# RUN-004 — Round 03 (v0.3): the five- and seven-link values reproduce; the seven-link family contains every five-link curve, so seven below five is a search's shortfall

**Arm:** 懸衡 (Xuán Héng) / AMRAL Research Lab
**Date:** 2026-10-19
**Subject:** `Moser_Skew_Lab_v0.3.zip` (SHA-256 `9f4a3644…`, pinned in [`corpus.py`](../code/corpus.py)): its README,
`reports/ROUND_03_REPORT.md`, `src/METHOD.md`, and its data (`round03_summary.json`, `resolution_stability.json`,
`best_5link_curve.csv`, `best_7link_curve.csv`), with Round 02's `best_congruence_three_link_curve.csv` for the
baseline, read from the package bytes
**Tools:** [`run004_round03_symmetric_links.py`](../code/run004_round03_symmetric_links.py) — RUN-003's exact
three-normal formula for T/1.0048, its minima over all phases (a 3,600-phase scan, every local minimum refined by
golden section), this run's own five- and seven-link families and searches; [`gate.py`](../code/gate.py) for the drill
**Log:** [`run004-round03-symmetric-links.json`](../data/gate-logs/run004-round03-symmetric-links.json)

**Verdict: PARTIAL** — in RUN-002's sense: what the round computes reproduces, and one of its two conclusions does
not hold as stated. Reproduced: the three critical scales to 1.4×10⁻¹³ and their phases to 5.3×10⁻¹³; the logged
curves as their stated parameters, each its own mirror image; the contact ledgers and the four-contact skeleton; all
18 rows of the resolution table (local-minimum counts 7, 8 and 7 on every grid from 2,048 to 65,536 phases); the
metrics blocks, as an 8,192-phase grid's unrefined values; the contact-pressure entropy, under one reading of its
definition; and the values, gains, phases, lengths, angles and ledgers the README and the report state. Not as
stated: the seven-link family contains every five-link curve — set γ = β and split the second side in two — so its
optimum is at least the five-link's. The reported seven-link value, 8.0×10⁻⁵ below the five-link's, is where the
round's search stopped, not a property of the family; it is also 2.4×10⁻⁵ below the equal-thirds three-link of
RUN-003, which both families contain. "增加到七段並未繼續改善" (going to seven links brought no further improvement)
describes the search, and the degeneration the README and §6 point to — added links turning into near-parallel
pieces — cannot explain a shortfall of the family; at most it bears on why the search stalled. This run's own
searches go higher in both families: 0.998903093 for five links and 0.998904534 for seven, 1.4×10⁻⁶ above the five.
At the round's curves the least scale is reached on a single phase branch; at this run's, three branches are level.
The drill planted 16 defects; every one was caught by exactly the checks named for it, and the 4 controls disturbed
nothing.

---

## What Round 03 claims

Round 03 keeps Round 02's exact formula for the certified triangle (RUN-003),

  s(φ) = max_i (x_i/A + y_i/B) − min_i x_i / A − min_i y_i / B,

and searches two mirror-symmetric families of unit polylines: five links with directions (α, β, 0, −β, −α) and
lengths (l₁, l₂, l₃, l₂, l₁), and seven links with (α, β, γ, 0, −γ, −β, −α) and (l₁, l₂, l₃, l₄, l₃, l₂, l₁), every
length positive. A mirror-symmetric curve is congruent to its own reflection, so its two handednesses agree and
Round 02's chirality question does not arise inside these families ("手性假警報因此在參數層被排除", the false chirality
alarm is excluded at the level of the parameters). The objective is the family's largest least scale over rotations,
max over curves Γ of min_φ s_Γ(φ).

It reports a three-link baseline s₃ = 0.998574374711 — Round 02's corrected curve, re-evaluated on phase grids of up to
65,536 phases (§3) — a best five-link s₅ = 0.998754371668 and a best seven-link s₇ = 0.998674157956,
each with its parameters, phase and contact ledger; a resolution table on six grids; per-curve "metrics"; and a
contact-pressure entropy of the container, H_λ = 0.963370649827. Both best curves touch the triangle at four points —
both endpoints on the hypotenuse, one interior vertex on each leg — and §7 singles out the hypotenuse half:
"曲線可以讓同一斜邊約束由兩個端點同時飽和。這是第 3 輪最重要的結構發現" (a curve can saturate the same hypotenuse
constraint with both endpoints at once; this is round 3's most important structural finding).

