# RUN-048 — UESFCM Cycles 12–14: root 6217's three certificates replay and hold in exact arithmetic, and the symmetric triple contact reproduces to 80 digits; but the ρ₅ = 0 reduction drops the F3 constraint, so Cycle 13's 2-D certificate leaves the arc side of the slice uncovered — there the least area is again the contact point, so Gate 6217-A holds numerically and is not yet proved

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-09-27
**Subject:** UESFCM Cycles 12–14 (all dated 2026-09-20), each package read from the archive:

- Cycle 12 — root 6217: formal batch closure and symmetric triple-contact reduction. The package holds the document,
  the 124-root certificate and its verifier, the 76 remaining subroots, the rotation profile (27 samples), the
  contact data, the resolution summary, the state crystal and VALIDATION.
- Cycle 13 — formal closure of root 6217 at ρ₅ = 0. The package holds the document, the 2-D certificate and its
  verifier, the state crystal and VALIDATION, and copies of Cycle 10's verifier and of two Cycle 12 files.
- Cycle 14 — rotation-tail closure and boundary locking. The package holds the document, the eight-root certificate
  and its verifier, the closure summary, the profile structure, root 93's search frontier (30 subroots), the state
  crystal and VALIDATION, and a raw copy of Cycle 12's profile.

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

**Tools:** [`run048_uesfcm_cycles_12_14.py`](../code/run048_uesfcm_cycles_12_14.py)

- RUN-047's replay: the trees decoded, every leaf's witness bound rebuilt, and every area leaf certified in exact
  arithmetic with this tree's exact eroded cores;
- this tree's exact hull (RUN-009, with RUN-031's segments) and its envelope's pieces (RUN-046);
- the contact at 80 digits (mpmath);
- Nelder–Mead and golden-section searches on the exact hull.

**Logs:** [`run048-uesfcm-cycles-12-14.json`](../data/gate-logs/run048-uesfcm-cycles-12-14.json)

**Verdict: PASS for the three certificates and the contact; the rotation profile holds as a search. NOT
ESTABLISHED for Gate 6217-A: the ρ₅ = 0 reduction of Cycles 12–13 does not keep the F3 constraint. So Cycle 13's
certificate covers only part of the slice, and so does the "Gate 6217-A: CLOSED" that Cycle 14 and both crystals
carry.**

- All 66,927 area leaves of the three certificates hold in exact arithmetic with this tree's exact cores; the least
  margin is 2.60×10⁻⁵.
- The symmetric triple contact reproduces to 80 digits, and this tree's own minimum on the symmetric slice sits on it.
- On the part of the slice the reduction leaves out, the least area is again the contact point's, 2.05×10⁻⁴ above
  A3+. So the gate's statement holds numerically, but nothing in the packages proves it there.
- The profile's minima are search values, up to 8.3×10⁻⁸ above what this tree finds. Its two generic charts are not
  reproduced: every optimizer sits at near-ties, and the charts turn on hull runs shorter than 10⁻⁶ rad.

