> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# WSFC Round 01
## Polygonal $\omega$-Hierarchy, Finite-Stage Insufficiency, and Broadworm Breadth Sequence Closure

**English Title:** *Polygonal Omega-Hierarchy, Finite-Stage Insufficiency, and Broadworm Breadth Sequence Closure*  
**Document ID:** `WSFC-R01-v0.1`  
**Version:** v0.1  
**Date:** 2026-09-22  
**Author:** Neo.K  
**AI Research Collaborator:** Aletheia / OpenAI GPT-5.6 Sol  
**Series:** Worm Sequence Freedom Closure  
**Prerequisite:** `WSFC-00-v0.1`  
**Status:** `FIRST CONCRETE HIERARCHY / OMEGA-STAGE CLOSURE / BREADTH-PROBE BASELINE`  
**Canonical Source:** UTF-8 Markdown

---

# Abstract

WSFC-00 reframes the worm problem as the evolution and limit closure of finite-degree-of-freedom possibility spaces:

$$
\Gamma_0
\subseteq
\Gamma_1
\subseteq
\cdots
$$

as displayed above.

This round moves beyond the abstract level and adopts one of the most natural concrete hierarchies in the worm-problem literature:

$$
\boxed{
\Gamma_n
=
\{\text{planar polygonal arcs of total length }1\text{ with at most }n\text{ segments}\}.
}
$$

This hierarchy has four important properties.

First, after quotienting out translations and global rotations, a generic $n$-segment chain can be described by:

$$
(\ell_1,\ldots,\ell_n),
\qquad
\sum_{i=1}^{n}\ell_i=1,
$$

and:

$$
(\delta_1,\ldots,\delta_{n-1})
$$

so its generic freedom dimension is:

$$
\boxed{
d_n
=
(n-1)+(n-1)
=
2n-2.
}
$$

Thus:

$$
\boxed{
d_{n+1}-d_n=2.
}
$$

Each additional segment gives birth, generically, to exactly two new freedoms: a length-allocation freedom and a relative-turn freedom.

Second, Panraksa, Wetzel, and Wichiramala proved in 2007 that, for every fixed finite $n$, there exists a compact convex region $R_n$ that accommodates all $n$-segment unit polygonal arcs but not all unit arcs. Hence:

$$
\boxed{
\forall n<\infty,
\qquad
Q_n
\not\Rightarrow
Q_\infty.
}
$$

In other words, no fixed finite-degree-of-freedom stage can directly replace the full worm problem.

Third, the covering theorem of Wetzel and Wichiramala proves that a compact set accommodating every simple polygonal unit arc also accommodates all unit arcs. Thus, when all finite polygonal stages are considered together:

$$
\boxed{
\left(
\forall n,\;
Q_n(C)
\right)
\Longrightarrow
Q_\infty(C).
}
$$

Combining this with the trivial reverse inclusion yields:

$$
\boxed{
Q_\infty(C)
\iff
\forall n\,Q_n(C)
}
$$

for compact covers, while at the same time:

$$
\boxed{
\forall N<\infty,\;
\exists C_N:
Q_N(C_N)
\land
\neg Q_\infty(C_N).
}
$$

This is WSFC's first genuine instance of **$\omega$-stage task sufficiency without finite stabilization**.

Fourth, this round uses broadness / minimum width as its first scalar probe. Let:

$$
t_n
=
\sup_{\gamma\in\Gamma_n}
b(\gamma),
$$

and denote the maximal breadth of the broadworm by:

$$
b_\star
\approx
0.43893.
$$

The existing literature gives:

$$
t_n
\le
t_{n+1}
<
b_\star
$$

for all finite $n$, and:

$$
t_2
=
\frac{\sqrt3}{4}
\approx
0.433012701892,
$$

$$
t_3
\approx
0.435112923275.
$$

This paper further derives, from polygonal approximation / compact-cover closure:

$$
\boxed{
t_n
\uparrow
b_\star.
}
$$

Broadworm breadth therefore provides a particularly clean template for sequence freedom:

$$
\boxed{
\forall n<\infty,\;t_n<b_\star,
\qquad
\lim_{n\to\infty}t_n=b_\star.
}
$$

That is:

> Every finite-degree-of-freedom state falls strictly short of the final extremum, but the sequence limit of the possibility spaces reaches it.

