# RUN-014 — Round 13 (v0.13): the five-parameter event-KKT point lies 6×10⁻⁸ from the stated one and is a strict local maximum, and the gain over Round 08 is certified between 4.04×10⁻⁹ and 4.25×10⁻⁹; but the Jacobian's stated spectrum is not the system's — its condition is 2.1×10³, not 3.8×10⁷ — §7's projection is not reproduced, and three of the four other minima are not the curve's

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-20
**Subject:** `Moser_Skew_Lab_v0.13.zip` (SHA-256 pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`reports/ROUND_13_REPORT.md`, `ROUND_13_KKT_AUDIT.md`, `ROUND_14_CURVATURE_DISTRIBUTION_HANDOFF.md`, the replay script
(read, not run), and its data (`round13_summary.json`, `smooth_kkt_candidate.json`, `full_phase_minima.json`,
`branch_gradient_matrix.csv`, `basin_restarts.csv`), with Round 09's constants and Round 05's event root, read from the
package bytes
**Tools:** [`run014_round13_smooth_kkt.py`](../code/run014_round13_smooth_kkt.py) — this run's own solution of the
round's twelve equations at 60 digits, its own phase map of the stated curve, and RUN-011's interval certificate with a
target in place of s(3π/2); RUN-010's high-precision candidate (60 and 200 digits) and RUN-009's wing for the values,
derivatives and ledger; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run014-round13-smooth-kkt.json`](../data/gate-logs/run014-round13-smooth-kkt.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes holds, and four things it states do not. Holding: at
the stated curve the four branch values hold to 7×10⁻¹⁷ and the low phases to 5×10⁻¹⁸, and so do the spread, the common
scale, the improvement and the distance to one; Round 08's reference is Round 09's decimal curve, with its s(3π/2) to
the double; the gradients hold to 1.6×10⁻⁷, and the residual is the stated pressures times them. This run's own solution
of the twelve equations, six chord steps from the stated point to |F| < 10⁻³⁸, lies within 6.1×10⁻⁸ of the stated
parameters and 6.8×10⁻⁸ of the low phases, its pressures (all positive) within 1.6×10⁻⁶ of the stated ones, its scale
5.4×10⁻¹⁴ above the stated one. Its Jacobian has rank 12, and on the plane the four gradients leave free the
Lagrangian's Hessian is negative definite, so the solution is a strict local maximum of the least of the four branches,
on a neighbourhood of unestablished size — the "極小極大候選" (minimax candidate) the round names. The restarts' records, the
curvature ledger (the peak exactly, the width to 1.5×10⁻¹¹), the count of 8 local minima and the four controlling ones
hold too. Beyond what the round claims, the stated curve's least scale is certified here in interval arithmetic at no
less than s₁₃ − 1.5×10⁻¹⁰, so its gain over Round 08's least lies between 4.0451×10⁻⁹ and 4.2486×10⁻⁹, around the stated
4.2129×10⁻⁹. Not as stated: the Jacobian's singular values, 4.29×10⁸ down to 11.4 with condition 3.77×10⁷ — the exact
Jacobian's run from 5.40 down to 2.54×10⁻³, condition 2.13×10³, so "系統滿秩，但高度病態" (full rank but highly ill-conditioned)
does not describe it; §7's projection 2.7×10⁻⁹, which is 6.0×10⁻⁹ with the stated gradients and 1.7×10⁻⁹ with exact
ones; three of the four "other" minima — those at 1.73608 and 3.02827 are phases 7.6×10⁻⁵ and 5.7×10⁻⁵ from the curve's
minima, with scales above them, and the minimum near 3.028 is smooth, not the contact switch it is called, while the one
at 5π/3 is the cusp with a scale 1.6×10⁻⁵ too high; and the printed digits of the pressures (ten decimals, holding five
to seven), the gradients and the width. The conclusions the round draws — a four-way tie, a gain over Round 08 of about
4.21×10⁻⁹, a numerically isolated minimax point, no low branch outside the four — hold. The drill planted 22 defects;
each tripped exactly the checks named for it, and the 4 controls disturbed nothing.