- **The roots (W1).** Root 6217's box is Cycle 07's row as Cycle 12 prints it, with ρ5 ∈ [0, π/120] exactly.
  - The 200 subroots are Cycle 11's. Cycle 12's certificate takes 124 of them, box for box, and the saved 76 are
    the rest, row for row.
  - Cycle 14's eight (4, 5, 48, 50, 84, 96, 112, 113) are among the 76, box for box, which leaves 68 open, root 93
    among them.
  - The 2-D rectangle is the root's x3 range cut at x3⁰ − s, with its x5 range, to 10⁻¹⁸.
  - Root 93's 30 subroots are each reached from root 6217 by the split, through subroot 93, at their stored depths.
- **The structure (W2).** The 200 centres fall into five oriented body skeletons (103, 63, 25, 5, 4) and three
  reflection classes, 166, 25 and 9. So Cycle 12's "three skeletons" are counted up to reflection. The median
  half-widths of the 76 over the 124 are 2, 4, 4, 4 and 6 times, the stated ratios exactly.
- **The certificates (W3).** Cycles 12 and 14 use Cycle 10's split; Cycle 13 has its own two-dimensional rule (the
  wider of x3 and x5, x3 on a tie), read from its verifier. As in RUN-047, the counts do not tell rules apart; the
  leaf bounds do.
  - Cycle 12: 124 trees, 32,568 nodes, 16,151 area leaves, 195 NOT-F3 leaves, depth 15.
  - Cycle 13: one tree on the rectangle, 299 nodes, 150 leaves, depth 10.
  - Cycle 14: eight trees, 102,062 nodes, 50,626 area leaves, 409 NOT-F3 leaves, depth 19 (not stated).
  - Every tree uses exactly its bits, and every stored count is this tree's.
  - The rebuilt witness bound passes on every leaf. The least slacks are 3.7845403×10⁻⁸ (Cycle 12, subroot 181),
    3.1819403×10⁻⁶ (Cycle 13) and 6.3395956×10⁻⁹ (Cycle 14, subroot 112). Against the documents that is 8.6×10⁻¹⁶,
    2.2×10⁻¹⁶ and 7.5×10⁻¹⁶ apart; at 50 digits all three agree with both to 10⁻¹⁵.
  - Every tree's stored minimum agrees with this tree's to 7×10⁻¹⁶ (Cycle 12), 2×10⁻¹⁶ (Cycle 13) and 4×10⁻¹⁶
    (Cycle 14). That is what confirms the two split rules.
  - The float split tiles exactly at 943 of Cycle 12's 16,222 splits, 4 of Cycle 13's 149 and 4,460 of Cycle 14's
    51,027; the widest gap is 3.5×10⁻¹⁸, covered by continuity as in RUN-047.
- **Exact certification (W4).** Every area leaf certifies at 1,024 directions, none needs more, and no candidate
  fails its membership proof.
  - Least margins: 2.597×10⁻⁵ (Cycle 12, subroot 181), 3.456×10⁻⁵ (Cycle 13) and 2.610×10⁻⁵ (Cycle 14,
    subroot 96).
  - All 195 and 409 NOT-F3 boxes lie inside the F3 ball; the nearest edges are 1.31×10⁻⁵ and 1.93×10⁻⁶ away.
- **The contact (W5).**
  - x3* = x3⁰ − s is the document's −0.02097883209899.
  - θ* = arccos(1/(2c3)) and x5* agree with the stored 80-digit strings to their last digits.
  - The exact hull at the contact is 0.8349861105592238 (Cycle 12: 0.834986110559224), 2.049196×10⁻⁴ above A3+.
  - **This tree's own minimum over x5 on the symmetric slice sits at x5* to 3×10⁻¹⁴**; Cycle 12's scalar optimizer
    stopped 1.5×10⁻¹⁰ short, as §9 says.
  - At ±θ* and π ± θ* the disc, B3 and B5 all support at 1/2, to one ulp.
  - The switch is as §11 describes. Below x5* the hull exposes two more corners of B5 (v2, v3); above it, B5's arcs
    a2 and a3 each hold one more interval.
  - The one-sided gradients a little either side of x5* are (−0.03296, −0.03573) and (−0.21262, +0.40369), against
    the stored (−0.03307, −0.03542) and (−0.21251, +0.40323). The mix with λ = 0.919254 is the stored (−0.04756, 0).
  - On the symmetric slice alone, the KKT argument of §16 is sound: F is convex there (Fáry–Rédei horizontally), and
    (−c, 0) with c > 0 meets the normal cone of x3 ≤ x3*.
- **The ρ₅ = 0 reduction (W6).**
  - Cycle 12 §13 and Cycle 13 §2 use reflection in the x-axis and Fáry–Rédei convexity along the vertical motion
    t ↦ t(y3, y5). That gives A(x3, 0, 0, x5, 0) ≤ A(x3, y3, 0, x5, y5), for every placement.
  - But the motion ends at y3 = 0, and F3 asks d3 = hypot(x3 − x3⁰, y3) ≥ s. Where |x3 − x3⁰| < s the end point is
    no longer in F3. There the inequality bounds a far placement by a near one, and a near one can lie below A3+ (the
    seed does).
  - Cycle 13 §3 keeps only x3 ≤ x3⁰ − s: "the root does not intersect the right F3 branch". But root 6217's box runs
    to x3 = −0.012122, 0.00886 to the right of x3⁰ − s, and to y3 = 0.012122.
  - So its ρ₅ = 0 slice also holds F3 placements with x3⁰ − s < x3 ≤ −0.012122 and y3 ≥ √(s² − (x3 − x3⁰)²): the
    arc side. No step of Cycle 13 bounds them, and its verifier checks only the rectangle's leaves.
  - **The reduction can be completed.** For fixed (x3, x5) on the arc side, A is even and jointly convex in
    (y3, y5). On {|y3| ≥ r} its minimum therefore lies on |y3| = r, the arc d3 = s. The moved point stays in the
    box, since the box's y5 range holds 0. What is left is (α, x5, y5), with x3 = x3⁰ + s cos α and y3 = s sin α.
  - **Numerically the arc side is not lower.** Minimized over B5's translation in the box, at 25 angles from π to
    where the arc leaves the box (α = 1.5666):
    - at α = π, Cycle 12's contact point, the area is 0.834986110559;
    - it rises to 0.835484 at α = 1.632, then eases to 0.835483 at the box's edge.
    - So on the arc side too the least area is the contact point's, 2.05×10⁻⁴ above A3+. Gate 6217-A's statement,
      m₆₂₁₇(0) > A3+, holds numerically with the same margin. What is missing is a certificate for that side.
  - **Who covers the arc side.** Of 20,000 random F3 placements in the ρ₅ = 0 slice of the box, 10,856 lie on the
    arc side:
    - 145 (1.3 %) fall in subroots Cycle 12 certified, and none in Cycle 14's;
    - 6,812 (62.7 %) fall in the 68 subroots still open;
    - 3,899 (35.9 %) fall in none of the 200: that part of the box was closed only in Cycle 11's search logs.
  - The rectangle side's 9,144 are covered by Cycle 13's certificate through the symmetrization, which is valid
    there.
  - Cycle 14 §2 repeats that Cycle 13 "closed the entire ρ = 0 slice", and its crystal, like Cycle 13's, records
    Gate 6217-A as CLOSED.
- **The rotation profile (W7).**
  - The 27 stored minimizers re-evaluate to their stored areas, to 10⁻¹⁵, and lie on d3 = s to 2.4×10⁻⁸.
  - The stored profile is non-decreasing, and its own numbers give the stated quadratic ratios, 0.173793–0.245696.
    Cycle 14's raw copy holds exactly these samples.
  - **They are search values.** Searched again on the circle inside the root's box, all 27 come out lower, by at
    most 8.3×10⁻⁸. The profile stays non-decreasing, from the contact's 0.834986110559 to 0.8351052259 at π/120,
    and the quadratic ratios become 0.1738–0.2254. From ρ ≈ 0.014 on, the minimum over the box sits on its face
    y3 = 0.
  - **The unconstrained diagnostic.** Without the F3 constraint, the minimizers over the four translations lie
    outside the root's box at 20 of the 27 samples (y3 < 0), up to 2.2×10⁻³ from the anchor, a quarter of s.
  - Kept in the box they reach 9.1×10⁻⁴, 10 % of s, against the stated 6.96×10⁻⁴ and 7.9 %. Either way they stay
    well inside the circle. So the premise of Cycle 14's conditional theorem survives, with less room than stated.
  - **The skeletons depend on resolution.** Every optimizer sits at near-ties of D, B3 and B5: at each sample some
    hull run is shorter than 5×10⁻⁸ rad, and the shortest is 2×10⁻¹³.
    - At full resolution samples 0 and 14 differ from the rest, as Cycle 14's two degenerate states do. But the
      other 25 share one sequence.
    - Dropping runs under 10⁻⁹ or 10⁻⁶ rad gives three or four sequences, grouped otherwise than the cycle's charts
      A (samples 1–8) and B (9–13, 15–26).
    - So the two generic charts are not reproduced.
  - **Convexity along lines.** Second differences of the hull area are positive along all 600 random lines tried:
    200 vertical (least 2.8), 200 horizontal (3.0), and 200 moving B3 in x and B5 in y (3.1). The last kind is not
    a common direction, so the theorem does not cover it. Cycle 14 rightly does not claim it, and these samples show
    no failure either.
- **The crystals (W8)** agree with the numbers rebuilt here, but for two: the gate's status, and Cycle 14's four
  profile skeletons (two here at full resolution).
- **The verifiers (W9).** Read after the rebuild, never run.
  - Cycle 12's is Cycle 11's, bar its default certificate name. It loads Cycle 10's verifier from an absolute path,
    and its package does not ship that file.
  - Cycle 14's is Cycle 11's byte for byte, so its VALIDATION digest is Cycle 11's, and its default certificate is
    still Cycle 11's.
  - Cycle 13's loads Cycle 10's verifier from its own package (`dependencies/`, shipped) and splits with its 2-D
    rule. It checks the rectangle's leaves and nothing of the reduction that makes the rectangle suffice.
- **The package (P1).** Every VALIDATION digest matches the archive, and the replays' counts are this tree's.
  - Cycle 12's `summary_sha256` is its resolution summary.
  - Cycle 13's three dependency copies are byte-identical to their originals.
  - Cycle 14's `Rotation_Profile_raw.npz` is filed under Cycle 12's name, but it is not in Cycle 12's package. Its
    arrays are exactly the JSON's 27 samples.

