# RUN-016 — Round 15 (v0.15): the eight-mode candidate stands — exact to 2×10⁻¹⁴ and certified in interval arithmetic at least 1.3725×10⁻⁷ above Round 13's — with nine minima and a latent margin of 3.0046×10⁻⁹; but Round 13's point is not stationary in the modes, so the Hessian decides nothing there, the base's low-branch gradients are off by 10⁻⁵, the ledger's takeover opens and closes before the stated window, and the headline figures are printed past their digits

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-21
**Subject:** `Moser_Skew_Lab_v0.15.zip` (SHA-256 pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`reports/ROUND_15_REPORT.md`, `ROUND_15_BRANCH_OPENING_AUDIT.md`, `ROUND_16_FIVE_BRANCH_HANDOFF.md`, the model file
(read, not run: a docstring), and its data (`round15_summary.json`, `final_candidate.json`, `final_candidate_curve.csv`,
`mode_profiles.json`, `newton_line_full_phase.csv`, `pressure_hessian.csv`, `pressure_hessian_eigenvectors.csv`,
`reoptimized_multiplier_ledger.csv`), with Round 13's package for the mode coordinate and the reference, read from the
package bytes
**Tools:** [`run016_round15_functional_modes.py`](../code/run016_round15_functional_modes.py) — an exact support for the
mode-deformed curve at 40 digits (panels of 24-point Gauss–Legendre with each panel's Legendre interpolant of the
density integrated exactly), interval boxes for it (Simpson's rule for the density's integral with fourth-derivative
bounds from interval Taylor arithmetic, trapezoid positions with local bounds), RUN-014's solution, branches and least
bound, RUN-010's candidate at 60 digits; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run016-round15-functional-modes.json`](../data/gate-logs/run016-round15-functional-modes.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces where it can be rebuilt, and some of what
it states does not hold as stated. Holding: the mode basis is the Legendre modes in a curvature coordinate; the curve
file is the final candidate's curve — the density's layer at the candidate's own parameters, the coordinate from Round
13's curve or a layer within 10⁻⁷ of it — and at 40 digits the candidate's four branches and its latent fifth one are
the stated ones to 2×10⁻¹⁴: its least is the 120° cusp, 0.99891448071696448, 1.3741947945×10⁻⁷ above Round 13's least,
the latent branch 3.0045884×10⁻⁹ above it, and the nine listed minima are the curve's; the mode gradients hold to
8.1×10⁻⁸; the compensation, the Hessian's eigenpairs, the profiles (along the compensated paths) and the Newton line
(the first-order path, rebuilt at two multipliers to 10⁻¹⁵) are their data's; the ninth minimum opens at m = 2.2; and
the Newton direction is the Newton step on the Hessian's six most negative directions, near enough (cosine 0.99999996).
Beyond what the round claims — "尚未進行區間或 Arb 證書" (no interval or Arb certificate yet) — the candidate's least is
certified here in interval arithmetic at no less than its exact value less 1.5×10⁻¹⁰: its gain over Round 13 is at least
1.3725×10⁻⁷, and, against RUN-014's certified lower bound on Round 13's least, at most 1.3758×10⁻⁷. Not as stated: the
round infers "並非此八模態空間中的嚴格局部極大" (not a strict local maximum in this eight-mode space) from one positive Hessian
eigenvalue, while the point is not stationary — at RUN-014's solution the pressure-weighted mode gradients reach
1.98×10⁻⁶ — so the conclusion holds for a first-order reason and the Hessian there decides nothing; the base's parameter
gradients hold for the cusps to 3×10⁻¹⁰ but for the low branches only to 1.02×10⁻⁵, and its low branches are tied with
the cusps only in the round's own evaluator (they sit 9.7×10⁻¹⁰ above them); the ledger's latent branch is already below
the four at m = 3.2235 to 3.2245, the four control again to 3.2275, and the takeover comes only at 3.228; and the least,
the gain and the latent margin are printed to 15 decimals, 13 and 13 significant digits where they hold 13 decimals, 7
and 5 significant digits. The drill planted 25 defects; each tripped exactly the checks named for it, and the 4 controls
disturbed nothing.

