# RUN-015 — Round 14 (v0.14): the single peak is a saddle against splitting, not a local maximum — split into two halves with its five parameters re-solved, the curve rises, certified 1.19×10⁻⁷ above Round 13's at separation 0.03 — while the chirality modes are stable for the congruent scale, as stated; the search's evaluator puts the low branches 10⁻⁸ low, its profile holds the parameters fixed, and its minima lists are phase samples

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-20
**Subject:** `Moser_Skew_Lab_v0.14.zip` (SHA-256 pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`reports/ROUND_14_REPORT.md`, `ROUND_14_SEARCH_AUDIT.md`, `ROUND_15_CURVATURE_MODE_HANDOFF.md`, the model file (read, not
run: a docstring), and its data (`round14_summary.json`, `double_peak_search.json`,
`double_peak_high_resolution_audit.json`, `asymmetry_congruence_audit.json`, the three profile CSVs and the two curve
files), with Round 13's package for the reference curve, read from the package bytes
**Tools:** [`run015_round14_double_peak.py`](../code/run015_round14_double_peak.py) — an exact support for curves whose
wings carry a mixture of tanh layers, or different layers left and right, at 40 to 50 digits; RUN-014's solution of
Round 13's twelve equations, phase map and interval certificate, the last on interval boxes for the mixture; RUN-009's
polyline for the curve files; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run015-round14-double-peak.json`](../data/gate-logs/run015-round14-double-peak.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces where it can be rebuilt, and its central
conclusion does not hold. Holding: the two curve files are the reference curve and the search candidate's mixture; the
search candidate's high-resolution cusp values are exact to 3.4×10⁻¹⁵ and its low values sit 2.1×10⁻¹⁰ below the exact
ones, a polyline's deficit; the separation profile is Round 13's curve split into two equal halves, its cusp values
exact, its low values 1.8 to 2.7×10⁻⁹ low; its coefficient −0.0338 is half the low branches' second derivative along the
separation with the five parameters held fixed; the chirality profiles, census and fits are their data's, but for one
fit; and the mirror symmetry is a strict local maximum of the congruent (weaker-handedness) scale in both chirality
modes, as stated. Not as stated: "單峰曲率層對峰分裂方向局部穩定" (the single-peak curvature layer is locally stable against peak
splitting), which the round draws as "有限數據支持：" (what the finite data support), and its "沒有可辨識的上升方向" (no
identifiable ascent direction) within the tested families — at Round 13's solution the Lagrangian of its four branches
has curvature +1.587×10⁻⁴ along the separation, so the single peak is a saddle; split into two equal halves at
separation d and with the five parameters re-solved, the four branches tie 1.28×10⁻⁹ higher at d = 0.004 and 1.19×10⁻⁷
higher at d = 0.03, and that curve's least is certified in interval arithmetic at least 1.1924×10⁻⁷ above Round 13's
candidate's — inside the very family the round searched. Also not as stated: the double-minus-single difference
−6.55×10⁻¹⁰ compares a polyline's low branch with an exact cusp (the curves' difference is −8.60×10⁻¹⁰); from separation
0.028 the profile's low phases are the ends of a search interval, not minima, and its scale overstates the least; the
centre-mode fit is not the shipped profile's, and the congruent scale has a corner at zero that no quadratic coefficient
describes; and six of the eight full-phase entries are phase samples, the 120° one 3.4×10⁻⁵ above the cusp. The drill
planted 21 defects; each tripped exactly the checks named for it, and the 4 controls disturbed nothing.