---

## What Round 13 claims

Round 13 frees the layer width ε beside Round 08's four curve parameters and writes a twelve-equation system in (w, β,
δ, c, ε, φ₁, φ₂, s, μ₁…μ₄): the two smooth low branches and the 120° and 270° cusps all equal to s, the low branches
stationary in phase, Σ μᵣ ∇ₚmᵣ = 0 and Σ μᵣ = 1. Its solution has ε = 0.036823344353076 and the common scale
0.998914343297485, 4.2128524×10⁻⁹ above Round 08's — "其意義是峰值位置修正，而不是新的幾何級突破" (its meaning is a correction of the peak's
position, not a new geometric breakthrough). It gives the pressures (0.0282523300, 0.0283757600, 0.7360431600,
0.2073287400), the 4×5 gradient matrix, and a weighted stationarity residual of norm 4.025×10⁻⁸, which
"受有限差分與曲線積分插值限制，不能視為區間 KKT 證明" (is limited by finite differences and curve-integration interpolation, and cannot be
taken as an interval KKT proof). On the two-dimensional tangent space of the equal-height set the common gradient's
projection is about 2.7×10⁻⁹, so "目前沒有可辨識的一階上升方向" (there is at present no identifiable first-order ascent direction).
The 12×12 Jacobian has numerical rank 12, smallest singular value 11.4 and condition 3.77×10⁷: "系統滿秩，但高度病態" (the system
is of full rank but highly ill-conditioned), which "支持數值孤立，不支持穩健的區間可逆性結論" (supports numerical isolation, not a robust
conclusion of interval invertibility). Twenty restarts "全部返回同一尺度盆地" (all return to the same scale basin), with scales in
[0.998914343297213, 0.998914343297252]. The curvature ledger gives the turn δ, the peak 2.101101621113 and the 5–95 %
width 0.036441801352. The full phase circle has 8 local minima; outside the four controlling branches the lowest is
about 1.02935, and "沒有發現五參數修正引入的新低分支" (no new low branch introduced by the five-parameter correction was found). The
verdict: a five-parameter event-KKT candidate, "數值孤立、" (numerically isolated) "但高度病態的極小極大候選" (but a highly
ill-conditioned minimax candidate) — not an interval KKT or Arb certificate, and no new Moser bound.

## The checks

| id | guards | result |
| --- | --- | --- |
| X1 | Round 08's reference is Round 09's decimal curve, its scale the double nearest that curve's s(3π/2) (200 digits); the candidate's centre length, α, degrees and parameter shifts are its own arithmetic | **PASS** |
| X2 | at 60 digits: each low phase is a smooth minimum of s with the supports L\|p0\|p2 (to 10⁻¹⁵); the cusp phases are 2π/3 and 3π/2 within an ulp; the four branch values (to 2×10⁻¹⁶); the spread, the common scale (the least), the improvement over the reference and the distance to one | **PASS** |
| X3 | the stated gradients are the branches' partial derivatives at fixed phase (central differences at 60 digits, step 10⁻²⁰), to 2×10⁻⁷ | **PASS** |
| X4 | the residual is the stated pressures times the stated gradients, and its norm its norm; the pressures sum to one within their rounding; this run's own solution of the twelve equations, by chord steps from the stated point, converges (\|F\| < 10⁻³⁵) with positive pressures, within 10⁻⁷ of the stated parameters and phases, 2×10⁻⁶ of the pressures and 10⁻¹³ of the scale | **PASS** |
| X5 | at that solution the Jacobian (second differences at 60 digits, steps 10⁻¹⁵ and 10⁻¹², agreeing to 10⁻⁹) has rank 12, its smallest singular value above 10⁻³; the stated singular values and condition number are its own; the Lagrangian's Hessian is negative definite on the plane the four gradients leave free | **FAIL** |
| X6 | the data files hold the summary's records; the restarts' count, range and largest distance; the manifest's figures and files | **PASS** |
| X7 | the total turn is δ; the peak is the exact peak, to 10⁻¹⁴ relative; the 5–95 % width the exact width, to 2×10⁻¹¹ | **PASS** |
| X8 | this run's own phase map of the stated curve: 8 local minima, as stated; the controlling four are the branches; each of the other four is the exact minimum near it, to 5×10⁻⁶ in phase and 10⁻⁹ in scale, of the stated kind; none outside the four is as low | **FAIL** |
| X9 | the README, the report's figures and tables (§2–§6, §8–§12), the audit note and the handoff state the data's; §7's projection is the stated gradients'; the printed digits of the pressures, gradients and width are the quantities' | **FAIL** |

