# RUN-044 — UESFCM Cycles 01–03: the B7 separator LP reproduces to the last digit and the domain constants to the ulp; the "sharp root domain" is Mishra's Remark 20, which he notes saves no boxes; s7, the replacement family and the sublevel's thinness come out different from Cycle 03's figures

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** UESFCM Cycles 01–03 (all dated 2026-09-19), each package read from the archive:

- Cycle 01 — Mishra-seed expansion and the B7 separator gate: the document, `B7_Separator_Search.py`, the state
  crystal and VALIDATION;
- Cycle 02 — the certified minimizer atlas and exact witness-exchange equivalence (`Atlas_Compiler_Scaffold.py`);
- Cycle 03 — orbit localization, scalar separator closure and the sharp-minimum route (`Scalar_Gate_Utilities.py`).

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

- What they attribute to Mishra is read in his full text (arXiv 2608.30538v2).
- The placement they use is the one in his repository's ceiling log, at the commit the paper pins (RUN-010's
  snapshot).

**Tools:** [`run044_uesfcm_cycles_01_03.py`](../code/run044_uesfcm_cycles_01_03.py)

- this tree's exact hull (RUN-009, with RUN-031's verified segments);
- the support-direction LP (HiGHS);
- the continuous margin: the LP plus golden-section refinement of every peak, run until its two bounds agree,
  then over orientation;
- the domain constants by Brent's method;
- the outer polygons with a cancellation-free edge formula.

**Logs:** [`run044-uesfcm-cycles-01-03.json`](../data/gate-logs/run044-uesfcm-cycles-01-03.json)
**Snapshot:** [`uesfcm-c01-c03-mishra.json`](../data/external/uesfcm-c01-c03-mishra.json) — fifteen short phrases of
Mishra's full text; nothing else of the page is kept.

**Verdict: PASS for the three cycles' theorems and for every number they compute from definitions. Three things
differ from what the cycles report:**

- the sharp root domain presented as new is Mishra's own Remark 20;
- Cycle 03's continuous margin s7 is above the true minimum;
- its replacement-family candidate is well above a better placement.

The sublevel's reported thinness is the slack of the ceiling's construction, and along B5's turn the sublevel is
about 10⁻³ long, not thin.