---

## What Round 14 claims

Round 14 asks whether Round 13's single-peak, mirror-symmetric candidate is a false isolated point of a restricted
parametrization. It splits each wing's curvature density into a mixture (1 − a)ρ(c₁, ε₁) + aρ(c₂, ε₂) of two normalized
sech² layers — eight parameters, the single peak at c₁ = c₂, ε₁ = ε₂ — and breaks the mirror symmetry by two modes, the
two local centres moved alike (c + η_c on the left, 1 − c + η_c on the right) and the widths split (ε e^{±η_ε}), scoring
a curve by the weaker of its two handednesses. The best eight-parameter mixture has peak separation 3.3255×10⁻⁶ —
"重新塌縮為近單峰" (collapsed back to nearly one peak) — and lies 6.55×10⁻¹⁰ below the single peak "同一高解析求值器下" (under
the same high-resolution evaluator). A separation profile falls as Δs(d) ≈ −0.033794468d², "二次係數為負" (the quadratic
coefficient is negative). Both chirality modes peak at zero offset, with quadratic coefficients −3.674929940 and
−0.003092904, and the best of 80 random two-dimensional offsets is 2.81×10⁻⁵ below zero. Under "有限數據支持：" (what the
finite data support) it boxes two verdicts, "單峰曲率層對峰分裂方向局部穩定" and "鏡像對稱對所測中心與寬度手性模態局部穩定" (the mirror
symmetry is locally stable for the tested centre and width chirality modes), and concludes "在所測曲率分裂與手性破缺族內，"
(within the tested curvature-splitting and chirality-breaking families) "第 13 輪單峰鏡像對稱候選沒有可辨識的上升方向。" (the Round
13 single-peak mirror-symmetric candidate has no identifiable ascent direction); its summary records
`single_peak_local_stability_supported: true` and `double_peak_escape: false`. It lists what it cannot prove — that the
single peak is best among all curvature densities, that all double-peak models are worse — and hands Round 15 a
functional mode analysis.

## The checks

| id | guards | result |
| --- | --- | --- |
| Y1 | the reference is Round 13's common scale; the two curve files are RUN-009's polyline (12,000 wing steps, 160 centre steps) of Round 13's curve and of the search candidate's mixture, centred at their mean, to 10⁻¹⁴; the candidate's centres, widths and weight are its parameters' arithmetic | **PASS** |
| Y2 | at 40 digits, with an exact support for the mixture: the search candidate's high-resolution cusp values are exact (to 10⁻¹³) and its low values below the exact ones by no more than 10⁻⁹; its least is the exact least; its differences are their figures' arithmetic; the double-minus-single difference is the curves' | **FAIL** |
| Y3 | the separation profile is Round 13's curve split into two equal halves: at seven separations its cusp values are exact (to 10⁻¹²) and its low values below the exact ones by no more than 10⁻⁸; the losses are the scales less the first; the fit is the profile's own (a quadratic in d² over d ≤ 0.01); the low phases are stationary points, not the ends of a search interval | **FAIL** |
| Y4 | the profile's coefficient is half the low branches' second derivative along d with the five parameters fixed (to 1 %); at RUN-014's solution the Lagrangian of the four branches is negative semidefinite in the double-peak directions (separation, width split) | **FAIL** |
| Y5 | the chirality profiles' congruent scale is the weaker handedness and reflection exchanges them; the census and the refined candidates; at RUN-014's solution the two modes' gradients have rank 2; each fit is its window's fit of the shipped profile; a quadratic coefficient describes the congruent scale at zero | **FAIL** |
| Y6 | the listed local minima of the base curve are its minima (to 10⁻⁹ in scale); the double-peak list is at the base list's phases; the sampled global minimum is the 270° cusp | **FAIL** |
| Y7 | the summary holds the data files' records; the CSVs are the JSON profiles; the manifest | **PASS** |
| Y8 | the README, the report's figures and tables (§3, §4, §6–§8), the search audit and the handoff state the data's | **PASS** |