Drill: 22 / 22 plants caught by exactly the named checks, 3 / 3 controls undisturbed, 249 / 249
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 12.**
   - Root 6217's 200 subroots: three centre skeletons, 124 closed by a 2,000-node erosion certificate each, 76 left.
     Their persistence is resolution debt.
   - A multistart search over the whole root with d3 ≥ s ends at one symmetric structure: y3 = ρ5 = y5 = 0 and
     d3 = s. That is a four-direction triple contact with θ* and x5* in closed form, at a contact-switch cusp.
   - With Fáry–Rédei / Rogers–Shephard convexity, ρ5 = 0 reduces to a convex problem in (x3, x5) on x3 ≤ x3*. The
     contact's KKT condition would make it that problem's minimum, 2.05×10⁻⁴ above A3+.
   - Gates: A (ρ5 = 0) and B (0 < ρ5 ≤ π/120, a one-dimensional profile, searched at 27 points).
2. **Cycle 13.** Vertical symmetrization reduces Gate A "exactly" to the rectangle x3 ∈ [−0.024244, −0.020979],
   x5 ∈ [0.012320, 0.024640], closed by a 299-node certificate: "Gate 6217-A: CLOSED".
3. **Cycle 14.**
   - Eight of the 76 boxes, all in the rotation tail, close by a 102,062-node certificate: 76 → 68. Root 93 is
     localized in search to 30 subroots.
   - On the 27 samples the constrained minimizers sit on d3 = s, while the unconstrained ones lie within 7.9 % of s
     of the anchor. A conditional theorem turns that into boundary locking.
   - The profile shows four skeletons: two generic charts and two degenerate states.

