# RUN-011 — Round 10 (v0.10): the phase-contact map reproduces to 10⁻¹³, but one of its four cusp minima is a smooth minimum it misses, and at the wing switches one of each pair of stated derivatives is not the derivative; the least at 3π/2, which the round left open, is certified here

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.10.zip` (SHA-256 pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`reports/ROUND_10_REPORT.md`, `ROUND_10_GLOBAL_CERTIFICATE_INTERFACE.md`, `src/phase_contact_map.py` (a docstring), and
its data (`round10_summary.json`, `round10_global_exclusion_audit.json`, `phase_contact_intervals.csv`), with Round 08's
best candidate, Round 09's decimal constants and Round 05's event root, read from the package bytes
**Tools:** [`run011_round10_phase_map.py`](../code/run011_round10_phase_map.py) — RUN-010's high-precision candidate and
exact support (the four corners, and on each wing the one point whose tangent is perpendicular to the functional) at 40
and 60 digits; the envelope derivative on a fixed signature; a double-precision exact-support scan of its own; mpmath's
interval arithmetic for the certificate; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run011-round10-phase-map.json`](../data/gate-logs/run011-round10-phase-map.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and two things it states do not hold.
Reproduced: the 18 intervals and their signatures (the exact supports at every midpoint); the 17 switches' phases, to
5.22×10⁻¹⁴, and scales, to 1.04×10⁻¹³; the 12 stated stationary points, to 7.88×10⁻¹³ in phase and 9.58×10⁻¹⁴ in scale,
with their kinds and signatures; the one-sided derivatives at the five switches between two corners and at the two
special phases, to 1.11×10⁻⁸; the counts at every resolution, by a scan of this run's own; and the documents' tables.
Not as stated: at the twelve switches where a wing's interior support leaves or reaches a corner, s is continuously
differentiable, but the round states two one-sided derivatives there, apart by up to 0.21 — one of them the derivative at
a sample inside, not at the switch; and the cusp minimum it places at 3.028 rad is not one — the derivative is +0.173 on
both sides — while the minimum there is a smooth stationary point 5.44×10⁻⁹ rad before the switch, 6.16×10⁻¹¹ lower,
which the stationary list (13 points, not 12), interval 9's classification and root guards at all three densities, the
ranking and the ledger all miss: the hidden root its own limitation names. The global conclusion is untouched. Finding:
what the round left open, "s(φ) ≥ s(3π/2)" for every φ, is certified here in interval arithmetic: the candidate's least
scale is s(3π/2), reached there alone, and lies in [0.998914339066915, 0.998914339102351] — more than the event
five-link's by at least 1.0581934×10⁻⁵. The drill planted 18 defects; each tripped exactly the checks named for it, and
the 4 controls disturbed nothing.

---

## What Round 10 claims

Round 10's aim is to "將整個相位圓拆成固定活動支撐身份區間，枚舉全部局部極小並檢查 $270^\circ$ 是否仍為全域最低" (split
the whole phase circle into intervals of fixed active support identity, enumerate every local minimum, and check whether
270° is still the global minimum). For each phase it records the signature (I_min x, I_min y, I_max d) — p0 to p3 the
ends and junctions, L and R a wing's interior support — and on a fixed signature differentiates s by the envelope
theorem. It finds 18 intervals with 18 signatures; its §3 table is "本輪的核心「全域排除帳本」" (this round's core
"global exclusion ledger"), since "每個區間的最小值只可能出現在" (each interval's minimum can only occur at) its two
ends or an enumerated stationary point. The map is the same from 32,768 to 262,144 phases — "接觸區間圖沒有隨相位解析度增加而產生新的微小區間或新局部極小" (the map produced no new small interval or new local minimum as the resolution grew). Its §5
lists 12 smooth stationary points, the derivative's root count in each interval being the same on 257, 1,025 and 4,097
samples, so that "目前沒有發現隱藏於粗網格之間的額外導數換號" (no extra sign change of the derivative hidden between coarse
grid points has been found so far). Its §6 lists the 17 switches with their one-sided derivatives, of which
"四個接觸切換形成尖點局部極小" (four contact switches form cusp local minima); its §7 ranks the 8 local minima, the least at
3π/2, the 120° cusp 1.6383×10⁻⁹ above it. It concludes that no phase branch lies below 270°, but "不能把這句話替換成嚴格定理，因為導數符號仍以高密度浮點取樣與數值求根確認，而非區間算術包絡" (this sentence cannot be replaced by a rigorous theorem, since
the derivative signs are still confirmed by dense floating-point sampling and numerical root finding, not interval
envelopes); among its limits, "導數無隱藏根依賴多網格穩定性，而非嚴格界" (that the derivative has no hidden root rests on the
stability across grids, not on a rigorous bound). Its interface note sets the final target, s(φ) ≥ s(3π/2) for every φ,
and says "本輪尚未聲稱完成此嚴格證書" (this round does not yet claim this rigorous certificate).

