# RUN-018 — Round 15: the capacity algebra is exact, but the local proof's capacity is 0.835, not 0.8350003545 — its B7 lift root was sized for 0.835

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-25
**Subject:** `AMRAL_Lebesgue_Finite_Closure_Round_15_Proof_Capacity_Ladder_v0.1.md` (AMRAL-LUC-FC-R15, dated 2026-09-19) and its package, read from the archive:

- `PROOF_CAPACITY_LADDER_PROTOCOL_v0.1.md`, `VALIDATION.json`, `State_Crystal`;
- `master_domain_08365.json` and `reference_capacity_snapshot.json`;
- `reference_capacity_planner.py`, read only after this tree's reconstruction, and never run.

**Tools:** [`run018_round15_capacity_ladder.py`](../code/run018_round15_capacity_ladder.py) — capacity recursion and repair DP against brute force; Round 14's hot core; Round 10's manifest; Round 07's domain rule; RUN-009's exact hull
**Logs:** [`run018-round15-capacity-ladder.json`](../data/gate-logs/run018-round15-capacity-ladder.json)

**Verdict: PASS for the capacity algebra; PARTIAL for the reference planner and the ladder automation.**

- **Round 15's capacity algebra is exact.**
  - A node's capacity is the better of its best direct claim and the weaker of its two
    children: `C(p) = max(D(p), min(C(p0), C(p1)))`. On 200 random trees it equals the
    best of all proof routes, found by exhaustion.
  - A target is feasible exactly when it is at most `C(ε)`.
  - On 120 random trees the minimum-cost repair DP equals the cheapest plan, also
    found by exhaustion.
  - The reference's leaf and node capacities, breakpoints, deficit frontiers and
    limiting claims all rebuild, and so does its root, 0.8350003545377508.
- **The reference's capacity is still wrong.** Round 15's own first principle (§1–2)
  is that a root has a capacity. Shard slack cannot raise a target past the a-priori
  exclusion that justified the root.
  - The planner applies this to the base root only, which it records as
    `INFINITE-WITHIN-FIXED-LOCAL-SCOPE`.
  - The local proof has a second root: each base leaf's B7 lift square,
    `|x_7|, |y_7| ≤ t_7(0.835)` (Round 10). Placements outside that square are excluded
    only by the a-priori bound, which is exactly 0.835 at the square's edge.
  - For any target `T > 0.835`, the translations just beyond the edge, out to
    `t_7(T)`, are proved by neither the lift trees nor the exclusion. At `T = 0.83500035`
    that sliver is `1.06×10⁻⁶` wide, and the exclusion there gives only 0.83500017.
  - So the local proof's capacity is **0.835**. Its "free promotion" to 0.83500035
    (§15) is unsupported, and so is Round 14's scenario at 0.8350001, which RUN-017
    passed and now corrects.
  - The statement itself is almost certainly true: at the sliver's middle, the actual
    hull is about 1.005 at every one of 32 orientations tried. It is simply not proved
    by the recorded evidence.
- **The phase master root is rebuilt exactly.** At `T_M = 0.8365`: `d⋆ = 0.699860544061`,
  `t_3, t_5, t_7 = 0.2010235290, 0.2026182862, 0.2014734458`, a box volume up 11.66%
  and a `B_7` square up 4.66%.

  The root includes `t_7(0.8365)`, so the global design sizes the lift roots correctly.
  But the four-body optimum, 0.836494901 logged and ≤ 0.8364914 by RUN-010, lies below
  0.8365. In the B7 phase the evidence can never reach the root's capacity, so the root
  never binds.

Drill: 10 / 10 plants caught by exactly the named checks, 4 / 4 controls undisturbed,
115 / 115 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 Round 15 claims

1. **Root capacity** (§1–2): the root and its a-priori exclusion support targets only
   up to `C_root`, and shard slack cannot raise the target past it.
2. **Evidence capacity** (§3–7).
   - `D(p)`: the best compatible direct claim for `B(p)`.
   - `C(p) = max(D(p), min(C(p0), C(p1)))`.
   - Thm 6.1: every `T ≤ C(p)` is proved on `B(p)`.
   - Relative optimality within the registered grammar.
3. **The global capacity and the ladder** (§8–12).
   - `C_global = min(C_root, C_evidence)`.
   - Free promotion up to it.
   - The capacity staircase, feasibility `F_T`, and the deficit frontier `Δ_T`.
4. **The reference** (§13–18). Leaf capacities at `e = 10⁻⁸`, and the root 0.8350003545.

   | target | deficit |
   | ---: | --- |
   | 0.83500035 | free promotion |
   | 0.83500036 | `{10}` |
   | 0.83500039 | `{00, 10, 11}` |
   | 0.83500095 | `{00, 10, 11}` |
   | 0.83500096 | all four |
5. **Repair and phases** (§19–27).
   - The repair DP `K_T(p) = min(J_T(p), K_T(p0) + K_T(p1))` (Thm on §20).
   - Planned ancestor contraction; cost stays out of the proof.
   - A phase-master root at `T_M = 0.8365` and its overhead.
   - The root-capacity wall.
6. **Automation** (§28–36): the ladder loop, target selection, bottleneck groups,
   alternative routes, the snapshot, a root-capacity contract, capacity status labels,
   C15-1 … 4 deferred, Round 16's topic.