## Findings

### 1. What holds

The reference scale is Round 13's common scale. The base curve file is RUN-009's polyline of Round 13's curve — 12,000
steps per wing, 160 across the centre — less its mean, to 1.0×10⁻¹⁵; the double-peak file is the same polyline of the
search candidate's mixture, to 2.0×10⁻¹⁵ (`curve_files_largest_difference`), and the mixture's centres, widths and weight
are exactly c ∓ d/2, ε e^{∓r} and the logistic of its last parameter (Y1). With an exact support for the mixture — the
support point on a wing found where the tangent is perpendicular to the functional, by inverting the mixture's F — the
search candidate's four branches are 0.9989143567270, 0.9989143562161 (the low pair) and 0.9989143424378,
0.9989143424543 (the cusps) (`search_exact`): the high-resolution cusp values are exact to 3.4×10⁻¹⁵, its low values
2.09×10⁻¹⁰ and 2.17×10⁻¹⁰ below, a polyline's deficit; the least is the 120° cusp, 8.597×10⁻¹⁰ below Round 13's least
(`high_resolution_stated_minus_exact`, `double_minus_single_exact`) (Y2). The separation profile is Round 13's stated
curve split into two equal halves at c ∓ d/2: at the seven separations checked its cusp values are exact to 1.5×10⁻¹⁴,
its low values 1.75 to 2.70×10⁻⁹ low (`profile_stated_minus_exact`); its losses are its scales less the first, and its
fit is a quadratic in d² over the 21 points with d ≤ 0.01, which reproduces the stated coefficients (Y3). The
coefficient −0.0337945 is what it measures: the low branches' second derivatives along d at fixed parameters are
−0.067619 and −0.067561 at Round 13's curve (`splitting_stated_curve_low_m_dd`), so they fall as −0.0338 d² (Y4). The
chirality profiles, the census and the refined candidates are consistent — the congruent scale is the weaker
handedness, reflection exchanges the two — and the width-mode fit is its window's fit of the shipped profile (Y5); the
copies, the manifest and the documents state the data's figures (Y7, Y8).

### 2. The single peak is a saddle against splitting

The test for a maximin point is not the scale along a direction with the other parameters held fixed, but the
Lagrangian's curvature on the directions its branches leave free (RUN-006, RUN-014). At RUN-014's solution of Round 13's
twelve equations, split the curve's layer into two halves at separation d, with widths ε e^{∓r} and the second half's
weight the logistic of q. Exchanging the two halves maps (d, r, q) to (−d, −r, −q) and leaves the curve unchanged; at
q = 0 it maps (d, r) to (−d, −r). So every branch's first derivatives in d and r vanish — exactly, by this symmetry: the
run's curves at ±d are one curve with its halves listed in the other order — and so do the cross derivatives with the
five parameters and with the phase: (d, r) lie in the critical cone, uncoupled from the five-parameter plane RUN-014
tested. On them, the four branches' second derivatives are (`splitting_branches`):

| branch | m_dd | m_dr | m_rr |
| --- | ---: | ---: | ---: |
| B1 low | −0.067619 | −0.0035250 | −0.00046074 |
| B2 low | −0.067561 | +0.0035268 | −0.00046040 |
| 120° | +0.0042856 | −2.9×10⁻⁷ | +2.49×10⁻⁵ |
| 270° | +0.0040120 | +7.8×10⁻⁷ | +2.33×10⁻⁵ |

The low branches fall as the peak splits; the cusps rise. Weighted by the pressures (0.028, 0.028, 0.736, 0.207), the
Lagrangian's block is [[1.5867×10⁻⁴, 4.2×10⁻⁷], [4.2×10⁻⁷, −2.89×10⁻⁶]], with eigenvalues +1.5867×10⁻⁴ and −2.90×10⁻⁶
(`splitting_lagrangian`, `splitting_eigenvalues`) (Y4). The positive one means the single peak is not a local maximum of
the least scale in the double-peak family: splitting the peak and re-balancing the five parameters raises all four
branches together, to second order by ½·1.587×10⁻⁴·d². The weight adds nothing at second order: with the two halves
identical it does not move the curve, and its cross terms with d and r are, branch by branch, a quarter of the
c-gradient and half of ε times the ε-gradient, whose pressure-weighted sums the first-order conditions make zero. The
mixture code reproduces RUN-014's solution to 4×10⁻⁴¹ (`splitting_reproduces_solution`).

