# RUN-045 — UESFCM Cycles 04–06: Gate A's dual certificate holds under this tree's own integer intervals, but its verifier skips two premises; the D3 quotient, the symmetric anchor and the box bounds are right; the far-branch minima reported are search values that sit above better points

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** UESFCM Cycles 04–06 (all dated 2026-09-19), each package read from the archive:

- Cycle 04 — Gate A closure and the far-set certificate: the document; the dual certificate (1,024 witnesses);
  `Verify_GateA.py`; the Gate B scaffold;
- Cycle 05 — the symmetry-quotiented two-colour Gate B certificate: the document, the certificate spec, the
  emitter and the verifier skeleton, and copies of Cycle 04's certificate and verifier;
- Cycle 06 — the symmetry-fixed anchor and the exact far-branch decomposition, with its branch scaffold;
- each cycle's state crystal and VALIDATION.

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

**Tools:** [`run045_uesfcm_cycles_04_06.py`](../code/run045_uesfcm_cycles_04_06.py)

- 400-bit integer intervals (RUN-030) with Machin's π (RUN-031) for every witness;
- the exact hull and the continuous margin of RUN-044;
- differential evolution and Nelder–Mead on the far branches' boundaries;
- random boxes for the bounds.

**Logs:** [`run045-uesfcm-cycles-04-06.json`](../data/gate-logs/run045-uesfcm-cycles-04-06.json)

**Verdict: PASS for every theorem of the three cycles and for Gate A's certificate. Two premises of that
certificate are left unchecked by the verifier shipped with it. The far-set minima the cycles report are search
values, and each sits above a lower point found here.**

- **Gate A holds (W1).** B7 does not fit the seed's hull, with M_U0(B7) > 0.0088366789091384. The check:
  - Every witness is re-enclosed in 400-bit integer intervals, from the placement's decimals, Machin's π and
    Taylor sums. The widths are below 10⁻¹⁰².
  - All 1,024 orientations are present.
  - Every triple of directions has all its gaps below π, so every weight is positive.
  - The least lower bound is 0.009274959134248587, at p = 767.
  - Less the bridge π/7168, that gives 0.008836678909138404. The certificate records 0.0088366789091384.
  - The certificate's bound is conservative. At 33 sampled orientations the true margin (B7 itself, exact
    supports, translation optimized) is above it by 2.2×10⁻⁷ to 1.8×10⁻⁴ (W2).
  - At p = 767 the true margin is 0.009458708251.
- **What the certificate records, and what its verifier checks.**
  - **The recorded intervals are points.** Its "intervals" have lower = upper at all 1,024 witnesses: they are
    20-digit decimals of mpmath intervals at 60 digits. On 76 witnesses the recorded lower end is above the
    enclosure's, by at most 3×10⁻¹⁹.
  - **The verifier trusts none of these values.** It recomputes them.
  - **It never checks two premises the bound rests on.**
    - That each triple's gaps are all below π, i.e. that the sine-gap weights are positive. With a gap past π a
      weight turns negative, and the dual bound fails.
    - That all 1,024 orientations are present. The Lipschitz bridge assumes the full grid.
  - Both premises hold here, as checked above.
  - **Float conversions.** It takes its maxima through float-keyed comparisons, and rounds the lower end to the
    nearest double before comparing. Both effects stay below 10⁻¹⁷.
- **The quotient is right (W3).**
  - The six D3 images of 40 random placements have the same hull area, to 1×10⁻¹⁵.
  - The turn of B5 moves by 2π/3 ≡ 4π/15 (mod 2π/5). With the reflection this gives the fundamental interval
    [0, π/15].
  - The branch that holds the seed is first ORBIT-COVERED at depth 31, as Cycle 05 reports.
  - The safe constant's arithmetic holds: 0.00927495 − 0.000438281 = 0.008836669, leaving 6.9×10⁻⁸ above
    s_A = 0.0088366.
  - The spec's root half-widths T3 and T5 are 22 and 12 ulps smaller than the exclusion bounds. They are rounded
    inward, where a certificate root should round outward. The spec says itself that the tree has not been emitted.
- **The anchor is right (W4).**
  - The seed's residual turn is 2.4129×10⁻⁵.
  - Cycle 06 bounds d_H(U0, U_sym) by 1.66×10⁻⁵. The true value is 7.1×10⁻⁶.
  - U_sym's area is 0.834780945826, slightly below the seed's.
  - The new radius is 0.0088366 − 0.0000166 = 0.00882.
  - The nearest orbit copy is 2R5 sin(π/30) = 0.109907730484 away.
- **The far branches (W5).** Searched with the exact hull on their boundaries:
  - F3's minimum is at most 0.834968955147, 1.1×10⁻⁸ below Cycle 06's 0.834968965913.
  - F5's minimum is at most 0.834804244389, 1.0×10⁻⁹ below Cycle 06's 0.834804245410.
  - So F5's margin over A3+ is at most 2.30534×10⁻⁵, not 2.30544×10⁻⁵.
  - For Cycle 04's own far boundary (the seed's orbit at radius 0.0088366789), its 0.834804245159 is 7×10⁻¹⁰
    below the best point found here.
  - Every value stays above A3+ = 0.834781191.
- **The box bounds (W6).**
  - Cycle 06's bounds on d3 and d5 held at all 12,000 sampled points of 400 random boxes.
  - The farthest-corner area bound never exceeded the exact hull.
- **The scripts (W8).** Read after the rebuild, never run.
  - Only Gate A's verifier executes a proof.
  - The Gate B emitter, the scaffolds and the branch scaffold take the area lower bound as an injected callback,
    and none is supplied.
  - Cycle 05's "verifier skeleton" contains no executable line.
- **The package (P1).** Every VALIDATION digest matches the archive. Cycle 05's two dependency copies are
  byte-identical to Cycle 04's files.

