# RUN-007 — Round 06 (v0.6.1): the escape, the census and the audit reproduce; the event root is a strict local maximum of the eight-parameter model, and so of the congruent least scale among nearby five-links on a neighbourhood of unestablished size

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.6.1.zip` (SHA-256 `6fddf556…`, pinned in [`corpus.py`](../code/corpus.py)): its README,
`reports/ROUND_06_REPORT.md`, `ROUND_06_INTERPRETATION.md`, `ROUND_06_LOCAL_ESCAPE_AUDIT.md`, `src/METHOD.md`, and its
data (`round06_summary.json`, `antisymmetric_mode_profiles.json`, `topology_census.json`, `local_escape_audit.json`,
`topology_escape_curve.csv`), with Round 05's summary for the root, read from the package bytes (the documents passed
through RUN-001's `restore_escapes`, which changes none of them)
**Tools:** [`run007_round06_chirality_escape.py`](../code/run007_round06_chirality_escape.py) — the general five-link
of Round 06's §1; its four branch values at their event phases as closed forms in all eight parameters, with
complex-step derivatives; RUN-003's exact formula and minima over phases, both handednesses; a vectorized grid of exact
scales for the coarse values; RUN-006's identity margins; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run007-round06-chirality-escape.json`](../data/gate-logs/run007-round06-chirality-escape.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 reference as Round 05's root, level in both handednesses; the 404 mode-profile rows,
as 2,048-phase grid minima of each curve and its mirror image; the escape curve and its refined values; the twenty
census rows, coarse and refined, with their winning branches and contact signatures, five distinct; the fifteen logged
audit candidates; the stability draft's base margins, and, in this run's own samples of its boxes, the four
signatures; and the documents' numbers. The draft's Jacobian figures and least slope are Round 05's, re-printed with
the last digits RUN-006 §4 found not right; that is charged to Round 05 there and not counted again. Not as stated: the modes table's "掃描內最大手性差" (largest chirality gap in the scan) are
differences of grid samples — the package never says the scans used a grid, and METHOD.md defines s± as minimized
over rotation — not of the curves' least scales, and differ from the true largest gaps by up to 3.7×10⁻⁴; and the interpretation note's reading that the eight-dimensional search's best candidate "重新收斂到近鏡像對稱"
(re-converged to near mirror symmetry) fails the report's own test — its antisymmetric lengths are 5.6 % and 8.1 % of
the symmetric ones, and it lies 3.47×10⁻⁴ below the event root, which is itself a member of the family searched.
Findings: the four branch values of the general five-link are closed forms in all eight parameters; their
antisymmetric gradients are nonsingular, so the eight-dimensional congruent problem has the same single critical
direction as the symmetric one and the same negative curvature, −0.1227 — the event root is a strict local maximum of
the eight-parameter model, and so, in double precision, of the congruent least scale among all unit five-links near it
up to rigid motion, on a neighbourhood of unestablished size. That is what the round's single-mode scans, census and
6,300-sample audit were gathered to suggest; the single-mode peaks at zero are forced by the mirror symmetry, unless
every branch is stationary in that mode, and could not show it alone. The drill planted 21 defects; each tripped exactly the
checks named for it, and the 4 controls disturbed nothing.

---

## What Round 06 claims

Round 06 leaves Round 05's mirror-symmetric five-link (RUN-006) for the general five-link near it. Four symmetric
parameters (l₁, l₂, β, δ) are joined by four antisymmetric ones (d_l₁, d_l₂, d_α, d_β): lengths (l₁ + d_l₁, l₂ + d_l₂, l₃,
l₂ − d_l₂, l₁ − d_l₁) and directions (α + d_α, β + d_β, 0, −β + d_β, −α + d_α). Every curve is scored by its congruent
scale s_cong = min{s₊, s₋}, the lesser of its two handednesses, "所以任一單手性提升，都必須由鏡像分支共同承擔，否則不算突破"
(so any gain in one handedness must be borne by the mirror branch too, or it is no breakthrough). It reports:

