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

# WSFC Round 02
## Freedom Birth Rate, Broadworm Residual Dynamics, and Reproduction of the $n=4,5,6$ Polygonal Extremizers

**English Title:** *Freedom Birth Rate, Broadworm Residual Dynamics, and Numerical Reproduction of the $n=4,5,6$ Polygonal Extremizers*  
**Document ID:** `WSFC-R02-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  
**Prerequisites:** `WSFC-00-v0.1`; `WSFC-R01-v0.1`  
**Status:** `LITERATURE RECOVERY / NUMERICAL REPRODUCTION / FREEDOM-DYNAMICS BASELINE`  
**Canonical Source:** UTF-8 Markdown

---

# Abstract

Round 01 established the polygonal hierarchy:

$$
\Gamma_n
=
\{\text{unit polygonal arcs with at most }n\text{ segments}}\}
$$

and obtained the generic quotient dimension:

$$
d_n=2n-2,
$$

Thus:

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

The original goal of Round 02 was to compute $n=4,5,6,8$ directly. A renewed literature search revealed an important detail that first needs correction: in their 2007 paper *The broadest three-segment unit arc*, Panraksa, Wetzel, and Wichiramala not only proved the $n=3$ case, but also explicitly listed seven conjectures for $n>3$ at the end and supplied computer-generated vertex coordinates, breadths, and endpoint distances for $n=4,5,6$.

Their published four-decimal-place candidates are:

$$
t_4\approx0.4382,
\qquad
t_5\approx0.4384,
\qquad
t_6\approx0.4387.
$$

This round uses those tabulated coordinates as seeds, rebuilds a minimum-width calculator, and performs multi-start Nelder–Mead local optimization of the scale-invariant breadth / chain-length ratio.

This yields research-level numerical candidates:

$$
\boxed{
t_4^{\rm cand}
\approx
0.438207692919662,
}
$$

$$
\boxed{
t_5^{\rm cand}
\approx
0.438438009670890,
}
$$

$$
\boxed{
t_6^{\rm cand}
\approx
0.438694694629164.
}
$$

These results agree with the four-decimal-place data from 2007, but still do not constitute a global optimality proof.

Meanwhile, using the analytic constants for the Zalgaller / Schaer broadworm:

$$
\beta
=
\left[
2\left(
\frac{\pi}{2}
-\varphi
-2\psi
+\tan\varphi
+\tan\psi
\right)
\right]^{-1},
$$

where:

$$
\varphi
=
\arcsin
\left(
\frac16
+
\frac43
\sin
\left(
\frac13\arcsin\frac{17}{64}
\right)
\right),
$$

$$
\psi
=
\arctan
\left(
\frac12\sec\varphi
\right),
$$

we obtain:

$$
\boxed{
\beta
\approx
0.438925369259466.
}
$$

Thus the candidate residual:

$$
r_n^{(b)}
=
\beta-t_n^{\rm cand}
$$

has, from $n=4$ to $6$, fallen to approximately:

$$
7.18\times10^{-4},
\quad
4.87\times10^{-4},
\quad
2.31\times10^{-4}.
$$

The most important WSFC conclusion of this round is:

$$
\boxed{
\text{Freedom Birth Rate}
\neq
\text{Theorem-Residual Contraction Rate}.
}
$$

The polygonal hierarchy gives birth to exactly two generic freedoms at each step, but the decreases in breadth residual are highly uneven.

Round 02 also performs a minimum-cost stress test for $n=8$ by subdividing the $n=6$ extremizer and then locally optimizing; the result does not reliably activate new effective freedom, so it is retained only as an Async Note, with no claim for the value of $t_8$.

---

# 1. Correction to the Status in Round 01

Round 01 abbreviated:

```text
n>=4 breadth
```

as `OPEN`.

This was not wrong at the level of an exact optimum proof, but the literature information was incomplete.

The more precise status is:

```text
n>=4 exact broadest-arc proof        OPEN
n>=4 uniqueness proof                OPEN
n>=4 full structure theorem          OPEN
n=4,5,6 numerical candidates         AVAILABLE SINCE 2007
```

Accordingly, this round does not overwrite Round 01, but records an append-only correction.

---

# 2. The Full Sequence Conjecture Was Already Proposed in 2007

The original paper proposes seven conjectures for $n>3$:

1. $\Gamma_n$ is unique up to rigid motions;
2. $\Gamma_n$ is convex;
3. $\Gamma_n$ is a drape;
4. even $n$ has bilateral symmetry, while odd $n>3$ has no symmetry;
5. $t_n$ strictly increases to the broadworm breadth;
6. the extremizer endpoint distance strictly decreases to the broadworm endpoint distance;
7. the arclength-parametrized $\Gamma_n$ uniformly converges to the broadworm.

