# RUN-012 — Round 11 (v0.11): the exact boundaries and the 120°/270° enclosures hold, and the 579 derivative boxes and the root boxes hold where sampled; the switch neighbourhoods' derivative intervals leave out the 2×10⁻¹⁰ rad beside each switch, and three exclusions rest on them

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-20
**Subject:** `Moser_Skew_Lab_v0.11.zip` (SHA-256 pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`reports/ROUND_11_REPORT.md`, `ROUND_11_CERTIFICATE_AUDIT.md`, `ROUND_12_BALL_ARITHMETIC_HANDOFF.md`, `src/METHOD.md`,
`src/special_branch_verify.py` (read, not run), and its data (`round11_summary.json`, `exact_contact_boundaries.json`,
`special_branch_interval_certificate.json`, `phase_derivative_box_certificate.json`, `stationary_root_boxes.json`,
`boundary_neighborhood_audit.json` and the two CSVs), with Round 10's map, Round 09's decimal constants and event
branches and Round 05's event root, read from the package bytes
**Tools:** [`run012_round11_box_certificate.py`](../code/run012_round11_box_certificate.py) — the exact support in double
precision (RUN-009's wing, positions exact to rounding) for sampling derivatives and scales; RUN-010's and RUN-011's
high-precision candidate (40 to 200 digits) for the boundary conditions, the switches' neighbourhoods, the special values
and the stationary points;
[`gate.py`](../code/gate.py) for the drill
**Log:** [`run012-round11-box-certificate.json`](../data/gate-logs/run012-round11-box-certificate.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes mostly holds, and two things it states do not.
Holding: the 19 boundary formulas are exact — 0, 2π and the 17 switches, each of which, evaluated at 60 digits,
satisfies its switch's condition to 6.0×10⁻⁶⁰; the s₂₇₀, s₁₂₀ and s₁₂₀ − s₂₇₀ intervals contain the exact values, s₂₇₀
8.88×10⁻¹² from either end, and their error bounds are the composite-midpoint bound with the verifier's constants; the
gain interval is the s₂₇₀ bounds less Round 09's event branch; each of the 579 derivative boxes contains the derivative
at 13 points of it, with the stated sign, and the boxes, root boxes and guards tile every interval; each of the 12 root
boxes holds one stationary point of its kind, with second-derivative and scale intervals that contain s″ and s; the
neighbourhoods' scale intervals contain s, and their classifications are the derivative's signs. Not as stated: each
neighbourhood's one-sided derivative intervals cover [φ ± 2×10⁻¹⁰, φ ± r], not [φ, φ ± r] — the 2×10⁻¹⁰ rad beside
every switch is enclosed by nothing in the certificate — and at 14 of the 17 switches the derivative there falls outside
them: by 0.022 to 0.157 at the 12 wing switches, where it is steep, and by 1.9×10⁻¹⁰ to 2.1×10⁻¹⁰ at π/2, π and one side
of 5π/3 − α; at switches 1 to 3, where the scale bound does not exclude, the exclusion rests on these intervals' signs,
which are the derivative's but which the intervals do not show. And §6 counts 12 smooth stationary points in all: there
are 13, the thirteenth 5.44×10⁻⁹ rad before π/2 + α, inside the neighbourhood the round leaves unresolved and excludes
by its scale — so the scheme survives there, the count does not. The round's qualitative conclusions — the least at
3π/2, s₁₂₀ above s₂₇₀, the candidate above the event five-link — hold, as RUN-011 certified with bounds of its own; its
sharper figures for them hold at high precision. The drill planted 17 defects; each tripped exactly the checks named for
it, and the 4 controls disturbed nothing.