This round also proposes an elementary quantitative approximation construction, yielding the coarse but explicit bound:

$$
\boxed{
0
\le
b_\star-t_n
\le
\frac{1}{n-1},
\qquad
n\ge2.
}
$$

This bound does not aim for numerical competitiveness; its significance is that it gives WSFC its first concrete prototype of a sequence-closure modulus.

---

# 1. The Question for This Round

WSFC-00 asks:

$$
\Gamma_n
\rightarrow
\Gamma_{n+1}
$$

what, exactly, is being expanded?

This round chooses the simplest answer, closest to the definition of the original worm problem:

$$
\boxed{
\text{segment-count hierarchy}.
}
$$

We do not begin by choosing Fourier modes or spline knots.

Instead, we ask directly:

> If each step allows a unit worm one more segment, how does the possibility space grow?

---

# 2. Canonical Polygonal Hierarchy

Define:

$$
\Gamma_n
=
\{
\gamma:
\gamma
\text{ is a polygonal arc of total length }1\text{, with segment count}\le n
\}.
$$

Then:

$$
\boxed{
\Gamma_n
\subseteq
\Gamma_{n+1}.
}
$$

Any polygonal arc with $m<n$ segments can be represented with $n$ segments by collinearly subdividing some of its segments, without changing its trace.

---

# 3. Generic Freedom Count

An ordered $n$-segment polygonal chain can be described by segment lengths:

$$
\ell_i>0,
\qquad
\sum_{i=1}^{n}\ell_i=1
$$

as above.

The length vector therefore lies in:

$$
\Delta_{n-1}^{\circ}
$$

and has:

$$
n-1
$$

degrees of freedom.

For directions, there are $n$ absolute segment angles.

After quotienting out global rotation, only:

$$
n-1
$$

relative directional freedoms remain.

Translation has also been eliminated by fixing the first vertex.

Thus the generic quotient dimension is:

$$
\boxed{
d_n
=
2n-2.
}
$$

---

# 4. What Does Each Freedom Expansion Add?

From:

$$
d_n=2n-2
$$

we obtain:

$$
\boxed{
d_{n+1}-d_n=2.
}
$$

So each step:

$$
n
\rightarrow
n+1
$$

generically adds:

1. a new length-allocation coordinate;
2. a new relative-turn coordinate.

These two freedoms can be written as:

$$
(\lambda,\beta).
$$

where:

$$
\lambda
$$

controls the length share assigned to the new segment,

and:

$$
\beta
$$

controls the direction / turning angle of the new segment relative to the preceding segment.

Thus the smallest freedom-birth unit is:

$$
\boxed{
\Delta d=2.
}
$$

---

# 5. This Does Not Mean That Two Theorem-Effective Freedoms Are Truly Added Each Time

The count above concerns only generic representation / geometry dimension.

We must still distinguish:

$$
d_n^{\rm rep},
\qquad
d_n^{\rm geo},
\qquad
d_n^{\rm eff}(Q).
$$

It is possible that:

$$
d_{n+1}^{\rm rep}
=
d_n^{\rm rep}+2
$$

while:

$$
d_{n+1}^{\rm eff}(Q)
=
d_n^{\rm eff}(Q).
$$

That is, the new segment adds only redundant geometry and does not change the specified theorem task.

Thus:

$$
\boxed{
\Delta d=2
}
$$

is only a freedom generation rule, not a theorem novelty guarantee.

---

# 6. The First Hard External Result: No Fixed Finite $n$ Is Sufficient

Panraksa, Wetzel, and Wichiramala (2007) consider:

$$
F_n
=
\{\text{$n$-segment unit arcs}\}.
$$

They prove:

> For every positive integer $n$, there exists a compact convex region $R_n$ containing a congruent copy of every $n$-segment unit arc, yet still missing some unit arc.

Their construction uses the breadth of the broadworm.

Let:

$$
t_n
=
\sup
\{
b(\gamma):
\gamma\in F_n
\}.
$$

Then:

$$
t_n
<
b_\star
$$

for all finite $n$.