## Findings

### 1. What holds

At the stated curve — the summary's parameters, as their shortest decimals — the four branch values at 60 digits are
0.998914343297489195 (B1), 0.998914343297485032 (B2), 0.998914343297542111 (120°) and 0.998914343297542440 (270°); the
stated values are within 7×10⁻¹⁷ of them, the low phases within 5×10⁻¹⁸ of the two smooth minima, both with the supports
L|p0|p2 (`branch_values_stated_minus_exact`, `low_phases_stated_minus_exact`). The spread is 5.741×10⁻¹⁴, the least is
B2, and its excess over Round 08's s(3π/2) is 4.2128523149×10⁻⁹ (`improvement_exact`), which the stated
4.2128524003×10⁻⁹ holds to 9×10⁻¹⁷ (X2). Round 08's reference parameters are Round 09's decimals, and its scale is the
double nearest that curve's 200-digit s(3π/2) (X1). The stated gradients are the partial derivatives at fixed phase to
1.6×10⁻⁷ (X3), and the stated residual is the stated pressures times the stated gradients, to the rounding (X4). The
restarts' file holds the summary's count, range and largest distance (X6); the total turn is δ, the peak is the exact
peak δ/(ε(T₀ − T₁)w) to the double, and the width is the exact width to 1.5×10⁻¹¹ (X7). The full phase circle has 8
local minima — the two smooth low branches, the 2π/3 and 3π/2 cusps, the cusp at 5π/3, and smooth minima near 1.736,
3.028 and 3.779 — and none outside the four is lower than 1.0293 (X8). The tolerances are not drawn from a mechanism:
they sit 1.25 to 2 times above the deviations measured, which the log records (`gradients_stated_minus_exact_largest`,
`kkt_stated_minus_solution`, `curvature_exact`).

### 2. The KKT point exists, near the stated one, and is a maximum

The stated point does not solve the twelve equations exactly: with exact gradients its pressures leave a residual of up
to 2.9×10⁻⁸ (`residual_of_stated_pressures_with_exact_gradients`). This run solves them itself — values, phase
derivatives and gradients at 60 digits, the gradients by central differences with step 10⁻²⁰, chord steps with the
Jacobian at the stated point — and reaches |F| = 9.5×10⁻³⁹ in six steps (`kkt_residual`). The solution (`kkt_solution`)
has ε = 0.0368234056750, 6.1×10⁻⁸ above the stated ε, the other parameters within 6.8×10⁻⁹, the low phases within
6.8×10⁻⁸, the pressures (0.0282538923, 0.0283741595, 0.7360431486, 0.2073287995) — all positive, the first two 1.6×10⁻⁶
from the stated ones — and the common scale 0.99891434329753911, 5.4×10⁻¹⁴ above the stated
(`kkt_stated_minus_solution`). So the stated point is the system's solution to about 6×10⁻⁸ in its parameters; the
printed fifteen decimals define the curve this run certifies (§8 below), but they hold the solution only to 1.7×10⁻¹⁰ in
w, 7×10⁻⁹ in β, δ and c, and 6.1×10⁻⁸ in ε.

