# RUN-009 — Round 08 (v0.8.3): the smooth curvature layer's gain over the event five-link is real — 1.0581952124×10⁻⁵, confirmed in independent code and in 40-digit arithmetic — but its published curve file is not the candidate

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.8.3.zip` (SHA-256 `26ea8d85…`, pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`src/METHOD.md`, `reports/ROUND_08_REPORT.md`, `ROUND_08_FINITE_WIDTH_AUDIT.md`, `ROUND_08_DENSE_POINTCLOUD_CROSSCHECK.md`,
`ROUND_08_PEAK_DENSE_POINTCLOUD_CROSSCHECK.md`, and its data (`round08_summary.json`, `round08_focused_equalization.json`,
`round08_peak_scan.json`, the two point-cloud records, the eleven `smooth_eps_*.csv` and
`best_finite_width_smooth_curve.csv`), with Round 05's summary for the event five-link, read from the package bytes (the
documents passed through RUN-001's `restore_escapes`, which restores the three audit notes' TeX)
**Tools:** [`run009_round08_smooth_layer.py`](../code/run009_round08_smooth_layer.py) — this tree's own construction
of the tanh-layer curve (the exact tangent angle, positions by composite 20-point Gauss–Legendre quadrature) and its
own exact support (junctions, and the one wing point per functional whose tangent is perpendicular to it, found by
inverting the tanh); minima over phases by a 3,600-phase scan with golden refinement and the exact symmetric phases;
mpmath for the value at the least phase in 40-digit arithmetic; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run009-round08-smooth-layer.json`](../data/gate-logs/run009-round08-smooth-layer.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, including its headline, and four things
it states do not hold as stated. Reproduced: the best candidate, at layer width ε = 0.037, has least scale
0.998914339084632 — this tree's independent double-precision code gives exactly that, and 40-digit arithmetic at its
least phase, 3π/2, gives 0.99891433908463271725…, so its gain over the event five-link (a double-precision value) is
1.0581952124×10⁻⁵; the fifteen re-equalized widths' scales to 6.3×10⁻¹³; the eleven sweep curves and their values (the
sweep's to 1.1×10⁻⁸); the point clouds' figures; the curvature ledgers, as a grid's central differences and quantized
widths; and the documents' numbers. Not as stated: the best candidate's curve file is not the candidate — its centre
stops at 99 % of its length, a chord cuts the corner to a right wing sampled two-thirds of a step off, without its
endpoint, and read as a curve its congruent least scale is 0.998794830, 1.09×10⁻⁴ below the event five-link it is
published as beating; the report's curvature figures for the candidate, printed to twelve digits, are a 12,000-step
grid's central difference and quantized width (3.7×10⁻⁶ and 3.7×10⁻⁵ from the closed forms); the finite-width audit
calls a gain of 1.05×10⁻⁵ "約 $10^{-6}$"; and the ε = 0.04 cloud note's "兩種方法差" (difference between the two methods),
7.7×10⁻⁹, is the gap between a phase just beside the 120° cusp and the least, not between methods — its own record puts
the method difference at −2.0×10⁻¹⁴. Findings: the event five-link is not a local maximum among curves, which the
round's own caveat, "不是完整曲線空間的局部最優證明" (not a proof of local optimality in the full curve space), left open — the tanh
family contains it as ε → 0, and at the event's own parameters a layer of width ε raises its scale by about 2.0×10⁻⁴ ε;
and at the candidate the four branches are not level: their true values span 4.7×10⁻⁸. The drill planted 17 defects;
each tripped exactly the checks named for it, and the 4 controls disturbed nothing.

---

## What Round 08 claims

Round 08 replaces each wing's inner corner of the event five-link with an endpoint-normalized tanh layer
("端點正規化的雙曲正切層"): along a wing of length w, at u ∈ [0, 1], the tangent angle is