Thus a rectangle of the form:

$$
R_n
=
[0,1]\times[0,t_n]
$$

can accommodate all $n$-segment unit arcs, but the broadworm cannot fit because:

$$
b_\star>t_n
$$

as above.

---

# 7. WSFC Interpretation: Non-Finite Sufficiency Theorem

Define:

$$
Q_n(C)
$$

to mean:

> $C$ covers $\Gamma_n$.

Define:

$$
Q_\infty(C)
$$

to mean:

> $C$ covers all unit arcs.

The 2007 result gives:

$$
\boxed{
\forall n<\infty,
\quad
\exists R_n:
Q_n(R_n)
\land
\neg Q_\infty(R_n).
}
$$

Thus there is no fixed:

$$
N<\infty
$$

such that:

$$
\boxed{
\forall C,\;
Q_N(C)
\Rightarrow
Q_\infty(C).
}
$$

This directly rules out:

$$
\boxed{
\text{finite-stage exact collapse}.
}
$$

---

# 8. The Second Hard External Result: All Finite Polygonal Stages Together Are Sufficient

Wetzel and Wichiramala (2010) prove:

> If a compact set contains a congruent copy of every simple polygonal unit arc, then it contains a congruent copy of every unit arc.

Thus, provided that:

$$
Q_n(C)
$$

holds for all finite $n$,

$C$ certainly covers every finite simple polygonal unit arc,

so:

$$
Q_\infty(C)
$$

holds.

The reverse implication:

$$
Q_\infty(C)
\Rightarrow
Q_n(C)
$$

is trivial.

Therefore:

$$
\boxed{
Q_\infty(C)
\iff
\forall n\,Q_n(C).
}
$$

---

# 9. The First Core Conclusion: $\omega$-Stage Task Sufficiency

We obtain:

$$
\boxed{
Q_\infty(C)
\iff
\bigwedge_{n<\omega}
Q_n(C),
}
$$

but:

$$
\boxed{
\forall N<\omega,
\qquad
Q_N
\not\equiv
Q_\infty.
}
$$

Thus:

$$
\boxed{
\text{task sufficiency first appears at the }\omega\text{-stage}.
}
$$

This is not a philosophical metaphor.

It is an actual structure obtained by combining known theorems about the worm problem.

---

# 10. Direct Relation to SDRPC Quantifier Descent

The original problem:

$$
\forall\gamma\in\Gamma,
\qquad
\exists g\in G:
g\gamma\subseteq C.
$$

can now be rewritten as:

$$
\boxed{
\forall n\in\mathbb N,
\quad
\forall\gamma\in\Gamma_n,
\quad
\exists g\in G:
g\gamma\subseteq C.
}
$$

This has not yet made all infinite quantifiers finite.

But it descends the:

$$
\text{function-space universal quantifier}
$$

to a:

$$
\boxed{
\text{countable sequence of finite-dimensional universal tasks}.
}
$$

Thus this round obtains:

$$
\boxed{
\text{Quantifier Stratification}
}
$$

rather than completed finite quantifier elimination.

---

# 11. An Explicit Approximation Modulus: From a Unit Arc to an $n$-Segment Polygonal Worm

We now give an elementary construction.

Let:

$$
\gamma:[0,1]\rightarrow\mathbb R^2
$$

be parametrized by arclength.

Choose:

$$
m=n-1.
$$

Divide:

$$
[0,1]
$$

into $m$ equal subintervals.

Take the vertices:

$$
v_i
=
\gamma(i/m),
\qquad
i=0,\ldots,m.
$$

Connect adjacent vertices by straight segments to obtain the polygonal path:

$$
\pi_m.
$$

---

# 12. One-Sided Trace Approximation

For any:

$$
t\in[i/m,(i+1)/m]
$$

the arclength distance to the nearer endpoint is at most:

$$
\frac{1}{2m}.
$$

Euclidean distance does not exceed arclength distance.

Hence:

$$
\boxed{
\gamma
\subseteq
U_{\varepsilon_m}(\pi_m),
\qquad
\varepsilon_m
=
\frac{1}{2m}.
}
$$

That is:

$$
\boxed{
\varepsilon_n
=
\frac{1}{2(n-1)}.
}
$$

---

# 13. Completing the Polygonal Approximation to Unit Length

The polygonal length of $\pi_m$ satisfies:

$$
L_m
\le1.
$$

If:

$$
L_m<1,
$$

append, from the terminal vertex, another segment of length:

$$
1-L_m
$$

as above.