---

## What Round 15 claims

Round 15 turns to the curvature function itself. With t(u) = 2∫₀ᵘρ₀ − 1 the curvature coordinate of the base
("基準曲率累積座標"), it uses the bounded modes ψₙ = √(2n+1) Pₙ(t), n = 1…8, orthonormal for ρ₀ du = dt/2 (Gram error
2.220×10⁻¹⁶), and deforms the density to ρ_p e^{Σaₙψₙ}, normalized. At a re-solved base its §1 table gives the
pressure-projected first derivative of each mode — nonzero, −1.978316×10⁻⁶ for P4 — then it compensates each mode by the
five parameters so that the four branches move equally to first order, and builds a Hessian it defines as the "分支壓力加權
Lagrangian，沿一階等高補償路徑的有限差分 Hessian" (pressure-weighted Lagrangian, a finite-difference Hessian along first-order
equal-height compensated paths), "它不是完整無限維 Hessian，也不是區間 Hessian" (not the full infinite-dimensional Hessian, nor an
interval one). Seven eigenvalues are negative and one, 1.329×10⁻⁶, positive: "所以第 13 輪單峰候選並非此八模態空間中的嚴格局部極大" (so Round
13's single-peak candidate is not a strict local maximum in this eight-mode space). A four-branch optimum at
0.998914692247305 is excluded by a hidden middle branch at φ ≈ 0.1390815. Along a pressure-projected Newton direction,
the five parameters re-equalized at each multiplier m, the four branches rise, a ninth minimum appears after m ≈ 2.2,
and near m ≈ 3.228 "新分支開始接管" (the new branch begins to take over): "因此存在一個由分支開啟界定的有限上升窗" (so there is a finite ascent
window bounded by branch opening). At m = 3.2275 the candidate's least is 0.998914480716946, 1.374194610326×10⁻⁷ above
Round 13's, with nine local minima and the latent branch 3.004601123457×10⁻⁹ above the least —
"一條通過完整相位數值審計的八模態有限曲率候選，位於第五分支接管之前的狹窄上升窗口內" (an eight-mode finite curvature candidate that passes the full-phase
numerical audit, in a narrow ascent window before the fifth branch takes over); no interval or Arb certificate, no Moser
bound.

## The checks

| id | guards | result |
| --- | --- | --- |
| Z1 | the eight modes named; their weighted means, norms and maxima are the Legendre modes' (0, 1, √(2n+1)); the Gram matrix and its stated error | **PASS** |
| Z2 | the curve file is the candidate's curve: the density on 96,000 steps with F and positions by cumulative trapezoid, t from Round 13's curve and ρ_p at the candidate's parameters, 320 centre steps, centred at its mean, to 10⁻¹⁰ | **PASS** |
| Z3 | the base curve's least is the stated base scale (to 10⁻¹²) and the Newton line's first row (to 10⁻¹⁵); the pressures sum to one; the residual is the pressures times the parameter gradients, which are the branches' partial derivatives at the base, to 2×10⁻⁷ | **FAIL** |
| Z4 | the mode gradients are the branches' derivatives in the modes at the base (40 digits), to 2×10⁻⁷; the weighted gradient is the pressures times them; the compensation is the minimum-norm equal-height solution | **PASS** |
| Z5 | the Hessian is symmetric, its eigenpairs reproduce it, one eigenvalue is positive, its file is the summary's; the Newton direction lies in the span of the six most negative eigen-directions | **PASS** |
| Z6 | at RUN-014's solution of Round 13's system the pressure-weighted mode gradients vanish, as a Hessian test of local maximality presupposes | **FAIL** |
| Z7 | the coefficients are 3.2275 times the Newton direction; at 40 digits the four branches and the latent one are the highest-resolution audit's (to 10⁻¹³); the least is the 120° cusp; the margin, the gains and the audit rows are their figures'; the nine listed minima are the curve's (phase to 10⁻⁷, scale to 10⁻¹³); a dense scan finds nine | **PASS** |
| Z8 | the Newton line's and the ledger's norms are 0.04 m and their gains the scales less the base; the Newton line is the first-order path, its least at m = 1 and 2 the curve's (to 10⁻¹³); the ninth minimum opens at m = 2.2; the ledger's four-branch and full scales agree (to 10⁻¹³) below the takeover window and part past it | **FAIL** |
| Z9 | each profile row's minimum, pressure scale and spread are its branches'; four rows' cusp values are the curve's along the stated equal-height path (to 10⁻¹³) | **PASS** |
| Z10 | the copies, the manifest, the README, the report's tables (§1, §3, §6–§8), the audit note and the handoff state the data's; the least, the gain and the latent margin are printed to the digits the curve holds | **FAIL** |