This is almost naturally isomorphic to WSFC.

The sequence proposed in Round 01:

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

already appeared in the 2007 literature in the form of an extremizer-sequence conjecture.

WSFC's addition is not being the first to think of $n\to\infty$, but decomposing it into:

- freedom birth;
- effective freedom;
- residual contraction;
- task quotient;
- certificate;
- active refinement;
- limit closure gate.

---

# 3. The Broadworm's Exact Numerical Anchor

Recent literature restates the Zalgaller broadworm escape length:

$$
\ell_0
=
2\left(
\frac{\pi}{2}
-\varphi
-2\psi
+\tan\varphi
+\tan\psi
\right).
$$

This round calculates:

$$
\ell_0
\approx
2.278291641440437.
$$

Thus the breadth of the unit-length broadworm is:

$$
\boxed{
\beta
=
\ell_0^{-1}
\approx
0.438925369259466.
}
$$

This value is a better residual anchor than the value used in Round 01:

$$
0.43893
$$

as above.

---

# 4. The $n=2$ and $n=3$ Anchors

$$
t_2
=
\frac{\sqrt3}4
\approx
0.433012701892219.
$$

For $n=3$, use the known formula:

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

Recomputation in this round gives:

$$
\boxed{
\theta_3^\star
\approx
63.831641742942^\circ
}
$$

and:

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

---

# 5. Numerical Reproduction Method

For an open polygonal chain with $n$ segments:

$$
P_0,P_1,\ldots,P_n,
$$

fix:

$$
P_0=(0,0),
\qquad
P_n=(1,0)
$$

only as gauge / scale normalization.

Compute the convex hull of all the vertices.

The breadth of the polygonal trace equals the minimum width of its convex hull.

For a convex polygon, the minimum width must be attained in the normal direction of some hull edge, so compute the projection range for each edge:

$$
w_e
=
\max_i\langle P_i,n_e\rangle
-
\min_i\langle P_i,n_e\rangle.
$$

Take:

$$
w
=
\min_e w_e.
$$

Since endpoint scale is fixed, use the scale-invariant objective:

$$
\boxed{
J
=
\frac{w}{L}
}
$$

where:

$$
L
=
\sum_{i=0}^{n-1}
\|P_{i+1}-P_i\|.
$$

Finally, rescale the chain to:

$$
L=1.
$$

The 2007 table gives coordinates to only four decimal places, so this round uses them as initial values and then performs multi-start local refinement.

---

# 6. Reproduction for $n=4$

2007:

$$
t_4\approx0.4382.
$$

The best local candidate in this round:

$$
\boxed{
t_4^{\rm cand}
\approx
0.438207692919662.
}
$$

Its gap from the broadworm:

$$
\boxed{
\beta-t_4^{\rm cand}
\approx
7.176763398041297e-04.
}
$$

Multi-start summary:

```text
best   = 0.438207692919662
median = 0.438201174878379
worst  = 0.438161642907312
```

The candidate is almost bilaterally symmetric within numerical tolerance.

This agrees with the 2007 even-$n$ symmetry conjecture, but is not a proof.

---

# 7. Reproduction for $n=5$

2007:

$$
t_5\approx0.4384.
$$

This round:

$$
\boxed{
t_5^{\rm cand}
\approx
0.438438009670890.
}
$$

Gap:

$$
\boxed{
\beta-t_5^{\rm cand}
\approx
4.873595885760418e-04.
}
$$

This candidate clearly lacks bilateral symmetry.

This agrees with the 2007 conjecture that odd $n>3$ has no symmetry, but likewise is not a proof.

---

# 8. Reproduction for $n=6$

2007:

$$
t_6\approx0.4387.
$$

This round:

$$
\boxed{
t_6^{\rm cand}
\approx
0.438694694629164.
}
$$

Gap:

$$
\boxed{
\beta-t_6^{\rm cand}
\approx
2.306746303028517e-04.
}
$$

The candidate returns to near bilateral symmetry.

---

# 9. Equal-Support Flattening Signal

For the refined candidates at $n=4$ and $n=6$, the optimizer has nearly equalized the widths in the normal directions of the convex hull edges.

That is:

$$
w_1
\approx
w_2
\approx
\cdots
\approx
t_n^{\rm cand}.
$$

This is an important signal in extremal geometry.

It suggests that the broadest polygonal arc may be reformulated as a:

$$
\boxed{
\text{support-width equalization problem}
}
$$

rather than relying solely on raw vertex search.

This may provide an entry point for future dimension reduction at $n\ge7$.

---

# 10. Freedom Birth Rate

We still have:

$$
d_n=2n-2.
$$

Thus:

$$
\boxed{
\beta_n^{\rm freedom}
=
d_{n+1}-d_n
=
2.
}
$$

This sequence is extremely regular.