### 3. The ascent, solved and certified

For each separation in turn this run solves Round 13's twelve equations for the split curve — the four branches equal,
the low ones stationary in phase, the pressures balancing the five parameter gradients, these by central differences
with step 10⁻¹² — by quasi-Newton steps (RUN-014's Jacobian at the single peak, updated by Broyden's rule) from the
previous solution, at 40 digits (`split_solutions`):

| d | the four branches' common scale less the single peak's | ε | steps, residual |
| ---: | ---: | ---: | --- |
| 0.004 | +1.2790×10⁻⁹ | 0.036726 | 7, 2×10⁻²⁹ |
| 0.008 | +5.2352×10⁻⁹ | 0.036431 | 9, 3×10⁻²⁹ |
| 0.016 | +2.3051×10⁻⁸ | 0.035196 | 11, 7×10⁻³⁰ |
| 0.03 | +1.1942×10⁻⁷ | 0.030113 | 14, 2×10⁻²⁸ |

At small d the rise is ½·1.587×10⁻⁴·d² (1.269×10⁻⁹ at 0.004); further on it grows faster, while the re-solved layers
narrow. The pressures barely move (0.0283, 0.0284, 0.7360, 0.2073 at d = 0.03). That the four branches stay the least is
not assumed: the d = 0.03 curve — its parameters to 30 digits (`certificate.curve`) — is certified with RUN-014's
interval machinery, on boxes built for the mixture (the trapezoid error bound now from both layers' derivatives): every
phase holds s ≥ 0.998914462565054, the four branches' common scale less 1.5×10⁻¹⁰ — beside 2π/3 and 3π/2 by the
corners' one-signed derivatives and values, elsewhere by 1,595 cover intervals, the worst 0.998914462589249
(`certificate`). Round 13's candidate's least is at most 0.998914343315417, its s(3π/2) enclosed from RUN-014's boxes for
its own curve. So the split curve's least exceeds Round 13's candidate's by at least 1.1924×10⁻⁷ — about twenty-eight
times Round 13's own gain over Round 08 — inside the double-peak family Round 14 searched. This run did not look for the
family's maximum, and claims nothing about it.

### 4. Why the round did not see it

Its separation profile holds the five parameters fixed at Round 13's values, so it measures the low branches' fall
(−0.0338 d²) and not the trade with the rising cusps that the pressures weigh; along that profile the scale does fall,
and the coefficient is right for what it is (§1). Its search could have found the trade, but at the search candidate the
search's low branches are 1.22×10⁻⁸ and 1.32×10⁻⁸ below the exact values and its cusps 6×10⁻¹⁰ above
(`search_stated_minus_exact`) — errors ten times the whole gain at d = 0.004, 1.3×10⁻⁹; this run did not measure how the
errors change with d. The search stopped at a separation of 3.3×10⁻⁶. And its headline difference, −6.55×10⁻¹⁰
"同一高解析求值器下", sets the evaluator's single peak — whose least is a low branch, a polyline's, 2.05×10⁻¹⁰ low
(`high_resolution_base_minus_exact`) — against the double peak's, whose least is an exact cusp; the curves differ by
−8.597×10⁻¹⁰, the figure the round also gives as the difference from Round 13's stored scale (Y2).

### 5. The profile at larger separations

From d = 0.028 on, the profile's first low phase is 0.14500000, and from d = 0.032 its second is 0.13699999, at every
separation — the ends of an interval the minimizer was confined to; at d = 0.028 the second, 0.13676, is neither an end
nor a minimum. At d = 0.028 and 0.04 the low branch has one minimum, at 0.14085 and 0.13939 (the two have merged), and
the derivative at the stated phases is up to 8.9×10⁻⁴; at d = 0.04 the least of the low branch is 0.9988420155, the
stated scale 0.9988424470, 4.3×10⁻⁷ higher (`profile_merged_low_minimum`) (Y3). It changes nothing about the
fixed-parameter fall, which is steep there anyway.

