# RUN-008 — Round 07 (v0.7): the curvature wings reproduce; the arc wing's deficit stands, every segmented family reaches the event five-link in a limit, and the m = 6 signature is one of Round 05's four events

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.7.zip` (SHA-256 `61b02f8d…`, pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`reports/ROUND_07_REPORT.md`, `ROUND_07_INTERPRETATION.md`, `ROUND_07_SEARCH_AUDIT.md`, and its data
(`round07_summary.json`, `segmented_results.json`, `circular_result.json`, `circular_equalized_result.json`, the five
`segmented_m*.csv`, `event_five_link_control.csv`, `continuous_circular_wing_sample.csv`), with Round 05's summary for
the control, read from the package bytes (the documents passed through RUN-001's `restore_escapes`, which changes none
of them)
**Tools:** [`run008_round07_curvature_wings.py`](../code/run008_round07_curvature_wings.py) — the segmented wing
family as its curves show it to be built; the arc wing with an exact support (junctions and tangent points); RUN-003's
exact formula and minima over phases; RUN-007's vectorized phase grid; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run008-round07-curvature-wings.json`](../data/gate-logs/run008-round07-curvature-wings.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and one of its readings does not hold
as stated. Reproduced: the control as Round 05's event five-link, with its nine local minima; the five segmented
candidates as their parameters, their least scales, phases and counts; the m = 6 candidate's contact signature and
resolution audit; both arc wings — before and after equalization — their exact least scales, local minima and the
equalized arc's three tracked branches; the arc sample; and the derived figures and the numbers the consistency check
reads. Not as stated: Round 07's report reads the m = 6 candidate's signature as "另一種局部放置機制" (another kind of
local placement mechanism); at its least phase one wing link is exactly parallel to the leg x = 0, with the endpoint on
the bottom leg and the centre's far end on the hypotenuse — the event of Round 05's branch B2, one of the "四種不同放置機制"
(four different placement mechanisms) Round 04 named, not another kind. Findings: the round's comparison that matters —
the equalized arc wing 4.14×10⁻⁵ below the event five-link — stands at the strength of a local search: from the arc a
local search does not move, and of eight nearby starts none ends higher. And the round's own search audit for m = 2
extends to every m: with its skew and power unrestricted, each segmented family contains the event five-link as a
limit (the power to infinity), reaching it to 10⁻¹³ at power 160, so every segmented candidate's deficit is the
search's. The drill planted 20 defects; each tripped exactly the checks named for it, and the 4 controls disturbed
nothing.

---

## What Round 07 claims

Round 07 asks whether the near-parallel pairs of wing links in Rounds 03 to 06 are a coarse discretization of a curved
wing or a finite contact skeleton that needs its corners. Round 05's event five-link, 0.998903757132509, is the
explicit control. Against it the round sets "兩類延拓" (two continuations), all mirror-symmetric with a horizontal centre
segment:

- **segmented wings** of m = 2, 3, 4, 6, 8 pieces per side, directions passing monotonically from α to β, piece
  lengths set by an exponential bias ("段長使用指數偏置參數分配") and the turning by a curvature power
  ("方向過渡使用曲率冪次控制"), each found by one heuristic search; the best, m = 6, at 0.998839177646286, whose contact
  signature the report reads as
  "此簽章不同於第 5 輪的四事件骨架，表示多段曲率翼確實進入另一種局部放置機制；但目前壓力仍較低"
  (this signature differs from Round 05's four-event skeleton, showing that multi-piece curvature wings really enter
  another kind of local placement mechanism; but the pressure is still lower for now), with a resolution audit on grids
  of 4,096 to 131,072 phases;
- **constant-curvature arc wings**: each side an arc of length w whose tangent turns linearly from α to β, the centre
  segment 1 − 2w; the outer search uses "高密度圓弧採樣" (dense arc sampling), and the final scales are an analytic
  support computation over the arcs' end points, their tangent points and the centre segment's ends
  ("最終固定相位尺度是圓弧模型內的解析支撐計算"); equalized on three branches (a low phase, 120° and 270°) to
  9.99×10⁻¹⁶ at s_arc = 0.998862397495093, 4.135963741603×10⁻⁵ below the event control.