## Findings

### 1. The candidate at 40 digits

The round's summary calls the mode coordinate the "base curvature CDF"; the curve file says which. Built on the file's
grid — the density on 96,000 steps, F and positions by cumulative trapezoid, 320 centre steps, centred at the mean —
with t from Round 13's stated curve and ρ_p the tanh layer at the candidate's own five parameters, the file is
reproduced to 4.3×10⁻¹² (`curve_file_largest_difference`); with t from the round's base or from RUN-014's solution,
layers within 10⁻⁷ of Round 13's, to 5.0×10⁻¹¹ and 3.9×10⁻¹¹, and with t from the candidate's own layer only to
1.06×10⁻⁷ (`curve_file_t_from_other_layers`) (Z2). So ρ_p is the candidate's, and t is Round 13's layer or one
indistinguishable from it at this grid; this run takes Round 13's. With an exact support for that curve at 40 digits —
the density's integral and the positions from high-order quadrature on 189 panels, the support points by inverting F —
the candidate's branches are (`final_exact`, `final_branches_above_least`):

| branch | value | above the least |
| --- | --- | ---: |
| 120° cusp | 0.99891448071696448 | 0 |
| 270° cusp | 0.99891448071696486 | 3.8×10⁻¹⁶ |
| B1 low | 0.99891448112086535 | 4.039×10⁻¹⁰ |
| B2 low | 0.99891448173927110 | 1.0223×10⁻⁹ |
| latent | 0.99891448372155293 | 3.0045884×10⁻⁹ |

The stated values hold to 1.84×10⁻¹⁴; the least is 1.3741947945×10⁻⁷ above Round 13's (`gain_over_round13_exact`). The
nine listed minima are the curve's: the smooth ones to 4×10⁻⁹ in phase, all to 1.84×10⁻¹⁴ in scale — the one at 3.0282
the smooth minimum just before the switch at π/2 + α, as on Round 08's and Round 13's curves (RUN-011, RUN-014) — and a
dense double-precision scan of the curve file at 65,536 phases finds nine (`listed_minima_stated_minus_exact`,
`dense_scan_local_minima`) (Z7). Of the five resolutions the round audits, the highest, 96,001 points, is the one
furthest from the exact least: 1.84×10⁻¹⁴ below it, where the others lie within 6.8×10⁻¹⁵ (`final_least_minus_stated`).

### 2. The certificate

The round leaves its candidate without an interval certificate; this run gives it one, with RUN-014's machinery on
interval boxes built for the mode curve. The wing's points at 262,144 steps are enclosed by integrating the density with
Simpson's rule, each step within h⁵/2880 times a bound on the density's fourth derivative over its block of 64 steps —
the bound from interval Taylor arithmetic of order 4 through the tanh layers, the Legendre polynomial and the
exponential — and the positions with the trapezoid rule, each step within w·h³/12 times a local bound on θ′² + |θ″|, the
error terms rounded up. The boxes are at most 4.8×10⁻¹² wide. Every phase then holds s ≥ 0.998914480566964, the exact
least less 1.5×10⁻¹⁰ — beside 2π/3 and 3π/2 by the corners' one-signed derivatives and values, elsewhere by 1,598 cover
intervals, the worst 0.998914480841346 — and s(3π/2) lies in [0.998914480714660, 0.998914480719270] (`certificate`).
Round 13's least is at most its s(3π/2), 0.998914343315417, and at least 0.998914343147485, RUN-014's certified bound,
recomputed here; so the candidate's least exceeds Round 13's by at least 1.3725×10⁻⁷ and at most 1.3758×10⁻⁷. With
RUN-015's split curve's least at most 0.998914462741709 (its own certificate), the candidate's is at least 1.78×10⁻⁸
higher than that curve's too.