## W — rebuilt

| check | cycle | rebuilt | result |
| --- | --- | --- | --- |
| W1 | 12–14 | root 6217, the 200 / 124 / 76 / 8 / 68, the rectangle, root 93's 30 | exact |
| W2 | 12 | the 200 centres' skeletons; the resolution ratios | 3 reflection classes 166 / 25 / 9 |
| W3 | 12–14 | the three trees decoded; the witness bound on every leaf; the tiling | every count; least slacks to 9×10⁻¹⁶ |
| W4 | 12–14 | exact certification of every leaf | 66,927 / 66,927, least 2.60×10⁻⁵; 604 / 604 NOT-F3 |
| W5 | 12 | the contact at 80 digits; the slice minimum; supports; switch; gradients | agree |
| W6 | 12–13 | the reduction against F3; the arc profile; coverage | the arc side is left out; numerically not lower |
| W7 | 12, 14 | the 27 samples; locking; the searches; skeletons; convexity | search values, 8.3×10⁻⁸ high; charts not reproduced |
| W8 | 12–14 | the crystals | agree, but for the gate's status and the skeleton count |
| W9 | 12–14 | the verifiers, read | two need a file outside their package |

## P — the package

**P1.** Every VALIDATION digest matches the bytes hashed here and the archive's manifest, and the replays' counts
are this tree's. Cycle 13's dependency copies are byte-identical to Cycles 10 and 12's files. Cycle 14's verifier
digest is Cycle 11's, because the file is.