At the solution the four gradients have rank 3 (singular values 4.03, 0.312, 0.0132 and 5×10⁻³¹), so with every pressure
positive the critical directions are the plane the gradients leave free. On that plane the Lagrangian's Hessian — Σ μᵣ
∇²mᵣ, with each low branch's Hessian reduced by its phase, m_pp − m_pφ m_pφᵀ/m_φφ (m_φφ = 0.344) — has eigenvalues
−0.113 and −0.023 (`reduced_hessian_eigenvalues`): negative definite. The solution is a strict local maximum of the
least of the four branches over the five parameters, on a neighbourhood of unestablished size, and — the cusps'
one-sided slopes strict, the low branches' phase curvature positive, the other minima at 1.0293 and above — of the
curve's least scale near it. That is the second-order condition the round does not test; its "沒有可辨識的一階上升方向" is true at
the solution, where the projection it speaks of is zero.

### 3. The Jacobian

The 12×12 Jacobian of the round's system — unknowns and equations in the order of its §1, unscaled — built at the
solution from second differences at 60 digits, has singular values 5.40, 4.29, 2.15, 1.36, 1.18, 0.861, 0.433, 0.0887,
0.0663, 0.0236, 0.0135 and 0.00254 (`jacobian_singular_values`), the same to every printed digit with steps 10⁻¹⁵ and
10⁻¹² (`jacobian_steps_relative_difference`). Rank 12 holds. The stated values — 4.29×10⁸, 1.36×10⁸, 2.81×10⁷, … , 12.7,
11.4 — are not these: the largest is 8×10⁷ times larger, the smallest 4,500 times larger, and the condition, 3.77×10⁷
stated, is 2.13×10³ (`jacobian_condition`). The exact Jacobian's entries are of order one — gradients up to 2.9, mixed
phase–parameter derivatives up to 2.4, ±1 for the scale and the pressures' sum — and its Frobenius norm √Σσ² is 7.5,
which bounds every singular value: 4.29×10⁸ is not a singular value of the exact Jacobian of the system as §1 writes it,
unscaled. A Jacobian assembled from finite differences of a noisy evaluation can have almost any spectrum, and this run
did not find what the stated values are the spectrum of. A condition of 2×10³ is not, on any usual reading, a high one,
and the round's "高度病態" (highly ill-conditioned), repeated in §15 and the README, with §13's "系統條件數高" (the system's
condition number is high), rests on the stated numbers, not on the system. Its own caution stands for another reason:
rank 12 in double precision is not an interval proof of invertibility (X5).

### 4. Section 7's projection

The projection of the common gradient onto the plane on which the four branch gradients agree is 5.985×10⁻⁹ with the
stated gradients (`projection_stated_gradients`) and 1.651×10⁻⁹ with exact gradients at the stated point
(`projection_exact_gradients_at_stated_point`); at this run's solution it is zero. Neither is §7's 2.7×10⁻⁹, and this
run did not find the computation that gives it (X9). The conclusion drawn from it holds, as §2 shows.

### 5. The restarts

The 20 restarts' scales lie between 2.33×10⁻¹³ and 2.72×10⁻¹³ below the candidate's own common scale
(`restart_scales_below_candidate`), and so further below the solution's: none of them reaches the candidate's scale, let
alone the solution's. The package ships neither the restarts' parameters nor the program that made them, so this run
cannot say how their scales were evaluated or what "返回同一尺度盆地" (return to the same scale basin) meant in practice; the
records are what the summary says (X6), and §2's second-order test is this run's evidence of a basin.

### 6. The full phase audit

This run maps the stated curve's phase circle itself — a 2¹⁸-phase scan, 17 of the 18 switches fixed by bisection on the
changing support's label at 60 digits and the one at 0 exactly (the bottom side is level there), the zeros of each
interval's derivative found by bisection on a scan with geometric steps toward the ends (`switches`,
`stationary_points`, `local_minima`). Of the four "other" minima the round lists, one — 3.77896, 1.1275893354 — is the
curve's minimum, to 1.3×10⁻⁶ in phase and 2×10⁻¹¹ in scale. The other three are not:

| stated | the curve's minimum | s at the stated phase |
| --- | --- | --- |
| 1.73608, 1.029349885, smooth | 1.7361562900, 1.0293495204, smooth, 2.9×10⁻⁵ rad before the switch at π − β | 1.0293499079 |
| 3.02827, 1.0998191738, contact switch | 3.0282125272, 1.0998092540, smooth, 5.0×10⁻⁹ rad before the switch at π/2 + α | 1.0998192045 |
| 5.23599, 1.0814431618, contact switch | 5π/3 = 5.2359877560, 1.0814272247, the cusp | 1.0814284921 |

