# RUN-026 — Round 23: the strict-incidence grammar is sound and the B7 edges rebuild exactly; with exact cores B7 closes all twelve cells, and no harmonic can close any

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-25
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_23_Portfolio_Closure_v0.1.md` (AMRAL-LUC-FC-R23, dated 2026-09-19) and its package, read from the archive:

- `PORTFOLIO_MATRIX.json` (12 cells × 5 witnesses), `SET_COVER_SPEC.json`, `LONG_RUN_SPEC.json`,
  `VALIDATION.json`, `State_Crystal`;
- no code is shipped.

**Tools:** [`run026_round23_portfolio_closure.py`](../code/run026_round23_portfolio_closure.py) — Round 19's depth-40 segmentation rebuilt (RUN-022, RUN-023); one lift for B7 and the harmonics, with the pilot's sampled hull at any number of directions; RUN-023's exact-core B7 lift; RUN-025's bracketed smooth-witness hull, with each cell's erosion cores as the base
**Logs:** [`run026-round23-portfolio-closure.json`](../data/gate-logs/run026-round23-portfolio-closure.json)

**Verdict: PASS for the strict-incidence grammar and its theorems, the twelve cells, every B7 edge, the depth-22 crossover as computed, and the depth-16 infeasibility; PARTIAL for the harmonic edges, whose open counts rebuild but whose node counts do not. The routing the round draws does not survive exact cores: no harmonic can close any of the twelve cells at any depth, while B7 closes all twelve by lift depth 35.**

- **The grammar is sound.**
  - Strict incidence asks for a whole certified placement root over the whole cell.
    Theorem 5.1 is monotonicity of the hull: adding a witness never lowers the area. On 24
    random placements of B7 and B11 together, the joint hull exceeds each single one by at
    least 5.0×10⁻⁴.
  - The weighted set cover is feasible exactly when the residual frontier R_d is empty.
    Enumeration on 200 random incidence matrices agrees.
  - Partial edges do not compose, because each witness has its own placement adversary.
    Budget monotonicity is immediate, and the counterexample effect is Round 20's negative
    filter.
- **The twelve cells are Round 19's twelve lowest outer brackets.** They are among the 85
  depth-40 cells below T, 77 of them with outer brackets below T, as the long-run spec says.
  Paths and marker areas agree to 4.5×10⁻¹⁴.
- **Every B7 edge rebuilds — at 96 directions.**
  - The pilot's sampled hull now takes 96 directions, not the 40 of Rounds 16–22. Nothing
    says so.
  - At 96 directions, all twelve B7 edges agree in nodes and open boxes. At 40, none does:
    cell 0 gives 3,031/559 against 2,393/420.
  - The harmonic root is also new: |x|, |y| ≤ d* − ½ + |a|, the radius past which the
    witness's nearest-reaching point leaves B_{d*}. It is sound, and tighter than Round 22's
    0.21. The recorded root volumes are these.
- **The harmonic edges rebuild only in part.**
  - Open boxes at depth 16 agree on 42 of the 48 harmonic edges, and are within 2 on the rest.
    The unresolved volumes, the medians and the routing follow: H19 has the least open volume
    on all twelve cells.
  - Node counts run 26–64 above the package's on every harmonic edge. No convention tried
    closes that gap: tie order, pruning to the root disc, legality by the support's motion,
    a disc fallback, the body frame, the motion radius, the pad.
- **The depth-22 crossover rebuilds.**
  - On the four hardest cells, B7's and H19's open volumes at depth 22 agree to 10⁻¹⁵.
    B7 has less on cells 0–2, H19 on cell 3, as §17 says.
  - On cell 0, the whole portfolio ranks B7 < H19 < H17 < H13 < H11, as §18 says. B7's
    counts are exact, 12,515/1,488.
- **Exact cores change the picture.**
  - With exact cores, B7 closes every one of the twelve cells, at lift depths 30–35 with
    18,509–45,401 nodes. At depth 16 each still has 342–379 open boxes, so §13's
    infeasibility at depth 16 stands. A strict cover exists at depth 35, by B7 alone —
    in floating arithmetic, with this tree's kernel.
  - No harmonic can close any of these cells, at any depth, with any core. With each cell's
    erosion cores in place of the base bodies, the best a harmonic reaches is below T by its
    upper bound, on all 48 pairs. The highest upper ends are H11 0.8342519, H13 0.8339969,
    H17 0.8334024, H19 0.8333507.
  - So §16's routing sends all twelve cells to a witness that cannot finish any of them.
    §17's crossover compares B7 with a tree whose open volume has a floor. §22's "no strict
    edge yet" is a statement about the sampled pilot.