- **Cycle 01, the separator LP, reproduces to the last digit (W1).**
  - The representative U0 is the placement in Mishra's repository, and its exact hull is 0.834780945912.
  - Cycle 01 takes 2048 support directions and 256 orientations over one period of B7. Rebuilt here, the grid
    minimum is **0.009382983648263432**, at orientation 193 — the crystal's number to all seventeen digits.
  - The orientation loss is π/(7·256) = 0.0017531209004407328, and the structural lower bound is 0.007629862747822699.
  - The loss uses R_K ≤ 1. B7's circumradius is 0.5129, which would halve it to 0.00090.
  - **The ranking table.** The same LP for B9 … B41 gives every entry of the table to within one unit of its last
    printed digit: B11 (0.004727) and B15 (0.002646) are printed a unit low. The shipped script prints B7 alone,
    so the ranking has no shipped driver.
  - The headroom 0.834781191 − 0.8344 = 0.000381191 is exact.
- **Cycle 02 (W2).**
  - The hull Hausdorff radius τ(B) = max(δ3, δ5) holds on 300 random boxes around the seed: the worst ratio of
    d_H to τ is 0.92.
  - δ3 and δ5 are Mishra's Lemma 22, and R5 = 0.525731112119 is his.
  - The confining box [−0.70, 0.70]² × [0, 2π/5] × [−0.70, 0.70]² is Mishra's Corollary 19, stated for area below
    0.8344. It covers the A3+-sublevel too, since f(0.70) > 0.836548794 > A3+.
  - The exchange equivalence and the atlas theorem are correct as proved.
  - The difference 0.836494901 − 0.834780947 = 0.001713954 is exact. But the first of those numbers, Mishra's
    located four-body optimum, is itself above a four-body placement this tree found in RUN-010 (0.8364914416).
  - The emitting run's counts are consistent with Mishra's forest identity N = 2L − s:
    - 2 · 245,496,952 − 486,799,600 = 4,194,304 = 2²² roots;
    - that is the repository's seed depth of 22.
- **The ceiling is the source of the "2.45×10⁻⁷ headroom" (W2).**
  - A3+ = 0.834781191 is the area of an outer polygon of 10⁶ uniform supporting lines at the seed. That is
    2.45×10⁻⁷ more than the seed's exact hull: rebuilt here without cancellation 2.450×10⁻⁷, the repository's own
    2.450×10⁻⁷.
  - For 10⁶ directions the excess of a smooth body is about 10⁻¹¹. This one is first order: it sits at the hull's
    17 straight bridging edges, whose normals the uniform grid misses (the loss RUN-032 found in Round 29's
    polygons).
  - With those 17 directions added to 10⁴ uniform ones, the excess falls from 3.4×10⁻⁶ to 1.2×10⁻⁸.
  - So Cycle 03's "the family ceiling and the representative differ by only 2.45×10⁻⁷" (§13, §16) measures the
    ceiling's construction, not the minimum.
- **Cycle 03's domain (W3).**
  - d* = 0.694929333199893, T3 = 0.193949748872303 and T5 = 0.197117585445617 match the text to the ulp.
  - The four-dimensional factor 164.27178468 is right, and B3 and B5 placed just beyond T_n already span more than
    A3+.
  - **This is Mishra's Remark 20.** He applies Lemma 18 to the farthest corner with the same bound, to the digit,
    and at his own threshold 0.8344 obtains |t3| ≤ 0.192366, |t5| ≤ 0.195891 and a factor of 169. Rebuilt here:
    0.19236641, 0.19589071 and 169.1. He adds that it recovers Brass and Sharifi's box.
  - Cycle 03 presents it as `DOMAIN-STRENGTHENING`, verdict `SHARP ROOT DOMAIN: PROVED`, without citing him.
  - His Remark 34 reports that the smaller domain does not reduce the cost: at τ = 0.833, 25,425,568 boxes over
    the larger domain, 27,350,260 over the smaller. The next cycles still count it as a volume reduction
    (164.3×, then 985.6×).
- **Cycle 03's margin s7 (W4).**
  - M_U0(B7) is minimized continuously here:
    - the inner problem is solved until the LP's lower value and the true maximum at its translation agree to
      10⁻¹²;
    - a continuous scan of 2,048 turns follows, with every local minimum refined by golden-section search.
  - **s7 = 0.009384687909, at φ = 0.676813.**
  - The mirror basin at φ = 0.220843 gives 0.009385623167. The seed is only nearly symmetric, so the two differ.
    The other two basins are near 0.0121 and 0.0127.
  - Cycle 03's 0.00938484 (the crystal's 0.009384839564890224) is **1.52×10⁻⁷ above s7**, and not the value at
    either basin's minimum. A candidate for a lower bound should not lie above the minimum. It is harmless here:
    Gate A certifies 0.0088367 by its own means (RUN-045).
  - Cycle 01's grid value 0.009382983648 lies below s7, as the coarser problem allows.
  - This is a numerical minimum. Between scanned turns the margin can fall by up to 1.1×10⁻⁴. The 63 scan points
    within that of the best all sit within 16 steps of the two deep basins.
- **Sharpness, and the grid trap (W5).**
  - Every one of 210 sampled directions out of the seed raises the area: the ten coordinate directions, and
    200 random ones. At both step lengths the least slope is 0.0084.
  - **Minimized over all directions, the least one-sided slope is 1.12×10⁻⁵.** The direction is 98.7 % in B5's
    turn. Along it the area rises at 1.1×10⁻⁵ one way and 6.9×10⁻⁵ the other.
  - So the minimum is sharp, but in that direction barely. The sublevel A ≤ A3+ reaches 9.9×10⁻⁴ and 7.8×10⁻⁴
    along it, against at most 2.9×10⁻⁵ along the axes.
  - Cycle 05 §20's "weakest single-axis slope about 8×10⁻³" is right for the axes. §16's radius bound
    2(A3+ − A(v0))/c becomes 0.044 with the true c.
  - None of this threatens Gate B: 10⁻³ of B5's turn moves it by 5×10⁻⁴, far inside 0.0088.
  - **The grid trap of §14 is not reproduced.** This tree's grid-only hull follows every hand-over it sees.
    Minimizing it from the seed at 32, 64 and 128 angles returns the true local minimum: the grid area equals
    the exact area there to 5×10⁻¹⁶.
- **The replacement family D, B3, B7 (W6).**
  - Searched here, its minimum is at most **0.8305410847**, 9.8×10⁻⁵ below Cycle 03's candidate 0.830638626.
  - Its outer polygons of 1,000 and 10⁶ half-planes have areas 0.8309189 and 0.8305415.
  - So the rejection Cycle 03 draws, that this family's lower bound cannot pass 0.8344, stands with more room.
- **The scripts (W8).** Read after the rebuild, never run.
  - Cycle 01's script uses the repository placement and the grids rebuilt here, and prints only B7.
  - Cycle 02's scaffold takes both the box lower bound and the B7 margin as injected callbacks, and ships neither.
  - Cycle 03's docstring says it imports Mishra's `geom.py`. It imports only `math`.
  - None of the three ships the code for s7, the replacement search, the outer polygon, the sharpness probe or the
    grid trap.
- **The package (P1).** Every VALIDATION digest and document record matches the bytes hashed here and the
  archive's manifest.