Drill: 14 / 14 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 152 / 152
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 cycles claim

1. **Cycle 04.**
   - Gate A by a finite dual certificate:
     - B7 is replaced by the heptagon of its corners;
     - each Reuleaux body by the intersection of its unit discs;
     - three directions per orientation with sine-gap weights;
     - interval arithmetic;
     - an orientation-Lipschitz bridge.
   - A component-motion metric q.
   - Gate B written as a far-set minimum.
2. **Cycle 05.**
   - B5's turn is quotiented by the D3 symmetry to [0, π/15].
   - Two leaf predicates, ORBIT-COVERED and AREA-PRUNED, with a one-bit tree format.
   - Finite termination if Gate B is true.
   - A safe constant s_A = 0.0088366.
3. **Cycle 06.**
   - A reflection-symmetric anchor U_sym within 1.66×10⁻⁵ of U0, and the new radius 0.00882.
   - The far set splits exactly into F3 ∪ F5.
   - Branch box bounds and a farthest-corner area bound.
   - Search minima for both branches.

## W — rebuilt

| check | cycle | rebuilt | result |
| --- | --- | --- | --- |
| W1 | 04 | 1,024 witnesses in 400-bit intervals | PASS; the verifier skips two premises |
| W2 | 04 | the certificate against the true margin at 33 orientations | conservative by 2.2×10⁻⁷ to 1.8×10⁻⁴ |
| W3 | 05 | D3 on random placements; the orbit-basin depth; the constants | PASS; T3, T5 rounded inward |
| W4 | 06 | the anchor's distance, area, radius and orbit copies | PASS; the true d_H is 7.1×10⁻⁶ |
| W5 | 04, 06 | the far branches searched | lower than reported; all above A3+ |
| W6 | 06 | the box bounds on 400 random boxes | PASS |
| W7 | 04–06 | the state crystals against the numbers above | agree; the far candidates as in W5 |
| W8 | 04–06 | the five scripts, read | only Gate A's verifier executes a proof |

## P — the package

**P1.** Every VALIDATION digest and document record matches the bytes hashed here and the archive's manifest.
Cycle 05's dependency copies are byte-identical to Cycle 04's files.

## The cycles, read

| cycle | statement | verdict | note |
| --- | --- | --- | --- |
| 04 | Gate A: M_U0(B7) > 0.0088366789091384 (`CERTIFIED-FINITE-DUAL`) | PASS | W1 |
| 04 | the conservative target and cover supports; the three-active balance | PASS | positive weights needed (all gaps < π): checked here, not by the verifier |
| 04 | the far-set minimum, searched ≈ 0.834804245159 | PASS as search | W5 |
| 05 | the D3 quotient to [0, π/15] | PASS | W3 |
| 05 | the two-colour theorem; finite termination if Gate B is true | PASS | read |
| 05 | the safe constant s_A = 0.0088366 | PASS | W3 |
| 05 | the orbit-basin depth 31 | PASS | W3 |
| 05 | the runtime probe (320,000 nodes …) | not reproduced | it rests on the unshipped erosion prototype |
| 06 | d_H(U0, U_sym) < 1.66×10⁻⁵; the radius 0.00882 | PASS | W4 |
| 06 | the far set is F3 ∪ F5; the nontrivial orbit copies are far | PASS | W4 |
| 06 | the F3 and F5 search minima | PASS as search | both above better points (W5) |
| 06 | the branch box bounds; the farthest-corner bound | PASS | W6 |
| 06 | the F3 runtime probe (100,000 nodes …) | not reproduced | the same prototype |

## The status registry

Not touched: the cycles are not canonical documents.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| Cycle 04's least sample lower in the text | 1 | A0 |
| a witness missing; a weight made negative | 2 | W1 |
| a true margin below the certificate | 1 | W2 |
| D3 invariance broken | 1 | W3 |
| the orbit-basin depth | 1 | W3, W7 |
| U_sym's area | 1 | W4, W7 |
| Cycle 06's Hausdorff bound in the text | 1 | A0, W4 |
| a far minimum below A3+ | 1 | W5, W7 |
| a box-bound violation | 1 | W6 |
| a crystal's radius | 1 | W7 |
| the verifier's precision | 1 | W8 |
| a VALIDATION digest; a dependency copy's bytes | 2 | P1 |

Three controls:

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

**14 / 14 plants, 3 / 3 controls, 152 / 152 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 2 | 2 |
| W1 | 2 | 2 |
| W2 | 1 | 1 |
| W3 | 2 | 2 |
| W4 | 2 | 2 |
| W5 | 1 | 1 |
| W6 | 1 | 1 |
| W7 | 4 | 4 |
| W8 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Gate B.** No two-colour tree is emitted in these cycles. The cycles that follow attempt one.
- **The runtime probes.** Their counts depend on an erosion prototype that is not shipped.
- **The far minima as minima.** Each value found here is a placement in its branch, so it is an upper bound on
  the branch minimum. No search can establish the minimum itself.