Drill: 15 / 15 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
194 / 194 must-stay-green assertions held. Each of the eleven checks was blinded in turn, and
each was reported blind in exactly as many cases as plants name it.

---

## What Round 23 claims

1. **Strict incidence** (§1–4). I_{jK} = 1 only if witness K's whole relevant placement root is
   certified for every q in cell C_j. Partial trees, small pending or volume, search minima,
   the absence of counterexamples, fast tails and success elsewhere do not count.
2. **Portfolio closure** (§5). If every necessity cell has a strict edge, the chosen witnesses
   close those cells: Λ(F₃ ∪ B) ≥ T.
3. **Set cover** (§6–8).
   - min Σ f_K y_K + Σ c_{jK} x_{jK}, subject to Σ_K x_{jK} ≥ 1 and x_{jK} ≤ I_{jK} y_K.
   - Budgeted incidence I^{(d)}, and the residual frontier R_d: the cover is feasible at
     budget d iff R_d is empty.
4. **Partial edges** (§9–11). Quantitative edges (N, P, V, t, m, ρ) serve the scheduler only.
   PARTIAL + PARTIAL ⇏ STRICT COVER; synergy needs a joint certificate.
5. **The pilot** (§12–18).
   - The 12 strongest of 77 necessity cells × 5 witnesses: 60 edges at depth 16, none
     complete, so R₁₆ is all twelve and the strict cover is infeasible.
   - Mean nodes and open volume: B7 2401 and 9.42×10⁻⁴; H11 3749.5 and 4.77×10⁻⁴; H13 3644.2
     and 3.41×10⁻⁴; H17 2362.5 and 2.66×10⁻⁴; H19 2286.3 and 2.29×10⁻⁴.
   - H19 has the least volume on 12/12 cells. At depth 22, B7 overtakes it on 3 of the 4
     hardest cells. On cell 0: B7 5.17×10⁻⁵, H19 5.43×10⁻⁵, H17 6.27×10⁻⁵, H13 9.24×10⁻⁵,
     H11 1.09×10⁻⁴.
6. **Routing** (§19–27). A crossover depth d*; horizon-aware cost-to-complete; proof and
   routing kept apart; a 77-cell long run; append-only incidence; DISQUALIFIED edges; Round 24
   on the residual core and joint witnesses.

## A — sources

**A0.** 47 anchors. Round 23 has no reference list.

## W — rebuilt

| check | Round 23 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | §3–11, §24–25 | monotonicity; the cover and its frontier; the specs | PASS |
| W2 | §12 | the twelve cells from Round 19's segmentation | PASS |
| W3 | §13, §15–16 | sixty depth-16 edges; aggregates; routing | B7 exact; harmonics' open counts 42/48 |
| W4 | §17–18 | depth 22 on the four hardest cells; cell 0 | PASS |
| W5 | — | B7 with exact cores on each cell | all twelve close, at depths 30–35 |
| W6 | — | each harmonic's least hull with each cell's cores | below T on all 48 pairs |
| W7 | — | what is recorded and shipped | no code; the conventions unstated |

**W1 — the grammar.**

- Monotonicity: for 24 random base configurations and placements, the exact hull of the base,
  B7 and B11 exceeds the larger of the two single hulls by at least 5.04×10⁻⁴. That is
  theorem 5.1's only step.
- The set cover: on 200 random 6 × 5 incidence matrices, the optimum by enumeration of the
  witness set exists exactly when every row has an edge.
- The set-cover spec's constraints and the long-run spec's "PARTIAL edges never count as
  cover" are §6's and §10's.

**W2 — the twelve cells.** RUN-023's rebuild of Round 19's depth-40 refinement of the tracked
depth-31 cell has 85 cells below T. RUN-022's 20,000-direction brackets put 77 of their outer
areas below T. The twelve lowest outer brackets are the matrix's rows 0–11, in order, marker
areas to 4.5×10⁻¹⁴. The first four are also Round 20's four marked cells.