This gives a unit polygonal path:

$$
\eta_n
$$

whose segment count is at most:

$$
m+1=n.
$$

and:

$$
\pi_m
\subseteq
\eta_n.
$$

Therefore:

$$
\boxed{
\gamma
\subseteq
U_{\varepsilon_n}(\eta_n).
}
$$

$\eta_n$ is not required to be simple here.

---

# 14. Approximate Cover Modulus

If a compact region $C$ covers all of:

$$
\Gamma_n,
$$

there exists a congruent placement:

$$
g\eta_n
\subseteq C.
$$

By the preceding section:

$$
g\gamma
\subseteq
U_{\varepsilon_n}(C).
$$

Thus:

$$
\boxed{
Q_n(C)
\Rightarrow
Q_\infty^{(\varepsilon_n)}(C),
\qquad
\varepsilon_n
=
\frac{1}{2(n-1)}.
}
$$

where:

$$
Q_\infty^{(\varepsilon)}(C)
$$

means:

> Every unit arc can be placed in $C$'s $\varepsilon$-neighborhood.

This is WSFC's first explicit sequence modulus.

---

# 15. Why Does This Not Contradict the Insufficiency of Every Finite $n$?

Because:

$$
\varepsilon_n>0
$$

for every finite $n$.

Thus finite stages yield only:

$$
\text{approximate universality}.
$$

Only when:

$$
n\rightarrow\infty
$$

do we have:

$$
\varepsilon_n\rightarrow0.
$$

Hence:

$$
\boxed{
\text{finite exact insufficiency}
}
$$

and:

$$
\boxed{
\text{finite approximate sufficiency}
}
$$

can perfectly well hold simultaneously.

---

# 16. Completing the Limit by Compactness

If a fixed compact $C$ satisfies:

$$
Q_n(C)
$$

for all $n$,

then, for any unit arc $\gamma$, there exist:

$$
g_n\gamma
\subseteq
U_{\varepsilon_n}(C),
$$

with:

$$
\varepsilon_n\rightarrow0.
$$

Since the rotation group is compact and the translations are confined by a bounded neighborhood of $C$,

we can take a subsequence of congruences:

$$
g_{n_k}
\rightarrow
g_\star.
$$

Passing to the limit gives:

$$
g_\star\gamma
\subseteq C.
$$

Thus:

$$
Q_\infty(C)
$$

holds.

This supplies another WSFC-style proof skeleton:

$$
\boxed{
\text{finite-stage exact cover}
\rightarrow
\text{global approximate cover}
\rightarrow
\varepsilon_n\to0
\rightarrow
\text{compactness}
\rightarrow
\text{exact global cover}.
}
$$

---

# 17. Breadth Probe

For a compact set $K$, define its breadth by:

$$
b(K)
=
\min_{u\in S^1}
\left(
h_K(u)+h_K(-u)
\right).
$$

For an arc $\gamma$:

$$
b(\gamma)
=
b(\operatorname{conv}\gamma).
$$

Let:

$$
t_n
=
\sup_{\gamma\in\Gamma_n}
b(\gamma).
$$

The broadworm in the literature gives:

$$
\boxed{
b_\star
\approx
0.43893.
}
$$

and:

$$
\boxed{
t_n
\le
t_{n+1}
<
b_\star.
}
$$

---

# 18. Known Low-Stage Anchors

For:

$$
n=2,
$$

we have:

$$
\boxed{
t_2
=
\frac{\sqrt3}{4}
=
0.433012701892\ldots
}
$$

The optimal shape consists of two sides of an equilateral triangle with side length:

$$
\frac12
$$

as above.

---

For:

$$
n=3,
$$

the literature proves:

$$
t_3
=
\max_{\theta\in[\pi/3,\pi/2]}
\frac{
\sin\theta+\sin2\theta
}{
2\cos\theta+3
}.
$$

Numerical recomputation in this round yields:

$$
\boxed{
\theta_3^\star
\approx
63.8316417429^\circ
}
$$

and:

$$
\boxed{
t_3
\approx
0.435112923275303.
}
$$

consistent with the literature's approximate value:

$$
0.43511
$$

as above.

