# RUN-005 — Round 04 (v0.4.1): the equalized five-link reproduces; its four branch minima sit at exact phases, and re-equalized there they level to 10⁻¹⁵, 6.7×10⁻⁹ higher

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.4.1.zip` (SHA-256 `acc4235c…`, pinned in [`corpus.py`](../code/corpus.py)): its README,
`reports/ROUND_04_REPORT.md`, `reports/ROUND_04_CUSP_AUDIT.md`, `src/METHOD.md`, and its data
(`round04_summary.json`, `branch_sensitivity_matrix.csv`, `equalized_5link_curve.csv`), with Round 03's summary for
the starting curve and round v0.5's README for one cross-check, read from the package bytes (every document passed
through RUN-001's `restore_escapes`, which changes only the audit)
**Tools:** [`run005_round04_branch_equalization.py`](../code/run005_round04_branch_equalization.py) — RUN-003's exact
formula and minima over phases, RUN-004's five-link family, branch minima and contacts; central differences of the
true branch minima; SciPy's SLSQP for this run's own equalization; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run005-round04-branch-equalization.json`](../data/gate-logs/run005-round04-branch-equalization.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and two of the things it states do not
hold as stated. Reproduced: the equalized curve as its parameters; its least scale over all phases, 0.998903750476,
at φ = 3π/2, to 5×10⁻¹⁵; the starting curve as Round 03's five-link; all nine local minima over phases; the
resolution table's counts; the four branches' contact ledgers; the pressure vector's arithmetic; and the documents'
numbers. Not as stated: the README's and the main report's four branches level to 2.734812376559×10⁻¹² are the
tracker's values — the true branch minima span 1.30×10⁻⁸, and the round's own cusp audit withdraws the claim; and the
sensitivity matrix, printed to eight decimals, differs from the branches' derivatives by 2.3×10⁻⁵, 2.4×10⁻⁵,
1.3×10⁻⁴ and 1.5×10⁻⁴ in its four rows (E7). Findings: the four branch minima lie at exact phases — where link 1 or
link 2 is parallel to the leg x = 0, and at 2π/3 and 3π/2, where the curve's symmetry ties two vertices — so each
branch value is a closed form in the curve's parameters, smooth wherever the contact identities hold; with these,
the round's epigraph problem levels the four branches to 1.1×10⁻¹⁵ at 0.998903757133, 6.7×10⁻⁹ above the round, the
value the next round reports. The drill planted 17 defects; each tripped exactly the checks named for it, and the 4
controls disturbed nothing.

---

## What Round 04 claims

Round 04 starts from Round 03's five-link (RUN-004) and tracks the four lowest local minima of its phase landscape
— B1 and B2 at low phases, B3 at 120°, B4 at 270° — as functions m_r(η) of η = (l₁, l₂, β, δ). It solves the epigraph
problem

  max z over (η, z), subject to m_r(η) ≥ z, r = 1, …, 4,

with SLSQP, and reports the four tracked values level to 2.734812376559×10⁻¹² ("四分支等高誤差", the four-branch
level error, in the README), a critical scale s₅⁽⁴⁾ = 0.998903750476 from a full rescan of phases,
1.493788082036×10⁻⁴ above Round 03, and new parameters (l₁, l₂, l₃) = (0.1943225812, 0.1415865167, 0.3281818042),
(β, α) = (80.68621471°, 83.41615042°). It gives a finite-difference sensitivity matrix G_rj = ∂m_r/∂η_j, from which
it reads "不同分支對同一參數的反應方向互相衝突。沒有一個局部方向能讓四個分支全部上升，因而形成 max-min 等高點" (the
branches respond to the same parameter in conflicting directions; no local direction raises all four, so a max-min
level point forms); a "第二層分支壓力" (second-level branch pressure) μ on the simplex with Σ_r μ_r ∇_η m_r ≈ 0
(residual 1.840159765877×10⁻⁷); and per-branch contact ledgers. Its conclusions (§7, §10): "不增加線段數，只做最弱相位分支等高"
 (without adding links, only levelling the weakest phase branches) the five-link family's critical scale
"仍能繼續提高" (can still be raised), and a curve's difficulty is "一組競爭放置分支的多帳本平衡" (a balance of several
ledgers among competing placement branches). Its limits (§9) include floating-point one-dimensional minimization,
finite differences, no global guarantee for SLSQP, no interval certificate, and no proof that the active identities
are stable in a whole neighbourhood.