The round audits its own search: the m = 2 family contains the event five-link, yet the search found 0.998530736905809,
3.73×10⁻⁴ lower — "只能解釋為搜尋不足，不能解釋成數學劣勢" (it can only be explained as insufficient search, not as a
mathematical disadvantage); so the m > 2 results are "只稱為目前最佳候選" (only called the current best candidates), and
the reliable comparison is the equalized arc against the explicit control. Its "暫定工作命題" (provisional working
proposition), limited to "本輪研究族" (this round's families): "常曲率平滑化略微降低合同容納壓力" (constant-curvature
smoothing slightly lowers the congruent containment pressure), so the effective factor may be where curvature
concentrates. §8 lists what it cannot infer, among them that every smooth curve does worse, or that the m > 2
candidates are their families' optima; §10 proposes a family θ(s) = β + (α − β)F(s) with the curvature concentrated in
a narrowing layer, and asks whether its scale tends to the event scale as the layer shrinks. No bound is claimed.

## The checks

| id | guards | result |
| --- | --- | --- |
| C1 | the control's parameters are Round 05's and the logged control curve is its five-link; its least scale is the event scale, at a stated phase that reaches it, with the nine stated local minima | **PASS** |
| C2 | each segmented curve is its parameters: piece lengths in ratio exp(skew/(m − 1)), directions β + δ(1 − k/(m − 1))^power, wing and centre lengths as stated, total length 1 | **PASS** |
| C3 | each segmented candidate's least scale, phase and local-minimum count on 32,768 phases are the stated ones; the summary holds the same results | **PASS** |
| C4 | the m = 6 candidate is the summary's best, with its contact signature at its least phase and each row of its resolution audit | **PASS** |
| C5 | both arc wings: curvature −δ/w and centre 1 − 2w; their exact least scales, a stated phase that reaches it, and the seven local minima; the equalized arc's tracked branches and their spread; the summary holds both arcs | **PASS** |
| C6 | the logged arc sample is the chord polyline of the arc wing before equalization, rescaled to length 1, point for point to 10⁻¹⁴ | **PASS** |
| C7 | the reference is Round 05's event scale; the gains, distances, the ranking and the manifest's figures are the values they name | **PASS** |
| C8 | the README, the report, the interpretation note and the search audit state the data's numbers, tables and signature | **PASS** |

## Findings

### 1. What reproduces, and how exactly

- **The control.** The control parameters are Round 05's event solution, the logged control curve is its five-link,
  and its least scale is the event scale to 4.4×10⁻¹⁶; its nine stated local minima reproduce to 4.5×10⁻¹⁴ (C1,
  `reproduction_deviations`).
- **The segmented family.** The documents name an exponential length bias and a curvature power but give no formulas;
  the five logged curves fix them. Piece k of m (k = 0 at the endpoint) has length proportional to exp(skew·k/(m − 1))
  and direction β + δ·(1 − k/(m − 1))^power, the pieces summing to the wing length w and the centre being 1 − 2w —
  the report's §10 template θ = β + (α − β)F with F(t) = (1 − t)^power. Every logged curve is its parameters under this
  rule to 1.1×10⁻¹⁶ (C2). For m = 2 the rule gives directions (α, β) and any length ratio, so the family is the
  symmetric five-link, as the round says.
- **The candidates.** Each segmented candidate's least scale, phase and local-minimum count reproduce, the scales to
  4.1×10⁻¹³ (C3). The m = 6 candidate's signature is its contacts at its least phase, and its resolution audit's six
  rows hold to 9.7×10⁻¹⁴ (C4).
- **The arcs.** For an arc, the extreme of each of the triangle's three linear functionals lies at a junction or at
  the arc point whose tangent is perpendicular to it; with that support both arc wings reproduce — the earlier one
  (0.998860080009) and the equalized one (0.998862397495) — their least scales to 6.7×10⁻¹⁶, their seven local minima
  to 3.2×10⁻¹³, and the equalized arc's three tracked branches and their spread (C5).
- **The sample.** The summary's `continuous_circular_wing` entry and the logged arc sample are the earlier arc, not
  the equalized one the README and the report quote (the sample lies 3.7×10⁻⁴ from the equalized arc sampled the same
  way). The sample is that arc's chord polyline — 179 equal-arc chords per wing, 31 equal steps on the centre — whose
  chords total 0.9999999961, rescaled by 1 + 3.9×10⁻⁹ to length 1: it matches that to 1.1×10⁻¹⁵, the unrescaled chords
  to 1.6×10⁻⁹ (C6, `reproduction_deviations`).
- **The figures.** The gains, distances, the ranking and the manifest's two scales are the values they name (C7), and
  the README's three scales and gap, the report's §2 table, its m = 6 and arc numbers, its ranking and resolution
  tables and its signature block, the interpretation note's and the search audit's numbers state them (C8). The other
  numbers the report repeats — its §1, §3, §6, §7 and §12 restatements — agree on reading.