---

## What Round 11 claims

Round 11 writes Round 10's switches as exact angles — π/2 − α, π/2 − β, β − π/3, α − π/3, π/2, π − α, π − β, 2π/3,
π/2 + β, π/2 + α, π, 5π/3 − α, 5π/3 − β, π + β, π + α, 3π/2, 5π/3 — so that "相位分割不再依賴二分所得的小數，而是由有限個精確角度事件決定" (the phase partition no longer rests on bisected decimals but on finitely many exact angle events). It encloses
s₂₇₀ in [0.9989143390757518, 0.9989143390935133] and s₁₂₀ − s₂₇₀ in [1.634956007484312×10⁻⁹, 1.641712345829326×10⁻⁹] by
a composite midpoint rule on 262,144 parts with a global second-derivative bound and a float padding — "這是本系列第一次對 $120^\circ$／$270^\circ$ 的 $10^{-9}$ 級競爭給出顯式誤差盒" (the first explicit error box in the series for the 10⁻⁹-level
competition between 120° and 270°) — and subtracts the event five-link's first branch to enclose the gain. It covers each
interval's interior with adaptive derivative boxes, 579 in all, none unresolved, so that "除已隔離的駐點根盒與接觸邊界盒外，$18$ 個區間內部的導數符號全部被盒化排除" (apart from the isolated root boxes and boundary boxes, the derivative signs in the 18
intervals' interiors are all excluded by boxes); it puts 12 smooth stationary points in root boxes whose second-derivative boxes
exclude zero; and it treats each switch's neighbourhood — of radius 2×10⁻⁶, or 2×10⁻⁸ where both supports are corners — by
one-sided derivative intervals or a lower
bound of the scale — of 17, 16 by the derivative signs, and the last, π/2 + α, by the scale. It calls the result "一份具解析誤差界、浮點外擴與逐區盒化排除的半驗證計算證書" (a semi-validated computational certificate with analytic error bounds,
float padding and interval-by-interval box exclusion), since "NumPy 的基本運算不是全程 directed-rounding" (NumPy's basic
operations are not directed-rounding throughout). Its handoff asks the next round to recompute everything in ball
arithmetic. Its README: "本封裝不構成 Moser 問題的新上下界".

## The checks

| id | guards | result |
| --- | --- | --- |
| U1 | each of the 19 boundary formulas, at 60 digits, is its stated phase (to 10⁻¹⁵) and degrees, and the 17 between the ends satisfy their switches' conditions (to 10⁻⁴⁰); the largest difference from Round 10's switches | **PASS** |
| U2 | the s₂₇₀, s₁₂₀ and s₁₂₀ − s₂₇₀ intervals contain the exact values; the error bounds are M₂/(24N²) plus the stated padding; the centres are the midpoint sums | **PASS** |
| U3 | the gain interval is the s₂₇₀ bounds less Round 09's event branch B1; its lower end is positive | **PASS** |
| U4 | 579 boxes; each contains the derivative at 13 points of it (to 10⁻¹²) and has its interval's sign; per-interval counts and signs; boxes, root boxes and guards tile each interval | **PASS** |
| U5 | each root box holds one exact stationary point of its kind; its second-derivative interval contains s″ at four points inside it (60 digits, to 10⁻¹⁰) and excludes 0; its scale interval contains s; the margins; the boxes are all the stationary points | **FAIL** |
| U6 | each neighbourhood's scale intervals contain s; its one-sided derivative intervals contain the derivative over that side, the switch included (60 digits); its classification is the derivative's signs; the safe lower bounds, margins and flags (a neighbourhood with a special certificate excluded by it) | **FAIL** |
| U7 | the summary, the CSVs and the global candidate hold the data files' figures | **PASS** |
| U8 | the report's tables (§2, §5–§7) and numbers (§3, §4), the README, the audit note, the handoff and the manifest | **PASS** |
| U9 | the verifier's constants are Round 09's decimals; its bounds are |θ′| and |θ′|² + |θ″|; the method note states them | **PASS** |

## Findings

### 1. What holds

Every boundary formula, evaluated at 60 digits, satisfies its switch's own condition — two corners level under the
changing functional, or the wing's perpendicular point at the wing's end — to 6.0×10⁻⁶⁰ (U1,
`boundary_condition_residue`); the largest difference from Round 10's bisected switches is the stated 5.240×10⁻¹⁴. The
special enclosures contain the exact values: s₂₇₀ lies 8.88×10⁻¹² inside each end of its interval, and s₁₂₀ − s₂₇₀ =
1.63833406919315×10⁻⁹ inside [1.634956×10⁻⁹, 1.641712×10⁻⁹] (`s270_minus_interval_ends`, `s120_minus_s270_exact`); the
bounds are M₂/(24N²) + 2×10⁻¹⁵ for the coordinates and M_q·M₂/(24N²) + 2×10⁻¹⁵ for the difference, with M₂ =
15.128024568137338 = |θ′|² + |θ″| (U2, U9, `recomputed_bounds`). Round 11's s₂₇₀ interval lies inside RUN-011's certified
one, and its lower bound for s₁₂₀ − s₂₇₀, 1.634956×10⁻⁹, exceeds RUN-011's, 1.630111×10⁻⁹. The gain interval is its s₂₇₀
bounds, as shortest decimals, less Round 09's B1, cut to 84 decimals (U3). Each of the 579 derivative boxes contains
the exact derivative at nine equally spaced points and four near its ends, by at least 8.9×10⁻¹² (`boxes_least_slack`),
and its sign is its interval's; with the root boxes and the guards of 2×10⁻¹⁰ or 2×10⁻⁸ they tile every interval (U4).
Each root box holds one stationary point of its kind; its second-derivative interval contains s″ at four points inside
it, by at least 1.4×10⁻⁸ (`root_second_derivative_margin`), and excludes zero (U5). The neighbourhoods' scale intervals
contain s on either side, the switch included, and every classification is the sign pattern of the derivative over the
neighbourhood (U6). These are sampled checks — the derivative at points of each box — not a re-verification of the
boxes in interval arithmetic.