- **four antisymmetric modes**, each scanned alone over 101 values: in all four the best value is zero perturbation,
  with no gain over the centre, and the scan's largest chirality gaps are 1.686×10⁻³, 2.448×10⁻³, 1.329×10⁻² and
  5.370×10⁻³; it concludes "鏡像對稱不是任意美學限制" … "而是合同極小值對手性分支的自然平衡點" (mirror symmetry is no
  arbitrary aesthetic restriction but the natural balance point of the congruent minimum between the chirality
  branches);
- **an eight-dimensional search**, its best candidate checked on 131,072 phases: congruent 0.998557207698881, 3.47×10⁻⁴
  below the event scale, chirality gap 3.30×10⁻⁵, with d_l₁ = 1.11×10⁻², d_l₂ = −1.11×10⁻², d_α = −0.357°, d_β = 0.368°; the
  report's test: "若這些值重新收斂到零，表示八維搜尋沒有找到真正的手性越界，而是返回鏡像對稱平台" (if these values
  re-converge to zero, the search found no true chirality escape but returned to the mirror-symmetric platform);
- **a census** of 1,600 general five-links near the root, the top twenty re-checked at high resolution, five distinct
  winning contact signatures among them;
- **a local stability draft**: the root's inactive-vertex margins, Round 05's Jacobian figures and least one-sided
  slope, and 500 random samples in each of twelve symmetric parameter boxes from 10⁻⁷ to 5×10⁻⁴, all keeping the four
  signatures — "局部數值證書草案，不是區間證明" (a draft local numerical certificate, not an interval proof);
- **a local escape audit**: 6,300 general five-links at seven radii, the top thirty re-checked, none above the root,
  the best non-base 1.13×10⁻⁴ below it — "這不是一般五連桿局部最優證明" (this is not a proof of local optimality among
  general five-links).

The interpretation note reads the whole: the search produced no reproducible positive escape, "最佳候選重新收斂到近鏡像對稱"
(the best candidate re-converged to near mirror symmetry), the four single modes all reach their congruent peak
"在零擾動附近" (near zero perturbation), and
"因此目前資料支持五連桿事件根對小型手性破缺具有局部穩定性"
(so the present data support the event root's local stability against small chirality breaking). No bound is claimed.

## The checks

| id | guards | result |
| --- | --- | --- |
| A1 | the reference is Round 05's event solution; its two handednesses are the event scale, each stated phase a least-scale local minimum, with nine local minima on 131,072 phases and no chirality gap; the logged escape curve is the stated escape parameters in the general family | **PASS** |
| A2 | each of the 404 mode rows: its plus and minus values are the 2,048-phase grid minima of the curve and of its mirror image, its congruent value and gap are theirs; each mode's summary is its rows' | **PASS** |
| A3 | in every mode the plus value at t is the minus value at −t | **PASS** |
| A4 | the escape curve's two handednesses, their phases and their local-minimum counts on 131,072 phases are the stated ones; the improvement and the distance to 1 are the differences they name | **PASS** |
| A5 | each of the twenty census rows: coarse values are the 2,048-phase grid minima, refined values and phases this run's, the winning branch and its contact signature this run's; five distinct signatures | **PASS** |
| A6 | each of the fifteen logged audit candidates' values reproduce; gains, antisymmetric norms, the best non-base gain and the count above the root are theirs; 7 × 900 = 6,300 | **PASS** |
| A7 | the base margins are the root's identity margins; the Jacobian figures and least slope are Round 05's; this run's own box samples keep the four signatures, and each stated box minimum lies between the base margin and its first-order lower bound | **PASS** |
| A8 | the README, the report, the audit note and METHOD.md state the data's numbers, tables and method; the interpretation note still carries the reading §4 examines | **PASS** |