### 6. Chirality

The round's chirality verdict holds for the scale it scores, the congruent one, at first order. At RUN-014's solution
the four branches' derivatives in the centre mode are (−0.00673, −0.02511, −0.00200, +0.01761) and in the width mode
(−0.000286, −0.000286, −0.0000068, −0.0000033) (`chirality_gradients`) — the cusps taken where their two sides tie, which
moves as the symmetry breaks, at rates −0.0116 and −0.0000396 (`chirality_cusp_phase_rates`); the 4×2 matrix has rank
2 (singular values 0.0315 and 0.000284). A handedness and its mirror image move the branches by ±g·η, so no direction
can raise all eight together, and none outside the symmetric plane is critical: the symmetric point is a strict local
maximum of the congruent scale in both modes, the five parameters re-solved or not (RUN-007's lesson) (Y5). The same
gradients show what the round's fits miss. In the centre mode the least of the four branches falls on both sides of
zero, at different rates; in the width mode all four derivatives are negative, so one handedness's least rises for
η_ε < 0 while its mirror's falls. Either way the congruent scale, the weaker of the two, has a corner at zero, with
one-sided slope −0.0251 in the centre mode and −0.000286 in the width mode (`congruent_one_sided_slope`) — the profile's
own values at η_c = ±0.002, −5.18×10⁻⁵, follow the slope, not a quadratic. The stated "零點附近二次係數" (quadratic
coefficients near zero) are fits of even polynomials to a function with a corner. The width-mode fit is the shipped
profile's fit in its window; the centre-mode fit is not reproduced by the same fit — the shipped profile's 13 points in
the stated window 0.012 give −3.6255, not −3.6749 (`mode_fits_from_shipped_profiles`) (Y5). The congruent scale is the
round's objective, a curve being allowed to sit in the triangle reflected; were placements rotations and translations
only, the width mode would raise one handedness at first order — the shipped profile's rises by 3.2×10⁻⁶ at
η_ε = −0.45 (`width_mode_one_handedness_rise`) — and the symmetric point would not be a local maximum.

### 7. The full-phase lists are phase samples

Six of the base curve's eight listed minima are the curve's values at phases near its minima, not the minima: the
phases are off by up to 1.28×10⁻⁴, and the scales are s at those phases to 3.5×10⁻¹¹ but above the minima by up to
3.42×10⁻⁵ — the 120° entry, 2.0942673 with 0.9989485504, one grid step before the cusp; the 270° entry and the one at
3.7790 are minima to 4×10⁻¹² (`base_minima_against_exact`) (Y6). The double-peak list is evaluated at the base list's
phases. This also answers what RUN-014 could not trace: Round 13's "other minima" are four of these entries
(`round13_other_minima_against_list`), and its 5π/3 scale 1.0814431618 is s at 5.235859924, a sample 1.28×10⁻⁴ before
the cusp. The sampled global minimum is the 270° cusp, and no low branch is new.

### 8. No code

`src/curvature_distribution_models.py` is a docstring: the model definitions, no code. None of the package's numbers can
be reproduced from it; they can be checked, as here.

## At what strength each result stands

| result | strength |
| --- | --- |
| the curve files, the copies, the documents | exact (double precision for the polylines) |
| the search candidate's and the profile's exact branch values | high precision at 40 digits, the support exact for the mixture |
| the Lagrangian's curvature +1.587×10⁻⁴ along the separation (the saddle) | high precision: second differences at 50 digits around RUN-014's 60-digit solution; the first derivatives' vanishing by symmetry; not interval arithmetic |
| the split curves' common scales, 1.28×10⁻⁹ to 1.19×10⁻⁷ above the single peak | high precision: the four branches tie to the residual 10⁻²⁸ at 40 digits; the stationarity rows rest on central-difference gradients (step 10⁻¹²) |
| the d = 0.03 curve's least at least 1.1924×10⁻⁷ above Round 13's candidate's | interval arithmetic: trapezoid boxes for the mixture with the analytic error, mpmath's iv at 30 digits, RUN-014's cover; bounds cut toward their safe side |
| the chirality modes' strict local maximum of the congruent scale; the corner | high precision: first differences at 50 digits, the cusps at their moving ties; not interval arithmetic |
| the merged low minimum at d ≥ 0.028; the lists as samples | high precision at 40 to 60 digits |