---

# 19. Low-Stage Freedom Data

Use:

$$
b_\star\approx0.43893
$$

as an approximate anchor from the public literature:

$$
b_\star-t_2
\approx
0.005917298108,
$$

$$
b_\star-t_3
\approx
0.003817076725.
$$

Thus, from $n=2$ to $n=3$:

$$
\frac{
b_\star-t_3
}{
b_\star-t_2
}
\approx
0.64507.
$$

For this probe, the first addition of two generic freedoms has reduced the residual relative to broadworm breadth by approximately:

$$
35.49\%.
$$

This percentage is only a low-stage descriptive figure obtained using:

$$
b_\star\approx0.43893
$$

and is not an asymptotic law.

---

# 20. Derived Proposition: $t_n$ Must Converge to Broadworm Breadth

The literature establishes:

$$
t_n
\le
t_{n+1}
<
b_\star.
$$

Thus:

$$
t_n
$$

has a limit:

$$
L\le b_\star.
$$

Now use the approximate cover modulus above.

The rectangle constructed in 2007:

$$
R_n
=
[0,1]\times[0,t_n]
$$

covers every polygonal unit arc in $\Gamma_n$.

Thus the broadworm:

$$
\gamma_\star
$$

can be approximately covered within:

$$
U_{\varepsilon_n}(R_n)
$$

as above.

Meanwhile:

$$
U_{\varepsilon_n}(R_n)
$$

has width in the narrow direction at most:

$$
t_n+2\varepsilon_n.
$$

The breadth of the broadworm is:

$$
b_\star.
$$

Hence:

$$
b_\star
\le
t_n
+
2\varepsilon_n.
$$

Substituting:

$$
\varepsilon_n
=
\frac{1}{2(n-1)}
$$

gives:

$$
\boxed{
0
\le
b_\star-t_n
\le
\frac{1}{n-1}.
}
$$

Therefore:

$$
\boxed{
\lim_{n\to\infty}t_n
=
b_\star.
}
$$

---

# 21. A Genuine Case of Never Reaching at a Finite Stage, but Reaching in the Limit

Thus:

$$
\boxed{
\forall n<\infty,
\quad
t_n<b_\star,
}
$$

but:

$$
\boxed{
\sup_{n<\infty}t_n
=
b_\star.
}
$$

This is an ideal first example for WSFC:

$$
\boxed{
\text{finite freedom state}
<
\text{limit state}
}
$$

holds at every stage,

yet the entire sequence:

$$
\boxed{
\text{recovers the continuum extremum}.
}
$$

---

# 22. Broadworm Rectangle Ladder

Define:

$$
R_n
=
[0,1]\times[0,t_n].
$$

Then:

$$
R_n
\subseteq
R_{n+1}.
$$

For every finite $R_n$:

$$
Q_n(R_n)
=
\mathrm{true},
$$

but:

$$
Q_\infty(R_n)
=
\mathrm{false}.
$$

and:

$$
R_n
\rightarrow
R_\infty
=
[0,1]\times[0,b_\star].
$$

The limit rectangle has sufficient breadth and a unit-length diameter envelope, and thus forms a coarse universal rectangle cover for all unit arcs.

So we have not only:

$$
\Gamma_n
\rightarrow
\Gamma_\infty.
$$

The cover objects can also form:

$$
\boxed{
R_n
\rightarrow
R_\infty.
}
$$

---

# 23. This Sequence Does Not Aim for Optimal Area

We must be explicit:

$$
\operatorname{Area}(R_\infty)
=
b_\star
\approx0.43893.
$$

This is far larger than the current best convex-cover bounds.

The new preprint as of 2026-09-18 claims and formally verifies:

$$
0.239
\le
\alpha_{\rm convex}
\le
0.24633\ldots
$$

The Broadworm Rectangle Ladder is therefore not an area-optimal attack.

Its value is as a:

$$
\boxed{
\text{sequence-closure laboratory}.
}
$$

---

# 24. Relation to New Developments in 2026

Moser worm research in 2026 already makes extensive use of:

- finite polygonal witnesses;
- support inequalities;
- interval certificates;
- Lean formalization;
- continuum-to-polygonal reduction.