Two stated phases are 7.6×10⁻⁵ and 5.7×10⁻⁵ from the minima, with scales within 3.1×10⁻⁸ of s at those phases but
3.6×10⁻⁷ and 9.9×10⁻⁶ above the minima — samples near the minima, not the minima; the third phase is the cusp to its
five decimals, but its scale is 1.6×10⁻⁵ above the cusp's and 1.5×10⁻⁵ above s at its own phase, and this run did not
find where it comes from (`stated_other_minima_against_s`). And the minimum near 3.028 is not a contact switch: at π/2 +
α the support of min_x passes from the right wing's interior to its end p₃, where s is continuously differentiable, and
the minimum is smooth, just before it — as RUN-011 and RUN-012 found on Round 08's curve. None of this reaches the
round's conclusion: the lowest minimum outside the four is the curve's 1.0293495, which §11's 1.02935 holds, and none is
low (X8).

### 7. Printed digits

The report prints the pressures to ten decimals ("0.0282523300") from data with eight; against the solution's pressures
they hold five decimals for B1 and B2 and seven for the others. It prints the gradients to nine decimals; they hold six
to nine. It prints the width to twelve decimals, 0.036441801352; the width is 0.036441801337, and the figure holds ten
(`printed_digits`). The round states that its gradients are finite differences and its residual a numerical one, so
these are figures printed past what they were computed to, not claims of more (X9).

### 8. The certificate: the gain over Round 08

The round's improvement is a difference of two numerical scales. This run certifies it from both sides, in interval
arithmetic, with RUN-011's machinery: the curve's points at 262,144 steps of each wing in trapezoid boxes with the
analytic error, a bound beside each cusp (the active corners' sum moving away from its value on each side of 2π/3 and of
3π/2, within 0.05), and a second-order cover of the rest of the circle. With the target T = s₁₃ − 1.5×10⁻¹⁰ in place of
s(3π/2), every phase holds s ≥ T: the 2π/3 corners' value is at least 0.998914343284457, s(3π/2) at least
0.998914343279668, and the 1,587 cover intervals at least 0.998914343182500, the worst at the B2 minimum
(`certificate`). s(3π/2) is at most 0.998914343315417, an upper bound for the least. Round 08's least is its s(3π/2)
(RUN-011), here enclosed from the same boxes in [0.998914339066915, 0.998914339102351]. So the stated curve's least
exceeds Round 08's by at least 4.0451×10⁻⁹ and at most 4.2486×10⁻⁹ — the improvement is real, and the stated 4.2129×10⁻⁹
is inside its bounds.

### 9. The script makes none of the data

`src/smooth_kkt_replay.py` prints the summary's candidate, rank and residual norm; it computes nothing. The system's
solution, the gradients, the Jacobian, the restarts, the ledger and the phase audit come from code the package does not
ship, so the round's numbers can be checked, as here, but not reproduced.

## At what strength each result stands

| result | strength |
| --- | --- |
| the four branch values, the low phases, the improvement, the reference scale | high precision at 60 and 200 digits |
| the gradients to 1.6×10⁻⁷; the residual's arithmetic | high precision (central differences at 60 digits, step 10⁻²⁰) |
| the KKT solution, 6×10⁻⁸ from the stated point; its pressures positive | high precision: \|F\| < 10⁻³⁸ at 60 digits; not an interval Newton or Krawczyk proof |
| the Jacobian's rank and spectrum; the strict local maximum, on a neighbourhood of unestablished size | high precision: second differences at 60 digits, two steps agreeing; not interval arithmetic |
| the curve's least at least s₁₃ − 1.5×10⁻¹⁰; the gain over Round 08 in [4.0451×10⁻⁹, 4.2486×10⁻⁹] | interval arithmetic: trapezoid boxes with the analytic error, mpmath's iv at 30 digits, bounds cut toward their safe side |
| the 8 local minima and their kinds | high precision: a 2¹⁸-phase scan, switches and zeros by bisection at 60 digits; not an interval enclosure of the minima |
| the curvature ledger | exact formula for the peak and the width (double precision) |

