# RUN-006 — Round 05 (v0.5): the contact-event system reproduces; its root is a strict local maximum, of the four-branch model and so of the least scale on a neighbourhood of unestablished size

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.5.zip` (SHA-256 `2a86975f…`, pinned in [`corpus.py`](../code/corpus.py)): its README,
`reports/ROUND_05_REPORT.md`, `src/event_system.py` (a docstring, no code), and its data (`round05_summary.json`,
`event_candidate_curve.csv`, `event_branch_gradients.csv`, `event_kkt_jacobian.csv`, `perturbation_convergence.json`),
read from the package bytes (the documents passed through RUN-001's `restore_escapes`, which changes none of them)
**Tools:** [`run006_round05_contact_events.py`](../code/run006_round05_contact_events.py) — the report's four closed
forms written out, with complex-step derivatives; this run's own Newton iteration on the nine equations; an exact
Jacobian; RUN-003's exact formula and minima over phases; RUN-005's event-phase values; [`gate.py`](../code/gate.py)
for the drill
**Log:** [`run006-round05-contact-events.json`](../data/gate-logs/run006-round05-contact-events.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and two of its printed figures are
not right to the digits printed. Reproduced: the triangle; the logged curve as the event parameters; the four event
phases, exactly; the report's four closed forms, which equal the exact scale at the event phases to 3.3×10⁻¹⁶ and give
the stated branch values; the least scale over all phases, which is the common scale 0.998903757132509, reached at
all four events, with the next local minimum 0.030 higher; the logged gradients, to 8.9×10⁻¹⁶; the nine equations,
which hold at the stated solution to 5.7×10⁻¹⁶; the logged Jacobian's rank 9; the 40 perturbation trials; the contact
ledgers; the signs of the one-sided derivatives; and the documents' numbers. Not as stated: the one-sided derivatives
of §9, printed to nine decimals, are one-sided differences right to within 4×10⁻⁸; and the smallest singular value
and condition number, printed to thirteen and twelve significant digits, are a difference Jacobian's — the exact
Jacobian's are 3.8×10⁻¹⁰ and 1.7×10⁻⁶ away. Findings: the root satisfies the second-order sufficient condition for a
maximum of the four-branch max-min — the Lagrangian's curvature along the one critical direction is −0.123 — which is
more than the round's "數值孤立駐定點" (numerically isolated stationary point); and since every contact identity
holds there with a strict margin, the least scale over all phases equals the four-branch minimum on some
neighbourhood of the root, so the root is a strict local maximum of the curve's least scale too, on a neighbourhood
whose size is not established. This run's Newton iteration reaches the same root from Round 04's point and from
RUN-005's. The drill planted 19 defects; each tripped exactly the checks named for it, and the 4 controls disturbed
nothing.

---

## What Round 05 claims

Round 05 takes Round 04's four branches (RUN-005) and writes their phases from the active contact identities rather
than tracking them with a one-dimensional minimizer: φ₁ = π/2 − α (vertices q₀ and q₁ on the left leg), φ₂ = π/2 − β
(q₁ and q₂ on it), φ₃ = 2π/3 (both endpoints on the hypotenuse, "由 Wetzel 三角形" (from the Wetzel triangle) with
B/A = √3), and φ₄ = 3π/2 (both endpoints on the left leg). With α = β + δ, l₃ = 1 − 2(l₁ + l₂), q₂ = l₁v_α + l₂v_β,
q₃ = q₂ + l₃(1, 0) and the endpoints' horizontal span D = 2l₁cos α + 2l₂cos β + l₃, it writes the four branch values
as closed forms in p = (l₁, l₂, β, δ):

  m₁ = H_φ₁(q₃),  m₂ = H_φ₂(q₃) − x_φ₂(q₁)/A,  m₃ = −x_φ₃(q₃)/A − y_φ₃(q₂)/B,  m₄ = y₂/A + (D − x₂)/B,