Its conclusions: five links give "小幅但穩定的更高壓力" (a small but stable rise in pressure) over three (§6); seven
links do not improve further, "新增參數大量退化為近平行子段" (the added parameters largely degenerate into
near-parallel sub-segments; §6, boxed); §12 calls s₇ < s₅ "更重要的限制" (the more important limitation) and
concludes "解題方向不能簡化為「增加曲線複雜度」" (the way forward cannot be reduced to "increasing the curve's
complexity"). §11 lists the limits, among them "沒有證明五段或七段族的全域最優" (no proof that the five- or seven-link
optimum is global; item 2) and "七段低於五段可能是族本身、參數化或搜尋不足造成，尚不能作一般否定" (seven below five may
be due to the family itself, the parameterization or insufficient search, and cannot yet be read as a general
negative; item 7). No bound is claimed. The package ships no code; `src/METHOD.md` states the formula and the
objective, and says reflection is included because "the curve equals its mirror up to parameter reversal".

## The checks

| id | guards | result |
| --- | --- | --- |
| Z1 | each logged curve is its stated parameters, of length 1; its least exact scale over all phases is the stated s, at the stated phase, to 10⁻¹⁰; the five- and seven-link curves' mirror images give the same | **PASS** |
| Z2 | at each curve's minimizing phase the vertices on the two legs and the hypotenuse are the recorded ones, with the recorded multiplicity; for the five- and seven-link curves, both endpoints on the hypotenuse and one interior vertex on each leg | **PASS** |
| Z3 | the stated differences and distances to 1 are the differences they name | **PASS** |
| Z4 | each of the 18 resolution rows: the strict local minima on that phase grid number as stated, and the refined scale and phase are this run's to 10⁻¹¹ and 10⁻⁸; the summary holds the same table, and each curve's count is its finest row's | **PASS** |
| Z5 | each curve's metrics block is the unrefined exact scale on an 8,192-phase grid | **PASS** |
| Z6 | each family's reported optimum is at least the value of two curves it contains: the round's five-link written as a seven-link (γ = β, its second side split), and the equal-thirds three-link at RUN-003's turn, written in each family | **FAIL** |
| Z7 | the stated entropy is the normalized entropy of the three constraint rows' balance weights as the formula writes them, (1/A, 1/B, 1) | **PASS** |
| Z8 | the README and the report state the data's values, gains, phases, lengths, angles and contact ledgers | **PASS** |
| Z9 | METHOD.md and the report state the same formula; each logged curve is its mirror image traversed backwards | **PASS** |

## Findings

### 1. What reproduces, and how exactly

- **The values.** The least exact scale over all phases is 0.9985743747111 for the baseline, 0.9987543716677 for the
  five-link and 0.9986741579563 for the seven-link (`exact_minima`) — the stated values to 1.4×10⁻¹³, 6.4×10⁻¹⁴ and
  3.4×10⁻¹⁴ — at the stated phases to 5.3×10⁻¹³, 2.4×10⁻¹³ and 1.2×10⁻¹³ (Z1). The summary's s₃ is the table's
  32,768-phase row, not its 65,536-phase row (the five- and seven-link values are their 65,536-phase rows); the two
  differ by 1.3×10⁻¹³, below the twelve digits printed. The logged curves are their stated parameters, of length 1 to
  10⁻¹², and each is its own mirror image traversed backwards, as METHOD.md says (Z9); their mirror images give the
  same scales (Z1). The stated differences and distances to 1 are the differences of the stated values (Z3).
- **The contacts.** At each minimizing phase the vertices on the leg x = 0, the leg y = 0 and the hypotenuse are the
  recorded ones — [2], [1], [0, 3] for the baseline; [3], [2], [0, 5] for the five-link; [4], [3], [0, 7] for the
  seven-link — multiplicity 4 each, and for the two symmetric curves this is the stated skeleton (Z2). Why its
  hypotenuse half holds is §4 below.
- **The resolution table.** On every grid from 2,048 to 65,536 phases the exact scale has 7, 8 and 7 strict local
  minima for the three curves, as stated, and each row's refined scale and phase are this run's to 5.5×10⁻¹² and
  2.1×10⁻¹¹ (Z4, `resolution_max_deviation`).
- **The metrics.** Each curve's "metrics" block — minimum, maximum, barrier, local minima and their count — is the
  exact scale sampled on 8,192 phases, unrefined, to 10⁻¹² (Z5). Its "minimum" is therefore not the curve's scale:
  the true minimum is a sharp kink between grid points, and the sampled minimum lies 1.37×10⁻⁴ to 1.38×10⁻⁴ above it
  (`metrics_minimum_minus_refined`), about three quarters of the five-link's stated gain. The report does not use
  these blocks; a reader of the summary should not take `metrics.minimum` or `best_local_minima` for critical scales.
- **The entropy** reproduces under one reading of its definition (§5).
- **The documents.** The README and the report state the summary's values, gains, phases, lengths, angles and
  ledgers to the digits they print, at the first place each is stated (Z8); the repetitions in the report's §6, its
  N₃ = 7 and its count of four active contacts agree on reading. The report's §2 states METHOD's formula (Z9).

