# RUN-010 — Round 09 (v0.9): the 270° cusp holds to 120 digits and its Darboux bound is sound; the draft's target over the whole phase circle is certified here — the candidate's least scale exceeds the event five-link's

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.9.zip` (SHA-256 `2cc4d8b3…`, pinned in [`corpus.py`](../code/corpus.py)): its README, MANIFEST,
`reports/ROUND_09_REPORT.md`, `ROUND_09_ARBITRARY_PRECISION_AUDIT.md`, `ROUND_09_INTERVAL_CERTIFICATE_DRAFT.md`,
`src/high_precision_verify.py` (read only after the rebuild, to compare), and its data (`round09_summary.json`,
`branch_audit.json`, `darboux_cusp_enclosure.csv`, `precision_convergence.csv`, `epsilon_peak_fit.json`), with Round
08's summary and peak scan and Round 05's event root, read from the package bytes
**Tools:** [`run010_round09_precision_cusp.py`](../code/run010_round09_precision_cusp.py) — this tree's own
high-precision construction of the candidate (mpmath quadrature split at the layer, at 100 to 200 digits) and its own
exact support (the four corners, and on each wing the one point whose tangent is perpendicular to the functional, by
inverting the tanh); the event five-link from its six vertices; exact one-sided derivatives from the active supports;
mpmath's interval arithmetic for the bounds; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run010-round09-precision-cusp.json`](../data/gate-logs/run010-round09-precision-cusp.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and three things it prints do not hold to
their digits. Reproduced: the candidate's scale at 3π/2 to the 120 digits the summary, the audit note and the manifest
give — this run's two quadratures at 200 digits agree with each other to every digit — and each convergence row to its
working precision; the event five-link's four branches, their minimum and the difference
s₂₇₀ − s₀ = 1.0581952124042297834593636×10⁻⁵, to 120 digits; the Darboux sums to the 100 or so decimals they hold, and
the premise that makes them bounds; the branch audit's values at their phases to 70 digits and its contact labels; the
slopes as difference quotients; the peak fits; and the documents' tables. Not as stated: §4 says "採用 $120$ 位結果" (the
120-digit result is adopted) and the README and §4 print an 80-digit s₂₇₀ that holds to 72 of its decimals, and the
README and §5 an 80-digit event minimum that holds to 71; the branch audit's 75-digit values hold to 70 or 71 decimals,
and the two low branches' are values at the phases a minimizer returned, 6.6×10⁻⁹ and 5.8×10⁻⁹ rad from the stationary
points — read as the branch minima the §7 table presents, they hold to 17; and §8's 50-digit one-sided derivatives are
difference quotients at h = 10⁻⁹, 5.0×10⁻¹⁰ from the derivatives, so 9 of their digits are the derivative's. Finding:
what the round's comparison still needed — that no phase brings the candidate below the event five-link — is its own
draft's target, with a margin of 1.06×10⁻⁵, not the 10⁻⁹-level gap its §11 cites; this run certifies it in interval
arithmetic over the whole phase circle: the candidate's least scale is at least 0.99891425946, above the event
five-link's, at most 0.99890375714 — 99 % of the gain. The round's own open step, that the least is at 3π/2, is not
certified here. The drill planted 19 defects; each tripped exactly the checks named for it, and the 4 controls disturbed
nothing.