  θ_ε(u) = β + (α − β) F_ε(u; c),  F_ε(u; c) = (tanh((c − u)/ε) − tanh((c − 1)/ε)) / (tanh(c/ε) − tanh((c − 1)/ε)),

with a horizontal centre of length 1 − 2w and a mirrored wing; as ε → 0 the wing tends to two straight pieces (the
documents give F only in words — "Endpoint-normalized tanh tangent layer" — and the curves and the curvature ledger fix
it). Its control is the event five-link, which its §2 calls "固定接觸拓撲中的數值孤立折線候選，不是完整曲線空間的局部最優證明"
(a numerically isolated polyline candidate in a fixed contact topology, not a proof of local optimality in the full
curve space). It first expected s_ε → s_event, but found "有限寬度曲率層可略高於折線事件根" (a finite-width curvature layer
can be slightly above the polyline event root). Re-equalizing four branches over (w, β, δ, c) at nine widths from 0.008
to 0.05 and six more from 0.034 to 0.046, it reports the scale rising with width and falling back, with a best tested
width ε* = 0.037 and

  s* = 0.998914339084632,  s* − s₀ = 1.058195212322×10⁻⁵,

s₀ = 0.998903757132509 being the event five-link; tracked branches (B1, B2 at low phases, B3 at 120°, B4 at 270°) level to
3.3×10⁻⁸, with the true least at the 270° cusp and the remark that "追蹤局部最小器在非光滑尖點旁會產生約 $10^{-8}$ 的偏差"
(a tracking local minimizer beside a non-smooth cusp is off by about 10⁻⁸); a resolution table on integration grids of
3,001 to 12,001 points; and a dense point cloud of 240,017 points, which "不使用駐點反解" (does not use the stationary-point
inversion), giving 0.998914339084602 at the exact 270° cusp, 3.0×10⁻¹⁴ below. A curvature ledger gives ‖κ‖∞ = 2.092618923824,
a 5–95 % width 0.036579511462 and ∫|κ| ds ≈ 0.052047349063. Its finding: "五連桿折線平台可被一個有限寬度的平滑曲率層小幅超越"
(the five-link polyline platform can be slightly surpassed by a finite-width smooth curvature layer); its limits include
that "兩種支撐路徑仍共享曲線座標的數值積分" (the two support paths still share the numerical integration of the curve's
coordinates), and its README calls the result "候選數值突破" (a candidate numerical breakthrough),
"尚未經任意精度或區間證書確認" (not yet confirmed by arbitrary precision or an interval certificate). No bound is claimed.

## The checks

| id | guards | result |
| --- | --- | --- |
| L1 | each of the eleven sweep curves is its parameters' curve (2,400 equal wing steps, 49 centre steps), to 10⁻⁷ against the exact curve; mirror-symmetric, length 1 | **PASS** |
| L2 | the best candidate's curve file: its left wing is the candidate's, its centre has length 1 − 2w, it is its own mirror image, its length is 1 | **FAIL** |
| L3 | for each sweep width the full audit's least scale and the fixed-event-parameters audit are this run's, to 2×10⁻⁸; the local minima number as stated; the distances | **PASS** |
| L4 | for each of the fifteen re-equalized widths the best scale and each audit row are this run's least scale, to 10⁻¹⁰; the counts, gains and tracked spreads; the summary holds the same records | **PASS** |
| L5 | the best candidate's least exact scale is the stated stationary-support scale, to 10⁻¹³, at 3π/2; it is the peak scan's ε = 0.037 record; the gain and the manifest's figures | **PASS** |
| L6 | each point-cloud figure lies at or below the exact scale at its phase, within 10⁻¹¹; the stationary values, differences and gains | **PASS** |
| L7 | each curvature ledger is its curve's on an equal grid of the wing (2,400 steps; 12,000 for the candidate): the peak the grid's largest central difference, to 10⁻¹⁰ relative; the 5–95 % width the span of the grid points with F in [0.05, 0.95]; the turning δ, to 10⁻⁸; the centroid, to 10⁻⁶ | **PASS** |
| L8 | the README's figures, the report's §3–§8 tables and numbers, the finite-width audit's ε*, scale, gain and spread, and both cloud notes' scales, differences and gains | **PASS** |

## Findings

### 1. The headline reproduces, in independent code and in high precision

The round's two support paths share its integration of the curve; this run shares neither. It builds the curve from
the exact tangent angle with composite 20-point Gauss–Legendre quadrature (2,000 panels), and finds each linear
functional's extreme among the junctions and the one point per wing where the tangent is perpendicular to it — θ is
monotone, so that point comes from inverting the tanh exactly. With it the best candidate's least scale is exactly the
stated stationary-support value, 0.998914339084632 (L5), reached at 3π/2, where the two endpoints tie on the leg x = 0.
There the scale is y₁/A + (D − x₁)/B, with (x₁, y₁) the wing's end and D the endpoints' span — a single integral of the
tangent; in 40-digit arithmetic it is