## The checks

| id | guards | result |
| --- | --- | --- |
| S1 | the curve parameters are Round 08's best and Round 09's decimal constants; the source note states the labels and the envelope formula | **PASS** |
| S2 | 18 contiguous intervals from 0 to 2π, 17 boundaries; each interval's signature is the exact supports at its midpoint; each boundary within 10⁻¹² of the exact switch; the distinct signatures; the CSV and the summary hold the same intervals | **PASS** |
| S3 | each switch's scale is the exact one, to 2×10⁻¹³; at a switch between two corners the one-sided derivatives are the exact ones, to 10⁻⁷; at a switch to or from a wing's interior support, where s is continuously differentiable, the two stated derivatives agree; a cusp minimum has D₋ < 0 < D₊; the special phases | **FAIL** |
| S4 | each stated stationary point is an exact one, to 10⁻¹¹ in phase and 2×10⁻¹³ in scale, of its kind and place; the list is complete | **FAIL** |
| S5 | each interval's sampled derivative lies within the exact derivative's range; its classification and the guards' root counts are the exact derivative's sign changes | **FAIL** |
| S6 | the 8 minimum candidates and the 18 ledger rows are the exact minima of their kind and place, to 2×10⁻¹³; the margins; the global candidate | **FAIL** |
| S7 | this run's own scans at 32,768 to 262,144 phases give the stated cyclic switch counts, signature counts, sampled minima and global phase; the stated minimum is s(3π/2) to 2×10⁻¹³ | **PASS** |
| S8 | the report's tables (§3–§7) and figures, the README's counts and lines, the manifest and the interface note state the data's | **PASS** |
| S9 | the report's long figures of s₂₇₀, s₀ and their difference (§7, §11) hold to their digits | **PASS** |

## Findings

### 1. The map, the switches and the stated stationary points reproduce

This run takes the candidate from Round 09's decimal constants (S1) and evaluates the exact support at 60 digits. At
every interval's midpoint the active supports are the stated signature (S2). Each of the 17 switches, located by
bisection on the active labels, lies within 5.22×10⁻¹⁴ of its stated phase, and its stated scale is the exact one less
3.58×10⁻¹⁴ to 1.04×10⁻¹³ (`switches`) — the same few 10⁻¹⁴ the round's own double-precision value at 3π/2 carries,
8.38×10⁻¹⁴ below the exact one (`special_phases_minus_exact`; §7 prints 8.37×10⁻¹⁴, against a double reference). The 12
stated stationary points, found here at 40 digits, are exact ones of their kinds and signatures (S4). This run's scans at
32,768, 65,536, 131,072 and 262,144 phases give 18 cyclic switches, 18 signatures, 8 sampled local minima and the least
at 3π/2 every time (S7, `scans`). The switch count is 17 in the manifest and the §6 table and 18 in the README and §4;
the one omitted from §6 is the switch at φ = 0, where the two ends tie in y, and it is not a minimum (the derivative is
−0.654 before it and −0.214 after).

### 2. At a wing switch the derivative is continuous, and steep

At twelve of the 17 switches the support that changes is a wing's interior point reaching or leaving a corner — the
point where the wing's tangent is perpendicular to the functional runs into the wing's end. There the support point is
the corner itself and the other two functionals' supports are single corners, so s is continuously differentiable: this
run's one-sided derivatives at those switches agree to 4.6×10⁻⁹ (the residue of the 10⁻³⁰ tie tolerance that locates
them), while the round's two stated values are apart by 0.047 to 0.21 (`switches`; S3). One of each pair is the exact
derivative to 1.2×10⁻⁸; the other is in each case the derivative at a point inside the interval, because the derivative
is steep there: the tanh layer leaves each wing within 1.4×10⁻⁸ rad of α up to u = 0.3, and within 2.5×10⁻⁸ of β from
u = 0.85 on (`wing_angle_within_of_*`), so the interior support races along that straight stretch as the phase turns.
At interval 1's left switch the derivative is −0.3292 (switch 0 in `switches`); from the stated start, 7.5×10⁻¹⁶ rad
past it, it is −0.3239, at 10⁻⁹ of the interval's width −0.2087, at 10⁻⁶ −0.1305, at 10⁻³ −0.0523
(`interval_1_derivative`). At the five switches between two corners — π/2, 2π/3, π, 3π/2 and 5π/3 — and at the two
special phases, the stated one-sided derivatives are the exact ones to 1.11×10⁻⁸ (difference quotients).