### 2. The arc wing's deficit stands

The comparison the round relies on is the equalized arc against the control. The arc family has three parameters
(w, β, δ). From the equalized arc, a Nelder–Mead search on the least scale (a 360-phase scan with golden refinement)
does not move (`arc_local_search_from_equalized`). From eight starts at w ± 0.01 and β, δ ± 0.02 rad, four end within
1.1×10⁻¹⁰ of it (two to 10⁻¹⁵) and four stop 3.7×10⁻⁶ to 1.1×10⁻⁵ lower; none ends higher (`arc_multistart_*`). So the
deficit, 4.14×10⁻⁵, is the gap between the event five-link and a local maximum of the arc family, at the strength of a
derivative-free local search at a max–min kink; a higher arc elsewhere in the family is not excluded, and the round's
proposition is, as it says, provisional and limited to this family.

### 3. Every segmented family reaches the event five-link

The round's audit — the m = 2 family contains the event five-link, so its candidate's deficit is the search's —
extends to every m, provided the skew and the power may take any value (the documents state no search box; RUN-004 §2
made the same proviso about angle ordering). Under the family's rule, as the power grows the first piece keeps
direction α and all the others tend to β, so the wing tends to a two-direction wing. Set w, the centre, β and δ to the
event five-link's and choose the skew so that the first piece has its outer length l₁: the limit is the event five-link.
At power 160 every family's least scale is the event scale to 1.3×10⁻¹³, and at power 40 to between 7×10⁻¹⁵ (m = 3) and
1.5×10⁻⁵ (m = 8) (`segmented_limit_power_to_infinity`); the skews needed run from −1.43 to −6.04, far from the round's
candidates (−0.98 to 0.21, powers 0.82 to 2.09). Each segmented family's supremum is therefore at least the event
scale, 4.14×10⁻⁵ above the arc, and every candidate's deficit — 6.5×10⁻⁵ for m = 6, up to 3.7×10⁻⁴ for m = 2 — is the
search's, not the family's. This is the round's own §10.2 question — does a narrowing curvature layer tend to the event
scale? — answered yes for the segmented rule.

The round's ranking places the arc above "本輪最佳多段曲率候選" (this round's best multi-piece candidate); as a ranking of
candidates that is accurate, and the round says its m > 2 results are only candidates, so it is not counted here — unlike
RUN-004's seven-link item, where the round drew a conclusion about the family. Only the README's heading "可靠控制排序"
(reliable control ranking) sits uneasily with it, since the search audit names only the arc and the event as reliable.

### 4. The m = 6 signature is one of Round 05's four events

The m = 6 candidate's landscape has seven local minima (`m6_minima_with_signatures`). The least, at 7.957°, has link 3
of its six wing links exactly parallel to the leg x = 0 — the first four links' directions plus the phase are 91.55°,
91.06°, 90.54° and 89.99999999996° (`m6_least_phase_link_angles_plus_phase_degrees`) — with vertices 3 and 4 on that
leg, vertex 0 on the leg y = 0 and vertex 7, the centre's far end, on the hypotenuse. That is the pattern of Round 05's
branch B2 — an interior wing link parallel to the left leg, the endpoint on the bottom leg only, the centre's far end on
the hypotenuse (Round 05's contact ledgers, RUN-005 §1; its phase π/2 − β, RUN-005 §3) — with the parallel link further in
because the wing has more links; B1 differs in putting the endpoint at the corner, and this landscape has no B1-type
minimum. The other low minima are B3's and B4's: both endpoints on the hypotenuse at exactly 120° (0.998854829779) and
both on the leg x = 0 at exactly 270° (0.998854830818), 1.57×10⁻⁵ above the least; the remaining four lie above 1.028.