## A — sources

**A0.** 19 anchors. Round 15 has no reference list.

## W — rebuilt

| check | Round 15 | rebuilt | result |
| --- | --- | --- | --- |
| W1 | Thm 6.1, §7, §11–12 | 200 random trees, recursion against all routes; feasibility at the capacity | PASS |
| W2 | §19–20 | 120 random trees, DP against all repair plans | PASS |
| W3 | §13–18 | the reference from Round 14's hot core | PASS |
| W4 | §1–2, §15 | the B7 lift root's exclusion | the local capacity is 0.835 |
| W5 | §24–26 | the phase root's constants | PASS; the root never binds in the B7 phase |

**W1 — the recursion.** The proof routes for `B(p)` are:

- `p` directly;
- or a route for each child, combined.

Their best capacity, the largest over routes of the least `D` on the route, found by
listing every route, equals `C(p)` on 200 random trees up to depth 4. At `C(ε) ± 10⁻⁹`,
a route exists exactly on the feasible side.

**W2 — the repair DP.** On 120 random trees up to depth 3, with random direct capacities,
costs and targets, the DP's value equals the cheapest set of nodes whose
recertification makes the root feasible, found by trying every subset.

**W3 — the reference.** From Round 14's hot core at `e = 10⁻⁸`:

| seed | leaf capacity | claim |
| --- | ---: | --- |
| 00, 11 | 0.8350003869877956 | A-v1 |
| 01 | 0.8350009574949350 | B-v2 |
| 10 | 0.8350003545377508 | C-v1 |

- Node `0` has capacity 0.8350003869877956, node `1` has 0.8350003545377508, and the
  root has 0.8350003545377508.
- The breakpoints, and the five targets' deficits and limiting claims, are the
  snapshot's and the text's.
- B-v2's capacity rests on the slack audited under the rejected B-v1 (RUN-017).

**W4 — the second root.** Round 10's manifest records the B7 lift root as a square of
half-width `t_7(0.835) = 0.1969350467711682` in each of its four base leaves. Round 08
§6 excludes the placements outside that square by the a-priori bound: some corner of
`B_7` lies within `π/7` of the direction of its translation `t`, so the hull already has
area at least `Φ_{1/2}(√(|t|² + R_7² + 2|t|R_7 cos(π/7)))`. At `|t| = t_7(0.835)` that
is exactly 0.835.

Raise the target to `T`, and the translations along the square's axes with
`t_7(0.835) < |t| ≤ t_7(T)` lie outside the square, where no lift tree covers them.
Their exclusion falls short of `T`:

| target | `t_7(T)` | sliver width | exclusion in the sliver |
| ---: | ---: | ---: | ---: |
| 0.8350001 (Round 14) | 0.1969353504 | `3.04×10⁻⁷` | 0.83500005 |
| 0.83500035 (Round 15) | 0.1969361096 | `1.06×10⁻⁶` | 0.83500017 |

So the exclusion capacity of the lift root is 0.835, and the local proof's capacity is
`min(0.835, 0.8350003545) = 0.835`. Two conclusions follow:

- Round 15's `READY` at 0.83500035 is unsupported.
- Round 14's migration to 0.8350001, recorded ready, is unsupported for the same reason.

The fact behind it is not in doubt. With B7 in the middle of the sliver, at the worst
of 32 orientations, the actual hull still has area 1.0046, and the a-priori bound uses
one corner of B7 and nothing else. A sharper exclusion would close the gap. So would
lift roots sized for the phase target, as §24 does for `T_M = 0.8365`. Either is a new
proof step, not a free promotion.

The general lesson is Round 15's own, applied once more. Every a-priori exclusion in a
nested proof is a root with its own capacity. The base root has one; so does each
witness's lift root; so would any other excluded region. A root-capacity contract
(§33) needs one entry per exclusion.

**W5 — the phase root.** At `T_M = 0.8365`:

| | rebuilt | recorded |
| --- | ---: | ---: |
| `d⋆` | 0.6998605440613 | 0.6998605440613 |
| `t_3` | 0.2010235290074 | 0.2010235290074 |
| `t_5` | 0.2026182862304 | 0.2026182862304 |
| `t_7` | 0.2014734457863 | 0.2014734457863 |

- The half-width ratios to the `0.835` root are 1.03165, 1.02425 and 1.02305.
- The four translation coordinates' box grows by 11.66%, and the `B_7` square by 4.66%.
- The four-body optimum is 0.836494901 logged, and 0.8364914 by RUN-010.

Both lie below `T_M`. Any proof in the `B_3, B_5, B_7` phase can certify at most
`Λ(D, B_3, B_5, B_7) < 0.8365`. So the root capacity 0.8365 will never be the binding
limit in this phase. §26 calls the choice an engineering ceiling, and so it is: a
ceiling the phase cannot touch.

