# WSFC Round 27
## 2007 Conjecture Ledger / Authority Audit：Classical Closure、WSFC Conditional Proofs 與 Formal Debt

**English Title:** *The 2007 Conjecture Ledger: Classical Closures, WSFC Conditional Proofs, and Remaining Formal Debt*  
**文件代號：** `WSFC-R27-v0.1`  
**版本：** v0.1  
**日期：** 2026-09-24  
**作者：** Neo.K  
**AI 協力研究者：** Aletheia / OpenAI GPT-5.6 Sol  
**系列：** Worm Sequence Freedom Closure  
**前置：** `WSFC-R21`–`WSFC-R26`  
**狀態：** `AUTHORITY AUDIT / NO NEW GLOBAL OVERCLAIM / CONJECTURE-LEVEL LEDGER`  
**Canonical Source：** UTF-8 Markdown

---

# 摘要

2007 年 Panraksa–Wetzel–Wichiramala 在 *The broadest three-segment unit arc* 最後提出七條 general-$n$ conjectures：

1. $\Gamma_n$ unique up to rigid motions；
2. $\Gamma_n$ convex；
3. $\Gamma_n$ is a drape；
4. even $n$ bilateral symmetry，odd $n>3$ no symmetries；
5. $t_n$ strictly increases to broadworm breadth $t_0$；
6. endpoint distance $b_n$ strictly decreases to broadworm endpoint distance；
7. arclength-parametrized $\Gamma_n$ uniformly converges to the broadworm。

Round 27 的核心不是增加新 theorem，而是 authority normalization。

最重要的修正是：

$$
\boxed{(2)\text{ convexity and }(3)\text{ drape are classical-source consequences}.}
$$

Adhikari–Pitman 1989 已證：對任何 fixed segment cap $M$，minimal width-at-least-one polygonal arc 必 convex。將 broadest $n$-segment unit arc scale 到 breadth $1$，再利用 lower segment count 可由 collinear subdivision 嵌入 exact-$n$ representation，得到 conjecture (2)。

同一篇對 convex minimizer又指出：若 arc 越出 endpoints 的 perpendicular strip，可作 perpendicular truncation得到更短且 breadth 更大的 arc。Maki–Wetzel–Wichiramala 2005 的 drape 定義正是 convex arc 位於此 endpoint strip。因此 conjecture (3) 亦是既有 published theorem chain 的 consequence。

所以七條的正確 authority 不是「WSFC 全解」，而是：

$$
\boxed{2\text{ classical source closures}+5\text{ WSFC provisional research closures}.}
$$

其中 conjecture (6) 含 decisive computer-assisted interval certificate；目前沒有任何 WSFC-specific 新 claim 已 proof-assistant closed 或 external independent-reviewed。

---

# 1. Source Ledger

原始 2007 conjectures：

$$
\begin{array}{ll}
(1)&\Gamma_n\text{ unique up to rigid motions},\\
(2)&\Gamma_n\text{ convex},\\
(3)&\Gamma_n\text{ a drape},\\
(4)&\text{even bilateral symmetry / odd no symmetry},\\
(5)&t_n\uparrow t_0\text{ strictly},\\
(6)&b_n\downarrow b_0\text{ strictly},\\
(7)&\Gamma_n\to\Gamma_{\rm BW}\text{ uniformly}.
\end{array}
$$

source 同時已知 $n=2,3$ uniqueness，以及：

$$
t_n\le t_{n+1}<t_0.
$$

---

# 2. Conjecture (2)：Convexity — Classical Source Closed

Adhikari–Pitman 1989 Lemma 7：

$$
\boxed{\text{a minimal arc with at most }M\text{ segments is convex}.}
$$

breadth/length scaling duality將 broadest finite-$n$ unit arc轉為 breadth-one finite-$M$ minimal arc，因此 (2) 不應列為 WSFC novelty。

---

# 3. Conjecture (3)：Drape — Classical Source Closed

Drapeability 2005 的 drape 是：

> convex arc lying in the closed semi-infinite endpoint strip.

Adhikari–Pitman 的 convex minimizer strip reduction給這個更強條件，所以：

$$
\boxed{\Gamma_n\text{ is a drape}.}
$$

這不只是 generic “drapeable”。

---

# 4. Conjecture (1)：Uniqueness

R24 的兩個 strict-convexity engines控制 integer allocation與 continuous geometry；strict Jensen再使 tangent support partition unique。R23 排除 lower-stage optimizer。conditional on G0：

$$
\boxed{\operatorname{Argmax}(\mathcal F_n)/\operatorname{Isom}(\mathbb R^2)\text{ is a singleton}.}
$$

狀態：

```text
RESEARCH-PROOF-LEVEL PROVISIONALLY CLOSED
FORMAL OPEN
INDEPENDENT REVIEW OPEN
```

---

# 5. Conjecture (4)：Symmetry

Even stage：

$$
q_L=q_R,\qquad x=d/2,
$$

left/right support lists互為鏡射，所以 bilateral symmetry。

Odd stage：

$$
|q_L-q_R|=1.
$$

任何 trace symmetry必 preserve endpoint set。若固定兩 endpoints，nonidentity 候選是 endpoint-line reflection，但會把單側 convex arc翻到另一側；若交換 endpoints，則會交換 left/right support chains，迫使 $q_L=q_R$，矛盾。