---

## What Round 09 claims

Round 09 fixes Round 08's best candidate as "一條由明確十進位常數定義的曲線" (a curve defined by explicit decimal
constants): ε = 0.037, w = 0.3361057714712081, β = 1.405382794839393, δ = 0.05204734906280986, c = 0.5801781668857768,
with tangent angle θ(u) = β + δ (g(u) − g(1))/(g(0) − g(1)), g(u) = ½[1 − tanh((u − c)/ε)] — Round 08's layer, here
written out. At φ = 3π/2 the scale reduces to s₂₇₀ = y₁/A + (x₁ + l₀)/B, so "關鍵候選值只需兩個一維積分" (the key
candidate value needs only two one-dimensional integrals). Its §4 computes s₂₇₀ with mpmath's tanh–sinh and
Gauss–Legendre quadratures at 40, 60, 80 and 120 digits, which agree "在所有保存位數一致" (in every saved digit), and
"採用 $120$ 位結果" (adopts the 120-digit result). Its §5 recomputes the event five-link's four branches in high
precision — they differ at 10⁻¹⁶ because "第 5 輪只保存雙精度根" (Round 05 kept only a double-precision root) — and
takes their minimum s₀, so that s₂₇₀ − s₀ = 1.0581952124042×10⁻⁵. Its §6 notes that θ decreases inside (0, π/2), so
sin θ decreases and cos θ increases and left and right sums bound the two integrals: at 4,096 parts the lower bound
exceeds s₀ by more than 7.8×10⁻⁶ — a positivity that "來自被積函數的單調次序" (comes from the integrands' monotone
order) — with the limit that "Darboux 包絡只證明精確 $270^\circ$ 尖點值高於事件控制，不直接證明它是完整相位圓的全域最低值"
(the Darboux envelope proves only that the exact 270° cusp value is above the event control, not that it is the global
minimum over the full phase circle). §7 audits the candidate's four branches — two low-phase stationary minima, the
cusps at 2π/3 and 3π/2 — with the nearest competitor 1.6383×10⁻⁹ above s₂₇₀ at 120°; §8 estimates the cusps' one-sided
slopes with h = 10⁻⁹; §9 fits Round 08's peak scan, placing ε* between 0.0367 and 0.0371. §10 lists the full
phase-interval exclusion, a Krawczyk box and a formal proof as not done, and §11 says that with a global Lipschitz
constant and a uniform grid, "$120^\circ$ 與 $270^\circ$ 間的 $10^{-9}$ 級差距會迫使網格極度細密" (the 10⁻⁹-level gap
between 120° and 270° would force an extremely fine grid). The interval-certificate draft states the target as
s(φ) ≥ s₂₇₀ − η for every φ with η < s₂₇₀ − s₀. Its README: "本封裝不構成 Moser 問題的新上下界" (this package does not
constitute a new upper or lower bound for the Moser problem).

## The checks

| id | guards | result |
| --- | --- | --- |
| P1 | the decimal constants are Round 08's best parameters and Round 05's event root, as shortest decimals; the verifier script and the report's §2 use them; the g-form of the layer is RUN-009's F | **PASS** |
| P2 | this run's s₂₇₀ (two quadratures, 200 digits) holds the summary's, the manifest's and the audit note's 120 digits, and each convergence row to its working precision; the two methods' rows are identical | **PASS** |
| P3 | the event five-link's four branches from its vertices at 200 digits: the summary's values, minimum and difference, and the manifest's and the audit note's, to their digits (a difference of two 120-digit values to 120 decimals) | **PASS** |
| P4 | θ decreases inside (0, π/2), so the sums are bounds; each enclosure row's lower, upper, width and margin, and the audit note's, to 99 decimals | **PASS** |
| P5 | each branch-audit value is the exact scale at its phase, to 10⁻⁶⁸; its labels name active supports; each low phase lies within 10⁻⁷ rad of a stationary minimum; the margins; the cusp phases | **PASS** |
| P6 | each slope is the one-sided difference quotient at its h, to 10⁻⁴⁵, and within h of the exact one-sided derivative; D₋ < 0 < D₊ at both cusps | **PASS** |
| P7 | each fit is the least-squares polynomial of its degree through Round 08's six peak-scan scales: coefficients, ε*, scale, second derivative | **PASS** |
| P8 | the report's tables (§4–§10) and short figures, the README's lists and Darboux margin, the draft's open target, the manifest, and the summary's copies of the data files | **PASS** |
| P9 | every long figure of the README and the report, the branch audit's values and §8's derivatives hold to their printed digits | **FAIL** |

## Findings

### 1. The cusp value, the event control and their difference reproduce

This run builds the candidate from the decimal constants — P1: they are Round 08's best parameters and Round 05's
event root as shortest decimals, the verifier script and §2 use the same ones, and the g-form of the layer is RUN-009's
F to 10⁻⁴⁰ — and evaluates the two wing integrals with mpmath's tanh–sinh and Gauss–Legendre quadratures, split at the
layer's centre and 2, 6 and 12 widths either side (its own splits, not the verifier's), at 200 digits. The two agree to
every digit (`quadratures_differ_by`), and

  s₂₇₀ = 0.998914339084632717252964418536702246180192119314833519984798128351274681109990178989484072997947224021575344613070601820…

