# RUN-046 — UESFCM Cycles 07–09: the saved F3 frontier, its 24 skeletons and 1,332 signatures, the 12 reflection classes and the Taylor diagnostic all reproduce from this tree's own hull; the compilers they motivate ship as interfaces only

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** UESFCM Cycles 07–09 (all dated 2026-09-19), each package read from the archive:

- Cycle 07 — F3 runtime closure-failure localization and the active-envelope compiler. The package holds the
  document, the saved frontier (`F3_Unresolved_Frontier.npz`: 22,516 box centres and half-widths), the runtime
  probe and the scaffold.
- Cycle 08 — the two-level skeleton compiler. The package holds the body-skeleton catalog and a scaffold.
- Cycle 09 — reflection-quotiented skeletons and multi-chart Taylor closure. The package holds the reflection
  catalog, the Taylor diagnostic (npz and summary) and a scaffold.
- Each cycle's state crystal and VALIDATION, and the frontier's copies in the packages of Cycles 08 and 09.

They are evidence for UESFCM (RUN-043), not canonical documents.

**Tools:** [`run046_uesfcm_cycles_07_09.py`](../code/run046_uesfcm_cycles_07_09.py)

- this tree's exact hull (RUN-009, with RUN-031's verified segments), at every frontier centre;
- the envelope's own segments for the skeletons;
- central differences for the derivatives.

**Logs:** [`run046-uesfcm-cycles-07-09.json`](../data/gate-logs/run046-uesfcm-cycles-07-09.json)

**Verdict: PASS. Every number these three cycles compute on the saved frontier reproduces from this tree's own
hull: the membership, the centre areas, the skeleton catalog, the reflection classes and the Taylor margins. The
compilers they propose ship as interfaces with the proof steps not implemented, so the frontier and all the
diagnostics come from code that is not in the archive.**

- **The frontier (W1).**
  - The saved file holds 22,516 roots.
  - Their membership in F3 gives exactly the probe's fractions: 0.9204121513590335 fully inside,
    0.07718955409486587 straddling and 0.0023982945461005507 outside. A box counts as inside when its least
    distance d3 to the anchor reaches 0.00882, and outside when its greatest does not.
  - The exact centre areas give the least, 0.835188375167486 at root 21,409 (2 ulps from the probe's
    0.8351883751674862).
  - They also give the probe's eight quantiles, at levels 0, 0.001, 0.01, 0.05, 0.1, 0.5, 0.9 and 1, to its
    eight decimals.
  - This tree's kernel raised at 85 of the centres for the area and at 76 for the segments. There the chain of
    hand-overs inside one grid step exceeded its bound, where an arc meets its own corner at a noise-level tie. A
    different grid resolved each case, and the segments are checked in closed form either way.
- **The perimeter bound (W2).**
  - A(v) ≥ A(c) − P(c)·τ(B) closes none of the 22,516 roots.
  - P ranges over 3.33–3.40 and τ over 0.00024–0.031. The median penalty is 3.2 times the centre's slack over A3+.
  - Cycle 07's rejection, `REJECTED-AS-TOO-LOOSE`, stands.
- **The skeletons (W3).**
  - Walking the hull's envelope at every centre gives 24 body skeletons and 1,332 full piece signatures.
  - The coverage of the top 2, 5 and 10 is 0.6755196304849884, 0.7999200568484633 and 0.9184579854325813.
    These are the catalog's numbers, and the catalog's 24 sequences with their counts are this tree's.
  - **Corner stability.** Center and all 32 corners share a skeleton on 549 of a 2,000-root sample (0.275),
    against the cycle's 1,493 of 5,000 (0.2986). The gap is 2.3 standard errors of this sample; the cycle's
    sample and seed are not given.
- **The reflection quotient (W4).**
  - Reflection reverses the envelope's cyclic order. Under rotation and reversal the 24 skeletons fall into 12
    classes, equal to the catalog's with the same counts.
  - Every class's members are mutual reversals.
  - The dominant class is 6,979 + 8,231 = 15,210 roots.
- **The Taylor diagnostic (W5).**
  - The dominant-class roots within 10⁻³ and 5×10⁻⁴ of A3+ come out as the stored 649 and 27, index for index.
  - The stored centre areas agree to 10⁻¹⁵ and the half-widths exactly.
  - This tree's central differences agree with the stored gradients to 4×10⁻⁸ and the Hessians to 2×10⁻⁵.
  - First order: D1 = A(c) − A3+ − Σ|gᵢ|hᵢ is positive on 646 roots, and not on 3766, 6217 and 8001, as the
    cycle says. Its values agree with the stored ones to 3×10⁻¹¹.
  - Second order: D2 is positive on all 27, with least 1.655×10⁻⁴.
  - The summary is exactly its own arrays: D1 from its parts, D2 = D1 − ½Σ|Hᵢⱼ|hᵢhⱼ, and the quantiles at levels
    (0, 0.01, 0.05, 0.25, 0.5, 0.75, 0.95, 1) and (0, 0.1, 0.5, 1).
  - These margins are finite differences, and the cycle labels them `NUMERICAL-DIAGNOSTIC`.
- **The scaffolds (W7).** Read after the rebuild, never run.
  - Cycle 07's active-envelope compiler raises `NotImplementedError` in both proof steps: the signature
    stability and the interval area.
  - Cycle 08 implements the cyclic compression of labels. Its interval Newton step, winner gap and two-level area
    are not implemented.
  - Cycle 09 implements the arithmetic of the Taylor bounds, with the derivative bounds injected.
  - No shipped code produces the runtime probe, the frontier, the catalogs or the finite differences.
- **The package (P1).**
  - Every VALIDATION digest matches the archive.
  - The frontier's copies in Cycles 08 and 09 are byte-identical to Cycle 07's.

