# RUN-047 — UESFCM Cycles 10–11: both erosion certificates decode exactly and all 56,149 of their area leaves hold in exact arithmetic with this tree's exact cores, by at least 2.28×10⁻⁵; the least slacks and η_model reproduce, the float split leaves 10⁻¹⁸ gaps that continuity closes, and Cycle 11's verifier cannot run from its own package

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** UESFCM Cycles 10 and 11 (dated 2026-09-19 and 2026-09-20), each package read from the archive:

- Cycle 10 — finite closure of the 27 priority F3 roots. The package holds the document, the certificate
  (27 trees), a standalone verifier, the state crystal and VALIDATION, and copies of Cycle 07's frontier and
  Cycle 09's Taylor diagnostic.
- Cycle 11 — the 617-root batch and the two-root frontier. The package holds the document, the certificate
  (617 trees), a Numba verifier, the final frontier (225 subroots) and its summary, six search logs, the state
  crystal and VALIDATION.

They are evidence for UESFCM (RUN-043), not canonical documents. The floating-point model Cycle 10 takes from
Mishra is read in his full text (arXiv 2608.30538v2, §1 and §5.4).

**Tools:** [`run047_uesfcm_cycles_10_11.py`](../code/run047_uesfcm_cycles_10_11.py)

- the trees decoded with the package's split rule (below);
- the finite witness set rebuilt here from its description, and the two tightest leaves evaluated again at
  50 digits;
- every area leaf certified in exact arithmetic:
  - candidates: the contact points of the disc and of this tree's exact eroded cores (RUN-012) at 1,024
    directions (more only where needed), each moved 10⁻¹² toward its body's centre;
  - each candidate proved inside its body: the disc in exact rationals, a core against its corners enclosed in
    RUN-030's 400-bit integer intervals;
  - the exact hull of those that pass;
- exact rationals for the split's tiling and for the NOT-F3 leaves.

**Logs:** [`run047-uesfcm-cycles-10-11.json`](../data/gate-logs/run047-uesfcm-cycles-10-11.json)
**Snapshot:** [`uesfcm-c10-c11-mishra.json`](../data/external/uesfcm-c10-c11-mishra.json) — thirteen short phrases of
Mishra's §1 and §5.4; nothing else of the page is kept.

**Verdict: PASS. Both certificates decode exactly and every count they state reproduces. All 56,149 area leaves
also hold in exact arithmetic with this tree's exact cores: the tightest clears A3+ by 2.28×10⁻⁵, where the
certificates' own least slacks are 3.8×10⁻⁸ and 2.5×10⁻⁹. Three things are noted:**

- the split rule is in neither document; it was read from the verifier;
- the float split does not tile its boxes exactly: the gaps, at most 4.7×10⁻¹⁸, are closed by continuity, which
  neither the documents nor the verifiers state;
- Cycle 11's verifier loads Cycle 10's verifier from an absolute path on the machine that wrote it, and the package
  does not ship that file.