## Findings

### 1. What reproduces, and how exactly

- **The reference.** The reference parameters are Round 05's event solution, and both handednesses of its curve have
  least scale equal to the event scale 0.998903757132509 to 2.2×10⁻¹⁶ (`reference_handednesses`), with nine local
  minima each on 131,072 phases (A1). The four events
  are level at the root, so the "phase" of the least scale is any of the four; each stated phase is one of them (A1).
- **The family.** The logged escape curve is the stated escape parameters built as §1 defines the general five-link,
  with l₃ = 1 − 2(l₁ + l₂) (A1).
- **The mode profiles.** All 404 rows are the 2,048-phase grid minima of the curve and of its mirror image, to 6.7×10⁻¹⁶,
  and each mode's summary is its rows' (A2). Antisymmetric perturbation by −t is the mirror image of perturbation by
  +t, so the plus value at t must be the minus value at −t; the rows satisfy it to rounding (A3).
- **The escape.** Its two handednesses, 0.998557207699 and 0.998590245470, reproduce to 9.8×10⁻¹⁴, their phases to
  3.7×10⁻¹³, and their local-minimum counts, 8 and 9, exactly (A4, `reproduction_deviations`).
- **The census.** For each of the twenty logged rows the coarse values are the 2,048-phase grid minima, the refined
  values reproduce to 4.5×10⁻¹³ and their phases to 1.6×10⁻¹² (`reproduction_deviations`), and the winning branch
  and its contact signature are this run's; the twenty carry five distinct signatures (A5). The 1,600 samples
  themselves are not logged. The rows are ordered by the coarse score (`census_sorted_by_coarse`), which lies 5.7×10⁻⁵
  to 5.6×10⁻⁴ above the refined one, so the report's "排名" (rank) column is the coarse rank, as its procedure says
  ("先做雙手性粗相位評估", a coarse two-handedness phase evaluation first): by refined value its rank 2 is below its
  rank 3, its rank 9 above its rank 8, and its rank 10 would be sixth. The ten are still the top ten of the twenty.
- **The audit.** The fifteen logged candidates' handednesses reproduce to 4.6×10⁻¹³, and their gains, norms, the best
  non-base gain and the count above the root follow over the fifteen logged (A6); the audit note re-checked thirty, and
  the other fifteen, like the 6,300 samples, are not logged. The seven radii (0.02 to 1.0) are in units the documents do
  not state. Of the fifteen, fourteen have antisymmetric norms of 1.7×10⁻⁴ to 6.2×10⁻⁴; the fifteenth is the root itself
  (norm 0, gain +1.1×10⁻¹⁶, one rounding unit, which the stated count excludes).
- **The stability draft.** The base margins are the root's identity margins (RUN-006 §2) to 1.1×10⁻¹⁶
  (`base_margins_minus_root`); the Jacobian figures and the least one-sided slope are Round 05's, re-printed as
  "第 5 輪事件根的既有診斷" (the round-5 event root's existing diagnostics), so RUN-006 §4's finding on their last digits applies
  — the exact least slope is 0.0237736992, the printed 2.377371632889×10⁻² (an entry of G_d below);
  this run's own box samples, 100 in each of the twelve boxes, keep the four signatures, and each stated box minimum
  lies between the base margin and its first-order lower bound (A7).
- **The documents.** The README, the report, the audit note and METHOD.md state the data's numbers, tables and the
  congruent objective (A8); of the interpretation note A8 checks only that it carries the reading §4 examines.

### 2. The event root is a strict local maximum among all nearby five-links

The eight parameters are a chart of every unit five-link near the root, up to rigid motion: the map from (p, d) to the
five lengths and the four free directions, (L₁, …, L₅; θ₁, θ₂, θ₄, θ₅) with the middle link horizontal and L₃ fixed by the
total length, is linear and invertible — l₁ = (L₁ + L₅)/2, d_l₁ = (L₁ − L₅)/2, α = (θ₁ − θ₅)/2, d_α = (θ₁ + θ₅)/2, and so
on. And reflecting a curve in the x-axis, traversing it backwards and turning it by π — a reflection followed by a
rigid motion — takes the d-curve to the −d curve; so s₋(p, d) = s₊(p, −d), as the least scales confirm to
4.4×10⁻¹⁶ at random points (`mirror_of_d_minus_minus_d`).