But the breadth gain:

$$
\Delta t_n
=
t_{n+1}-t_n
$$

is not constant and has no simple monotone pattern.

For example, in this round's candidate sequence:

$$
t_3
\rightarrow
t_4^{\rm cand}
\rightarrow
t_5^{\rm cand}
\rightarrow
t_6^{\rm cand}.
$$

the breadth improvements obtained per addition of two generic freedoms are clearly different.

Hence:

$$
\boxed{
\text{representation freedom dynamics}
\neq
\text{task progress dynamics}.
}
$$

---

# 11. Broadworm Residual

Define:

$$
r_n^{(b)}
=
\beta-t_n.
$$

For the candidate sequence:

$$
r_4^{(b)}
\approx
7.176763398041297e-04,
$$

$$
r_5^{(b)}
\approx
4.873595885760418e-04,
$$

$$
r_6^{(b)}
\approx
2.306746303028517e-04.
$$

This sequence can now begin to serve as WSFC's first genuine:

$$
\boxed{
\text{task residual sequence}.
}
$$

It is not the full worm-cover residual.

It is only a scalar proxy for the breadth task.

---

# 12. Freedom-to-Residual Transfer

Define:

$$
\eta_n^{(b)}
=
\frac{
r_n^{(b)}-r_{n+1}^{(b)}
}{
2
}.
$$

The denominator $2$ is exactly the number of generic freedoms added in each round.

For the first time, this lets us place:

$$
\boxed{
\text{freedom birth}
}
$$

and:

$$
\boxed{
\text{theorem-sensitive residual contraction}
}
$$

in the same data sequence.

The current data are still too sparse to fit an asymptotic law.

---

# 13. Relation Between the 2007 Conjecture 5 and Round 01

The 2007 paper already conjectures:

$$
t_n
\uparrow
\beta.
$$

Round 01 obtains an elementary route through polygonal approximation / the approximate cover modulus that supports:

$$
0
\le
\beta-t_n
\le
\frac1{n-1}.
$$

Thus, at least for the breadth scalar, WSFC already has:

$$
\boxed{
\text{finite freedom}
\rightarrow
\text{explicit residual envelope}
\rightarrow
\text{continuum limit}.
}
$$

The role of Round 02 is to begin filling in real low-stage dynamics.

---

# 14. Why Can $n=8$ Not Be Obtained Directly by Subdividing $n=6$?

The cheapest strategy:

$$
\Gamma_6^\star
\rightarrow
\text{subdivide}
\rightarrow
\Gamma_8^{\rm seed}
\rightarrow
\text{local optimize}.
$$

Theoretically:

$$
t_8\ge t_6.
$$

But in the actual optimizer, the new collinear-subdivision freedom can easily remain:

$$
\text{task-null}.
$$

That is, new parameters exist, but the old extremizer basin has not truly been left.

This round's local stress test did not produce a result that can be regarded as a reliable $t_8$ candidate.

Thus $n=8$ is moved to:

```text
ASYNC / GLOBAL SEARCH NEEDED
```

rather than forcing a local-basin number into a claim for $t_8$.

---

# 15. This Precisely Confirms a Core WSFC Distinction

$$
\boxed{
\text{freedom available}
\neq
\text{freedom activated}.
}
$$

More fully:

$$
d_n^{\rm rep}
\uparrow
$$

does not imply:

$$
d_n^{\rm eff}(Q)
\uparrow.
$$

Sequence computation must therefore begin retaining:

- freedom birth;
- activation status;
- local sensitivity;
- residual reduction;
- basin identity.

---

# 16. A Better Next Method for $n\ge7$

Deng (2026) recently reformulated the dual unit-strip broadworm problem as a width-constrained shortest-path functional and noted that its finite-difference discretization can be written as convex optimization.

This suggests that WSFC's next step need not rely solely on raw-vertex Nelder–Mead.

Consider the dual of:

$$
\boxed{
\text{maximize breadth at fixed length}
}
$$

namely:

$$
\boxed{
\text{minimize length at fixed width}.
}
$$

If the fixed-width formulation can be converted to more convex support constraints, then:

$$
n=7,8,16,\ldots
$$

may be easier than direct nonsmooth minimum-width maximization.

---

# 17. Support-Space Freedom

Round 02 thus adds another possible freedom hierarchy:

Originally:

$$
\boxed{
\text{vertex-space freedom}.
}
$$

Next, we can compare:

$$
\boxed{
\text{support-space freedom}.
}
$$

That is, instead of letting vertices move arbitrarily, let the active support directions, support distances, and path order evolve.

If all critical widths of the broadest candidate tend toward equalization, support space may be a representation closer to the theorem-relevant quotient.