- **The split rule.** The documents say only "deterministic split rule".
  - Two guesses of this tree's decode trees that parse, but leave boxes that fail even the exact-core bound:
    - Mishra's weights (1, 1, R5, 1, 1) throughout: 36 such leaves, the worst 5.8×10⁻⁵ short;
    - Cycle 06's F3 weights (1.5, 1.5, R5, 1, 1) throughout: 27, the worst 4.9×10⁻⁵ short.
  - The verifier, read after that, splits the widest coordinate by weights that change with membership:
    (1.5, 1.5, R5, 1, 1) while the box may still meet the F3 ball; (0.7, 0.7, 1.5R5, 1.5, 1.5) once it lies wholly
    outside. The first index wins a tie.
  - Cycle 07's text describes this adaptivity (B3's translation first until membership is settled, B5's turn and
    translation first after), but not the weights 0.7 and 1.5.
  - The counts cannot tell rules apart: the bits alone fix every tree's nodes, leaves and depth. What confirms this
    rule is that every leaf then passes, and that Cycle 11's 617 per-tree minima agree to 7×10⁻¹⁶ (W3).
- **The roots (W1).**
  - Cycle 10's 27 roots are Cycle 09's 27 closest, in its order.
  - Cycle 11's 617 are the other 622 near-ceiling roots less 3766, 6217, 8001, 8015 and 8016.
  - Every root box is Cycle 07's saved row to the bit.
- **The trees (W2).** Bits high first, preorder, 1 = split. Every tree uses exactly its bits.
  - Cycle 10: 2,699 nodes, 1,363 area leaves, no NOT-F3 leaf, depth 13. Root 8006 alone takes 2,153 nodes.
  - Cycle 11: 109,037 nodes, 54,786 area leaves, 41 NOT-F3 leaves, depth 16. Root 8018 takes 13,181.
  - Every tree's stored counts are this tree's.
  - **The tiling.** A child's centre c ± h/2 is rounded, so the two children need not cover the parent exactly.
    In exact rationals:
    - Cycle 10: 280 of 1,336 splits tile exactly; the widest gap is 2.5×10⁻¹⁸, the widest overlap 3.3×10⁻¹⁸;
    - Cycle 11: 15,915 of 54,210; gap 4.7×10⁻¹⁸, overlap 3.3×10⁻¹⁸.
  - A placement in a gap lies within 4.7×10⁻¹⁸ of a certified box. Moving the bodies by d changes the hull area by
    at most Pd + πd², with P ≤ 8.026 (Mishra's bound). That is 3.7×10⁻¹⁷, far below every leaf's slack, so the
    gaps are covered. Neither verifier checks coverage, and neither document says it.
- **The witness bound (W3).** Rebuilt here from its description: the disc's inscribed 256-gon, and on each eroded
  core its corners and round(256·span/total) − 1 points per arc.
  - It passes on every area leaf after the 10⁻⁹. Lemma 18's a-priori term decides none.
  - Cycle 10's least raw slack is 3.8179494×10⁻⁸ at root 8006: 8.9×10⁻¹⁶ above the certificate's
    3.817949290×10⁻⁸.
  - Cycle 11's least is 2.5376479×10⁻⁹ at root 21309. Less 10⁻⁹ that is 1.5376479×10⁻⁹, 3.6×10⁻¹⁶ from the
    certificate's.
  - **At 50 digits** the same leaves give 3.81794932×10⁻⁸ and 2.53764884×10⁻⁹. The certificates' doubles sit
    2.7×10⁻¹⁶ and 6.1×10⁻¹⁶ below, this tree's 5.9×10⁻¹⁶ above and 9.7×10⁻¹⁶ below. Cycle 11 prints its slack to
    seventeen digits (1.5376482309648054×10⁻⁹); six of them carry.
  - Cycle 11's per-tree minima agree with the stored ones to 7×10⁻¹⁶ on all 617 trees.
  - **η_model.** It is Mishra's §5.4 model with his constants unrounded:
    - ε0 = 10⁻¹² + 4u + 8uR and P = 2πR, with R = 0.70 + 1/√3 and u = 2⁻⁵³;
    - γ_N·NR², with N = m + 2K = 768.
  - That gives the certificate's 1.1488312266895049×10⁻¹⁰ to every digit, and the raw slack over it exactly the
    stored 332.33. Mishra's printed, rounded constants give 3.4×10⁻¹⁶ more.
  - **Hull cross-check.** On Cycle 10's leaves the double-precision hull differs from the exact hull of the same
    points by at most 1.0×10⁻¹⁵. The document's "≤ 1.2×10⁻¹⁵" between two double-precision routines is of that
    size.
- **Exact certification (W4).**
  - Every area leaf certifies at 1,024 directions; none needs more, and no candidate fails its membership proof.
  - Cycle 10: 1,363 leaves; the least margin is 2.352×10⁻⁵, at root 8006.
  - Cycle 11: 54,786 leaves; the least is 2.277×10⁻⁵, at root 20561.
  - All 41 NOT-F3 boxes lie inside the F3 ball in exact arithmetic. The nearest is 1.31×10⁻⁵ from its edge.
  - So the witness set costs about 2.3×10⁻⁵ against the exact cores. The trees are much deeper than an exact-core
    bound needs.
- **Cycle 11's search (W5).**
  - **The 5,000-node batch.**
    - It ran 622 roots, the 649 less Cycle 10's 27, and closed 616; the six others are 3766, 6217, 8001, 8015,
      8016 and 8018.
    - The 616 closed rows equal their certificate trees field for field. Root 7989 is a single NOT-F3 leaf: the
      certificate writes no minimum, the log the non-standard JSON token `Infinity`.
    - The certificate is those 616 trees and 8018's 13,181 nodes.
  - **The targeted layers.** Each is fed exactly by the last one's unfinished nodes, parent by parent:
    - five roots at 20,000 nodes leave 9 + 11 + 7 + 11 + 13 = 51;
    - Round 2 closes 40 of 51 and leaves 91;
    - Round 3 closes 63 of 91 and leaves 242: 3766 32, 6217 200, 8001 3, 8016 7;
    - the non-6217 pass closes 39 of 42 and leaves 31, all under 3766;
    - the 3766 pass closes 28 of 31 and leaves 25.
  - Every log's closed, hard, new-frontier and visited counts are its own rows'.
  - **The frontier.** 225 subroots, 200 under 6217 and 25 under 3766.
    - Each is reached from its parent root by the split rule, at depths 4 to 28, exactly the stored depths.
    - Each lies inside its parent to 2.3×10⁻¹⁸, and none contains another.
    - 489 pairs share a face to within 5.9×10⁻¹⁸, which is the split's rounding.
  - **The centre diagnostic** reproduces: 0.8349695459155 under 6217 and 0.8352612051955 under 3766, to
    2×10⁻¹⁶. The exact centre areas are 0.8350052734 and 0.8353009037.
- **The crystals (W6)** agree with the numbers rebuilt here. Cycle 11's packaging debt is the five hard roots less
  the two parents.
- **The verifiers (W7).** Read after the rebuild, never run.
  - Cycle 10's is standalone. It asserts the certificate's constants, takes η from it and accepts a witness point
    within 10⁻¹² of its core, as §5.4 allows.
  - Cycle 11's loads Cycle 10's verifier from an absolute path on the machine that wrote it and swaps in a
    Numba-compiled hull. The Cycle 11 package does not ship that file; Cycles 13 and 17 carry copies.
  - It also fixes η at 10⁻⁹ and does not check the certificate's constants.
- **The paper (X1).** Mishra's §5.4 is as cited:
  - acceptance within ϑ = 10⁻¹², the distance's 4u and the corners' 8uR, so ε0 = 1.0016×10⁻¹²;
  - hull excess ε0P + πε0² with P ≤ 8.026;
  - shoelace γ_N·S with N ≤ m + 2K = 3072, and η = 1.72×10⁻⁹.
  - His own numbers follow from his constants: 8.04×10⁻¹², 1.7095×10⁻⁹ and 1.7176×10⁻⁹. His "S ≤ NR² = 5012" is
    5012.35 before rounding, and 1.71×10⁻⁹ still holds.
- **The package (P1).** Every VALIDATION digest matches the archive, and the replay counts are this tree's.
  - Cycle 10's copies of Cycle 07's frontier and Cycle 09's diagnostic are byte-identical to theirs.
  - Cycle 10's `verifier_stub_sha256` matches no file in the package.
  - The six search logs carry no digests.