### 2. The 2×10⁻¹⁰ rad beside each switch

Each switch has a neighbourhood of radius r — 2×10⁻⁶, or 2×10⁻⁸ at the five switches between two corners — with an
interval for the derivative on each side. On every side, the interval's end nearest the switch is the derivative
2×10⁻¹⁰ rad from the switch, less the round's padding (`near_end_minus_derivative_at_2e-10`): the interval covers
[φ ± 2×10⁻¹⁰, φ ± r], not [φ, φ ± r]. The derivative boxes' guards are 2×10⁻¹⁰ on the corner intervals and 2×10⁻⁸ on
the wing intervals, so those 2×10⁻¹⁰ rad beside every switch are enclosed by nothing. Whether the omission shows depends
on how much the derivative moves there (`move_within_2e-10`, at 60 digits; `neighbourhoods`):

- **At the 12 wing switches**, on the side next to the wing's interior, the derivative is steep — the tanh layer leaves
  the wing nearly straight near its ends — and moves by 0.022 to 0.157 within the 2×10⁻¹⁰ rad, far beyond the padding of
  2.2×10⁻⁶ to 3.0×10⁻⁶. At π/2 − α the right-hand interval is [−0.19350, −0.08917]: its lower end is the derivative
  2×10⁻¹⁰ rad past the switch, −0.193496, less 2.2×10⁻⁶ of padding, while at the switch the derivative is −0.329242 —
  its value on the other side too, s being continuously differentiable there. The shortfalls are 0.136 at π/2 − α and
  π/2 + α, 0.157 at α − π/3 and 5π/3 − α, 0.078 at π − α and π + α, and 0.038, 0.044 and 0.022 at the β-end switches.
- **Where both supports are corners** the derivative moves by |s″| · 2×10⁻¹⁰, about 2×10⁻¹⁰. A padding of 8×10⁻¹⁰ or more
  hides it at 2π/3, 3π/2 and 5π/3 and on the corner sides of the wing switches; at π/2, π and the left side of 5π/3 − α
  the padding is 0.9×10⁻¹¹ to 2.0×10⁻¹¹, and the intervals miss by 1.9×10⁻¹⁰ to 2.1×10⁻¹⁰.

The signs the round reads from these intervals are the derivative's on every side, the switch included — the
classifications pass — and at the 11 failing switches where the scale bound also excludes, nothing rests on them. At
π/2 − α, π/2 − β and β − π/3 it does not: the neighbourhoods' scale bounds lie 2.69×10⁻², 2.30×10⁻² and 7.7×10⁻⁴ below
s₂₇₀, and the round excludes them — its audit note's rule is that "全部邊界均通過導數分類、尺度鄰域下界或特殊分支證書" (every
boundary passes by the derivative classification, the neighbourhood's scale lower bound, or a special-branch
certificate) — as decreasing or increasing through the switch, on the strength of intervals that leave out the very
stretch where the derivative is steep, on the side that carries the wing (switch 1 on the right, 2 on the left, 3 on
the right). The conclusion is true there — the derivative keeps its sign — but the certificate does not show it. The
round's own handoff asks for these objects to be recomputed in ball arithmetic; this is where it matters. (At 3π/2 the
scale margin is −1.8×10⁻¹¹, its lower bound being s₂₇₀'s own; that neighbourhood is excluded by the special certificate,
so of the 14 the summary counts as passing by scale, 13 pass by their margins.)

### 3. Thirteen stationary points, not twelve

The 12 root boxes are 12 of the 13 stationary points RUN-011 found; the thirteenth, the smooth minimum 5.44×10⁻⁹ rad
before π/2 + α (`stationary_outside_root_boxes`), lies inside that switch's neighbourhood. Round 11 leaves that
neighbourhood "unresolved" — its left derivative interval, [−0.06691, 0.03742], straddles zero because the root is
there, while the derivative on that side runs up to +0.17317 at the switch — and excludes it by the scale, 7.39×10⁻²
above s₂₇₀. So the exclusion holds, and the report's account of the unresolved switch — a smooth support point leaving
the wing at its end, with a wider derivative box — is right as far as it goes; but §6's count of 12 smooth stationary
points and §11's 12 "個唯一駐點根盒" (unique stationary-point root boxes), as the full count, are not, and the summary's 4
smooth minima are 5 (`smooth_minimum_count`).