The package also holds a **cusp audit**. The tracker's bounded one-dimensional minimization stopped beside a cusp;
a full rescan, on grids up to 65,536 phases with every local minimum refined, puts the true global scale at
0.998903750476358, at φ = 4.712388980384675, "即接近精確的" (that is, close to the exact) 3π/2, 1.301910645957×10⁻⁸
below "追蹤器的最低值" (the tracker's lowest value). It withdraws the level claim — "不能宣稱四個真實局部極小已等高到 $10^{-12}$"
 (it cannot be claimed that the four true local minima are level to 10⁻¹²) — and keeps three statements:
the tracker's four epigraph constraints are level to 2.73×10⁻¹²; the four branches' true heights lie in "約 $10^{-8}$ 的窄帶"
 (a band of about 10⁻⁸); the true global scale is 0.998903750476358. It attributes the gap to the
270° branch, a cusp where the active vertex switches. And it says "下一輪不可只使用" (the next round must not use
only) ∇_η m_r: for the cusps it lists one-sided derivatives, Clarke subdifferentials, switching equations,
"精確特殊相位" (exact special phases) and "分段接觸事件系統" (a piecewise contact-event system). The README states the
tracked spread and the audit's numbers side by side. No bound is claimed.

## The checks

| id | guards | result |
| --- | --- | --- |
| E1 | the logged curve is the after parameters (l₃ = 1 − 2l₁ − 2l₂), of length 1, its own mirror image traversed backwards; the before parameters are Round 03's five-link | **PASS** |
| E2 | each curve's least exact scale over all phases is its stated global scale, to 10⁻¹⁰, and the after curve's is at the audit's phase, to 10⁻⁸ | **PASS** |
| E3 | the after curve's local minima over phases are the nine stated, values to 10⁻¹⁰ and phases to 10⁻⁸, and there are no others | **PASS** |
| E4 | each resolution row: the strict local minima on that grid number as stated, and its global scale is the least exact scale, to 10⁻¹² | **PASS** |
| E5 | each tracked phase (after) lies within 10⁻⁶ rad of a true local minimum, and each tracked value (before and after) above that minimum by at most 2×10⁻⁸; the stated spread is the tracked values' | **PASS** |
| E6 | the improvement, the distance to 1 and the README's statement of the audit's tracker overestimate are the differences they name | **PASS** |
| E7 | the summary's matrix is the csv's; each entry is this run's central difference of the true branch minimum in η = (l₁, l₂, β, δ), δ = α − β, to 2×10⁻⁴, and to 10⁻⁶ | **FAIL** |
| E8 | μ is on the simplex, μG is the stated residual with the stated norm, and this run's own G gives a least-norm simplex combination within 10⁻³ of μ | **PASS** |
| E9 | at each branch's true local minimum the vertices on the two legs and the hypotenuse are the recorded ones | **PASS** |
| E10 | the README, the report and the audit state the data's values, parameters, tables and ledgers; METHOD.md states the report's epigraph problem and pressure condition | **PASS** |