Drill: 18 / 18 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 190 / 190
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 10.**
   - A binary-subdivision certificate over Cycle 09's 27 closest roots: root box, deterministic split, one bit per
     node in preorder.
   - At a leaf, either the box lies inside the F3 ball, or the hull of the finite witness set, less 10⁻⁹, exceeds
     A3+. The witness set lies in every placement the box allows, so its hull is at most inf A over the box.
   - The 10⁻⁹ covers Mishra's floating-point model, recomputed at m = K = 256 as 1.149×10⁻¹⁰.
   - Result: the 27 roots hold no placement in F3 with A ≤ A3+.
2. **Cycle 11.**
   - The same primitive on the other 622 near-ceiling roots: 616 close at 5,000 nodes, 8018 at 13,181, giving a
     617-root certificate.
   - Five roots (3766, 6217, 8001, 8015, 8016) go through four targeted layers of search. That leaves 225 subroots
     under 6217 and 3766, and exhausts 8001, 8015 and 8016 in search only: "packaging debt", not certificates.
   - 6217 is named the primary persistent family, with the smaller centre margin.

## W — rebuilt

| check | cycle | rebuilt | result |
| --- | --- | --- | --- |
| W1 | 10, 11 | the root sets and boxes | Cycle 09's 27; the 622 less five; boxes to the bit |
| W2 | 10, 11 | both trees decoded; the tiling | every count exact; gaps ≤ 4.7×10⁻¹⁸, covered |
| W3 | 10, 11 | the witness bound on every leaf; 50 digits on two | least slacks to 9×10⁻¹⁶; η_model exact |
| W4 | 10, 11 | exact certification of every leaf | 56,149 / 56,149, least 2.28×10⁻⁵; 41 / 41 NOT-F3 |
| W5 | 11 | the batch, the layers, the frontier | consistent throughout; the centre values to 2×10⁻¹⁶ |
| W6 | 10, 11 | the crystals | agree |
| W7 | 10, 11 | the verifiers, read | Cycle 11's needs a file outside its package |

## P — the package

**P1.** Every VALIDATION digest matches the bytes hashed here and the archive's manifest, and the replay counts
are this tree's. Cycle 10's dependency copies are byte-identical to Cycles 07 and 09's files. Cycle 10's recorded
verifier-stub digest matches nothing in the package.