So the round's first clause is literally true — the candidate's least is a single event, not Round 05's "四事件骨架"
(four-event skeleton) of four level branches — but the inference fails: its events are the B2, B3 and B4 that Round 04
already counted among its "四種不同放置機制", not another kind.

## At what strength each result stands

| result | strength |
| --- | --- |
| the segmented rule | inferred from the five logged curves, which it reproduces to 1.1×10⁻¹⁶; the documents give no formula |
| the arc wing's support | exact (the extreme of a linear functional over arcs and segments) |
| the least scales, phases, minima, counts, signatures | numerical, double precision; minima over phases by a scan with every local minimum refined |
| the m = 6 least at a link parallel to the leg x = 0 | numerical, to 4×10⁻¹¹ degrees |
| the arc family's local maximum at 0.998862397495 | numerical: a derivative-free local search that does not move and eight multistarts; not proved global in the family |
| each segmented family's supremum at least the event scale | exact as a limit of the rule with skew and power unrestricted; numerical at power 160, to 10⁻¹³ |

## The drill

| | |
| --- | ---: |
| planted defects | 20 |
| tripping exactly the checks named for them | **20** |
| controls (the package unmodified, the report with CRLF endings, the curves copied, the README with trailing spaces) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **154/154** |

Plants include a triangle leg, a control local minimum and a control-curve point, a segmented power, a segmented value,
a count and a gain, the best candidate's signature and a resolution count, an arc curvature, an arc local minimum and a
tracked branch, the manifest, a curve point, an arc sample point, a report digit, a README digit, an interpretation
digit, a search-audit digit, and a term of the exact formula. The expected sets were written after a dry run with every
set empty, each trip reasoned before it was named, so 20 of 20 says the reasoning matched the harness, not that the
plants were chosen blind. Some trips are weak: the segmented plants trip C3, and the gain plant also C4, only through
the summary's identity with the results file. The control-minimum plant was enlarged after the referee read from 10⁻¹²
to 10⁻¹⁰, because it tripped C1 at 1.01×10⁻¹² against a tolerance of 10⁻¹²; the sample check has a single plant. Many
branches have no plant of their own — the control parameters' identity with Round 05's, the segmented curves' piece
counts and lengths, the arcs' reach and count branches, the reference's identity with Round 05's, and most of the
consistency check's patterns. Of the four controls, the unmodified package exercises nothing by construction, the
copied curves re-run the segmented checks, and the other two only the consistency check. Each check, replaced in turn by
one that always passes, was reported blind in exactly as many cases as plants name it — C1 4, C2 2, C3 8, C4 6, C5 5, C6 1, C7 3, C8 9 — which by construction
shows only that the harness counts as described (eight 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 one way worth recording. The sample check first compared the sample
with the arc's chords at a tolerance of 2×10⁻⁹, which the unexplained 1.6×10⁻⁹ systematic deviation just passed; the
deviation is the rescaling to length 1, and the check now applies it and compares to 10⁻¹⁴. An independent reading of
this report before it was committed found that, the m = 6 signature attributed to B1 as well as B2 and to the wrong
skeleton, a miscount of the wing's links, the multistart results misstated, the limit argument's parameter proviso
missing, the round's ranking wrongly implied to be a misstatement, a dropped hedge, and the drill's weak points
undisclosed; all are corrected here, and the m = 6 landscape and the sample's deviations are now logged.

## What this run does not establish

Any bound for Moser's problem. That the equalized arc is the arc family's global maximum. That any segmented family's
supremum exceeds the event scale (only that it reaches it), or that the round's search domain contained the skews and
powers the limit needs. Anything about the package's five figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run008_round07_curvature_wings.py --drill
```

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