### 3. The minimum at 3.028 rad is smooth, and missed

The round's switch 9, at 3.028226470697 rad, where min x passes from the right wing's interior to the end p₃, is listed
as a cusp minimum with derivatives −0.006893 and +0.173166. The exact derivative is +0.173166 on both sides, so the switch
is not a minimum. Interval 9's derivative runs from −0.3817 at its start (switch 8's exact value) to +0.1732 at its end,
and crosses zero 5.44×10⁻⁹ rad before the switch, at

  φ* = 3.028226465255688362712441…,  s(φ*) = 1.0998102257068941255…,

6.16×10⁻¹¹ below the stated cusp value (`interval_9_minimum`; RUN-009's golden refinement had found this minimum,
1.0998102257069 at 3.0282264653). The round's sampling reaches neither: interval 9 is classified "strictly decreasing",
its guards find no root on 257, 1,025 or 4,097 samples (the largest derivative they see is −0.006893), the stationary
list has 12 points where this run finds 13 (`stationary_count`), the ranking lists the minimum as a cusp at the switch,
and the ledger gives interval 9 the switch's value (S3–S6). So there are three cusp minima, not four, and five smooth
ones. On the twelve intervals whose three supports are all corners, s is a fixed sum of sinusoids with s″ = −s < 0, so
each has at most one stationary point, a maximum, which the signs at its ends decide; the count on the six wing
intervals rests on this run's scan of the exact derivative, 95 points an interval with 15 geometric steps to 10⁻¹⁵ of
the width at each end. The missed root is the hidden root of the round's own limitation, sitting in the steep stretch of
§2; it is 0.10 above the least and changes nothing about the global minimum.

### 4. The least at 3π/2, certified

The interface's target — s(φ) ≥ s(3π/2) for every φ — is certified here, in mpmath's interval arithmetic at 30 digits:

- **At 3π/2.** A/B = 1/√3 = tan(π/6) exactly, so with β > π/6 and α < π/2 (checked in interval arithmetic) the curve
  rises and falls, runs right throughout, and peaks in y/A − x/B at the wing's end p₁; hence s(3π/2) = y₁/A + (x₁ + l₀)/B,
  which RUN-010's trapezoid boxes (262,144 steps a wing; boxes at most 3.7×10⁻¹¹ wide in x and 1.3×10⁻¹¹ in y) put in
  [0.998914339066915, 0.998914339102351].
- **Beside 3π/2 and 2π/3.** On each side of each cusp, over a range that runs 0.05 rad (widened by 10⁻¹², to overlap the
  cover below) from the far end of the cusp's interval enclosure, s(φ) ≥ g(φ), the three functionals on the corners
  active on that side; and g′ has one sign there — negative before the cusp, positive after — on each of 64
  sub-intervals in interval arithmetic (`derivative_signs`). So beside 3π/2, s(φ) ≥ g(φ) ≥ g(3π/2) = s(3π/2), strictly
  away from it. Beside 2π/3, s(φ) ≥ g(2π/3), and g(2π/3) − s(3π/2), written as a linear form in x₁ and y₁ so that each
  box enters once, is at least 1.630111×10⁻⁹ on either side (`s_2pi_3_corners_minus_s270_at_least`) — the 120° margin of
  1.6383×10⁻⁹, resolved.
- **Elsewhere.** For a phase interval of half-width h about m, and the points qₖ chosen at m (translated, with all the
  boxes, to one fixed centre), G(φ) = Σₖ nₖ(φ)·qₖ ≤ s(φ) and G(φ) ≥ G(m) − |G′(m)|h − Lip h²/2, since |G″| = |G| ≤
  Σ|aₖ||qₖ| ≤ Lip = 1.3306 (an interval bound); a floating-point search chooses 1,558 intervals covering the rest of the
  circle, and on each the bound, taken in interval arithmetic, exceeds the upper end of s(3π/2) — by at least
  1.855678×10⁻⁹, the tightest beside the low branch at 0.124 rad, 4.69×10⁻⁸ above the least.

So the candidate's least scale is s(3π/2), reached at 3π/2 alone, in [0.998914339066915, 0.998914339102351]; against
the event five-link's least, at most 0.998903757132508674955129824903 (RUN-010), the gain is at least 1.0581934×10⁻⁵
(`certificate`). Every figure of a bound is cut toward its safe side.