(`s270_120_decimals_cut`) holds the summary's, the audit note's and the manifest's 120 digits (P2). Each convergence
row is right to its working precision — the 40-, 60- and 80-digit rows to 41, 61 and 81 decimals, the 120-digit rows to
120 (`convergence_rows_digits_held`) — though each prints 125; the two methods' rows are identical at every precision,
as "在所有保存位數一致" says. The event five-link's four branches, from its six vertices at 200 digits, hold the summary's
120 digits (P3); B1 is the least, and B2, B3, B4 lie 8.2×10⁻¹⁷, 2.3×10⁻¹⁶ and 5.7×10⁻¹⁷ above it
(`event_branches_minus_minimum`). So

  s₂₇₀ − s₀ = 1.05819521240422978345936361457659629238926026414161953594484…×10⁻⁵,

which the summary, the manifest and the audit note print to 124 decimals; a difference of two 120-digit values, it holds
to 121 of them (`differences_digits_held`).

### 2. Printed digits

Figures printed past the precision they were computed at — the convergence rows, the enclosure rows, the differences —
are read here to that precision, and hold. Three groups are of another kind: they print more digits than they hold where
the round states a precision, or presents a value as something it is not (P9).

- **The adopted value.** §4 says "採用 $120$ 位結果" and boxes an 80-digit s₂₇₀, which the README repeats; it holds to 72
  decimals, and the 80-digit event minimum of §5 and the README to 71 (`long_figures_digits_held`). The 70-digit s₂₇₀
  and s₀ of §13, and the 64- and 49-digit differences of §5, §13 and the README, hold to every digit.
- **The branch audit.** Its 75-digit values are the exact scale at their phases to 70 or 71 decimals
  (`branches.*.digits_held`). The two low branches' printed phases, 0.15469913367438023 and 0.12408080074251457, lie
  6.6×10⁻⁹ and −5.8×10⁻⁹ rad from the stationary points; the phase derivative at the printed phases is 2.2×10⁻⁹ and
  −1.9×10⁻⁹, and their values exceed the branch minima by 7.3×10⁻¹⁸ and 5.6×10⁻¹⁸. Read as the branches' values, as §7's
  table presents them, they hold to 17 decimals.
- **The slopes.** §8 prints the cusps' one-sided slopes to 50 digits. They are the difference quotients at h = 10⁻⁹ —
  P6 reproduces every quotient in the branch audit to 10⁻⁴⁵ — and the exact one-sided derivatives, from the active
  supports by the envelope rule, are

  120°: D₋ = −0.267658287627918161991503299…, D₊ = 0.612416405839174998855469498…
  270°: D₋ = −0.329480283321469717992727949…, D₊ = 0.432686758448835705462640528…

  Every quotient lies 0.4995 h from its derivative, at every h (`quotient_minus_exact`): §8's figures hold to 9 of their
  50 decimals. The signs D₋ < 0 < D₊, which is what §8 concludes, hold.

### 3. The Darboux bound is sound

θ decreases (δ > 0) from α = 1.4574 to β = 1.4054, inside (0, π/2), so sin θ decreases and cos θ increases: on each part
the right value of sin θ and the left value of cos θ are the smaller, and the sums built from them bound y₁ and x₁ from
below, the other two from above. Both enter s₂₇₀ with positive coefficients, so the sums bound s₂₇₀ (P4); that s(3π/2) is
y₁/A + (x₁ + l₀)/B is the support identity at that phase — θ in (0, π/2) makes the wing rise to the right, so the ends
tie on the leg and the wing's end p₁ carries the hypotenuse — derived here, not checked in interval arithmetic. Every
row's bounds, width and margin hold to 100 to 102 decimals of the 104 to 110 printed (`darboux_rows_digits_held`); P4
asks for 99, the package verifier's working precision being 100 digits. Redone in interval arithmetic, the 4,096-part
lower bound is 0.9989115747096505504998971, the stated bound less 8.8×10⁻²⁹ of rounding allowance, and exceeds the event
five-link's interval upper bound (§4 below) by more than 7.8175771418×10⁻⁶ (`darboux_4096_interval_*`): the round's
statement that s(3π/2) exceeds s₀ by more than 7.8×10⁻⁶ is certified. It concerns one phase, as the round says.