**W3 — the matrix.**

- One lift serves every witness: RUN-023's root and split, a box's bound the pilot's hull of
  the disc, the cell's two cores and the witness core, less the 10⁻⁹ pad.
  - B7's core is the Reuleaux erosion at the vertex motion.
  - A harmonic's core is h − δ at the boundary motion |Δt| + 2(½ + |a|) sin(Δφ/2) (RUN-025),
    with the bound π/4 while δ > σ.
- The root: t₇ for B7; d* − ½ + |a| for a harmonic. All five root volumes are the matrix's to
  10⁻¹⁵. Every recorded volume is pending × root / 2¹⁶ to 10⁻¹².
- At 96 directions:

  | witness | edges exact | open boxes exact | this tree's mean nodes | package | mean open | package |
  | --- | ---: | ---: | ---: | ---: | ---: | ---: |
  | B7 | 12 / 12 | 12 / 12 | 2401.3 | 2401.3 | 423.75 | 423.75 |
  | H11 | 0 / 12 | 11 / 12 | 3810.7 | 3749.5 | 330.17 | 330.33 |
  | H13 | 0 / 12 | 11 / 12 | 3707.0 | 3644.2 | 282.00 | 282.08 |
  | H17 | 0 / 12 | 9 / 12 | 2392.8 | 2362.5 | 290.17 | 290.50 |
  | H19 | 0 / 12 | 11 / 12 | 2317.8 | 2286.3 | 280.83 | 280.92 |

- At 40 directions, B7 matches no cell. On cell 0 it gives 3,031 nodes and 559 open.
- The routing by least open volume is H19 on all twelve cells, here and in the package. No
  edge is complete at depth 16. Every aggregate is its rows' mean or median.

**W4 — depth 22.**

| cell | B7 volume | H19 volume | less | package |
| ---: | ---: | ---: | --- | --- |
| 0 | 5.17036×10⁻⁵ | 5.42745×10⁻⁵ | B7 | B7 |
| 1 | 5.56995×10⁻⁵ | 5.59072×10⁻⁵ | B7 | B7 |
| 2 | 5.42053×10⁻⁵ | 5.50781×10⁻⁵ | B7 | B7 |
| 3 | 5.85487×10⁻⁵ | 5.67873×10⁻⁵ | H19 | H19 |

Every volume is the package's to 10⁻¹⁵. On cell 0 at depth 22, this tree has B7 12,515/1,488
(the package's), H11 19,457/4,835, H13 19,095/4,887, H17 16,309/4,385 and H19 15,785/4,255.
The package has 19,419/4,838, 19,055/4,888, 16,287/4,387 and 15,759/4,255. The ranking is the
same.

**W5 — B7 with exact cores.** RUN-023's exact-core lift, to depth 36:

| cell | closes at | nodes | open at depth 16 |
| ---: | ---: | ---: | ---: |
| 0 | 30 | 18,555 | 342 |
| 1 | 31 | 22,009 | 361 |
| 2 | 31 | 26,189 | 363 |
| 3 | 34 | 36,193 | 379 |
| 4 | 31 | 21,271 | 342 |
| 5 | 30 | 18,509 | 343 |
| 6 | 33 | 31,209 | 358 |
| 7 | 35 | 45,401 | 378 |
| 8 | 32 | 26,439 | 364 |
| 9 | 34 | 41,077 | 377 |
| 10 | 33 | 31,793 | 359 |
| 11 | 30 | 21,895 | 365 |

Cells 0–3 close exactly as Round 20's four marked cells did in RUN-023.

**W6 — the harmonics' limits.**

- As a box shrinks, a harmonic's core tends to the placed witness. So no lift of a harmonic
  can close a cell where, at some placement inside the root, the hull of the disc, the
  cell's cores and the witness itself is below T. It stays below T in a whole neighbourhood
  of that placement.
- Searched from 14 starts per pair and bracketed at 64,000 directions, every one of the 48
  pairs has such a placement. The upper ends:

  | cell | H11 | H13 | H17 | H19 |
  | ---: | ---: | ---: | ---: | ---: |
  | 0 | 0.8342392 | 0.8339887 | 0.8333754 | 0.8333377 |
  | 1 | 0.8341674 | 0.8339184 | 0.8333036 | 0.8332660 |
  | 2 | 0.8341740 | 0.8339186 | 0.8333187 | 0.8332722 |
  | 3 | 0.8341029 | 0.8338449 | 0.8332473 | 0.8332006 |
  | 4 | 0.8342519 | 0.8339969 | 0.8334024 | 0.8333507 |
  | 5 | 0.8342392 | 0.8339896 | 0.8333755 | 0.8333377 |
  | 6 | 0.8341881 | 0.8339243 | 0.8333559 | 0.8332911 |
  | 7 | 0.8341118 | 0.8338460 | 0.8332722 | 0.8332128 |
  | 8 | 0.8341744 | 0.8339175 | 0.8333187 | 0.8332722 |
  | 9 | 0.8340995 | 0.8337978 | 0.8332266 | 0.8331726 |
  | 10 | 0.8341870 | 0.8339252 | 0.8333557 | 0.8332913 |
  | 11 | 0.8341673 | 0.8339170 | 0.8333001 | 0.8332659 |

- The depth-40 cells' cores cost more than a harmonic's forcing. On the tracked depth-50 cell
  (RUN-025), H11 and H13 could still close. Here none can.