### 3. The base and its gradients

The round's base is not Round 13's stated point nor RUN-014's solution: its parameters lie up to 6.53×10⁻⁸ from the
solution and its first two pressures 1.13×10⁻⁵ (`base_minus_solution_parameters`, `base_pressures_minus_solution`). At
60 digits its cusps are the stated ones to 3.7×10⁻¹⁶, and its least is the cusps', within 2.84×10⁻¹³ of the stated base
scale (`base_exact`, `profile_zero_row_minus_base_exact`) (Z3). But its low branches lie 9.66×10⁻¹⁰ above the cusps,
while the round's single-mode profiles at amplitude 0 have them tied with the cusps, 9.7×10⁻¹⁰ below the curve's own:
the round's evaluator puts the low branches low, as Round 14's did, and the base is a four-way tie only in it. The same
evaluator makes the low branches' parameter gradients: the cusp rows hold to 3.0×10⁻¹⁰, the low rows only to 1.02×10⁻⁵
(`base_parameter_gradients_stated_minus_exact_largest`) (Z3) — errors of the size of the pressures' 1.13×10⁻⁵ offset
from RUN-014's, which they would explain. The mode gradients are better: they hold to 8.1×10⁻⁸, the weighted ones to
2.1×10⁻⁹ (`base_mode_gradients_stated_minus_exact_largest`, `base_weighted_mode_gradient_exact`), the compensation is
the minimum-norm solution of the equal-height conditions, and the single-mode profiles run along those compensated
paths: their cusp values are the curve's there to 1.7×10⁻¹⁵ (`profile_cusps_stated_minus_exact`), and differ from the
fixed-parameter curve's by the compensation's change, to first order (Z4, Z9).

### 4. Not stationary, so the Hessian decides nothing

At a point where the pressure-weighted gradient in a direction is not zero, moving along the compensated path raises the
four branches together at that rate, to first order; second-order tests — the Hessian's signs — apply only where the
first-order conditions hold. The round's own §1 table shows nonzero first derivatives at its base; they persist at
RUN-014's solution of Round 13's twelve equations, stationary in the five parameters
(`solution_weighted_mode_gradient`):

| mode | P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 |
| --- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| gradient | −1.9×10⁻¹⁴ | 4.81×10⁻⁷ | 1.49×10⁻⁸ | −1.979×10⁻⁶ | 5.7×10⁻⁹ | 5.42×10⁻⁷ | 1.3×10⁻⁹ | 3.95×10⁻⁷ |

P1, close to the layer's translation, is in the five parameters' span and vanishes; the others do not, the even ones —
shapes the tanh layer cannot take — most (Z6). So Round 13's point is not stationary in the eight-mode space, and the
round's conclusion, not a strict local maximum, is true for this first-order reason; its §3 draws it from the Hessian's
one positive eigenvalue, which at a non-stationary point decides nothing. The candidate bears this out: its coefficients
against the gradients give a first-order term of 1.558×10⁻⁷ at the solution (1.559×10⁻⁷ with the round's own gradients
at its base), and the stated Hessian's quadratic term is −1.09×10⁻⁸ — without the first-order term the model would
predict a loss — a model total of 1.45×10⁻⁷ against a gain of 1.3747×10⁻⁷ over the base on the re-equalized path
(`first_order_term_at_solution`, `first_order_term_at_base`, `second_order_term_stated_hessian`).