### 2. The seven-link family contains the five-link family

Set γ = β in the seven-link family and take its lengths (l₁, l₂, l₃, l₄) to be the five-link's (l₁, l₂/2, l₂/2, l₃):
the seven-link's second and third links then point the same way, so the curve is the five-link curve
(α, β, 0, −β, −α), (l₁, l₂, l₃, l₂, l₁), with its second side marked at the midpoint. Every five-link curve is thus a
seven-link curve, with every length positive, and the seven-link family's optimum is at least the five-link family's.
The round's own five-link, written this way, has least scale 0.998754371668 as a seven-link curve; the reported
seven-link optimum is 0.998674157956, below it (Z6). If the round's search kept its angles strictly ordered, γ = β
lies on the boundary, and the supremum is still at least s₅, since the least scale over rotations depends
continuously on the curve.

So of the three causes §11 item 7 allows, the family itself is excluded, and the comparison is a statement about the
search. The report's §5 puts it that way ("七段族在本輪搜尋中沒有超越五段族", the seven-link family did not surpass
the five-link family in this round's search). The README does not: "七段未超越五段，顯示新增線段大量退化為近平行子段"
(seven links did not surpass five, which shows that the added segments largely degenerate into near-parallel
sub-segments); nor do §6's boxed second conclusion or §12's "更重要的限制" and the lesson drawn from it. Degeneration
cannot lower a family's optimum: a seven-link curve whose added links are exactly parallel is a five-link curve, with
that curve's value. What near-parallel pieces can do is make a search stall.

Both families also contain the equal-thirds three-link of RUN-003 §3 — directions (α, 0, −α), lengths 1/3 each,
α = 82.435472°, least scale 0.998697791610 — as (α, α, 0, −α, −α) with lengths (1/6, 1/6, 1/3, 1/6, 1/6) and as
(α, α, α, 0, −α, −α, −α) with lengths (1/9, 1/9, 1/9, 1/3, 1/9, 1/9, 1/9); written either way it gives the same value
to 10⁻¹² (Z6). The reported seven-link is 2.36×10⁻⁵ below it; the reported five-link is 5.66×10⁻⁵ above it.

The round's five-link is not where its own family stops either. Moving l₁ up by 10⁻⁴ and l₃ down by 2×10⁻⁴ — the
parameters of drill plant M15 — already raises it by 2.67×10⁻⁵ (`five_link_nudged`). This run's own searches —
Nelder–Mead from two starts per family, each candidate's least scale over phases taken on a 360-phase scan with
golden refinement, each result re-evaluated on 3,600 phases and the winner on 65,536, where it did not move — go
higher in both families (`five_link_search`, `seven_link_search`):

- five links, from the round's optimum: **0.998903093**, at (l₁, l₂, l₃) = (0.194587, 0.141402, 0.328021),
  α = 83.4458°, β = 80.6281° — 1.49×10⁻⁴ above the round's s₅; from the equal-thirds three-link, 0.998832412;
- seven links, from that five-link with its second side split: **0.998904534**, at (l₁, l₂, l₃, l₄) = (0.193781,
  0.071306, 0.070830, 0.328167), α = 83.4190°, β = 80.7988°, γ = 80.5838° — 2.30×10⁻⁴ above the round's s₇ and
  1.44×10⁻⁶ above this run's five-link; from the round's own seven-link, 0.998827199.

