# RUN-065 — BCODR LUC 03: the true overlap times, the modes and 127's one-dimensional reduction rebuild to 40 digits; but 101, 103 and 104 have no single provenance — four prune boxes tie exactly there — and the "reverse numerical error" is LUC 01's ray grid, not the optimizers

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-05
**Subject:** *BCODR-LUC-03 — True Common-Point Overlap and Provenance Resolution* (BCODR-LUC-03-v0.1, 2026-09-21), with its
ledger, gap CSV, plot, verifier, state record and VALIDATION. It is evidence, not a canonical document.

**Tools:** [`run065_bcodr_luc03.py`](../code/run065_bcodr_luc03.py)

- RUN-064's exact grid of root 6217's box and its arrival times (T₊ to 50 digits, T₋ exact);
- the dual of a convex problem for the true overlap, at 40 digits.

**Logs:** [`run065-bcodr-luc03.json`](../data/gate-logs/run065-bcodr-luc03.json)

**Verdict: PASS for the theorem, every true overlap time and gap, the modes, 131's midpoint, and 127's value, provenance
and reduction, and for the records. Not as stated: the provenance of 101, 103 and 104, where four boxes tie exactly.
Noted: the seven outward-limited gaps are exactly zero, and the ledger's negative ones come from LUC 01's ray grid; the
printed r\* and λ\* are right only to about 10⁻¹¹; at 127 the provenance 155 is right, but no flip led to it.**

- **The definitions and the theorem (§1–3).** T_true ≥ T_box holds by the argument the document gives, and every
  rebuilt value satisfies it.
- **The method.** Fix a ray point p and a prune box Q.
  - The least value over z in B of max(|z − p|, d(z, Q)) is a convex problem. Its dual, over a weight μ in [0, 1],
    separates coordinate by coordinate, with a closed form in each.
  - If μ = 1 at the optimum, the ray alone decides (outward-limited). If μ = 0, the box alone decides. Anything between
    means both are active (balanced).
  - T_true is the least of these values over 0 ≤ ρ ≤ 0.003 and over the five boxes. It is computed to 40 digits on
    RUN-064's exact grid.
- **The times (§5).**
  - T_true matches the ledger to 8×10⁻¹³ for all nine parents.
  - 131: 0.0473333498919450. The ledger's 0.0473333498927312 is 7.9×10⁻¹³ high.
  - 127: 0.12695366779092488, a gap of 0.00195366779092488 above 1/8.
  - For the seven other parents the gap is exactly zero: there T_true = T₊(B).
  - The ledger's raw gaps for those seven are negative, from −5×10⁻¹¹ to −1.3×10⁻⁹. To within 5×10⁻¹³, each is LUC 01's
    T_box minus the exact T₊, that is, the ray grid of LUC 01's ledger (RUN-063). They do not come from "independent
    numerical optimizers" (§3).
- **The modes (§4–6, §16).**
  - 131 and 127 are balanced, with μ = 1/2 and μ = 0.2394… At the meeting point d₊ = d₋.
  - The other seven are outward-limited (μ = 1), with the earliest box at least 0.026 inside the radius.
  - This is as the ledger and the state list them.
- **The provenance (§5, §10, §17).**
  - At 131, 166, 167, 164, 165 and 127, 155 alone is the earliest box. The margins match the ledger's to 2×10⁻¹².
  - **At 101, 103 and 104, four boxes tie exactly: 28, 68, 144 and 155.**
    - At the point where the ray touches the parent, all four are already within T₊(B), so each of them gives T_true.
    - The ledger's "28" is the first of the four in order. Its margins there (1.4×10⁻¹³, 2×10⁻¹³ and 4×10⁻¹⁶) are noise.
    - §17 calls these cells "close to" PROVENANCE_DEGENERATE; they are exactly degenerate.
    - The next box, 36, comes 0.18 to 0.24 later.
  - At 164, second place is a three-way tie between 28, 68 and 144, at √(5/32). The ledger names 28.
  - At 127, 155 is right, ahead of 144 by 0.04201490361714857. §10 says this means LUC 02's flip from 144 to 155 "still
    holds". But there was no flip: 127 is as far from 144 as from 155 (RUN-064).
- **Root 131 (§8).**
  - μ = 1/2 exactly. The meeting point is the midpoint of the ray point (at ρ = 0.002540688875) and that point's nearest
    point of 155. The midpoint lies inside 131, and T_true is half the ray's distance to 155.
  - The ledger's midpoint error of 2.8×10⁻⁸ belongs to its optimizer.
  - 144 comes next, with μ = 0, at the distance from 131 to 144 itself: √2/8, from a gap of 1/8 in ρ₅ and 1/8 in y₅.
  - The ledger's second time, 0.17677669529537773, is 1.3×10⁻¹² below √2/8. No point of 131 can reach that, so the
    optimizer left the box.
- **Root 127 (§9–15).**
  - Minimizing §12–13's λ(r) and T(r)² over r gives the 5-D value to 40 digits, with λ = 1 − μ.
  - Its active set holds:
    - the meeting point lies on 127's lower x₃ face, inside 127;
    - its y₃ and x₅ are the ray's, and lie inside 155's ranges;
    - the faces sit at −1/8 (127) and at −1/4, −7/16 and 3/8 (155), as printed.
  - The printed reduced T, 0.126953667790925, is right to all its digits. The 5-D value in §9, 0.126953667790942, is
    1.7×10⁻¹⁴ high, as §14 says.
  - The printed r\* = 0.0025452863775066 and λ\* = 0.760555242947675 are right only to 3.7×10⁻¹¹ and 7.1×10⁻¹². The
    rebuilt values are r\* = 0.0025452863403046 and λ\* = 0.76055524294058. The minimum is flat: T at the printed r\* is
    only 2×10⁻¹⁸ higher.
- **The records.**
  - The CSV holds the ledger's nine rows. It starts with a byte-order mark.
  - The state's lists and its numbers for 127 are the ledger's.
  - The ledger's "heuristic nearest prune" keeps LUC 01's 144 at 127, 103 and 104.
- **The verifier**, read after the rebuild and never run, has 46 lines.
  - It reads the ledger alone and checks its numbers against each other: T_true against T_box, the distances against
    T_true, and that one side is active.
  - It also checks five constants: [127, 131], 155, a margin above 0.04, 10⁻⁶ and 10⁻⁸.
  - It computes no distance.
- **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, 160 / 160 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 true overlap time of a box is the least radius at which the two fields cover one same point of it. It is never
   less than the container time T_box (§1–3).
2. For the nine parents: 131 and 127 are balanced and the other seven outward-limited, with gaps of zero within 2×10⁻⁹;
   131's true time is 0.0473, at a midpoint (§4–8, §16).
3. 127's true time is 0.126953667790942, from 155, ahead of 144 by 0.0420149036170295, so LUC 02's flip holds (§9–10).
4. 127's 5-D minimax reduces to one dimension: r\* = 0.0025452863775066, λ\* = 0.760555242947675,
   T = 0.126953667790925 (§11–15).
5. 101, 103 and 104 are close to provenance-degenerate; an information vector, a scheduler, new terms and LUC 04 follow
   (§17–23).