---

# 18. Round 02 Closure Ledger

```text
LITERATURE n4-n6 CANDIDATES        RECOVERED
BROADWORM EXACT ANCHOR             RECOVERED
n2                                 EXACT
n3                                 ANALYTIC FORMULA / REPRODUCED
n4                                 NUMERICAL REPRODUCED
n5                                 NUMERICAL REPRODUCED
n6                                 NUMERICAL REPRODUCED
EVEN-STAGE SYMMETRY SIGNAL         OBSERVED
ODD-STAGE ASYMMETRY SIGNAL         OBSERVED
SUPPORT-WIDTH EQUALIZATION         OBSERVED
FREEDOM BIRTH RATE                 EXACT GENERIC FORMULA
TASK RESIDUAL SEQUENCE             INITIALIZED
n8 GLOBAL CANDIDATE                OPEN
n4-n6 GLOBAL OPTIMALITY            OPEN
INTERVAL CERTIFICATE               OPEN
SUPPORT-SPACE FORMULATION          NEXT
```

---

# 19. Next Round

For Round 03, the recommendation is not to increase $n$ further by brute force.

First undertake:

## A. Support-Space Reparameterization

Convert from:

$$
P_0,\ldots,P_n
$$

to:

$$
(\theta_i,h_i,\text{contact order}).
$$

## B. Equal-Width Constraint Extraction

Investigate whether candidate extremizers satisfy:

$$
w_i=t_n
$$

for all active hull edges.

## C. Even-$n$ Symmetry Reduction

For:

$$
n=4,6,8
$$

first solve in the bilaterally symmetric subspace, then red-team in the full space.

## D. Odd-$n$ Symmetry-Breaking Coordinate

For:

$$
n=5,7
$$

find the smallest asymmetry variable.

## E. Dual Fixed-Width Formulation

Fix:

$$
b=1
$$

and minimize the chain length:

$$
L_n^\star.
$$

Then use:

$$
t_n
=
\frac1{L_n^\star}.
$$

If this dual is more stable, it will serve as the main runtime for $n\ge7$.

---

# 20. Conclusion

The true achievement of Round 02 is not merely computing three more decimal values.

Rather, it is the first time WSFC's three sequences are presented together:

$$
\boxed{
d_n
}
$$

represents state-space freedom growth;

$$
\boxed{
t_n
}
$$

represents breadth-task output;

$$
\boxed{
r_n^{(b)}=\beta-t_n
}
$$

represents theorem-sensitive residual.

We have now seen that:

$$
d_n
$$

increases regularly and linearly,

while:

$$
r_n^{(b)}
$$

shrinks in a highly uneven manner.

This is:

$$
\boxed{
\textbf{Sequence Freedom Dynamics}.
}
$$

What truly needs study next is no longer:

> Would two more degrees of freedom make things better?

Instead, it is:

> Which two degrees of freedom are actually activated? Through which support events do they eliminate how much residual? When does adding freedom merely reparametrize within the old basin?

This begins to transform the worm problem from high-dimensional geometric search into:

$$
\boxed{
\text{freedom birth}
\rightarrow
\text{freedom activation}
\rightarrow
\text{support event}
\rightarrow
\text{residual contraction}
\rightarrow
\text{limit closure}.
}
$$

---

# References

1. Chatchawan Panraksa, John E. Wetzel, Wacharin Wichiramala, *The broadest three-segment unit arc*, Periodica Mathematica Hungarica 55(2), 157–168 (2007). DOI: 10.1007/s10998-007-4157-9.
2. Chatchawan Panraksa, John E. Wetzel, Wacharin Wichiramala, *Covering n-Segment Unit Arcs Is Not Sufficient*, Discrete & Computational Geometry 37(2), 297–299 (2007). DOI: 10.1007/s00454-006-1258-7.
3. Zhipeng Deng, *Revisit escape path for infinite unit strip forest and unit broadworm*, arXiv:2607.18563 (2026).
4. Ethan Keller, *Improved bounds for universal convex covers of unit arcs*, arXiv:2609.21968 (2026).

---

# Canonical Source Declaration

The original file is the UTF-8 Markdown canonical source for `WSFC-R02-v0.1`.

The numerical grading in this round is:

```text
EXACT
ANALYTIC-FORMULA
LITERATURE-NUMERICAL
RESEARCH-NUMERICAL-CANDIDATE
ASYNC-LOCAL-STRESS
CERTIFIED
FORMAL
```

The refined breadths for $n=4,5,6$ belong only to:

```text
RESEARCH-NUMERICAL-CANDIDATE
```

They must not be treated as proven values of $t_n$.

The local result for $n=8$ belongs only to:

```text
ASYNC-LOCAL-STRESS
```

It must not be treated as an estimate of $t_8$.

===END===