In this run's searches, then, seven links do improve on five, by 1.44×10⁻⁶, with a β and γ 0.21° apart: near-parallel
pieces that raise the value. Neither search is proved maximal. Round v0.4.1's README and
`reports/ROUND_04_REPORT.md` report a five-link at 0.998903750476, 6.6×10⁻⁷ above this run's; RUN-005 will check it.

Where the round's search stopped shows in its curves' phase branches (`round_branches`). At the round's five-link
the least scale is reached on one branch only, at φ = 2π/3; the next lowest local minimum over phases is
0.999238612, 4.8×10⁻⁴ higher, and at the seven-link the next is 3.6×10⁻⁴ higher. The searches from those curves rise
until other branches come level: at this run's five-link three branches, at φ/π = 2/3, 0.0364 and 0.0521, agree to
1.2×10⁻¹¹; at its seven-link three, at φ/π = 3/2, 2/3 and 0.0511, to 10⁻¹⁴ (`best_branches`). Levelling the lowest
branches is what the report's §8 and §10 propose for the next round.

### 3. The baseline

Round 03's s₃ = 0.998574374711 is the orientation-preserving value of Round 02's corrected curve; that curve is not
mirror-symmetric, and its mirror image fits at 0.998573794671, its congruent value — the one Round 02's correction
prescribes for comparison with the certifying preprint. The five- and seven-link values are congruent values (their
curves are their own mirror images), so the stated gains compare a congruent value with an orientation-preserving
one; against the congruent baseline the five-link gain is 1.80577×10⁻⁴ instead of 1.79997×10⁻⁴. The difference is
small. The larger one is the baseline itself: the corrected curve is not the three-link family's best (RUN-003 §3),
and against the equal-thirds three-link the five-link's gain is 5.66×10⁻⁵, about a third of the stated 1.80×10⁻⁴,
and the seven-link's is negative. "三連桿不是本搜尋架構中的終點" (the three-link is not the end point in this search
architecture) still holds — the five-link is higher than every congruent three-link value Round 02 or RUN-003 found
(the highest, 0.998697792) — but by less.

### 4. Where the minimum falls, and why both endpoints touch the hypotenuse

The five- and seven-link curves are symmetric about the y-axis. Turned by φ = 2π/3, that axis becomes parallel to the
hypotenuse's outer normal (30°), with the endpoints on the hypotenuse's side; two points that are mirror images across
a line parallel to a normal have equal values along it, so the endpoints have equal hypotenuse values. Both curves'
minimizing phases are 2π/3 to the last bit of a double (the stated ones to 2.4×10⁻¹³), the endpoints' hypotenuse
values there agree to 2×10⁻¹⁶, and the minimum is a kink between the two endpoints' branches, with one-sided slopes
−0.269 and +0.615 (five-link) or +0.616 (seven-link) (`axis_phase`). The double contact on the hypotenuse is thus what
the imposed symmetry gives at the phase where the minimum falls; what the search found is that the minimum falls
there, with one interior vertex on each leg. The nearly symmetric baseline's minimum is 1.66×10⁻⁶ rad from 2π/3.

So the feature §7 calls the round's most important structural finding — two endpoints saturating the hypotenuse —
is the part of the skeleton that follows from the construction rather than from the search. And it is one placement
among several at this run's higher curves (§2), whose least scale is reached at three phases
(`best_level_branch_contacts`): for the five-link, at 2π/3 with the same four contacts, and at φ/π = 0.0364 and
0.0521 with a whole link lying along the leg x = 0 (vertices 0 and 1, then 1 and 2), vertex 0 on the leg y = 0 and
vertex 3 on the hypotenuse; for the seven-link, at 2π/3 likewise, at φ/π = 0.0511 with vertices 1 and 2 on the leg
x = 0, and at 3π/2, where the axis is parallel to that leg's normal and the two endpoints tie on it, by the same
symmetry.