## P — the package

**P1.** The protocol states the same algebra: `C_global`, the recursion, the DP, and
the phase root with its overheads rounded to 11.7% and 4.7%.

**P2.** `VALIDATION.json` describes the round's document, the state crystal and the
protocol correctly: SHA-256, bytes, `$$` count and parity, the two flags. Its digest of
the planner matches the archive, and its copies of the phase root and the snapshot are
the files.

**`reference_capacity_planner.py`, read after the reconstruction.**

- It computes the recursion, feasibility and deficits as W1–W3 do.
- It writes `root_domain_capacity` as the fixed string
  `INFINITE-WITHIN-FIXED-LOCAL-SCOPE`, and the free promotion compares the target with
  the evidence capacity only.
- It loads the hot core from an absolute path outside the package, so it does not run
  as shipped.

## RUN-017, revisited

RUN-017 passed Round 14's scalar margin migration and its four scenarios. The rule is
right for what it covers: every leaf's raw witness satisfies `L ≥ T_c + e_c + s_c`.

But a claim's proof also uses the a-priori exclusion outside its lift root, and that
has no slack. A claim's capacity is therefore the lesser of `T_c + s_c + e_c − e'` and
its exclusions' capacity. For the reference claims that is 0.835 at any budget.
Round 14's "higher target" scenario (0.8350001, ready) is unsupported. RUN-017 is
corrected accordingly, in its report and its registry reviews.

## The theorems, read

| Round 15 | statement | verdict | note |
| --- | --- | --- | --- |
| §1–2 | root capacity | PASS | right; it applies to every exclusion, the witness lift roots included |
| §3–5, Thm 6.1 | direct and recursive capacity | PASS | W1 |
| §7 | relative optimality | PASS | W1: the recursion is the best route |
| §8 | `C_global = min(C_root, C_evidence)` | PASS | provided `C_root` covers every exclusion (W4) |
| §9 | free promotion | PASS | as a theorem; the reference's promotion is not supported (W4) |
| §10–12 | staircase; feasibility; deficit frontier | PASS | W1, W3 |
| §13–18 | the reference capacities and targets | PARTIAL | the evidence figures are right (W3); the capacity and its free promotion ignore the lift root (W4) |
| §19–22 | repair DP; planned contraction; cost outside the proof | PASS | W2 |
| §23–27 | phase-master root; its overhead; the wall | PASS | W5; in the B7 phase the root never binds |
| §28–31 | ladder loop; target selection; groups; routes | PASS | as design; see W4 for what it must include |
| §32–34 | snapshot; root-capacity contract; status labels | PASS | the contract needs one entry per exclusion |
| §35–38 | deferred runs; Round 16; checklist; handoff | n/a | |

## The status registry

Round 15's 20 judgments are reviewed, and its 8 first-used phrases defined:
`FREE-PROMOTABLE`, `MIXED-BOTTLENECK`, `PASS / FAIL`, `PHASE-DESIGN-CHOICE`, `READY`,
`ROOT-BOTTLENECK`, `SHARD-BOTTLENECK` and `FORMALIZED + REFERENCE TESTED`. The first seven
are the labels of §0, §15, §26 and §34. Two judgments were added later: §0's boxed
"GLOBAL 0.8350 HEAVY CERTIFICATE: STILL COMPUTE-DEFERRED" by census v3, and §37's
"Capacity staircase: FORMALIZED + REFERENCE TESTED" (PASS) by census v4 (RUN-003).
The two PARTIALs are:

- "Reference capacity planner: PASS" — right on evidence, wrong on capacity;
- "THRESHOLD LADDER AUTOMATION: CLOSED" — it would promote past an exclusion's
  capacity.

| verdict | judgments |
| --- | ---: |
| PASS | 9 |
| PARTIAL | 2 |
| n/a | 9 |

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| the document's `C_ref` digit | 1 | A0 |
| the recursion with min and max swapped | 1 | W1, W3 |
| the repair always splitting | 1 | W2 |
| the text's deficit at 0.83500036 | 1 | W3 |
| the promotions assumed supported | 1 | W4 |
| `t_7(0.8365)` off in the eleventh digit; the four-body optimum assumed above 0.8365 | 2 | W5 |
| the protocol's DP term dropped | 1 | P1 |
| the planner's recorded digest | 1 | P2 |
| a review dropped | 1 | R1 |

Four controls:

- the unmodified inputs;
- the document with CRLF line endings;
- the capacity computed by listing routes;
- the repair computed by listing plans.

**10 / 10 plants, 4 / 4 controls, 115 / 115 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 | 2 | 2 |
| P1 | 1 | 1 |
| P2 | 1 | 1 |
| R1 | 1 | 1 |

## What this round does not establish

- **That the sliver's configurations reach the higher targets rigorously.** One exact
  hull of 1.0046 at the sliver's middle shows the gap is in the proof, not in the fact.
  It is not a certificate.
- **Any global capacity.** Everything here is Round 10's local box.