### 4. The whole phase circle, certified for the comparison

For its comparison the round needs that no phase brings the candidate below the event five-link. Its §11 frames the
open step as resolving the "$10^{-9}$ 級差距" (10⁻⁹-level gap) between the 120° and 270° branches — §7's 1.6×10⁻⁹ — that
is, as showing that the least is at 270°; the draft states the target the comparison needs, s(φ) ≥ s₂₇₀ − η for every φ
with η < s₂₇₀ − s₀, a margin of 1.06×10⁻⁵, about 6,500 times wider. This run certifies it.

For any point q of the curve, s(φ) = Σₖ maxₚ nₖ(φ)·p ≥ Σₖ nₖ(φ)·q, so points of the curve bound s from below, and
nₖ(φ)·q changes with φ no faster than |aₖ||q|. The candidate's points at u = j/65,536 on both wings are enclosed in boxes:
each step's integral of cos θ and of sin θ is its trapezoid value within h³M₂/12, where M₂ = 15.129 bounds the second
derivative of both (|f″| ≤ θ′² + |θ″|, from |F′| ≤ 1/(ε(T₀ − T₁)) and |F″| ≤ (4/(3√3))/(ε²(T₀ − T₁))), all in mpmath's
interval arithmetic at 30 digits. The widest box is 5.9×10⁻¹⁰ by 2.0×10⁻¹⁰, on the right wing (2.0×10⁻¹⁰ square on the
left), the mirror image carrying the end's width twice. With the curve centred (radius at most 0.2674; the three
functionals sum to zero, so a translation leaves s unchanged), the functionals' phase derivatives are at most
Lip = 1.3306, both bounds in interval arithmetic. A floating-point search bisects the phase circle, taken as
[0, 6.2832], until on each interval an estimate at its midpoint clears the target; on each of the 42,838 intervals, which
cover it, the bound — three chosen points' boxes at the midpoint, less Lip times the half-width — is then taken in
interval arithmetic. The least of the event five-link's scales at its four branch phases (enclosing intervals), from its
vertices in interval arithmetic, is at most 0.998903757132508674955129824903, which bounds its least scale from above.
So

  candidate's least scale ≥ 0.99891425946916933623 > 0.998903757132508674955129824903 ≥ the event five-link's least scale,

a certified margin of more than 1.0502336×10⁻⁵, 99 % of the round's gain (`certificate`); every figure of a bound is cut
toward its safe side. Floating-point arithmetic chooses only the intervals and the points. As a control, a target above
the candidate's own certified value at 3π/2 (at most 0.99891433937), asked on a neighbourhood of 3π/2, is refused there
(`control_beside_3pi_2_target_above_s270_refused`). The candidate's least scale lies between 0.99891425946 and
s₂₇₀, within 8.0×10⁻⁸. That it is reached at 3π/2 — §11's target, "s(φ) ≥ s₂₇₀" for every φ — is not certified here.

### 5. The branches and the peak fits

At the low branches the supports are the left wing's interior (min x), the start p₀ (min y) and the right wing's start
p₂ (max d), as labelled. At 120° the hypotenuse functional ties the two endpoints p₀ and p₃, and at 270° the leg
functional ties them; the cusps are those ties, and the labels name one point of each tied pair (`branches.*.active`).
The low branches' stationary phases are 0.154699127028335487754398589663 and 0.124080806550179655854701882269, their
minima 0.998914384873626650041301807… and 0.998914385986533194245715981…. The peak fits reproduce: ε* to 3.9×10⁻¹², the
scales to 2.2×10⁻¹⁶, the second derivatives to 5.2×10⁻¹¹ (P7, `peak_fit_minus_stated`).

## At what strength each result stands

| result | strength |
| --- | --- |
| s₂₇₀ to 120 digits, the event branches, s₀ and the difference | high precision: two quadratures at 200 digits, agreeing to every digit; the event's closed by its vertices |
| s(3π/2) − s₀ > 7.8×10⁻⁶ | certified: interval arithmetic, given the support identity at 3π/2 |
| the candidate's least scale above the event five-link's, by more than 1.050×10⁻⁵ | certified: interval arithmetic over the whole phase circle, for Round 09's decimal curves — the candidate and the event five-link alike |
| the least reached at 3π/2 | numerical (RUN-009's double-precision scan); certified only to within 8.0×10⁻⁸ |
| the exact one-sided derivatives, the stationary phases and minima | high precision, 100 digits |
| the printed-digit defects | exact as to the printed strings; high precision as to the values |