where H_φ is the rotated hypotenuse functional. It observes that m₁ = m₂ = m₃ = m₄ alone gives three equations in
four unknowns, "一般形成一維候選族" (in general a one-parameter family of candidates), and adds the branch pressures
— μ_r ≥ 0, Σμ_r = 1, Σμ_r∇_p m_r = 0 — to obtain nine equations in nine unknowns (p, s, μ₁, …, μ₄). Its solution:
s_event = 0.998903757132509, the four values level to 2.220446049250×10⁻¹⁶, 6.656150519646×10⁻⁹ above Round 04's
audited value, with residual ‖F‖ = 2.777480200316×10⁻¹⁶; pressures (0.0280, 0.0288, 0.7365, 0.2067); a 9×9 Jacobian
of numerical rank 9, smallest singular value 3.607219048175×10⁻², condition number 162.781733366; 40 of 40 perturbed
starts back at the same root, at most 8.3×10⁻¹⁵ away; one-sided derivatives with D₋s(φ_r) < 0 < D₊s(φ_r) at all four
events, so each is a kink minimum; and a 262,144-phase audit whose grid minimum is the event scale, at 3π/2. Its
reading, prefaced "這不構成解析唯一性證明，但強烈支持" (this is not a proof of analytic uniqueness, but strongly
supports): "目前候選是此固定接觸拓撲中的數值孤立駐定點" (the candidate is a numerically isolated stationary point in
this fixed contact topology); black-box search in this topology "已大致收束" (has largely converged). Its limits (§14)
include double precision, numerical rank, perturbation return not being uniqueness, no interval Newton or Krawczyk
certificate, no proof that this topology is optimal among five-links, and no bound. The package ships no code;
`src/event_system.py` is a docstring stating the unknowns and the nine equations.

## The checks

| id | guards | result |
| --- | --- | --- |
| V1 | the triangle's A and B are the certified legs over 1.0048, B/A = √3; the logged curve is the event parameters (from the origin, α = β + δ, l₃ = 1 − 2(l₁ + l₂), degrees the radians'), of length 1, its own mirror image | **PASS** |
| V2 | the event phases are π/2 − α, π/2 − β, 2π/3 and 3π/2, in radians and degrees, and the ledgers' phases are theirs | **PASS** |
| V3 | the report's four closed forms are the exact scale at the event phases, and both are the stated branch values, to 10⁻¹⁴; the common scale and the spread are theirs | **PASS** |
| V4 | the least exact scale over all phases is the common scale, reached at all four events, with no fifth local minimum within 10⁻⁶; the stated 262,144-phase audit is that grid's minimum, at its stated phase | **PASS** |
| V5 | the logged gradients are the closed forms' complex-step derivatives, to 10⁻¹² | **PASS** |
| V6 | μ is on the simplex; the nine equations hold at the stated solution to 10⁻¹⁴, and the stated residual is at that level; the improvement over Round 04 and the distance to 1 are the differences they name | **PASS** |
| V7 | the logged Jacobian is this run's difference Jacobian, to 10⁻⁷; its shape, rank, singular values and condition number are the stated ones | **PASS** |
| V8 | the 40 stated trials all succeed, at the stated largest distance; this run's Newton iteration from 40 starts, each unknown perturbed by a Gaussian of standard deviation 10⁻², returns to the stated root, to 10⁻¹² | **PASS** |
| V9 | at each event the contact ledger is the recorded one, and the exact one-sided derivatives are the stated ones to 10⁻⁶, with D₋ < 0 < D₊ | **PASS** |
| V10 | the README and the report state the data's numbers and tables; `event_system.py` states the report's nine equations and unknowns | **PASS** |