## At what strength each result stands

| result | strength |
| --- | --- |
| the map, the switch phases and scales, the stated stationary points | high precision: exact supports at 60 digits (stationary points at 40); switches by bisection on the labels |
| s continuously differentiable at the twelve wing switches | exact (the support coincides with a corner, the others are single corners); this run's one-sided values agree to 4.6×10⁻⁹ |
| the smooth minimum in interval 9, and 13 stationary points in all | exact in number on the twelve corner-only intervals (s″ = −s < 0); high precision on the six wing intervals — a dense, geometrically refined scan, then bisection, not an interval enclosure |
| the least at 3π/2, alone, in [0.998914339066915, 0.998914339102351] | certified: interval arithmetic over the whole phase circle, for Round 09's decimal curve |
| the gain over the event five-link, at least 1.0581934×10⁻⁵ | certified: a lower bound; its upper bound would need the event five-link's least from below, not done |

## The drill

| | |
| --- | ---: |
| planted defects | 18 |
| tripping exactly the checks named for them | **18** |
| controls (the package unmodified, the report with CRLF endings, the README with trailing spaces, the audit copied) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **174/174** |

Plants include a triangle leg, Round 08's layer centre, the source note, a switch's phase, a switch's scale, a corner
switch's derivative, one stationary point's scale and another's interval, a sampled derivative, a guard's root count, a
minimum candidate, a ledger margin, a resolution count, the report's word for the number of cusp minima, a README count,
a digit of §11, the manifest's count of stationary points and a CSV count. The expected sets were reasoned before the dry
run with every set empty was read; it tripped them all, and one more — moving switch 4 by 10⁻⁹ also tripped S8, since §6
prints the phase to 12 decimals, which the reasoning had missed — so that plant now moves §6's cell alike. So 18 of 18
says the reasoning matched the harness, not that the plants were chosen blind, and one plant was changed after the dry
run. Where a plant moves a figure the package restates — a switch's boundary, a sampled derivative in the CSV, a ledger
margin and a resolution count in the audit, a minimum candidate and a resolution count in the report — the restatements
are planted alike, so that only the comparison can see the plant; the switch's plant still trips S2 twice, through the
exact switch and through the boundaries' identity with the intervals, whose ends it leaves. S3 to S6 fail on the
package with fixed-text messages, so their plants trip them through a figure that holds. The triangle leg reaches nearly
every value test; some it alone reaches — S3's special phases, S4's phase match, S6's candidate count and ledger minima,
S7's global minimum and S9's figures. No plant reaches S2's counts, tiling, widths, signatures (distinct and at the
midpoints), switch–boundary identity or CSV row count; S3's agreement at wing switches or its cusp kinds, which already
fail; S4's completeness, which already fails; S5's classification; S6's global candidate or the audit's copy of the
ledger; S7's copy of the table, the double-minus figure or the reference; or most of S8's patterns. Margins
(`check_margins`): S2's switch test at 19 times (5.22×10⁻¹⁴ against 10⁻¹²), S4's phase test at 12, S3's corner-derivative
test at 9; the scale tests — S3's at 1.9 times (1.04×10⁻¹³ against 2×10⁻¹³), S4's and S6's at 2.0, S7's at 2.3 — are
thin, the round's own evaluator carrying a few 10⁻¹⁴, and 2×10⁻¹³ was set from that before the checks ran. Of the four
controls, the unmodified package exercises nothing by construction; the report with CRLF endings re-runs S8 and S9, the
README S8, the copied audit S5 to S7. Each check, replaced in turn by one that always passes, was reported blind in
exactly as many cases as plants name it — S1 2, S2 3, S3 3, S4 3, S5 3, S6 3, S7 2, S8 3, S9 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 probe took the derivative at a cusp
from the active set at a bisection root, where one side's tie is already broken, and saw jumps that were not there; the
one-sided derivatives now come from each side's own supports. A first comparison of the map's boundaries read a list of
records as a list of numbers. An independent reading of this report before it was committed confirmed both findings
with code of its own and found 13 stationary points too, and found that the certificate's sign checks began at the
40-digit values of 3π/2 and 2π/3 rather than at enclosures of them, leaving a sliver of about 2×10⁻⁴¹ rad unchecked
beside each cusp; they now run from the cusps' interval enclosures. It also found two derivative figures cited from the
wrong place, an incomplete list of unplanted branches, and roundings that went both ways; all are corrected here.

## What this run does not establish

Any bound for Moser's problem: the candidate's least scale is about 0.99891, below 1, and the README says the package
"不構成 Moser 問題的新上下界". The stationary points as an interval enclosure: on the corner-only intervals their number is
exact, on the wing intervals it rests on a dense and geometrically refined scan of the exact derivative, far finer than
the round's but not a proof — the certificate of §4 does not need it. An upper bound for the gain. Anything about the
curve Round 08 stored in double precision, as against Round 09's decimal one. Nor the package's six figures, which were
not examined.

## Re-running

```
python code/fetch_source.py
python code/run011_round10_phase_map.py --drill
```

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