  s* = 0.9989143390846327172529644…

(`best_high_precision`), 7.2×10⁻¹⁶ above the double-precision value. The event five-link's s₀ is itself known only in
double precision — its parameters are a double-precision root, and its four branches, evaluated in 40 digits, agree to
2.3×10⁻¹⁶ (`event_root_branches_40_digits`) — so the gain is s* − s₀ = 1.0581952124×10⁻⁵. Both curves are their own mirror
images, so both values are congruent least scales: the comparison is on equal footing. The candidate's eight local minima
over phases are 0.998914339085 at 3π/2, 0.998914340723 at 2π/3, 0.998914384874 and 0.998914385987 at the two low-phase
branches, and four above 1.029 (`best_minima`). So the least is the 270° cusp, and the gain over the event five-link holds
— at the strength of a double-precision scan with every local minimum refined and a 40-digit value at the least phase, not
of an interval certificate.

The fifteen re-equalized widths' scales are this run's to 6.3×10⁻¹³ (L4, `equalized_minus_exact`), and their peak —
0.037 of the widths tried — is as stated. The point clouds' figures lie at or just below the exact scales at their
phases (L6): at the smooth low-phase branches by 4.1×10⁻¹² to 4.7×10⁻¹² — a cloud of points misses a smooth extreme by a
little — and at the cusps, where every extreme is a junction the cloud contains, by 1.2×10⁻¹⁴ to 3.5×10⁻¹⁴, the cloud's
own integration error (`pointcloud_minus_exact`). The package publishes two gains: the cloud's, 1.058195209325×10⁻⁵
(README, summary, the report's §7, the peak note), and the stationary support's, 1.058195212322×10⁻⁵ (the report's boxed
figures in §4 and §12, the manifest), 3.0×10⁻¹⁴ apart.

### 2. The best candidate's curve file is not the candidate

`best_finite_width_smooth_curve.csv` has 8,034 points. Its left wing is the candidate's, 4,000 equal steps. Its centre
then runs 33 equal steps to 99 % of the candidate's centre length 1 − 2w, from where a 0.0033 chord at −0.96° jumps to a
right wing of only 4,000 points: they lie on the candidate's right wing, to 1.4×10⁻¹⁰, but sampled at
u = 1 − (k + ⅔)/4,000 — starting two-thirds of a step past the junction and stopping a third of a step short of the
endpoint, which is missing; the file's end is 2.8×10⁻⁵ above its start (`best_curve_file.right_wing`). The chord cuts
the corner between the centre and the right wing by 5.5×10⁻⁵ (`best_curve_file.corner_cut`). The polyline's length is
0.99992566 and it is not its own mirror image (L2). Read as a curve, as a reader of the package would read it, its
congruent least scale is 0.998794830 — 1.09×10⁻⁴ below the event five-link, 1.20×10⁻⁴ below the candidate it is meant to
show. With the candidate's right wing restored it is 0.99891433908463 again; with the candidate's corner inserted and
its endpoint added to the file's own points, 0.99891433908463 (`least_with_corner_and_endpoint_added`); with the corner
alone 0.998873602, with the endpoint alone 0.998823385 — the missing end and the cut corner together account for the
drop. The eleven sweep curve files, by contrast, are their parameters' curves, symmetric and of length 1, sampled from
an integration whose error grows as the layer narrows — 5.8×10⁻¹⁰ at ε = 0.2, 5.7×10⁻⁸ at ε = 0.002 (L1,
`sweep_curve_files_minus_exact`).

### 3. The event five-link is not a local maximum among curves

RUN-006 and RUN-007 showed the event five-link, numerically, to be a strict local maximum of the congruent least scale
among nearby five-links, symmetric or not, on a neighbourhood of unestablished size. Round 08's §2 already says it is no
proof of local optimality among curves, and its own fixed-event-parameter audits show why; this run confirms them (to
1.6×10⁻¹¹ where the layer is narrow). The tanh family contains the event five-link as ε → 0, and at the event's own
parameters a narrow layer raises the scale — by 1.6×10⁻⁶ at ε = 0.008, 1.0×10⁻⁶ at 0.005 and on down to 5.0×10⁻⁸ at
0.00025 — 1.9997×10⁻⁴ ε at every width from 0.005 down (`event_parameters_small_eps`,
`event_parameters_small_eps_rise_over_eps`) — returning to the event scale as the layer closes. Since these curves tend
to the event five-link uniformly (in Hausdorff distance), it is not a local maximum among curves in that topology;
re-equalized, the layer climbs to 1.06×10⁻⁵ at ε = 0.037.