## Findings

### 1. What reproduces, and how exactly

- **The setup.** A and B are the certified legs divided by 1.0048, to 1.1×10⁻¹⁶ (V1, `precision`). B/A = √3 holds
  exactly — 3(4 + √3) = √3(3 + 4√3) — and in floating point to 2.2×10⁻¹⁶. The logged curve starts at the origin and is
  the event parameters to 5.6×10⁻¹⁷, of length 1, its own mirror image (V1); the event phases are π/2 − α, π/2 − β,
  2π/3 and 3π/2 exactly (V2).
- **The closed forms.** Written out as the report gives them, the four closed forms equal the exact scale at the four
  event phases to 3.3×10⁻¹⁶ and give the stated branch values (V3) — so the report's formulas are the right ones for
  these contact identities, and the event phases are where Round 04's branch minima lie (RUN-005 §3).
- **The landscape.** The least exact scale over all phases is the common scale, 0.9989037571325088; all four events
  are local minima at that value, level to rounding, and the scan reports 3π/2 because it is a grid point; the fifth
  lowest local minimum is 0.0301 higher (V4, `fifth_minimum_above`). The 262,144-phase audit reproduces: 262,144 = 2¹⁸
  is a multiple of four, so 3π/2 is a grid point and the grid's minimum is the event scale exactly; none of the other
  three event phases is a grid point (`grid_holds_event_phase`), as the report's §11 says.
- **The equations.** The logged gradients are the complex-step derivatives of the closed forms to 8.9×10⁻¹⁶ (V5).
  At the stated solution the nine equations leave 5.7×10⁻¹⁶ (V6); the round's 2.8×10⁻¹⁶ is reproduced in order, not
  in digits. Every block of the logged Jacobian is differenced — its Σμ row, whose entries are exactly 1, reads
  1.000000000372, 1.0000000026549 and 0.99999999986574 — and it lies 2.65×10⁻⁹ from the exact Jacobian
  (complex-step first derivatives, Richardson-extrapolated second derivatives, self-consistent to 2.2×10⁻¹¹), this
  run's difference Jacobian 1.06×10⁻⁹ from it (`exact_jacobian`). Its rank 9 is not in doubt; its last singular
  values are §4's. The 40 trials all succeed, at most 8.3×10⁻¹⁵ from the root (V8).
- **The cusps.** At each event the vertices on the two legs and the hypotenuse are the recorded ones, and the
  one-sided derivatives have opposite signs, the smallest in size 0.0238 (V9).
- **The documents.** The README and the report state the summary's numbers, the gradient, pressure, contact and
  slope tables to the digits printed, and the docstring states the nine equations (V10).

### 2. The root is a strict local maximum

The round calls its root a numerically isolated stationary point; it is more. The equations come from the problem:
maximize z over (p, z) subject to m_r(p) ≥ z, r = 1, …, 4, with Lagrangian L = z + Σ_r μ_r (m_r(p) − z), μ ≥ 0. Write
G for the 4×4 matrix whose rows are the branches' gradients. At the root every μ_r is positive, so all four
constraints are strongly active, and the second-order sufficient condition for a strict local maximum asks that
dᵀ(Σ_r μ_r ∇²m_r)d < 0 for every nonzero (d, dz) in the critical cone {(d, dz) : Gd = dz·𝟙}. Here μᵀG = 0 is the
stationarity equation itself, so G has rank at most 3; its singular values are 5.68, 0.314, 0.0407 and 1.7×10⁻¹⁷
(`gradient_rank`). A critical direction then has dz = dz·(μ·𝟙) = μᵀGd = 0, so Gd = 0, and the cone is the line through
G's unit right null vector v. Along it

  vᵀ(Σ_r μ_r ∇²m_r)v = −0.1227,

by central differences of the complex-step gradients, and again from the closed forms' values (`second_order`). The
condition holds: the root is a strict local maximum of the four-branch max-min min_r m_r(p).

This is also what the round's isolation rests on. The KKT Jacobian is nonsingular exactly when the constraints'
gradients [G, −𝟙] have full rank — they do, since 𝟙 is not orthogonal to μ — and vᵀ(Σμ∇²m)v ≠ 0; the round's rank 9
reports that this scalar does not vanish, and this run adds its sign.

The same line has a plainer meaning. Because the differences m_r − m₄ have independent gradients at the root, the
level set m₁ = m₂ = m₃ = m₄ is a smooth curve there — the report's one-parameter family — with tangent v; along it the
common value has zero first derivative (μᵀGv = 0) and, at unit speed, second derivative −0.1227. The stationarity
equations pick a local maximum of the common value along the family.