## At what strength each result stands

| result | strength |
| --- | --- |
| the boundary formulas | high precision: each switch's condition at 60 digits, to 6.0×10⁻⁶⁰ |
| the special enclosures contain the exact values | high precision (the exact values at 100 to 200 digits); their own rounding model unverified, as the round says |
| the 579 boxes and 12 root boxes contain the derivative, s″ and s | numerical: sampled at points of each box (the derivative in double precision with the exact support, s″ at 60 digits); not an interval re-verification |
| the 14 neighbourhoods whose derivative intervals do not enclose, and the 2×10⁻¹⁰ rad they leave out | high precision: the derivative at the switch and at 2×10⁻¹⁰ rad from it, at 60 digits, lies outside |
| 13 stationary points | RUN-011: exact in number on the corner-only intervals, a refined scan on the wing intervals |
| the round's qualitative conclusions (the least at 3π/2; s₁₂₀ > s₂₇₀; the candidate above the event five-link) | certified by RUN-011, with its own bounds (s₁₂₀ − s₂₇₀ ≥ 1.630111×10⁻⁹, the gain ≥ 1.0581934×10⁻⁵); Round 11's sharper figures hold at high precision |

## The drill

| | |
| --- | ---: |
| planted defects | 17 |
| tripping exactly the checks named for them | **17** |
| controls (the package unmodified, the report with CRLF endings, the README with trailing spaces, the boxes copied) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **168/168** |

Plants include a triangle leg, a boundary's formula and another's phase, the special certificate's error bound, its s₁₂₀
interval and its event branch, a box's interval and an interval's box count, a root box's second-derivative interval, a
neighbourhood's scale interval and another's classification, a count in the summary, a sign in §5, a README count, the
handoff's root criterion, a verifier constant and the method note's bound. The expected sets were reasoned before the dry
run with every set empty was read, and it tripped exactly them; so 17 of 17 says the reasoning matched the harness, not
that the plants were chosen blind. Where a plant moves a figure the checks compare elsewhere — a formula or phase in §2,
the special certificate in the summary, a box count in the CSV and §5, a root box in the CSV, §6 and the summary's flag,
a classification in §7 — the restatements are planted alike, so that only the comparison can see the plant; the
formula's plant still trips U1 twice, through the phase and through the switch condition. U5 and U6 fail on the package
with fixed-text messages, so their plants trip them through a figure that holds. Branches with no plant of their own
include U1's degrees, count and largest-difference tests; U2's containment of s₂₇₀ and the difference and its centring,
U5's one-stationary-point test and U6's derivative-enclosure test, which only the triangle leg reaches; U3's arithmetic
and sign; U4's 579-and-resolved, sign and tiling tests; U5's scale, margin and count tests; U6's safe-bound and flag
tests; most of U7's and U8's identities; and U9's bound and fragment tests. Of the four controls, the unmodified package
exercises nothing by construction; the report with CRLF endings re-runs U8, the README U8, the copied boxes U4, U7 and
U8. Each check, replaced in turn by one that always passes, was reported blind in exactly as many cases as plants name
it — U1 3, U2 3, U3 1, U4 3, U5 2, U6 3, U7 1, U8 3, U9 2 — which by construction shows only that the harness counts as
described (nine runs; a blind run writes no log). Two runs of the drill made side by side wrote gate logs with the same
SHA-256.

The instrument was wrong before it was right, in ways worth recording. The first boundary check compared the formulas
with RUN-011's switches, which bisection on the labels locates only to about 5×10⁻²⁰ where a wing ends tangentially at
β; the formulas are now checked against each switch's own condition. The first gain check used the double's exact
binary value where the round subtracts its shortest decimal, and the first difference check expected 17 printed digits
where the report prints 16. An independent reading of this report before it was committed confirmed its findings with
an evaluator of its own and found that the neighbourhoods' derivatives had been sampled in double precision, which near
the α-end switches loses up to 3.3×10⁻³ in arctanh and cannot place a point within an ulp of a switch; they are now
taken at 60 digits, at the switch and 2×10⁻¹⁰ rad from it, which showed the single omission behind the 14 failures. It
also found a test of the second derivative too loose to see a displaced interval, a flag failure at 3π/2 left out, the
radius misstated at the corner switches, RUN-011's certificate credited with the round's sharper figures, and several
figures without a source in the log; all are corrected here.

## What this run does not establish

That the 579 boxes and 12 root boxes are valid enclosures: they contain the derivative wherever this run looked, which is
evidence, not a proof. A repair of the three neighbourhood exclusions inside Round 11's scheme: RUN-011's certificate
covers them by another route. Any bound for Moser's problem; the README says the package "不構成 Moser 問題的新上下界".
Nor the package's four figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run012_round11_box_certificate.py --drill
```

The run is correct when the gate log is byte-identical to the committed one.