## The cycles, read

| cycle | statement | verdict | note |
| --- | --- | --- | --- |
| 10 | the 27 roots are Cycle 09's closest | PASS | W1 |
| 10 | 2,699 nodes, 1,363 area leaves, depth 13, 2,153 on the hardest root, no NOT-F3 leaf | PASS | W2 |
| 10 | the witness hull is at most inf A over its box | PASS | up to the 10⁻¹² acceptance, which η covers; W4 certifies every leaf independently |
| 10 | η_model ≤ 1.149×10⁻¹⁰ from Mishra's model | PASS | X1; exact with his constants unrounded (W3) |
| 10 | least raw slack 3.8179493×10⁻⁸, 332× η_model, 3.7179493×10⁻⁸ after 10⁻⁹ | PASS | to 9×10⁻¹⁶ (W3) |
| 10 | the two hull routines agree to 1.2×10⁻¹⁵ | PASS | a double-precision hull against the exact one: 1.0×10⁻¹⁵ (W3) |
| 10 | the 27 roots hold no placement in F3 with A ≤ A3+ | PASS | certified here in exact arithmetic (W4), gaps by continuity (W2) |
| 11 | 622 inputs; 616 / 622 at 5,000 nodes; 8018 at 13,181 | PASS | W5 |
| 11 | the 617-root certificate: 109,037 / 54,786 / 41 / 16 | PASS | W2 |
| 11 | independent replay, least verified slack 1.5376482309648054×10⁻⁹ | PASS | six digits carry (W3) |
| 11 | the layers 51 → 91 → 242 → 31 → 25 and the 225-subroot frontier | PASS as search records | W5 |
| 11 | the centre diagnostic 0.8349695459154723 and 0.8352612051955018 | PASS | W5 |
| 11 | 8001, 8015, 8016 search-closed, not certified | as stated | nothing to replay |
| 11 | 6217 and 3766 persist under erosion subdivision | PASS as search, for the witness bound | exact cores not tried here |

## The status registry

Not touched: the cycles are not canonical documents.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| Cycle 10's node total in the text | 1 | A0 |
| ε0 dropped from Mishra's snapshot | 1 | X1 |
| two of Cycle 10's roots swapped | 1 | W1 |
| a flipped bit; a tree's stored leaf count; a split gap of 10⁻⁹ | 3 | W2 |
| Cycle 11's least witness slack lowered by 10⁻¹³ | 1 | W3, W6 |
| the certificate's η_model | 1 | W3 |
| an exact margin below zero; a NOT-F3 box outside the ball | 2 | W4 |
| a batch row's node count; a layer's frontier count | 2 | W5 |
| a subroot moved to the other parent | 1 | W5, W6 |
| the centre diagnostic | 1 | W5 |
| the crystal's packaging debt | 1 | W6 |
| the loaded verifier's path made relative | 1 | W7 |
| a VALIDATION digest; a dependency copy's bytes | 2 | P1 |

Three controls:

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

**18 / 18 plants, 3 / 3 controls, 190 / 190 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| X1 | 1 | 1 |
| W1 | 1 | 1 |
| W2 | 3 | 3 |
| W3 | 2 | 2 |
| W4 | 2 | 2 |
| W5 | 4 | 4 |
| W6 | 3 | 3 |
| W7 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **F3's closure.** The two certificates close 644 of Cycle 07's 22,516 roots. The rest, the five hard roots
  among them, are not addressed here.
- **The search-closed parts.** 8001, 8015 and 8016, and the parts of 6217 and 3766 outside the 225 subroots, are
  closed only in search logs. The trees are not shipped.
- **Cycle 07's frontier as a cover of F3.** It comes from the unshipped prototype (RUN-046). Which boxes are
  roots is taken from the file.
- **The continuity step as a formal lemma.** It closes the 10⁻¹⁸ gaps with Mishra's perimeter bound; it is standard
  and short, but written out here only in words.
- **Whether 6217 and 3766 persist under the exact cores.** They persist for the 256-point witness bound, which
  costs about 2.3×10⁻⁵ here. That question goes to Cycles 12–17 (RUN-048).