly its parent halved. The ledger's 288 children are
    these boxes.
- **The signatures (§4–7).**
  - Every one of the 288 is the ledger's: 187 MINUS_ONLY, 88 UNREACHED, 4 PLUS_ONLY and 9 BOTH. Root 131's children are
    2 PLUS_ONLY, 8 MINUS_ONLY, 22 UNREACHED and no BOTH.
  - T₊ matches to 10⁻⁹ (the ledger's ray grid) and T₋ to 4×10⁻¹⁶.
  - Every nearest prune box is the ledger's. Two UNREACHED children of 165 are equally far from 68 and 155.
  - Apart from the exact ties below, no child comes within 10⁻⁵ of its radius. Neither the ray grid nor rounding can
    change a signature.
- **What a BOTH child is (§4–6, §21).** At t_B = max(T₊, T₋), the field that arrives last has only just reached the
  parent. A child is never nearer than its parent, so that field reaches a child only when the child holds a point where
  the field touches the parent.
  - **At seven parents the ray arrives last.** It touches each of them at a single point, the one nearest the ray's end
    ρ = 0.003. Exactly one child holds that point, and that child is the BOTH child.
  - **At 127 the prune boxes arrive last** (T₋ = 1/8 > T₊). They touch 127 along part of its low-x₃ face, and 12 children
    lie on it. The two BOTH children are the ones the ray also reaches by 1/8.
  - So in every parent, the set both fields reach at t_B has no volume: it is a point, or part of a face.
  - The ledger's floats sit exactly on the radius at all 19 of these ties.
- **The provenance (§11, §17–19).**
  - **127 is exactly 1/8 of the span from 144 and from 155.** Each lies across 127's low-x₃ face, with no gap in the other
    four dimensions.
    - In floating point, both distances are the same number, 0.12499999999999978. The ledger's "coarse nearest 144" is
      the first of the two in the order 28, 36, 68, 144, 155.
    - So nothing flips at 127. Its BOTH children are nearest to 155 alone, which is one of the parent's two nearest boxes.
    - Inside 127 the nearest box does change: 8 children are nearest to 144 and 24 to 155. A switching surface runs
      through 127, but not between the coarse level and the cut.
  - 101's change is real: the parent is nearest to 28 alone, and its BOTH child to 144 alone.
- **The recomposition (§10–16, §24–27).**
  - The 17 classes are the ledger's, child for child: 7 BOTH, 6 MINUS_ONLY, 2 PLUS_ONLY and 2 UNREACHED. That holds when
    two closures count as adjacent whenever they meet, even at a corner, and every child keeps its parent's ρ class.
  - Counting faces only gives 19. Giving each child its own ρ class gives 22: 16 of 131's children lie wholly below
    ρ = 0.003, and 16 of 127's wholly above.
  - The seven BOTH classes are §16's. 166 ↔ 167 is the only merge across parents. No two BOTH classes meet, so the
    classes from 101, 103 and 104 stay apart.
- **The persistence (§20–23, §29).**
  - The second cut gives one BOTH grandchild for each of the seven and 6 + 3 for 127's two, as recorded. The 4.5, 9/64
    and 1/32 follow §29.
  - The same tie decides the second cut:
    - for each of the seven, one grandchild holds the ray's touching point;
    - for 127's two, 8 and 4 grandchildren lie on the face, and 6 and 3 of them are within the ray's reach.
  - 127's 4.5 is a face against a point. It records which field arrives last, not a broader interaction.
- **The records.**
  - The CSV is the ledger's 288 children, field for field. It starts with a byte-order mark.
  - The recomposition, the persistence record and the state agree with each other and with the rebuild.
  - The graph draws the nine parents, 131's three branches, the seven BOTH classes and §24's three families.
- **The verifier**, read after the rebuild and never run, has 40 lines.
  - It loads the three JSON files and asserts 15 reported numbers as constants.
  - It recomputes nothing, not even the signature counts from the children it loads.
- **The package (P1).** Every digest in VALIDATION matches its file and the archive's manifest. "graph_sha256" is the
  PNG's digest; the DOT is not digested.