Round 05's four branch identities stay meaningful for the general five-link: link 1, then link 2, parallel to the leg
x = 0, at φ = π/2 − (α + d_α) and π/2 − (β + d_β); the endpoints tied on the hypotenuse, at φ = 2π/3 − θ₅; and on the leg
x = 0, at φ = 3π/2 − θ₅, where θ₅ is the direction of the far endpoint from the near one. With these phases each branch
value is a closed form m_r(p, d) in the eight parameters; at the root they are the event scale to 3.3×10⁻¹⁶, and their
symmetric gradients are RUN-006's complex-step gradients of Round 05's closed forms to 4.4×10⁻¹⁶
(`eight_dim_closed_forms`). So near the root

  s_cong(p, d) = min over r of { m_r(p, d), m_r(p, −d) },

as long as the identities hold; sampled at 40 points each at Gaussian scales 10⁻⁵ and 10⁻⁴ in all eight parameters, the
least scale over all phases of the curve and its mirror image equals this minimum to 1.1×10⁻¹⁵, and every sample lies
below the root (`eight_dim_near_root`) — by first-order amounts, at least 4.1×10⁻⁶ and 2.5×10⁻⁵, which say nothing
about the critical line below; on that line the eight-parameter model is the symmetric one, and RUN-006's line
probes cover it.

Now the second-order argument of RUN-006 §2 carries over. Maximize z subject to m_r(p, d) ≥ z and m_r(p, −d) ≥ z.
At d = 0 the two sets of constraints share their symmetric gradients G_p and have opposite antisymmetric gradients ±G_d.
The 4×4 matrix G_d is nonsingular — its singular values are 2.52, 0.239, 0.135 and 0.0394 (`antisymmetric_gradients`) —
so stationarity in d forces equal multipliers on the two sets, μ/2 each, all positive. A critical direction (dp, dd, dz)
then satisfies G_p dp + G_d dd = dz·𝟙 and G_p dp − G_d dd = dz·𝟙: so G_d dd = 0, hence dd = 0, and G_p dp = dz·𝟙, hence,
as before, dz = 0 and dp along G_p's null vector v. The critical cone is the single line (v, 0), and along it the
Lagrangian's curvature is the symmetric problem's, −0.1227 (`eight_dim_curvature_along_critical_line`). The root is a
strict local maximum of the eight-dimensional model; and since its identity margins, its strictly signed one-sided
slopes and the 0.030 gap to every other local minimum (RUN-006 §2) hold for the curve and its mirror image alike, the
congruent least scale equals the model on some neighbourhood of the root, whose size is not established.

This is the statement the round's interpretation note reaches for — "事件根對小型手性破缺具有局部穩定性" (the event
root's local stability against small chirality breaking) — at the strength of a double-precision second-order
condition rather than of finite sampling, and stronger: on that neighbourhood no unit five-link, chiral or not, does
better.

### 3. What the single-mode scans can and cannot show

For a single antisymmetric mode, s₋(t) = s₊(−t), so s_cong(t) = min{s₊(t), s₊(−t)} is even in t, and it has a peak at 0
whatever the curve unless every branch is stationary in that mode. On the true least scales (not the grid), all four
modes peak strictly at zero, with one-sided slopes ±1.052, ±1.137, ±0.198 and ±0.251 (`mode_profiles_true`) —
difference quotients over the scan's step (5×10⁻⁴ in length, 0.1° in angle), each within 3×10⁻⁴ of the largest
entry in its column of G_d. So "零擾動附近形成尖頂或近尖頂" (a peak or near-peak forms near zero perturbation) is forced
by the symmetry — the true profiles show a strict peak, so the hedge was not needed — and four peaks along four axes do not exclude a gain along a combination of modes; that is what G_d's
nonsingularity (§2) excludes.

The profiles themselves are the 2,048-phase grid's minima, not the curves' least scales. At t = 0 the grid holds 3π/2
and the value is exact; elsewhere it lies above the least scale, by up to 5.6×10⁻⁴ (`grid_minus_true_max`). The modes'
best values are therefore right a fortiori, but the table's "掃描內最大手性差" are differences of grid samples; on the
least scales the largest gaps are 1.743×10⁻³, 2.083×10⁻³, 1.330×10⁻² and 5.069×10⁻³, against the printed 1.686×10⁻³,
2.448×10⁻³, 1.329×10⁻² and 5.370×10⁻³.