**From the model to the curve.** The curve's least scale is S(p) = min over φ of s(φ; p); the model is min_r m_r(p).
At the root every contact identity holds with a strict margin: in each event's ledger the terms not tied have a single
active vertex and the tied term exactly two, each at least 0.0067 from the next vertex (`identity_margins`); the
one-sided derivatives are strictly signed, |D±| ≥ 0.0238; and every other local minimum over phases is at least
0.030 higher. All of these move continuously with p, so on some neighbourhood of the root each branch's minimum stays
a kink at its event phase, with the same identities, and no other phase comes lower: S(p) = min_r m_r(p) there, and
the root is a strict local maximum of S. That holds at the strength of the margins, which are double-precision
numbers; how large the neighbourhood is would need the interval certificate the round plans
("局部不存在更高解的計算證書草案", a draft computational certificate that no higher solution exists locally) or its
"活動身份鄰域" (the active identities' neighbourhood). It is not attempted here.

Probes agree. Along the straight line p ± tv the least closed form falls by 2.04×10⁻⁹, 2.04×10⁻⁷ and 2.04×10⁻⁵ at
t = 10⁻⁴, 10⁻³ and 10⁻²; along the level curve, corrected back onto it by Newton's method, the common value falls by
6.1×10⁻⁸, 5.5×10⁻⁷ and 6.1×10⁻⁶ at t = 10⁻³, 3×10⁻³ and 10⁻² — both quadratic, with second difference −0.1227 — and at
every probe point the least scale over all phases equals the least closed form to 2.2×10⁻¹⁶
(`critical_line_probes`). Sixty random points around the root, each parameter perturbed by a Gaussian of standard
deviation 10⁻⁵ (distances 5.8×10⁻⁶ to 3.7×10⁻⁵), and sixty at 10⁻⁴ (3.7×10⁻⁵ to 3.4×10⁻⁴), give the same equality to
5.6×10⁻¹⁶ and all lie below the root's value (`identity_stability`); those drops are first-order, transverse to v,
and say nothing about the critical direction, which the line probes cover.

### 3. The root is reached from the earlier rounds

This run's Newton iteration on the nine equations — the closed forms, their complex-step gradients, and a difference
Jacobian — returns to the stated root to 2.2×10⁻¹⁶ from the root itself, and reaches it from Round 04's equalized point
(8.0×10⁻¹⁶) and from RUN-005's SLSQP point (3.1×10⁻¹⁵), each started with equal pressures (`newton_from_*`). RUN-005's
point is within 6×10⁻⁹ of the root in each parameter and 1.0×10⁻¹⁵ below it in value (`run005`). From 40 random
starts, each unknown perturbed by a Gaussian of standard deviation 10⁻³ or 10⁻², the iteration returns every time; at
3×10⁻², 32 of 40 (`newton_returns_of_40`). The round does not say how far its 40 starts were perturbed.

### 4. Printed digits that are not the quantities'

- **The one-sided derivatives.** The exact derivatives of the active pieces — for a fixed vertex, dx/dφ = −y and
  dy/dφ = x after the rotation — differ from the stated ones by 3.0×10⁻⁹ to 3.9×10⁻⁸
  (`one_sided_exact_minus_stated`). The stated values are consistent with one-sided differences of the exact scale at
  step 10⁻⁸: three of the eight agree bit for bit and the rest by one or two rounding units over the step
  (`one_sided_differences_step_1e-8`). The report's §9 table, printed to nine decimals, is right to within 4×10⁻⁸
  (B3's left slope prints −0.267655431; the derivative is −0.2676554550). V9 allows 10⁻⁶, as RUN-005's E7 did, so it
  passes; but by the standard RUN-002 and RUN-003 applied to printed last digits, these are not as stated.
- **The smallest singular value and the condition number** are the logged difference Jacobian's. The exact Jacobian
  gives σ_min = 3.607219010513×10⁻², 3.8×10⁻¹⁰ below the printed 3.607219048175×10⁻² — right to eight significant
  digits of thirteen — and κ = 162.781735059, 1.7×10⁻⁶ above the printed 162.781733366; the other singular values
  differ by up to 4.2×10⁻¹⁰ (`exact_jacobian`). None of this touches the rank or the isolation.
- **"約 $10^{-9}$"** (about 10⁻⁹), the report's §5 gloss on the gain over Round 04, sits beside the printed exact
  6.656150519646×10⁻⁹, nearer 10⁻⁸ on a log scale; a loose gloss, not counted.