## The cycles, read

| cycle | statement | verdict | note |
| --- | --- | --- | --- |
| 12 | root 6217's box; ρ5 ∈ [0, π/120] | PASS | W1 |
| 12 | three centre skeletons, 166 / 25 / 9 (83 %) | PASS | as reflection classes; five oriented (W2) |
| 12 | 124 / 200 close; 32,568 / 16,151 / 195 / 15; least 3.684540239579093×10⁻⁸ | PASS | W3, W4 |
| 12 | persistence is resolution-driven: median ratios 2, 4, 4, 4, 6 | PASS as a statistic | W2 |
| 12 | the constrained minimum is the symmetric contact (y3 = ρ5 = y5 = 0, d3 = s) | PASS as search | at ρ5 = 0 on both sides of the slice (W5, W6) |
| 12 | θ*, x5*, the four-direction triple contact, the switch | PASS | W5 |
| 12 | the optimizer 1.52×10⁻¹⁰ from the formula | PASS | this tree's slice minimum is 3×10⁻¹⁴ from it (W5) |
| 12 | vertical symmetrization at ρ5 = 0 | PASS as an inequality | it does not keep F3 (W6) |
| 12 | horizontal translation convexity | PASS | a known theorem; positive second differences on 200 lines (W7) |
| 12 | the symmetric F3 domain is x3 ≤ x3*, so ρ5 = 0 is a two-variable convex problem | FAIL | the arc side is left out (W6) |
| 12 | the contact KKT on the symmetric slice | PASS as evidence | sound on that slice (W5) |
| 12 | A* ≈ 0.834986110559224, 2.049×10⁻⁴ above A3+ | PASS | W5 |
| 12 | the 27-point profile, non-decreasing, ratios 0.1738–0.2457 | PASS as search | minima 8.3×10⁻⁸ lower here; ratios 0.1738–0.2254 (W7) |
| 13 | Gate A "reduced exactly" to the rectangle | FAIL | W6 |
| 13 | the 2-D certificate: 299 / 150 / 10, least 3.18194×10⁻⁶ | PASS | W3, W4 |
| 13 | m₆₂₁₇(0) > A3+: `GATE 6217-A: CLOSED` | NOT ESTABLISHED | holds numerically, 2.05×10⁻⁴ (W6) |
| 14 | "Cycle 13 closed the entire ρ = 0 slice" | FAIL | W6 |
| 14 | the eight-box certificate: 102,062 / 50,626 / 409, least 5.3395949×10⁻⁹ | PASS | W3, W4 |
| 14 | 76 → 68; root 93 left as 30 subroots, in search only | PASS | W1 |
| 14 | the 27 samples lie on d3 = s, within 2.365×10⁻⁸ | PASS | W7 |
| 14 | the unconstrained minimizers lie within 7.9 % of s of the anchor | PARTIAL | 10 % in the box, 25 % in the plane; still inside the circle (W7) |
| 14 | four profile skeletons, two generic charts, a transition near ρ ≈ 0.0141 | PARTIAL | samples 0 and 14 stand out; the charts rest on sub-10⁻⁶ rad runs (W7) |
| 14 | fixed-direction translation convexity; full 4-D joint convexity not claimed | PASS | W7 |
| 14 | the conditional boundary-reduction theorem | PASS | its hypotheses stated as such |

## The status registry

Not touched: the cycles are not canonical documents.

## The drill

| kind | cases | named checks |
| --- | ---: | --- |
| Cycle 12's certificate nodes in the text | 1 | A0 |
| two certificate trees' subroot indices swapped; root 93's stored depth | 2 | W1 |
| a remaining subroot's index | 1 | W1, W2 |
| a centre's skeleton | 1 | W2, W8 |
| a resolution ratio | 1 | W2 |
| a flipped bit in Cycle 13's tree; a stored leaf count | 2 | W3 |
| Cycle 14's least witness slack | 1 | W3, W8 |
| an exact margin below zero | 1 | W4 |
| a digit of θ* | 1 | W5, W8 |
| a stored branch gradient; a support value | 2 | W5 |
| a dip on the arc; a coverage count | 2 | W6 |
| a profile sample's area; the locking bound; a second difference | 3 | W7 |
| a crystal's remaining count | 1 | W8 |
| Cycle 13's split made strict | 1 | W9 |
| 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.

**22 / 22 plants, 3 / 3 controls, 249 / 249 must-stay-green assertions; no check without a plant.**

| check blinded | cases reported blind | plants naming it |
| --- | ---: | ---: |
| A0 | 1 | 1 |
| W1 | 3 | 3 |
| W2 | 3 | 3 |
| W3 | 3 | 3 |
| W4 | 1 | 1 |
| W5 | 3 | 3 |
| W6 | 2 | 2 |
| W7 | 3 | 3 |
| W8 | 4 | 4 |
| W9 | 1 | 1 |
| P1 | 2 | 2 |

## What this run does not establish

- **Gate 6217-A on the arc side.** The completed reduction leaves a three-parameter problem in (α, x5, y5), and it
  is only searched here, at 25 angles. A certificate over it, like Cycle 13's over the rectangle, is not built.
- **Gate 6217-B** (0 < ρ5 ≤ π/120), open in the cycles too. The 27-point profile is a search, here and there.
- **The 68 open subroots, and the part of the box outside the 200.** The latter was closed only in Cycle 11's
  search, whose trees are not shipped.
- **Fáry–Rédei and Rogers–Shephard themselves.** They are cited as known theorems; here they are only checked
  numerically along lines (W7).
- **Root 3766**, untouched by these cycles.