This means WSFC's sequence question is not an extra abstraction detached from modern proof architecture.

On the contrary:

$$
\boxed{
\text{finite polygonal stage}
\rightarrow
\text{continuum guarantee}
}
$$

is already one of the central bridges in current proofs.

WSFC's additional work is to make that bridge itself the primary research object, tracking:

- freedom birth;
- residual;
- modulus;
- event birth;
- task sufficiency;
- limit closure.

---

# 25. The First Closure Ladder Obtained in This Round

We can now make the WSFC closure ladder concrete:

### Level 0 — Parameterized Stage

$$
\Gamma_n
$$

has a finite parameter description.

### Level 1 — Nested Freedom

$$
\Gamma_n
\subseteq
\Gamma_{n+1}.
$$

### Level 2 — No Finite Sufficiency

$$
\forall n,\;
Q_n\not\Rightarrow Q_\infty.
$$

### Level 3 — Approximate Modulus

$$
Q_n
\Rightarrow
Q_\infty^{(\varepsilon_n)},
\qquad
\varepsilon_n
=
\frac{1}{2(n-1)}.
$$

### Level 4 — Omega Sufficiency

$$
\forall n\,Q_n
\Rightarrow
Q_\infty.
$$

### Level 5 — Scalar Probe Closure

$$
t_n
\uparrow
b_\star.
$$

### Level 6 — Future Optimal-Cover Closure

Not yet completed:

$$
A_n
\rightarrow
A_{\rm worm}.
$$

---

# 26. The Most Important Conceptual Distinction

This round shows:

$$
\boxed{
\text{sequence closure}
\neq
\text{finite stabilization}.
}
$$

A problem can perfectly well satisfy:

$$
\forall n:
\text{stage }n\text{ insufficient},
$$

while:

$$
\boxed{
\text{the full directed sequence is sufficient}.
}
$$

This is exactly the situation of the worm problem's polygonal hierarchy.

---

# 27. New Term: Omega-Stage Task Sufficiency

Definition:

If:

$$
Q_\infty
\iff
\forall n\,Q_n
$$

but:

$$
\forall N,\;
Q_N\not\equiv Q_\infty,
$$

then the hierarchy is said to have, for $Q$:

$$
\boxed{
\textbf{Omega-Stage Task Sufficiency}.
}
$$

The abbreviation may be:

$$
\boxed{
\mathrm{OSTS}.
}
$$

The worm problem's polygonal hierarchy is the first canonical example.

---

# 28. New Term: Freedom Birth Rate

If:

$$
d_n
$$

is the generic effective parameter-space dimension,

define:

$$
\boxed{
\beta_n
=
d_{n+1}-d_n.
}
$$

For the polygonal hierarchy:

$$
\boxed{
\beta_n=2.
}
$$

This is the most basic freedom birth rate.

But the theorem-effective birth rate:

$$
\beta_n^Q
=
d_{n+1}^{\rm eff}(Q)
-
d_n^{\rm eff}(Q)
$$

remains unknown.

The latter is the quantity truly worth calculating in the next phase.

---

# 29. New Term: Freedom-to-Residual Transfer

Each round adds:

$$
\beta_n
$$

freedoms,

while the residual:

$$
r_n^Q
$$

shrinks.

Define the exploratory ratio:

$$
\boxed{
\eta_n^Q
=
\frac{
r_n^Q-r_{n+1}^Q
}{
\beta_n
}.
}
$$

It measures:

> How much theorem-relevant residual is eliminated on average per additional degree of freedom.

For this round's breadth proxy,

if we use:

$$
r_n^{(b)}
=
b_\star-t_n,
$$

then:

$$
\eta_2^{(b)}
\approx
\frac{
0.005917298108
-
0.003817076725
}{
2
}
\approx
0.00105011.
$$

At present there are only low-stage anchors, so an asymptotic law cannot be extrapolated.

---

# 30. What Should the Next Round Actually Attack?

Exact / analytic results are now available for $n=2,3$.

The existing 2007–2008 literature explicitly states that, for $n>3$, the broadest $n$-segment arcs were still largely open at that time.

Thus a very natural next experimental line for WSFC is:

$$
\boxed{
t_4,t_5,t_6,\ldots
}
$$

But the aim is not merely to improve breadth values.