## Findings

### 1. What reproduces, and how exactly

- **The curves.** The logged curve is the after parameters to 10⁻¹⁷, with l₃ = 1 − 2l₁ − 2l₂ exactly, of length 1, and
  its own mirror image (E1, `curve_minus_parameters`). The before parameters are Round 03's five-link (E1), and its
  least scale is Round 03's, 0.998754371668 (E2). The parameter η is (l₁, l₂, β, δ) with δ = α − β, the outer turn's
  excess over the inner one, and l₃ = 1 − 2l₁ − 2l₂: with that reading the round's sensitivity matrix reproduces to
  its first three or four decimals (§4).
- **The global scale.** The equalized curve's least exact scale over all phases is 0.9989037504763532, the stated
  0.9989037504763583 to 5×10⁻¹⁵, reached at φ = 3π/2 to the last bit of a double (the audit's phase to 1.5×10⁻¹⁴)
  (E2, `global`). The stated improvement, distance to 1 and tracker overestimate are the differences they name (E6).
- **The landscape.** Its nine local minima over phases are the stated nine, values to 8.6×10⁻¹⁴ and phases to
  4.9×10⁻¹³ (E3, `minima_max_deviation`); the four lowest are B1 to B4. Each branch's contact ledger is the recorded
  one: vertices 0 and 1 on the leg x = 0 at B1, vertices 1 and 2 at B2 — the placements RUN-004 §4 found at its own
  five-link — the two endpoints on the hypotenuse at B3, the two endpoints on the leg x = 0 at B4 (E9). As the report
  says, the four branches are different contact topologies, not rotated copies of one placement.
- **The resolution table.** On grids of 2,048 to 65,536 phases the landscape has 8, 9, 9, 9, 9 and 9 strict local
  minima, as stated, and each row's global scale is the least exact scale to 6.0×10⁻¹⁵ (E4). The counts are a real
  statement about resolution and reproduce. The global-scale column is weaker evidence than it looks: under
  φ_k = 2πk/n, this run's convention (the package states none), every grid in the table has a multiple of four points,
  so 3π/2 is a grid point of each, and on each the grid minimum is at k = 3n/4 and equals the least scale exactly
  (`resolution_rows`). The column cannot tell refinement from coincidence. Its six values scatter by 3.3×10⁻¹⁵ and sit
  2.7×10⁻¹⁵ to 6.0×10⁻¹⁵ above the least scale, so the round did refine, as the audit says ("在所有局部極小附近細化").
- **The tracker.** Each tracked phase (after) lies within 3.0×10⁻⁸ rad of a true local minimum
  (`tracked_phase_minus_true`), and each tracked value lies above its minimum — by 3.2×10⁻¹¹ to 1.30×10⁻⁸ after the
  equalization, by 2.6×10⁻¹¹ to 1.28×10⁻⁸ before it (E5, `tracked_minus_true`); the stated spread is the tracked
  values' (E5).
- **The pressure vector.** μ lies on the simplex and μG is the stated residual, with the stated norm (E8).
- **The documents.** The README, the report — its branch, contact, sensitivity and pressure tables and its
  parameters — and the audit state the summary's numbers to the digits they print; METHOD.md states the report's
  epigraph problem and pressure condition (E10).

### 2. The level claim is the tracker's

The README's "四分支等高誤差：2.734812376559e-12" and the main report's §2 and §10 state that the four branches are
level to 2.73×10⁻¹². That is the spread of the tracker's four values. The four true branch minima are
0.99890376346193 (B1), 0.99890376346325 (B2), 0.99890375813127 (B3) and 0.99890375047635 (B4), spread 1.30×10⁻⁸
(`true_branch_spread`); the audit puts them in a band of about 10⁻⁸ and withdraws the claim, and the README carries
the audit's numbers beneath its own; a reader should take the audit's. The audit names one cusp, the 270° branch.
The 120° branch's tracked value is also 5.4×10⁻⁹ above its minimum, and every one of the four minima is a kink, with
one-sided slopes of opposite sign: −0.330 and +0.024 at B1, −0.024 and +0.234 at B2, −0.268 and +0.613 at B3, −0.330
and +0.432 at B4 (`one_sided_slopes`).

### 3. The four branch minima sit at exact phases

Each branch minimum is where the curve's contact with the triangle changes identity, and each such phase is exact
(`branch_minima_at_special_phases`; the stated minima agree with these phases to 2.8×10⁻¹³):

| branch | contact change | phase | true minimum's phase minus it |
| --- | --- | --- | ---: |
| B1 | link 1 (direction α) parallel to the leg x = 0: vertices 0 and 1 tie on it | π/2 − α | 2.2×10⁻¹⁵ |
| B2 | link 2 (direction β) parallel to the leg x = 0: vertices 1 and 2 tie on it | π/2 − β | −5.0×10⁻¹⁶ |
| B3 | the symmetry axis parallel to the hypotenuse's normal: the endpoints tie on the hypotenuse | 2π/3 | 4.4×10⁻¹⁶ |
| B4 | the symmetry axis parallel to the leg x = 0's normal: the endpoints tie on that leg | 3π/2 | 0 |