## W — rebuilt

| check | rebuilt | result |
| --- | --- | --- |
| W1 | T_box, T_true and the gaps | to 8×10⁻¹³; the seven gaps exactly zero |
| W2 | the modes | 131 and 127 balanced, the rest outward-limited |
| W3 | the earliest boxes and the margins | four-way ties at 101, 103 and 104 |
| W4 | root 131 | half the ray's distance to 155, at the midpoint |
| W5 | root 127 and its reduction | the 5-D value; r\* and λ\* to about 10⁻¹¹ |
| W6 | the CSV, state and ledger | consistent; the CSV has a BOM |
| W7 | the verifier, read | checks the ledger against itself |

## P — the package

**P1.** The paper's, the ledger's, the CSV'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 |
| --- | --- | --- | --- |
| §1 | the true overlap time | PASS | |
| §2 | T_true ≥ T_box | PASS | W1 |
| §3 | gaps within 2×10⁻⁹ are numerical | PASS | the negative ones are LUC 01's ray grid (W1) |
| §4 | balanced, outward-limited, inward-limited | PASS | from the optimal μ (W2) |
| §5 | the table | PASS / not as stated | every time and mode; "28" at 101, 103 and 104 is one of four (W3) |
| §6–7, §16 | 127 and 131 balanced, the rest outward-limited | PASS | W2 |
| §8 | 131: 0.0473333498927312, a midpoint | PASS | 7.9×10⁻¹³ high; the midpoint exact (W4) |
| §9 | 127: 0.126953667790942, balanced | PASS | the 5-D value, 1.7×10⁻¹⁴ high (W5) |
| §10 | 155 by 0.0420149036170295; the flip still holds | PASS / not as stated | the margin is right; there was no flip (W3, RUN-064) |
| §11–13 | the active face; the reduction | PASS | W5 |
| §14 | r\*, λ\*, T; 1.707e-14 | PASS | r\* and λ\* to 4×10⁻¹¹ and 7×10⁻¹² (W5) |
| §15 | a low-dimensional germ | PASS | the first common point lies on the face |
| §17 | 101, 103 and 104 close to degenerate; 127 resolved | not as stated | exactly degenerate, four boxes (W3) |
| §18–22 | the vector, the scheduler, the terms, LUC 04 | n/a | |
| §23 | the conclusions | PASS | |

## The status registry

Not touched: LUC 03 is not a canonical document.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the reduced T in the text | 1 | A0 |
| a true time; a raw gap | 2 | W1 (both also W6) |
| a mode in the ledger; a mode in the rebuild | 2 | W2 (the first also W6) |
| a provenance; a tie broken | 2 | W3 (the first also W6) |
| the ray's distance to 155; 131's box | 2 | W4 |
| the reduced T; an active-set fact | 2 | W5 |
| a CSV value; the state | 2 | W6 |
| the verifier computing a distance | 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.

**16 / 16 plants, 4 / 4 controls, 160 / 160 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 | 6 | 6 |
| W7 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Anything about areas.** The true overlap is still bookkeeping in a parameter space.
- **That the true overlap time is the better scheduler.** That is the document's argument. This run computes the time
  but does not test the argument.