## The drill

| | |
| --- | ---: |
| planted defects | 21 |
| tripping exactly the checks named for them | **21** |
| controls (the package unmodified, the report with CRLF endings, the README with trailing spaces, the chirality audit copied) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **176/176** |

Plants include a triangle leg, a point of each curve file, a component width, a high-resolution low value, the search
candidate's scale, a profile cusp value, a profile loss, the separation coefficient, a congruent value, the census count,
the width-mode fit, the sampled global minimum, a double-peak list phase, the separation and width CSVs, the manifest's
files, the README's separation, a report loss, the search audit's separation and the handoff's base scale. The expected
sets were reasoned before the dry run with every set empty was read, and it tripped exactly them; so 21 of 21 says the
reasoning matched the harness, not that the plants were chosen blind. Where a plant moves a figure the checks compare
elsewhere — a summary record and its copies in the high-resolution, search and chirality files and the CSVs; the
separation coefficient in the summary, the high-resolution file, the report and the search audit; the census count and
the width fit in the summary and the report — the restatements are planted alike, so that only the comparison with the
curves can see the plant. Y2 to Y6 fail on the package with fixed-text messages, so their plants trip them through a
figure that holds. The triangle leg, moved by 5×10⁻⁹, reaches Y2's exact values, Y3's cusp values and Y6's 270° cusp, and
stays inside Y4's one-percent tolerance; no plant reaches Y1's reference test, Y2's difference arithmetic, Y3's
interval-end test, Y4's Lagrangian test, Y5's rank, reflection and refined-candidate tests, Y6's sample test, Y7's search,
high-resolution and manifest-figure comparisons, or most of Y8's patterns. Of the four controls, the unmodified package
exercises nothing by construction; the report with CRLF endings re-runs Y8, the README Y8, the copied chirality audit Y5,
Y7 and Y8. Each check, replaced in turn by one that always passes, was reported blind in exactly as many cases as plants
name it — Y1 3, Y2 3, Y3 4, Y4 1, Y5 3, Y6 3, Y7 3, Y8 4 — 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 first replication of the separation fit took
the points up to d = 0.012 and did not reproduce it; the window that does is d ≤ 0.01, and the check now names it as
inferred. The first test of the sampled global minimum asked for the double nearest the exact cusp value; the stated
value is a double-precision evaluation 1.9×10⁻¹⁶ from it, and the test now allows 4×10⁻¹⁶. A first plant on the centre
profile set a congruent value to the weaker handedness, which it already was, and changed nothing; it now sets it to the
stronger one. An independent reading of this report before it was committed confirmed the saddle's curvature, two split
solutions, the certificate's logic and the d = 0.03 curve's least (by a dense scan of its own) with code of its own, and
found that the chirality derivatives had taken the cusps at fixed phases — on a broken curve the cusps move, and the
fixed-phase values are one side's — that the width mode does not lower the least on both sides, that the profile's
second phase reaches its interval end only from d = 0.032, that the first derivatives in d and r vanish by construction
and could not be measured, that the weight's cross term with d was missing, and that "chord steps", "the same boxes",
"curvature" for the half second derivative, the search's error, the quotes' hedge and the scope of the chirality verdict
were stated loosely; all are corrected here, the derivatives computed again at the cusps' moving ties.

## What this run does not establish

The double-peak family's maximum, or where the four-branch tie stops being the least as d grows: this run solved four
separations and certified one. Any interval enclosure of the saddle's curvature or of the split solutions. Whether a
wider family — Round 15's curvature modes — goes further. Any bound for Moser's problem. Nor the package's six figures,
which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run015_round14_double_peak.py --drill
```

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