### 4. The eight-dimensional search did not return to the platform

The event root is the member d = 0 of the family, with congruent scale
0.998903757; the search's best is 0.998557208, 3.47×10⁻⁴ lower (`escape_below_root`). Its antisymmetric parameters are
d_l₁ = 0.0111 and d_l₂ = −0.0111, 5.6 % and 8.1 % of l₁ and l₂, and its angles −0.357° and +0.368° (`escape_antisymmetric_relative`)
— not near zero, by the report's own test. Its small chirality gap, 3.3×10⁻⁵, is what a max–min over the two
handednesses balances toward, not a sign of symmetry. So the interpretation note's "最佳候選重新收斂到近鏡像對稱" does not hold: the search stopped at a worse
point of the family, away from the platform. That it found no escape is consistent with §2, but it is not evidence for
it.

## At what strength each result stands

| result | strength |
| --- | --- |
| the closed forms for the general five-link's four branches | exact (the contact identities); equal to the exact scale near the root to 1.1×10⁻¹⁵, and to the event scale at the root to 3.3×10⁻¹⁶ |
| the handednesses, phases, counts, grid minima, contacts | numerical, double precision; least scales by a 3,600-phase scan with every local minimum refined |
| the root as a strict local maximum of the eight-dimensional model | numerical: G_d's singular values and the second-order condition in double precision |
| the root as a strict local maximum of the congruent least scale among nearby five-links | numerical: the model's maximum, transferred by continuity from strict margins (RUN-006 §2), with sampling; the neighbourhood's size not established; no interval certificate |
| the census and the audit beyond their logged rows | the round's, unchecked (the samples are not logged) |

## The drill

| | |
| --- | ---: |
| planted defects | 21 |
| tripping exactly the checks named for them | **21** |
| controls (the package unmodified, the report with CRLF endings, the mode profiles copied, the README with trailing spaces) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **161/161** |

Plants include a triangle leg, a reference length, the reference mirror scale, two mode rows, a mode summary, the escape
inner angle, the escape's local-minimum count, its improvement, a census scale and a census signature, an audit
candidate and the audit's count, a base margin, a box minimum, the escape curve, a report digit, a README digit, an
audit-note digit, METHOD's minimum, 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; one plant, the reference mirror scale's, was enlarged
from 10⁻¹⁴ to 10⁻¹² because the smaller one sat below the reference check's 10⁻¹³. Of the four controls, the
unmodified package exercises nothing by construction, the copied profiles re-run only the two mode checks, and the
other two only the consistency check. Some branches of the checks have no plant of their own: the reference's event
scale against Round 05's, the escape parameters' membership of the family, the mode summaries' best and centre
fields, the summary's census against the census file, the audit's 7 × 900, the Jacobian figures' equality with Round
05's, and the interpretation note's phrase. A1's tolerance on the handednesses, 10⁻¹³, is some 500 times the actual
agreement. Each check, replaced in turn by one that always passes, was reported blind in exactly as many cases as
plants name it — A1 6, A2 6, A3 2, A4 5, A5 4, A6 4, A7 4, A8 8 — which by construction shows only that the harness
counts as described (eight 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 four ways worth recording. The reference check first required the
stated phase of the least scale to be this run's; at the root the four events are level, so the phase is any of the
four, and the check now asks only that the stated phase reach the least scale at a local minimum. Two tolerances of the
consistency check were set from the wrong order of magnitude — thirteen significant digits of a number near 10⁻² are
good to 5×10⁻¹⁵, not 5×10⁻¹⁶ — and now follow the printed precision. The grid of exact scales, looped phase by phase,
made a single pass of the checks take over three minutes; it is now computed for all phases at once, with the same
rotation, legs and B term, and agrees with the package's grid minima to 6.7×10⁻¹⁶ (`reproduction_deviations`). And
the first findings left the reproduction deviations out of the log; they are now in it. An independent reading of
this report before it was committed found three numbers misattributed or misstated, Round 05's re-printed digits not
accounted for, the title without its hedge, the chart and mirror identity of §2 asserted without argument, a hedge
dropped from a quotation, the census ranks half explained, the first-order nature of the samples' drops unstated, and
several gaps in the drill; all are corrected here, and the new figures are logged.

## What this run does not establish

Any bound for Moser's problem. That the root is the general five-link family's global maximum, or that no other
contact topology does better. The size of the neighbourhood of §2. Anything about the census's and the audit's
unlogged samples, or the package's seven figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run007_round06_chirality_escape.py --drill
```

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