B3 and B4 are RUN-004 §4's symmetry ties. At each such phase the active vertex of one term of the exact scale
switches, so the scale has a kink there and a smooth one-dimensional minimizer stops beside it — what the audit found
at B4. But the branch value is then the exact scale at a known phase: m_r(η) = s(φ_r(η); η) with φ_r one of π/2 − α,
π/2 − β, 2π/3, 3π/2. That is a closed form, smooth in η wherever the contact identities hold; its gradient equals the
central differences of the true minima to 1.5×10⁻⁹ (`sensitivity_closed_form_vs_true_minima`). The non-smoothness the
audit describes is in φ, not in η, and its methodological conclusion — that ∇_η m_r alone will not do — does not
follow at this point: ∇_η m_r is well defined here, and computed from the closed forms it is exact to that precision.

Solved with these four closed forms, the round's own epigraph problem (SLSQP from the round's point, 6.8×10⁻⁵ away in
η) levels the four branches to 1.1×10⁻¹⁵ at

  s = 0.998903757133, at (l₁, l₂, l₃) = (0.194313, 0.141593, 0.328188), α = 83.4151°, β = 80.6883°,

6.7×10⁻⁹ above the round's 0.998903750476. A full scan of the curve's phases, 3,600 and 65,536 of them with every
local minimum refined, gives the same least scale, and the 8,192-phase branch scan finds the four branches level
(`exact_phase_equalization`). SLSQP ended with status 8 ("Positive directional derivative for linesearch") rather
than reporting success, so the point was examined directly. There the four gradients have rank 3 (singular values
5.68, 0.314, 0.0407 and 2.7×10⁻¹¹), as stationarity requires: a least-norm multiplier on the simplex, (0.0280,
0.0287, 0.7365, 0.2067), is the left null vector, with residual 6.5×10⁻⁸. Along the right null direction, where all
four branches are stationary to first order, the lowest drops by 2.04×10⁻⁹ at ±10⁻⁴; along 200 random directions it drops at 10⁻⁵; and three restarts from 10⁻³ away
return to within 7×10⁻¹⁶. So the point is a local maximum of the four-branch model, by these probes; it is not proved
maximal in the family. It is one item of the audit's list for the next round, exact special phases, carried out.
The next round does it with contact-event equations: round v0.5's README gives an "事件共同尺度" (common event scale)
of 0.998903757132509, this run's value to 1.2×10⁻¹⁵; RUN-006 will check it.

The round's headline — levelling the weakest branches raises the five-link's scale without adding links — holds. But
RUN-004's search, which maximized the least scale directly without tracking branches, reached 0.998903093 from the
same start: 1.49×10⁻⁴ above Round 03 and 6.6×10⁻⁷ below Round 04, ending with three branches level. At a max-min
optimum the lowest branches come level however it is approached; what the round adds is the ledger of which
branches they are, and §3 adds their exact identities.

### 4. The sensitivity matrix and the pressure vector

This run's central differences of the true branch minima (step 10⁻⁶; steps of 10⁻⁵ and 10⁻⁷ agree to 3.7×10⁻⁹,
`sensitivity_step_stability`) reproduce the round's rows B1 and B2 to 2.3×10⁻⁵ and 2.4×10⁻⁵ — about the fourth
decimal — and its rows B3 and B4 to 1.3×10⁻⁴ and 1.5×10⁻⁴, about the third (`sensitivity_row_max_difference`). No
row is the curve's to the eight decimals printed (E7, which allows 10⁻⁶). B3 and B4, whose tracked values also carry
the tracker's larger error (5.4×10⁻⁹ and 1.3×10⁻⁸; E5), deviate about six times more; the package has no code, so the
cause is not established. The sign pattern the report reads — "反應方向互相衝突" (conflicting directions) — is the
same in both matrices.

The pressure vector's arithmetic holds exactly (E8). On this run's G at the round's point, the least-norm combination
on the simplex is (0.02802, 0.02871, 0.73655, 0.20672), within 2.9×10⁻⁴ of the round's μ, with residual 8.3×10⁻⁶; the
round's μ on this run's G leaves 2.0×10⁻⁴ (`pressure_this_run`). Read strictly, a nonzero residual means some
direction raises all four branches at once, at a rate of order 10⁻⁵ — and the round's point is not a level point
(only B4 is active there). The multiplier reading belongs at the levelled point of §3, where it holds: μ* > 0 and the
residual is 6.5×10⁻⁸. So the round's reading — no direction raises all four branches — holds for the curve to about
10⁻⁵ at its own point, and to 10⁻⁷ at the point it was converging to.