A polyline had in fact passed the five-link "平台" (platform) before any smoothing: RUN-004's best seven-link, 0.998904534,
is 7.77×10⁻⁷ above the event five-link (and 9.8×10⁻⁶ below the candidate). And polylines can come as close to the
candidate as one likes: inscribed polylines converge to it in Hausdorff distance, the least scale is continuous in that
distance, and rescaling them to length 1 multiplies by a factor tending to 1. The platform is the five-link's.

### 4. Discretizations in the stated figures

- **The sweep.** The eleven-width sweep's full audits and fixed-event-parameter audits are this run's least scales to
  1.1×10⁻⁸ (L3, `sweep_values_minus_exact`). The full audits deviate — by 1.1×10⁻⁹ to 4.9×10⁻⁹ — only where the least
  sits at a smooth low-phase branch, at 6.7° to 8.0° (ε ≥ 0.02), and agree to 1.8×10⁻¹² where it sits at the 120° cusp
  (ε ≤ 0.012; `sweep_least_phase_degrees`), where only junctions matter. The mechanism is not shown; the pattern fits
  an approximate stationary-point support, such as the "駐點角度反插值" (stationary-angle inverse interpolation) the ε = 0.04
  cloud note names. The report does not print these values.
- **The tracked branches.** At the candidate the tracker's B3 and B4 values are the exact scale at their own phases,
  to 1.4×10⁻¹⁵ — phases 2.1×10⁻⁸ and 1.7×10⁻⁸ rad beside the 120° and 270° cusps (`best_tracked_*`) — which is the
  round's own stated mechanism. Its B1 and B2 values are 6.5×10⁻⁹ below the exact scale at their phases, which no
  offset from a minimum can produce: they come from some other evaluation. So the "追蹤分支差距" (tracked branch spread),
  3.3×10⁻⁸, mixes the two; the true four span 4.7×10⁻⁸ and are not level.
- **The curvature ledgers.** The closed-form peak curvature is δ/(ε(T₀ − T₁)w), at u = c, and the 5–95 % width is
  w ε [atanh(T₁ + 0.95(T₀ − T₁)) − atanh(T₁ + 0.05(T₀ − T₁))]. The stated peaks lie below the former, by a
  fraction 1.9×10⁻⁶ at ε = 0.2 and 1.9 % at ε = 0.002 (`ledger_peak_stated_over_analytic`). The rule behind them, found by
  fitting and now tested by L7: each peak is, to 2.7×10⁻¹² relative, the largest central difference δ|F′|/w on an
  equal grid of the wing — 2,400 steps for the sweep, 12,000 for the candidate — and each width is exactly the span of
  that grid's points with F between 0.05 and 0.95, a whole number of steps (`ledger_grid`); the candidate's is 1,306
  steps of w/12,000. For the candidate the report prints ‖κ‖∞ = 2.092618923824 and W = 0.036579511462; the closed
  forms give 2.092622664354 and 0.036616788565 (`best_ledger_exact`) — the printed twelve digits are not the curve's.
- **The finite-width audit** says "提升量只有約 $10^{-6}$" (the gain is only about 10⁻⁶); its own figure, above it, is
  1.045806040811×10⁻⁵, and the main report's §10 says "提升量只有約 $10^{-5}$".
- **The ε = 0.04 cloud note** gives "兩種方法差" as 7.703293336547×10⁻⁹ and the gain as 1.046576370145×10⁻⁵. Both are taken
  from the cloud's value at the tracked B3 phase, 9.0×10⁻⁹ rad beside the 120° cusp; the exact scale there exceeds the
  least by 7.7033×10⁻⁹ (`cloud_eps_0.04_note`). The figure is a phase gap — 2.2×10⁻⁹ of it the 120° branch's own
  height above the least, 5.5×10⁻⁹ the 9.0×10⁻⁹ rad offset (`exact_at_2pi_3_minus_least`); the record's own method
  difference, at the exact cusp, is −2.0×10⁻¹⁴, and its gain 1.0458060388×10⁻⁵ (`cloud_eps_0.04_note`).