所以 odd $n>3$ 無 nontrivial Euclidean symmetry。

狀態：research-proof-level provisional，conditional on G0 + balancing。

---

# 6. Conjecture (5)：Breadth Sequence

R22：

$$
0\le t_0-t_n\le\frac1{\lfloor n/2\rfloor},
$$

故 $t_n\to t_0$。

R23 以 tangent-gap split 與 exact base 證 strictness：

$$
t_n<t_{n+1}.
$$

因此 (5) research-proof-level provisionally closed；strictness conditional on G0-lite。

---

# 7. Conjecture (6)：Endpoint Sequence

R25 修正：

$$
b_n=t_nd_n^\star.
$$

R26 將 competing elasticity 化為 kernel $\mathcal K>0$，並用 adaptive interval certificate覆蓋 balanced continuous support-count interpolation：

```text
processed boxes             28244
unresolved terminal boxes       0
minimum K lower bound        0.0006339178281571378
```

配 exact finite base與 R22 convergence：

$$
\boxed{b_n\text{ strictly decreases to }b_0.}
$$

狀態：research-proof-level provisional + decisive computer-assisted component；formal open。

---

# 8. Conjecture (7)：Uniform Convergence

原文若不固定 placement，$\Gamma_n$ 只定義到 rigid motion，所以 literal coordinate convergence需要 normalization。

最安全表述：

$$
\boxed{\Gamma_n\to\Gamma_{\rm BW}\quad\text{modulo Euclidean isometries}.}
$$

或固定：initial endpoint at origin、final endpoint on positive $x$-axis、reflection convention。

arclength parametrizations皆 1-Lipschitz。Arzelà–Ascoli給 subsequential uniform limit。R22給 $t_n\to t_0$，breadth continuity把 limit breadth送到 $t_0$。length lower semicontinuity給 $L\le1$；若 $L<1$，scale到 unit length會產生 breadth $>t_0$，矛盾。故 $L=1$，limit是 maximal-breadth unit arc。由 broadworm uniqueness，所有 subsequential limits都是 normalized broadworm，所以 full normalized sequence converges。

因此 (7) 在 normalized/mod-isometry form research-proof-level closed。

---

# 9. Authority Ladder

```text
S = published source theorem
R = WSFC analytic research proof
C = WSFC computer-assisted certificate
F = proof-assistant formal verification
I = independent reproduction/review
P = peer-reviewed publication
```

目前 strongest authority：

| Conjecture | Authority |
|---|---|
| (1) | R |
| (2) | S |
| (3) | S |
| (4) | R |
| (5) | R |
| (6) | R + C |
| (7) | R |

WSFC-specific：

```text
F = none
I = none completed
P = none
```

---

# 10. Literature Search Discipline

2026-09 targeted search涵蓋 original title、exact conjecture phrases、later broadest-$n$ phrases、2025 worm-problem review與 author publication lists。

沒有定位到一篇明確宣布 general-$n$ 七條 conjectures已整體被後續文獻解決的文章。

但：

$$
\boxed{\text{targeted search}\neq\text{priority proof}.}
$$

正式 novelty claim仍需 MathSciNet / zbMATH / citation-tree / author-contact level review。

---

# 11. Anti-Overclaim Rules

禁止：

1. 把 (2) convexity說成 WSFC 新 theorem；
2. 把 (3) drape說成 WSFC 新 theorem；
3. 把 (6) 說成 purely analytic formal proof；
4. 把 targeted search說成 novelty proof；
5. 把 (7) 不經 normalization寫成 literal coordinate convergence；
6. 把 odd mirror pair混成 orientation-preserving rigid-motion uniqueness。

---

# 12. Current Ledger

```text
(1) UNIQUE
    R provisional, conditional G0

(2) CONVEX
    S classical closed

(3) DRAPE
    S classical closed

(4) SYMMETRY
    R provisional, conditional G0

(5) BREADTH STRICT -> t0
    R provisional

(6) ENDPOINT STRICT -> b0
    R+C provisional

(7) UNIFORM -> BROADWORM
    R closed in normalized/mod-isometry form
```

---

# 13. 下一步

Round 28 應轉成：

$$
\boxed{\textbf{Independent Verification Pack}.}
$$

拆成：

```text
Packet A — G0 finite-stage exactness
Packet B — strict breadth monotonicity
Packet C — finite-stage uniqueness / symmetry
Packet D — full-domain coverage / convergence
Packet E — endpoint strictness interval certificate
```

每包只保留 theorem statement、minimal assumptions、proof skeleton、machine certificate、dependency與attack checklist。

---

# 結論

Round 27 的核心是 authority normalization：

$$
\boxed{2\text{ classical source closures}+5\text{ WSFC provisional research closures}.}
$$

下一階段不是再多堆 theorem，而是 authority elevation：formalization、independent replay、independent mathematical review、literature priority review。

===END===