Drill: 16 / 16 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 190 / 190
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 the cycles claim

1. **Cycle 01.**
   - B7 is a separator of U0, by a finite-grid LP.
   - A ranking of the regular Reuleaux odd-gons.
   - The gate δ7 = min over minimizers of M_U(B7) > 0.
   - Mishra's four-body optimum is quoted.
2. **Cycle 02.**
   - The hull Hausdorff radius of a placement box.
   - The certified minimizer-atlas theorem.
   - The exact witness-exchange equivalence and the single-witness corollary.
   - The semi-decision theorem and the trichotomy.
3. **Cycle 03.**
   - The farthest-corner root domain.
   - The separator–orbit radius inequality δ7 ≥ s7 − R⁺, and the scalar closure criterion.
   - The ORBIT-COVERED cell rule.
   - Numerical evidence for s7, a sharp minimum and a grid trap.
   - The rejection of D, B3, B7.
   - A new exact gate: s7 > R̄.

## A — sources

**A0.** The anchors in the three documents: the numbers and formulas above.

**X1.** Mishra's v2 full text supplies each fact the cycles attribute to him:

- the gauge ρ ∈ [0, 2π/5) and Lemma 18's f;
- Corollary 19, its box and f(0.70) > 0.836548794;
- the radii R3, R5 and Lemma 22's corner movement;
- the forest identity and the emitting run's 486,799,600 nodes and 245,496,952 leaves;
- the located optima 0.834780947 and 0.836494901, and the fourth body;
- the eight-dimensional cost, 8.6×10¹¹ boxes and about 880 days;
- the frozen commit;
- Remarks 20 and 34.

## W — rebuilt

| check | cycle | rebuilt | result |
| --- | --- | --- | --- |
| W1 | 01 | the grid LP for B7 … B41; U0; the headroom | reproduced to 17 digits; the table within one unit |
| W2 | 02 | τ(B) on random boxes; R5; the box; the ceiling's excess | PASS; A3+'s 2.45×10⁻⁷ is first-order polygon loss |
| W3 | 03 | d*, T3, T5 and the factor at A3+ and at 0.8344; the exclusion | to the ulp; it is Mishra's Remark 20 |
| W4 | 03 | s7 over every basin of orientation | 0.009384687909; the cycle's is 1.5×10⁻⁷ high |
| W5 | 03 | slopes at the seed; the least direction; the grid trap | sharp, but 1.1×10⁻⁵ along B5's turn; no trap |
| W6 | 03 | D, B3, B7 searched; outer polygons | 9.8×10⁻⁵ below the cycle's candidate |
| W7 | 01–03 | the crystals | the domain to 1–4 ulps; the rest exact |
| W8 | 01–03 | the three scripts, read | as above |

## P — the package

**P1.** VALIDATION's records of each document, script and crystal match the bytes hashed here and the archive's
manifest.