## At what strength each result stands

| result | strength |
| --- | --- |
| the tanh layer's form | inferred from the curve files and the ledger; the documents describe it in words |
| the curvature ledgers' rule (a grid's central difference, the band's span) | inferred by fitting; reproduced to 2.7×10⁻¹² relative and exactly, and tested by L7 |
| the candidate's least scale 0.998914339084632 and its gain 1.0581952124×10⁻⁵ | numerical: independent double-precision code, every local minimum refined, and 40-digit arithmetic at the least phase; not an interval certificate |
| the fifteen re-equalized scales, the sweep values, the point-cloud figures | numerical, double precision, to the stated tolerances |
| the best curve file's construction and its own least scale 0.998794830 | exact as to the file's points (lengths, symmetry); numerical as to the right wing's sampling (a fit to 1.4×10⁻¹⁰) and its least scale |
| the event five-link not a local maximum among curves | numerical: the scale at the event's parameters with layers 0.00025 to 0.008, rising as 2.0×10⁻⁴ ε; in Hausdorff distance |

## The drill

| | |
| --- | ---: |
| planted defects | 17 |
| tripping exactly the checks named for them | **17** |
| controls (the package unmodified, the report with CRLF endings, the sweep curves copied, the README with trailing spaces) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **140/140** |

Plants include a triangle leg, a sweep transition centre, a sweep full audit and a sweep event audit, the candidate's
ledger peak and its scale, a re-equalized scale, a peak-scan parameter, a point-cloud cusp, the manifest, a sweep curve
point, a point of the best curve file's left wing, a report digit, a README digit, an audit-note digit, a term of the
exact formula, and the summary's copy of a record. The expected sets were written after a dry run with every set empty,
each trip reasoned before it was named, so 17 of 17 says the reasoning matched the harness, not that the plants were
chosen blind. Where a plant moves a figure the package restates — the sweep audit with its two stated distances, the
candidate's scale with its peak-scan record, the summary's copy of it and the manifest, a re-equalized scale and a
peak-scan parameter with their gains and the summary's copies — every restatement is planted alike, so that only the
value comparison can see the plant; the summary's copy is planted alone in a plant of its own. Four plants were enlarged
because each sat within a tolerance: the sweep centre's from 10⁻⁶ to 10⁻⁴, the sweep event audit's and the sweep full
audit's from 10⁻⁸ to 10⁻⁷, and the candidate scale's from 10⁻¹³ to 10⁻¹². L2 fails on the package with fixed-text
messages, so its plant trips it through a new failure, the left wing's. Branches with no plant of their own: L1's family
test, L3's distances, L4's gains and tracked spreads, L5's phase, peak-scan record, gain and manifest ε and gain
tests, L6's differences, gains and point count, L7's width and turning, and most of L8's patterns. Margins are thin in places
(`check_margins`): L1 at 1.75 times the observed 5.7×10⁻⁸, L3 at 1.9 times 1.05×10⁻⁸ — a tolerance set from the observed
deviations, whose mechanism is not shown — and L6's lower bound at 2.1 times; L7's peak test has 37 times, and its width
test is exact. Of the four controls, the unmodified package exercises nothing by construction, the copied sweep curves
re-run L1, and the other two only L8. Each check, replaced in turn by one that always passes, was reported blind in
exactly as many cases as plants name it — L1 2, L2 1, L3 5, L4 5, L5 4, L6 3, L7 2, L8 6 — 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 ways worth recording. The sweep and ledger checks first assumed the
stated figures were the exact curve's, and failed. The ledger check then assumed the peaks were the closed-form
derivative sampled on the curve files' 2,400 steps, and fell back to a 3 % band; a second independent reading found the
rule — a central difference on the grid, and the band's span — which L7 now tests. A first pattern for the report's
peak-scan table matched its equalization table, whose header begins the same way. And the first comparison of the best
file's right wing was off by one point and reported the wing displaced by the centre's shortfall; it lies on the
candidate's wing, sampled off-step. Two independent readings of this report before it was committed found that
comparison, the tracked branches' two mechanisms run together, the cloud's cusp deviations misexplained, the sweep's
mechanism asserted, the gain printed beyond the event value's precision, RUN-004's seven-link and the round's own caveat
left out, the hedges of RUN-006 and RUN-007 dropped, the ε = 0.04 note's figure, the width's last digit, a sentence of
the verdict at odds with §2, the ledgers' rule, plants that tripped their checks through restated figures rather than
the comparison they were meant to test, and a misstated list of unplanted branches; all are corrected here, and the new
figures are logged.

## What this run does not establish

Any bound for Moser's problem: the candidate is below 1, as the round says. That the candidate is the tanh family's
maximum — its branches are not level — or that the least over phases is certified: the 40-digit value is at the least
phase, the phase found by a double-precision scan. Anything about the package's six figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run009_round08_smooth_layer.py --drill
```

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