Drill: 14 / 14 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 133 / 133
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 cycles claim

1. **Cycle 07.**
   - The erosion branch-and-bound on F3 leaves 22,516 roots after 490,166 nodes.
   - 92.04 % of them lie fully in F3.
   - The least centre area is 0.8351883751674862, 4.07×10⁻⁴ above A3+, so there is no second basin and the
     hardness is the bound's looseness.
   - The perimeter-Lipschitz bound is too loose.
   - Stable-signature analyticity and an active-envelope interval compiler.
2. **Cycle 08.**
   - 1,332 piece signatures but 24 body skeletons, with the given coverage.
   - Two-level skeleton sufficiency and finite adjacent-skeleton enumeration.
   - Corner stability of 29.86 %.
3. **Cycle 09.**
   - Reflection maps a skeleton to its reversal: 12 classes, one of 15,210 roots.
   - Single- and multi-chart Taylor bounds.
   - On the 649 near-ceiling roots, D1 > 0 on 646; on the closest 27, D2 > 0 on all.

## W — rebuilt

| check | cycle | rebuilt | result |
| --- | --- | --- | --- |
| W1 | 07 | 22,516 centre areas; membership | membership exact; the least to 2 ulps; the quantiles to 8 decimals |
| W2 | 07 | A(c) − P(c)τ at every root | closes none; penalty 3.2× the slack |
| W3 | 08 | skeletons and signatures at every centre; corner stability | 24 / 1,332 and the catalog exact; stability 0.275 on a sample |
| W4 | 09 | the reflection classes | 12, equal to the catalog |
| W5 | 09 | the near sets; gradients, Hessians, D1, D2 | the sets exact; derivatives to 4×10⁻⁸ and 2×10⁻⁵ |
| W6 | 07–09 | the crystals | agree |
| W7 | 07–09 | the scaffolds, read | the proof steps not implemented |

## The cycles, read

| cycle | statement | verdict | note |
| --- | --- | --- | --- |
| 07 | the frontier: 22,516 roots; the membership; the least centre area | PASS | W1 |
| 07 | the runtime probe (490,166 nodes …) | not reproduced | the erosion prototype is not shipped; its internal arithmetic holds (RUN-043) |
| 07 | the perimeter bound, rejected as too loose | PASS | W2 |
| 07 | stable-signature analyticity | PASS | read |
| 08 | 1,332 signatures, 24 skeletons, the coverage | PASS | W3 |
| 08 | two-level skeleton sufficiency; finite enumeration | PASS | read |
| 08 | corner stability 29.86 % | PASS as a sample | 27.5 % on 2,000 roots here |
| 08 | the translation-convexity route, below 0.83460 | not reproduced | its functional is not given |
| 09 | the reflection quotient: 12 classes, 15,210 roots | PASS | W4 |
| 09 | the Taylor bounds (single-chart, interval gradient, multi-chart) | PASS | read |
| 09 | D1 > 0 on 646 / 649; D2 > 0 on 27 / 27 | PASS as diagnostic | W5 |

## The status registry

Not touched: the cycles are not canonical documents.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| Cycle 07's visited nodes in the text | 1 | A0 |
| the probe's membership fraction | 1 | W1, W6 |
| the least centre area | 1 | W1, W5, W6 |
| a perimeter | 1 | W2 |
| a skeleton changed | 1 | W3, W4, W6 |
| a catalog count; the stability count | 2 | W3 |
| a reflection member's count | 1 | W4 |
| the stored gradients | 1 | W5 |
| the summary's exceptions | 1 | W5, W6 |
| a crystal's stability fraction | 1 | W6 |
| a scaffold function added | 1 | W7 |
| a VALIDATION digest; a frontier copy's bytes | 2 | P1 |

Three controls:

- the unmodified inputs;
- the documents and scaffolds with CRLF line endings;
- the VALIDATION records with the keys sorted.

**14 / 14 plants, 3 / 3 controls, 133 / 133 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 2 | 2 |
| W2 | 1 | 1 |
| W3 | 3 | 3 |
| W4 | 2 | 2 |
| W5 | 3 | 3 |
| W6 | 5 | 5 |
| W7 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **The erosion runs themselves.** The prototype that produced the frontier is not shipped, so which boxes it
  left is taken from the file.
- **The Taylor margins as bounds.** They rest on finite differences, as the cycle says; no interval derivative
  is computed here either.
- **Corner stability in the cycle's own sample.** Its sample and seed are not given.