## At what strength each result stands

| result | strength |
| --- | --- |
| the event phases and the closed forms for these contact identities | exact (the identities; RUN-005 §3) |
| the branch values, the common scale, the gradients, the residual | numerical, double precision; the gradients complex-step, exact to rounding |
| the least scale over all phases | numerical: a 3,600-phase scan and an 8,192-phase branch scan, every local minimum refined by golden section; not proved global |
| the Jacobian's rank 9 | numerical: singular values of a difference Jacobian and of an exact one |
| the root as a strict local maximum of the four-branch model | numerical: the second-order sufficient condition in double precision |
| the root as a strict local maximum of the curve's least scale | numerical: the model's maximum, transferred by continuity from strict double-precision margins, with line and random probes; the neighbourhood's size not established; no interval certificate |
| uniqueness of the root, optimality of this topology among five-links | not established (as the round says) |

## The drill

| | |
| --- | ---: |
| planted defects | 19 |
| tripping exactly the checks named for them | **19** |
| controls (the package unmodified, the report with CRLF endings, the curve copied, the README with trailing spaces) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **177/177** |

Plants include a triangle leg, the common scale, an event phase, an event phase's degrees, a length, a pressure, a
singular value, a contact ledger, a cusp slope, the improvement, the grid audit's minimum, a gradient entry, a
Jacobian entry, a trial's distance, a curve point, a report digit, a README digit, the docstring's equation count, and
a term of the exact formula. The expected sets were written after a dry run with every set empty, each trip reasoned
before it was named, so 19 of 19 says the reasoning matched the harness, not that the plants were chosen blind. Two
plants were enlarged after the dry run because their trips sat at the edge of a tolerance: the common scale's, from
10⁻¹² to 10⁻¹¹ (the Newton check's 10⁻¹²), and the grid minimum's, from 10⁻¹⁵ to 10⁻¹⁴ (the audit check's 10⁻¹⁵, which
the smaller plant crossed only through an exact-equality test on a stored difference). The curve-point plant trips the
setup check alone, since every other check rebuilds the curve from the parameters. Two of the four controls exercise
nothing by construction — the unmodified package, and the curve replaced by an identical copy — and the other two
exercise only the consistency check. Each check, replaced in turn by one that always passes, was reported blind in
exactly as many cases as plants name it — V1 3, V2 2, V3 4, V4 6, V5 4, V6 6, V7 5, V8 6, V9 4, V10 13 — 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 three ways worth recording. The Newton check first stopped the
drill: under the formula plant a perturbed start met a singular Jacobian and raised an error; a singular step now
counts as a start that does not return. The exact one-sided derivatives ignored the world's rotation sign and the
formula's B term, so the cusp check could not see the formula plant; they now follow the world, and that plant trips
it. And the first findings left several of the report's figures out of the log; they are now in `precision`,
`exact_jacobian`, `one_sided_differences_step_1e-8`, `critical_line_probes` and `identity_margins`. An independent
reading of this report before it was committed found the PASS verdict inconsistent with the earlier runs, the
singular value and condition number not right to their digits, the bridge from the model to the curve both overstated
and understated, the Jacobian's differencing misdescribed, the samples' distances misdescribed, an omitted hedge of the
round's, and the weak drill catches; all are corrected here.

## What this run does not establish

Any bound for Moser's problem. That the root is the five-link family's maximum, or the only root, or that this contact
topology is the best one. The size of the neighbourhood on which the root is a local maximum of the curve's least
scale. Any interval certificate. Anything about the package's eight figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run006_round05_contact_events.py --drill
```

The findings read RUN-005's committed gate log for its SLSQP point. The run is correct when the gate log is
byte-identical to the committed one.