Drill: 18 / 18 plants caught by exactly the named checks, 4 / 4 controls undisturbed, 199 / 199 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. The nine frontier parents of LUC 01 are cut along all five dimensions: 288 children. Each child is signed by the
   fields that reach it at its parent's meeting radius (§1–5).
2. The children split 187 / 88 / 4 / 9, so 9/288 = 3.125 % is the first two-way interaction region. Root 131 has no
   BOTH child: it is a coarse bicontact container, not a resolved overlap (§6–9, §31).
3. A strict merge (same signature, ρ class and prune provenance; adjacent closures) recomposes the 288 into 17 classes,
   7 of them BOTH; 166 ↔ 167 is the one merge across parents (§10–16).
4. The cut changes provenance: 127 from 144 to 155, and 101 from 28 to 144 (§17–19).
5. Cut again, most BOTH children keep one BOTH grandchild, but 127's two keep nine. 127 is a thick, branched,
   provenance-switching corridor and the next target. The spine has three families; the routes follow (§20–40).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | the parents and the halving | LUC 01's, exact |
| W2 | every child's arrival times and signature | 288 / 288 the ledger's |
| W3 | which children reach the later arrival time | one for the seven; twelve at 127 |
| W4 | 127 and 101 against the prune boxes | 127 a tie; 101 a change |
| W5 | the strict merge | 17 with corners and the parent's ρ class |
| W6 | the second cut | as recorded; a face against a point |
| W7 | the CSV, recomposition, persistence, state and graph | consistent; the CSV has a BOM |
| W8 | the verifier, read | asserts constants |

## P — the package

**P1.** The paper's, the ledger's, the recomposition's, the persistence record's, the CSV'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 | nine parents, 2⁵ children each, t_B from LUC 01 | PASS | W1 |
| §4–6 | the signatures; 187 / 88 / 4 / 9; 9/288 | PASS | the BOTH children are those on the later field's touching set (W3) |
| §7–9, §31 | 131: 2 / 8 / 22 / 0; a container, not a nexus | PASS | W2 |
| §10–13 | separate, then merge strictly | PASS | corners count as adjacency; the ρ class is the parent's (W5) |
| §14–16 | 17 classes; 7 BOTH; 166 ↔ 167 | PASS | W5 |
| §17 | 127: coarse 144, local 155 | FAIL | 127 is exactly as far from 155 as from 144 (W4) |
| §18 | 101: coarse 28, local 144 | PASS | W4 |
| §19 | a cut can change provenance | PASS | on 101's evidence (W4) |
| §20–21 | the second cut; one BOTH grandchild for most | PASS | forced wherever the ray arrives last (W3, W6) |
| §22–23 | 127: 2 → 9; 4.5; 9/64 against 1/32 | PASS | the numbers (W6) |
| §23, §32 | 127 a thick, provenance-switching corridor | not as stated | a face against a point, and a tie (W4, W6) |
| §24–27 | three families; grouping is not merging | PASS | W7 |
| §28–30 | H_prov, P_refine, the information vector | PASS / n/a | P_refine as §29 (W6); H_prov would mark 101 only |
| §33–35 | the routes | n/a | |
| §36–37 | not a global 195 → 7; 288 → 17 | PASS | |
| §38–40 | LUC 03; Results A, B and C | PASS / not as stated | Result C rests on §17 and §23 |

## The status registry

Not touched: LUC 02 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| a count in the text | 1 | A0 |
| a parent; a LUC 01 radius | 2 | W1 |
| a T₋; a signature | 2 | W2 (the second also W7) |
| a tie; a ledger value off its radius | 2 | W3 (the second also W7) |
| 127's nearest boxes; a flip flag | 2 | W4 |
| a class label; the own-ρ classes | 2 | W5 (the first also W7) |
| a second-level count; a grandchild's tie | 2 | W6 |
| a CSV value; a graph node | 2 | W7 |
| the verifier counting the children | 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 ledger with the keys sorted.

**18 / 18 plants, 4 / 4 controls, 199 / 199 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 | 2 | 2 |
| W7 | 5 | 5 |
| W8 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Anything about areas.** The cut is bookkeeping in a parameter space; no cell's area is computed here.
- **That 127 is not worth a deeper cut.** The switching surface between 144 and 155 inside 127 is real, and LUC 03 may
  find something there. This run says only why 127's counts are larger, and that its "flip" is a tie.