### 5. The entropy is a property of a row scaling, not of the curves

The report defines H_λ as the normalized entropy of "三角形三個法向的平衡權重" (the balance weights of the triangle's
three normals) and calls it "容器法向幾何決定的基準" (a baseline determined by the container's normal geometry). Its
value 0.963370649827 is the entropy of the weights (1/A, 1/B, 1), which balance the three constraint rows as the
formula writes them: −x ≤ 0, −y ≤ 0 and x/A + y/B ≤ s, with normals (−1, 0), (0, −1) and (1/A, 1/B), the last of
which is not a unit vector (Z7). Balance weights depend on the lengths given to the normals. With unit normals —
Round 02's convention, whose ledger has Σλ_j u(θ_j) = 0 for unit vectors u — the balancing weights are the side
lengths (B, A, √(A² + B²)), and the entropy is 0.965163253185. The round's reading also changes with the triangle's
size, not only its shape: for T undivided it gives 0.963701869, while the side-length reading gives 0.965163 for
both (`entropy_readings`). Either number belongs to the triangle and a choice of scaling; neither depends on any
curve, as the report itself notes.

## At what strength each result stands

| result | strength |
| --- | --- |
| the exact formula for the triangle | exact (RUN-003) |
| the three scales, phases, contacts, resolution rows, metrics, entropy and stated differences | numerical, the formula in floating point, to the stated tolerances |
| the minima over phases | numerical: a 3,600-phase scan, every local minimum refined by golden section; not proved global |
| the seven-link family contains every five-link curve, and both contain the symmetric three-links | exact (the construction in §2) |
| the five-link and seven-link curves' minima fall at φ = 2π/3 | numerical; the endpoints' tie at that phase is exact, by symmetry |
| this run's five-link 0.998903093 and seven-link 0.998904534, each with three branches level | numerical: this run's search; not proved maximal |

## The drill

| | |
| --- | ---: |
| planted defects | 16 |
| caught by exactly the checks named for them | **16** |
| controls (the package unmodified, the report with CRLF endings, the curves copied, the README with trailing spaces) | 4 |
| controls that disturbed nothing | **4** |
| must-stay-green assertions held | **146/146** |

Plants include a triangle leg, the five-link's scale, a seven-link contact, a hypotenuse contact added, a stated
gain, a metrics barrier, the entropy, the five-link's scale moved below the equal-thirds three-link's value, a
resolution count, a curve point, a report angle, a README digit, METHOD's formula, the metrics grid, a term of the
exact formula, and the five-link's parameters. Each check, replaced in turn by one that always passes, was reported
blind in exactly as many cases as plants name it — Z1 6, Z2 2, Z3 3, Z4 4, Z5 5, Z6 2, Z7 2, Z8 8, Z9 2 (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.

One of Z6's two catches is weaker than the rest. The harness counts a trip as a failure message the unplanted package
does not have, and Z6 already fails on the package; the five-link-parameters plant (M15) changes the curve Z6 writes
into the seven-link family, so Z6 reprints its existing failure with a new value (0.998781074920 for 0.998754371668)
rather than finding a new one. Its other plant, the five-link's scale moved below the equal-thirds three-link's
value, makes a new failure.

The instrument was wrong before it was right, in three ways worth recording. Its first reading of the summary's
metrics blocks was a 4,096-phase grid, which gives the baseline's and the seven-link's minimum and maximum but no
block's local minima; the 8,192-phase grid gives every entry of every block. Its first entropy check took the balance
weights with unit normals, gave 0.965163 and failed; the round's number is the formula's rows', and §5 now reports
both readings while the check tests the one the round used. And two plants — a triangle leg and a term of the formula —
first named the nesting check, and the formula's also the contact check: a larger triangle lowers every value, so
the nesting failures vanish rather than new ones appearing, and a set of contact vertices does not move with a small
shift of phase. Neither check is meant to catch those plants; the drill now names only checks that must trip, and the
blind counts above were measured after the correction. An independent reading of this report before it was
committed found an inexact quotation, the metrics gap and its comparison, the phases' precision, an overstated reason,
omitted hedges of the round, the attribution of §7's "most important" finding, and several numbers not yet in the
gate log; all are corrected here, and the numbers are now logged.

## What this run does not establish

Any bound for Moser's problem. That the minima over phases, the round's five-link value or this run's searches are
global. That seven links beat five in general: this run's 1.44×10⁻⁶ is the difference of two searches, and the
nesting gives only that the seven-link family's optimum is at least the five-link family's. Why the round's
seven-link search stopped where it did: the package has no code and no search log. Anything about the package's five
figures, which were not examined.

## Re-running

```
python code/fetch_source.py
python code/run004_round03_symmetric_links.py --drill
```

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