## The cycles, read

| cycle | statement | verdict | note |
| --- | --- | --- | --- |
| 01 | M_U0(B7) grid LP 0.00938298365, loss π/(7·256), lower 0.00762986275 | PASS | W1 |
| 01 | the ranking of B7 … B41 | PASS | two entries a unit low (W1) |
| 02 | hull Hausdorff radius τ(B) | PASS | W2; Mishra's Lemma 22 extended by hull nonexpansiveness |
| 02 | the minimizer-atlas theorem; exact exchange equivalence | PASS | read |
| 02 | 0.836494901 − 0.834780947 = 0.001713954 | PASS as arithmetic | the four-body optimum is not a minimum (RUN-010) |
| 03 | the sharp root domain, `PROVED` | PASS, not new | Mishra's Remark 20; his Remark 34 on its cost |
| 03 | δ7 ≥ s7 − R⁺; the scalar criterion; ORBIT-COVERED | PASS | read |
| 03 | s7 ≈ 0.00938484 | PARTIAL | the minimum is 0.009384687909; the figure is 1.5×10⁻⁷ above it (W4) |
| 03 | the sharp minimum | PASS | but the least slope is 1.1×10⁻⁵, along B5's turn (W5) |
| 03 | the sublevel is extremely thin | PARTIAL | about 10⁻³ along B5's turn; the 2.45×10⁻⁷ is the ceiling's slack (W2) |
| 03 | the grid trap | not reproduced | no false minimum with this tree's grid-only hull (W5) |
| 03 | D, B3, B7: min ≈ 0.830638626, outer 1000 ≈ 0.830957521 | PASS as search | a lower point exists (W6) |

## The status registry

Not touched: the cycles are not canonical documents.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| Cycle 01's orientation loss in the text | 1 | A0 |
| Remark 20 dropped from Mishra's snapshot; the seed depth | 2 | X1 |
| B7's grid values moved by 10⁻¹² | 1 | W1, W7 |
| a row of Cycle 01's ranking table | 1 | W1 |
| the Hausdorff ratio above 1; the ceiling's hand-over count | 2 | W2 |
| Cycle 03's T3 in the text | 1 | A0, W3 |
| an exclusion sample below zero | 1 | W3 |
| a far scan point within the margin's slack | 1 | W4 |
| a negative slope out of the seed | 1 | W5 |
| the replacement's area | 1 | W6, W7 |
| a crystal's orientation loss | 1 | W7 |
| Cycle 01's script given a closed direction grid | 1 | W8 |
| a VALIDATION digest; a crystal's bytes | 2 | P1 |

Three controls:

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

**16 / 16 plants, 3 / 3 controls, 190 / 190 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 2 | 2 |
| X1 | 2 | 2 |
| W1 | 2 | 2 |
| W2 | 2 | 2 |
| W3 | 2 | 2 |
| W4 | 1 | 1 |
| W5 | 1 | 1 |
| W6 | 1 | 1 |
| W7 | 3 | 3 |
| W8 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **s7 as a certified value.** The scan's Lipschitz slack between turns is 1.1×10⁻⁴. Cycle 04's dual certificate
  is the certified bound (RUN-045).
- **The minimizer set.** Its thinness depends on the true Λ(F3), which neither the cycles nor this run bound from
  below beyond Mishra's 0.8344.
- **The grid trap as a phenomenon.** The mechanism is real: a piece whose whole winning arc lies inside one grid
  step is lost. RUN-031 found it in this tree's own kernel. But the cycle's placement and evaluator are not given,
  and this tree's grid-only path, which follows every hand-over it sees, gives no false minimum from the seed.