### 5. The seven-link family

The round asks (§8, item 5) whether a new contact identity can break the current plateau ("可突破目前的 $0.9989$ 平台").
The seven-link family contains every five-link curve (RUN-004 §2), and RUN-004's seven-link search reached
0.998904534, 7.84×10⁻⁷ above the round's five-link and 7.77×10⁻⁷ above this run's equalized one (`run004_seven_link`).
That is above the round's five-link by less than 10⁻⁶, and RUN-004's search, like this one, is not proved maximal.

## At what strength each result stands

| result | strength |
| --- | --- |
| the exact formula for the triangle | exact (RUN-003) |
| the curves, the global scale, the nine minima, the resolution rows, the contacts, the stated differences | numerical, the formula in floating point, to the stated tolerances |
| the minima over phases | numerical: a 3,600- or 8,192-phase scan, every local minimum refined by golden section; not proved global |
| the four branch minima at π/2 − α, π/2 − β, 2π/3 and 3π/2 | the phases' identities exact (a link parallel to a leg, a symmetry tie); that the minima are there, numerical: this run's to 2.2×10⁻¹⁵, the stated ones to 2.8×10⁻¹³ |
| the sensitivity matrix | numerical: central differences of numerical minima, stable to 3.7×10⁻⁹ |
| this run's equalized five-link, 0.998903757133 | numerical: SLSQP on the closed forms (status 8), checked by a full scan of phases and by first- and second-order probes; a local maximum of the four-branch model, not proved maximal in the family |

## The drill

| | |
| --- | ---: |
| planted defects | 17 |
| tripping exactly the checks named for them | **17** |
| 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 | **166/166** |

Plants include a triangle leg, the global scale, a tracked value, a local minimum's phase, a resolution count, a
sensitivity entry, a pressure weight, a contact ledger, the equalized inner angle, a starting length, the
improvement, a curve point, a report digit, a README digit, the audit's phase, METHOD's inequality, and a term of the
exact formula. The expected sets were written after a dry run with every set empty, and each trip was reasoned
before it was named, so 17 of 17 says the reasoning matched the harness, not that the plants were chosen blind. One
trip looked wrong until the arithmetic was redone: the improvement plant's difference from the printed value is
3.6×10⁻¹⁵, not 3.6×10⁻¹⁷. Sets of contact vertices are discrete and do not move under a small change of a leg or a
single parameter, so the contact check is named only where a ledger or the formula changes. Two margins are thin:
the README-digit plant changes the overestimate by 10⁻²⁰ against a tolerance of 6×10⁻²¹, and the improvement plant
trips the derived check at 3.6×10⁻¹⁵ against 10⁻¹⁵. E7 fails on the package; its messages at the 10⁻⁶ level have
fixed text, so its three plants trip it through its coarse 2×10⁻⁴ part — new failures, not old ones reprinted with
new numbers. As in the earlier runs, each check was also replaced in turn by one that always passes; each such run
reported exactly the plants that name that check — E1 3, E2 5, E3 4, E4 4, E5 4, E6 4, E7 3, E8 3, E9 2, E10 12 — 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.

An independent reading of this report before it was committed found the matrix's precision misstated, the
resolution column overread, the multiplier read at the wrong point, an unlogged SLSQP exit status, the audit's
methodological claim left unchallenged, the round's headline unquoted, and several numbers not in the gate log; all
are corrected here, E7 now fails at 10⁻⁶, and the numbers are logged.

## What this run does not establish

Any bound for Moser's problem. That the minima over phases, the round's equalized curve or this run's are maximal in
the five-link family, or that the contact identities are stable in a whole neighbourhood. Why the round's sensitivity
rows differ from the derivatives: the package has no code. Anything about the package's eight figures, which were not
examined.

## Re-running

```
python code/fetch_source.py
python code/run005_round04_branch_equalization.py --drill
```

The findings read RUN-004's committed gate log for its seven-link parameters. The run is correct when the gate log is
byte-identical to the committed one.