## The drill

| | |
| --- | ---: |
| planted defects | 22 |
| tripping exactly the checks named for them | **22** |
| controls (the package unmodified, the report with CRLF endings, the README with trailing spaces, the restarts copied) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **209/209** |

Plants include a triangle leg, Round 08's ε, the centre length, a branch value, the distance to one, two gradients and a
pressure, the unknowns' count, the rank, the singular values, the gradient file, a restart's scale, the manifest's
files, the peak and the width, the minima's count and a minimum's kind, a report share, the README's gain, the handoff's
base scale and the audit note's restarts. The expected sets were reasoned before the dry run with every set empty was
read, and it tripped exactly them; so 22 of 22 says the reasoning matched the harness, not that the plants were chosen
blind. Where a plant moves a figure the checks compare elsewhere — a data record and its copies in the summary, the data
files and the audit note's block; a gradient or pressure and the residual and norm restated from it, in the summary, the
report and the audit note; the rank in the manifest, the README, the report and the audit note — the restatements are
planted alike, so that only the comparison with the curve can see the plant. X5, X8 and X9 fail on the package, X5 and
X9 with fixed-text messages, so their plants trip them through a figure that holds. The triangle leg, moved by 5×10⁻⁹,
reaches X1's reference scale, X2's low phases, values, spread, common scale and improvement, X4's solution and X8's
other minima, and stays inside X3's tolerance; no plant reaches X1's α and degrees tests, X2's cusp-phase test, X4's
residual arithmetic, norm and convergence, X5's second-order test, X6's candidate and minima files, X7's turn, X8's
controlling minima and its none-as-low test, or most of X9's patterns. Of the four controls, the unmodified package
exercises nothing by construction; the report with CRLF endings re-runs X9, the README X9, the copied restarts X6. Each
check, replaced in turn by one that always passes, was reported blind in exactly as many cases as plants name it — X1 3,
X2 3, X3 2, X4 3, X5 2, X6 3, X7 2, X8 3, X9 4 — 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 null space of the four gradients was
taken from an economy-size SVD, whose missing rows read as zeros: the projection came out 0 and one reduced eigenvalue
0, which looked like a degenerate maximum; with the full SVD both eigenvalues are negative and the projection is that of
§4. The first phase map bracketed each switch ±10⁻⁶ about a scan cell's midpoint, inside a cell 2.4×10⁻⁵ wide, and lost
most switches; it now bisects between the cell's own ends. The first test of the cusp phases asked for the doubles
nearest 2π/3 and 3π/2; the stated 2π/3 is the double that `2*math.pi/3` gives, 2.3×10⁻¹⁶ below the angle and not the
nearest double, and the test now asks for one ulp. An independent reading of this report before it was committed
confirmed the branch values, the Jacobian's spectrum (built anew as the derivative of the twelve equations), the
second-order test, the certificate's logic and the phase map with code of its own, and found the wrong functional named
at π/2 + α, the title's lower bound rounded up, a quote joined across lines, the second derivatives understated, a
manifest check claimed for X9, what the leg plant reaches understated, the local maximum's neighbourhood left unsized,
and the conditioning stated beyond the exact Jacobian; all are corrected here.

## What this run does not establish

An interval enclosure of the KKT solution, of its isolation or of its second-order condition: they rest on 60-digit
arithmetic, not on Krawczyk or interval Newton. Interval enclosures of the eight minima's places: the certificate bounds
the least scale, not where it lies. How the stated singular values, §7's projection, the 5π/3 scale or the restarts'
scales were computed: the package does not ship the code. 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/run014_round13_smooth_kkt.py --drill
```

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