### 5. The Hessian, the Newton direction and the Newton line

The Hessian is symmetric, its file is the summary's, its eigenpairs reproduce it, and one eigenvalue is positive (Z5).
The round calls its direction a "壓力投影 Newton 方向" (pressure-projected Newton direction) without saying how it is formed.
It is not −H⁻¹g (cosine 0.634), nor the step on the seven negative directions (0.99993): it is the Newton step on the
six most negative ones — its components along the weakest negative and the positive eigen-directions are below 2×10⁻¹⁷,
the cosine 0.99999996 — though not exactly a scaled one, its ratios to the Newton step running from 0.0433 to 0.04353
(`newton_direction_cosines`) (Z5). The candidate's coefficients are exactly 3.2275 times it (Z7). The Newton-line file
is not the re-equalized walk §5 describes but the first-order path p₀ + Q·m·a_N: at m = 1 and 2 its least, the 270°
cusp, is the curve's there to 1.1×10⁻¹⁵ and 5×10⁻¹⁶ (`newton_line_rows_rebuilt`) (Z8). On that path the least rises only
to 5.3×10⁻⁸ above the base and is still 4.76×10⁻⁸ above it at m = 3.2 — the audit note's "Newton 線存在真上升窗" (a true ascent
window on the Newton line) is this line's.

### 6. The window

On the Newton line, a ninth local minimum first appears at m = 2.2, as stated (Z8). The re-optimized ledger, which
re-equalizes the five parameters at each m, does not show a single takeover at m ≈ 3.228. Its full-phase scale falls
below the four branches' at m = 3.2235 (by 1.0×10⁻⁷), 3.224, 3.2244 and 3.2245 (by 1.5×10⁻⁹), agrees with them again
from 3.2246 to 3.2275, and falls below for good from 3.228 (`ledger_full_minus_four_branch`) (Z8). The latent branch
takes over, gives way and takes over again within the ledger. That its rows are not one smooth family is an inference —
the ledger's parameters are not shipped — from the rates: the gap between the latent branch and the four changes by
about 10⁻⁴ per unit of m across 3.2235 to 3.2246, against about 2.4×10⁻⁶ beyond 3.24. The candidate at 3.2275 is not
touched by this: its own full phase is audited above.

### 7. Printed digits

The report boxes s₁₅ = 0.998914480716946 (the curve's least is 0.998914480716964), s₁₅ − s₁₃ = 1.374194610326×10⁻⁷
(1.374194794×10⁻⁷) and the latent margin 3.004601123457×10⁻⁹ (3.0045884×10⁻⁹): they hold 13 decimals, and 7 and 5
significant digits (Z10). The README repeats them.

### 8. What cannot be checked

The excluded four-branch candidate is given by its scales and the hidden branch's phase only, without its parameters or
coefficients, so neither its 0.998914692247305 nor its hidden branch at 0.9989115500 can be rebuilt; this run checks
only that the documents state the summary's figures (Z10). The ledger's re-equalized curves are not shipped — only their
scales — so its rows are checked for their own arithmetic and against each other, not rebuilt. The model file is a
docstring: none of the numbers can be reproduced from the package, only checked.

## At what strength each result stands

| result | strength |
| --- | --- |
| the candidate's branches, least and latent margin; the listed minima | high precision at 40 digits, the support from high-order quadrature of the mode curve |
| nine local minima in all | a double-precision scan of the shipped curve file at 65,536 phases |
| the candidate's least at least 1.3725×10⁻⁷ (at most 1.3758×10⁻⁷) above Round 13's | interval arithmetic: Simpson boxes for the density's integral with interval Taylor bounds, trapezoid positions, mpmath's iv at 30 digits, RUN-014's cover on both curves; bounds cut toward their safe side |
| Round 13's point not stationary in the modes | high precision: central differences (step 10⁻⁸) at 40 digits around RUN-014's 60-digit solution; not interval arithmetic |
| the base's gradients; the profiles and the Newton line along their paths | high precision at 40 and 60 digits |
| the Hessian, the Newton direction, the compensation, the ledger | the data's own arithmetic, recomputed |
| the curve file | double precision, to 4.3×10⁻¹² |

## The drill

| | |
| --- | ---: |
| planted defects | 25 |
| tripping exactly the checks named for them | **25** |
| controls (the package unmodified, the report with CRLF endings, the README with trailing spaces, the profiles copied) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **259/259** |