What truly needs to be recorded together is:

$$
d_n,
\quad
t_n,
\quad
b_\star-t_n,
\quad
\text{hull event type},
\quad
\text{extremizer topology},
\quad
\text{representation sensitivity}.
$$

This would provide the first genuine test of:

$$
\boxed{
\text{freedom birth}
\rightarrow
\text{residual contraction}.
}
$$

---

# 31. Computational Design for the Next Round

Recommendations:

## Track A — Global Polygonal Search

For:

$$
n=4,5,6,8
$$

search for high-breadth candidates.

## Track B — Cross-Representation

Use at least two representations:

- segment length + turning angle;
- vertex / edge vector representation.

## Track C — Interval Replay

Produce rigorous enclosures for the best candidates.

## Track D — Event Ledger

Record:

- convex-hull vertex count;
- support-edge ordering;
- near self-contact;
- angle degeneracy;
- breadth-active edges.

## Track E — Broadworm Alignment

Compare candidates with the broadworm in terms of:

- endpoint distance;
- breadth;
- hull support signature;
- local turn pattern.

---

# 32. Round 01 Conclusion

The most important conclusion of this round is not a new Moser worm area bound.

Rather, it is the first identification of an existing mathematical skeleton almost perfectly isomorphic to WSFC:

$$
\boxed{
\Gamma_2
\subset
\Gamma_3
\subset
\Gamma_4
\subset
\cdots
\subset
\Gamma_\omega.
}
$$

At every finite stage:

$$
\boxed{
Q_n
\not\Rightarrow
Q_\infty.
}
$$

but for the entire sequence:

$$
\boxed{
\forall n\,Q_n
\iff
Q_\infty.
}
$$

The breadth probe also gives:

$$
\boxed{
t_n
<
b_\star
\quad
\forall n<\infty,
}
$$

and:

$$
\boxed{
t_n
\uparrow
b_\star.
}
$$

This means that the limit produced by the state-evolution sequence of possibility spaces is not merely abstract language in the worm problem.

It already has a very concrete instance:

$$
\boxed{
\text{Every finite-degree-of-freedom layer is strictly incomplete,
yet the }\omega\text{-closure of the directed sequence can recover the full task.}
}
$$

The next round should begin directly with:

$$
\boxed{
n\ge4
}
$$

computational experiments and the freedom-residual curve.

---

# References

1. C. Panraksa, J. E. Wetzel, W. Wichiramala, *Covering n-Segment Unit Arcs Is Not Sufficient*, Discrete & Computational Geometry 37 (2007), 297–299. DOI: 10.1007/s00454-006-1258-7.
2. J. E. Wetzel, W. Wichiramala, *A Covering Theorem for Families of Sets in R^d*, Journal of Combinatorics 1 (2010), 69–75. DOI: 10.4310/JOC.2010.v1.n1.a5.
3. C. Panraksa, J. E. Wetzel, W. Wichiramala, *The Broadest Three-Segment Unit Arc*, Periodica Mathematica Hungarica 55 (2007), 157–168. DOI: 10.1007/s10998-007-4157-9.
4. W. Wichiramala, C. Panraksa, *Wetzel's 30-60-90 Triangle Covers Unit Arcs*, arXiv:2606.14625, 2026.
5. Ethan Keller, *Improved Bounds for Universal Convex Covers of Unit Arcs*, arXiv:2609.21968, 2026-09-18. Preprint; current claim includes Lean-checked proof artifacts.

---

# Canonical Source Declaration

The original file is the UTF-8 Markdown canonical source for `WSFC-R01-v0.1`.

Only `$...$` and `$$...$$` are used as mathematical delimiters.

This round explicitly distinguishes:

- external existing theorems;
- WSFC reformulations derived from existing theorems in this round;
- the elementary modulus derived in this round;
- numerical reproduction in this round;
- the still-uncompleted $n\ge4$ computation.

This round does not claim:

- that Moser's worm problem is solved;
- a new optimal area bound;
- that $t_n$ has been determined for $n>3$;
- that broadness suffices to determine the Moser worm optimal cover;
- finite-stage sufficiency;
- that full formalization is complete.

===END===