## The drill

| | |
| --- | ---: |
| planted defects | 19 |
| tripping exactly the checks named for them | **19** |
| controls (the package unmodified, the report with CRLF endings, the README with trailing spaces, the branch audit copied) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **180/180** |

Plants include a triangle leg, Round 08's layer centre, a verifier constant, a digit of the summary's s₂₇₀, a
convergence row, an event branch, an enclosure bound, a branch margin, a contact label, a cusp slope, a fit coefficient,
a scale of Round 08's peak scan, a margin in §7's table, the README's enclosure margin, a digit of §13, a term of the
cusp formula, the manifest's difference, the summary's copy of the fits, and a digit of §8. The expected sets were
written down after the plants and before the dry run's result was read, and the dry run, with every set empty, tripped
exactly them; so 19 of 19 says the reasoning matched the harness, not that the plants were chosen blind. Where a plant
moves a figure the summary restates — a convergence row, an enclosure bound, the branch audit, the fits — the copy is
planted alike, so that only the comparison can see the plant; the summary's copy is planted alone in a plant of its own.
P9 fails on the package with fixed-text messages, one per figure, so its plants trip it through a figure that holds.
Branches with no plant of their own: P1's event constants, the verifier's formula fragments, the report's §2 and the
g-form; P2's agreement of the two quadratures; P4's premise; P5's cusp phases; P6's distance to the derivative; most of
P8's tables and figures; and P9's figures that already fail. Margins (`check_margins`): P3 and P4 hold one decimal
beyond what they ask (121 and 100 against 120 and 99) — tolerances set from the working precision the figures carry, not
from the deviations; P6's distance test at 2 times (0.4995 h against h) and its quotient test at 2.5×10⁵ times; P5's
value test at 540 times, P7's at 19 to 45 times. Of the four controls, the unmodified package exercises nothing by
construction; the report with CRLF endings re-runs P1, P8 and P9; the README with trailing spaces P8 and P9; the copied
branch audit P5, P6, P8 and P9. Each check, replaced in turn by one that always passes, was reported blind in exactly as
many cases as plants name it — P1 2, P2 4, P3 4, P4 3, P5 3, P6 2, P7 2, P8 4, P9 3 — 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 P3 and P4 asked the figures to hold to
every printed decimal and failed: the summary's difference is printed to 124 decimals from 120-digit terms, and the
enclosure rows to 104 to 110 from arithmetic good to about 100. The first P5 and P6 evaluated the cusps at their printed
60-digit phases, where the tie is broken at 10⁻⁶⁰ and the value moves by more than the 10⁻⁶⁸ asked; they now use the
exact cusp phases and check the printed ones against them. A first comparison of §7's low phases assumed rounding; the
report cuts them. The first reading of the README compared a 120-digit string at double precision. The certificate's
first figures were rounded to nearest, one lower bound ending a unit above what was proved; they are now cut toward
their safe side. An independent reading of this report before it was committed found that the first certificate's boxes
— from the monotone sums of §6 — were three times wider on the right wing than reported, and its radius and Lipschitz
constant taken in floating point with pads; the boxes are now trapezoid enclosures 300 times narrower, both constants
are interval bounds, and the margin certified rose from 95 % to 99 % of the gain. It also found the title claiming the
round's own open step, the refusing control passing for a reason other than its target, the 120-digit display rounded,
and several figures of bounds on the unsafe side; all are corrected here, and the new figures are logged.

## What this run does not establish

Any bound for Moser's problem: the candidate's least scale is about 0.99891, below 1, and the README says the package
"不構成 Moser 問題的新上下界". That the least is reached at 3π/2: the certificate bounds the least from below, within
8.0×10⁻⁸ of the value there, and the location rests on RUN-009's double-precision scan. That the candidate is the tanh
family's best, or a local maximum among curves. Anything about the curves Round 08 stored in double precision, as
against Round 09's decimal ones: their constants differ by at most 1.01×10⁻¹⁶ (`decimal_minus_double_constants_largest`),
which a displacement bound would carry far inside the certified margin, but the transfer was not carried out. Nor the
package's five figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run010_round09_precision_cusp.py --drill
```

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