Plants include a triangle leg, a mode's maximum and the Gram error, a point of the curve file and the file reversed, the
Newton line's first row, the stationarity norm, a mode gradient (the weighted gradient and the compensation restated
from it), a compensation entry, the Hessian file, the positive count, a coefficient, an audit row's spread, a listed
minimum and the minima's count, a Newton-line norm, a ledger row, the takeover window, a profile's cusp and another's
minimum, the README's gain, a report row, the audit note's hidden phase, the handoff's phase and the manifest's files.
Where a plant moves a figure the checks compare elsewhere — a candidate record in the summary, the candidate file and
the audit note's block; a mode gradient and the weighted gradient and compensation derived from it, with the report's
table; a profile's cusp and its pressure scale; the minima's count and the README's — the restatements are planted
alike. Z3, Z6, Z8 and Z10 fail on the package, Z6 and Z10 with fixed-text messages, so plants trip them through a figure
that holds. The expected sets were reasoned before a dry run with every set empty was read; it tripped them but for one
plant: the mode gradient also tripped Z9, because the profiles follow the compensation that the planted gradient
changes, a dependency Z9 acquired when it was corrected to the compensated paths; the set was revised. After the
independent reading Z5 and Z8 were rewritten, and the sets of the triangle leg, the mode gradient and the compensation
entry extended to Z8, which now rebuilds the Newton line from the base, the compensation and the triangle; the final
drill confirmed every set. No plant reaches Z6: it reads no package data, only RUN-014's solution, and the triangle leg
leaves its fixed-text failure as it is; Z6 stands on its computation alone. No plant reaches Z1's names test, Z3's
pressure sum, Z5's span and symmetry tests, Z7's coefficient norm and its least-is-the-cusp test, Z8's ninth-minimum
test, or most of Z10's patterns. Of the four controls, the unmodified package exercises nothing by construction; the
report with CRLF endings re-runs Z10, the README Z10, the copied profiles Z9 and Z10. Each check, replaced in turn by
one that always passes, was reported blind in exactly as many cases as plants name it — Z1 2, Z2 2, Z3 3, Z4 2, Z5 2, Z6
0, Z7 5, Z8 6, Z9 4, Z10 5 — which by construction shows only that the harness counts as described (ten 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 rebuild of the curve file took both the
coordinate and the density from the candidate's own layer and missed by 10⁻⁷; the file is matched only with the
coordinate from Round 13's curve. The first exact profiles held the five parameters fixed and missed the stated cusps by
up to 2.9×10⁻⁷, the compensation's change to first order: the profiles run along the compensated paths. The first
interval boxes bounded the density's integral by the trapezoid rule and came out 10⁻⁹ wide, too wide for a 4×10⁻¹⁰
margin; Simpson's rule with interval Taylor bounds brought the integral's error below 10⁻¹⁵. The first replication of
the Newton direction asked for −H⁻¹g, which the round never states. An independent reading of this report before it was
committed confirmed the certificate's machinery (Simpson's constant, the Taylor recurrences, the trapezoid bound, the
premise and the support identity, the boxes against exact positions), the candidate's cusps and the first-order gradient
at RUN-014's solution with code of its own, and found the gain's upper end taken against Round 13's s(3π/2) rather than
its least, the Newton direction misidentified and its −H⁻¹g test misplaced, the latent margin's digits understated, the
profiles' first-order agreement called exact, the Newton line called unbuildable, the curve file said to pin the
coordinate more than it does, a quote misattributed, several maxima rounded down, and the first-order evidence told by
one mode of eight; all are corrected here, the upper end against RUN-014's certified lower bound recomputed, the Newton
line rebuilt, the box error terms rounded up.

## What this run does not establish

An interval enclosure of the mode gradients or of the latent branch's margin by itself (the certificate bounds the least
over all phases, which needs no margin). Whether the eight-mode family, or a wider one, goes further; the five-branch
system Round 16's handoff sets out. The excluded candidate. Any bound for Moser's problem. Nor the package's eight
figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run016_round15_functional_modes.py --drill
```

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