**W7 — what is recorded.** Six files, no code. Neither the matrix nor the specs state the number
of directions, the harmonic root or any placement.

## P — the package

**P1.**

- §15's means, volumes and margins are the matrix's aggregates, rounded. The margins are
  Round 22's recorded worst values less T.
- §16–18's routing, crossover and cell-0 ranking are the matrix's.
- The specs' pilot result and portfolio agree with it. The crystal carries the same numbers.

**P2.**

- `VALIDATION.json` describes the document and the crystal correctly: SHA-256, bytes, `$$`
  202 and 6, both flags.
- Its digests of the matrix and both specs are the archive's.
- Its six checks follow from the matrix.

## Round 23, read

| Round 23 | statement | verdict | note |
| --- | --- | --- | --- |
| §0 | summary; boxed verdicts | PARTIAL | the grammar and the depth-16 infeasibility hold; the routing does not survive exact cores (W5, W6) |
| §1–11 | strict incidence; closure; set cover; non-composition | PASS | W1 |
| §12 | the pilot's scope | PASS | W2; the 96 directions are unstated (W3) |
| §13–14 | no complete edge at depth 16; do not force the cover | PASS | W3, W5 |
| §15 | the aggregates | PASS for B7; PARTIAL for the harmonics | node counts do not rebuild (W3) |
| §16–18 | shallow routing; crossover; cell 0 | PASS as computed | W3, W4 — but no harmonic can close these cells (W6) |
| §19–21 | crossover depth; horizon-aware scheduler; two layers | PASS as principles | here the cost-to-complete of every harmonic is infinite (W6) |
| §22 | the residual is all twelve | PARTIAL | at depth 16; with exact cores B7 empties it by depth 35 (W5) |
| §23–25 | long run; monotonicity; DISQUALIFIED | PASS | W1 |
| §26–27 | Round 24; handoff | n/a | |

## The status registry

Round 23's 7 judgments are reviewed, and its 3 first-used phrases defined: `PARTIAL`,
`DISQUALIFIED` and `INFEASIBLE`. Census v5 finds 3 of the 7: the set-off `INFEASIBLE` twice
and the set-off `STILL COMPUTE-DEFERRED`.

| verdict | judgments |
| --- | ---: |
| PASS | 6 |
| n/a | 1 |

The three INFEASIBLEs are right at depth 16. The note on the boxed one records what exact cores
add: B7 closes every cell by depth 35, and no harmonic ever can.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's cell-3 winner | 1 | A0 |
| a set-cover constraint | 1 | W1 |
| a marker area | 1 | W2 |
| a B7 edge's open count; the rebuild claim | 2 | W3 |
| a depth-22 B7 volume | 1 | W4 |
| an exact closing depth | 1 | W5 |
| a harmonic said closable; a limit | 2 | W6 |
| code said to be shipped | 1 | W7 |
| H19's recorded margin; a crystal value | 2 | P1 |
| the matrix's recorded digest; a VALIDATION check | 2 | P2 |
| a review dropped | 1 | R1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the matrix with its keys sorted;
- the set-cover spec with its keys sorted.

**15 / 15 plants, 4 / 4 controls, 194 / 194 must-stay-green assertions; no check without a
plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 1 | 1 |
| W2 | 1 | 1 |
| W3 | 2 | 2 |
| W4 | 1 | 1 |
| W5 | 1 | 1 |
| W6 | 2 | 2 |
| W7 | 1 | 1 |
| P1 | 2 | 2 |
| P2 | 2 | 2 |
| R1 | 1 | 1 |

## What this round does not establish

- **Any bound.** The exact-core closures and the limits are floating-point values of this
  tree's own kernel, not certificates.
- **The other 65 necessity cells.** Only Round 23's twelve are run.
- **The package's harmonic node counts.** Their convention was not found. The open counts, and
  everything that rests on them, rebuild.
- **Whether a finer base cell would let a harmonic close.** Not measured here.