> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # WSFC Round 22 ## G1 Full-Domain Coverage: Exact-Length Tent Approximation, a Uniform Cover Modulus and Broadworm Value Closure **English Title:** *G1 Full-Domain Coverage: Exact-Length Tent Approximation, a Uniform Cover Modulus, and Broadworm Value Closure* **Document ID:** `WSFC-R22-v0.1` **Version:** v0.1 **Date:** 2026-09-23 **Author:** Neo.K **AI Research Collaborator:** Aletheia / OpenAI GPT-5.6 Sol **Series:** Worm Sequence Freedom Closure **Prerequisites:** `WSFC-R20-v0.1`; `WSFC-R21-v0.1` **Status:** `G1/G2 QUANTITATIVE CLOSURE / VALUE CONVERGENCE / STRUCTURAL STRICTNESS OPEN` **Canonical Source:** UTF-8 Markdown --- # Abstract Round 21 promoted: $$ \boxed{ \text{G0 Finite-Stage Exactness} } $$ to research-proof-level provisionally closed. Round 22 tackles: $$ \boxed{ \textbf{G1 — Full-Domain Coverage}. } $$ External literature first provides two important anchors. Adhikari–Pitman already directly used the following in 1989: > An arbitrary arc can be approximated by arcs with finitely many straight segments. This was their step from finite-segment estimates back to arbitrary arcs. More explicitly, Wetzel–Wichiramala 2010 established the **Simple Approximation Theorem**: they regard curve, path and arc as synonymous, use the Hausdorff metric, and prove qualitative near-density of simple polygonal unit arcs among all unit arcs. On that basis, Round 22 adds an elementary but quantitative exact-length construction. Let: $$ \gamma:[0,1]\rightarrow\mathbb R^2 $$ be an arclength parametrization of a unit-length rectifiable path. Take: $$ N\ge1, $$ and divide the interval equally: $$ t_i=\frac{i}{N}, $$ so every block has arc length: $$ \Delta=\frac1N. $$ Let: $$ A_i=\gamma(t_i), \qquad B_i=\gamma(t_{i+1}). $$ Since a chord is no longer than its arc: $$ d_i=|A_i-B_i|\le\Delta. $$ There consequently exists a point: $$ M_i $$ such that: $$ \boxed{ |A_iM_i| = |M_iB_i| = \frac{\Delta}{2}. } $$ Specifically, if: $$ A_i\neq B_i, $$ take the chord midpoint $C_i$ and a unit normal $\nu_i$: $$ M_i = C_i + \frac12 \sqrt{ \Delta^2-d_i^2 } \,\nu_i. $$ If: $$ A_i=B_i, $$ choose any unit vector $\nu_i$ and set: $$ M_i=A_i+\frac{\Delta}{2}\nu_i. $$ Replace each original block with: $$ A_i\rightarrow M_i\rightarrow B_i $$ Concatenation gives the polygonal path: $$ P_{2N}. $$ Every block has length: $$ \frac{\Delta}{2} + \frac{\Delta}{2} = \Delta. $$ Therefore: $$ \boxed{ \ell(P_{2N})=1 } $$ **exactly**. Every block also satisfies: $$ \boxed{ d_H( \gamma_i, P_i ) \le \frac{\Delta}{2}. } $$ The reason is very simple. Any point of the original subarc is at most the following arc-length distance from its nearer endpoint: $$ \frac{\Delta}{2}. $$ Any point of the replacement tent is likewise at most the following distance along a half-edge from the corresponding endpoint: $$ \frac{\Delta}{2}. $$ The endpoints belong to both traces. Globally, therefore: $$ \boxed{ d_H( \operatorname{tr}\gamma, \operatorname{tr}P_{2N} ) \le \frac{1}{2N}. } $$ If the tent is parametrized synchronously on the same arclength blocks, we also have: $$ \boxed{ \|\gamma-P_{2N}\|_\infty \le \frac1N. } $$ This is a fixed-segment quantitative modulus **uniform over all unit rectifiable paths**. --- # 1. Literature Anchor: Simple Approximation Wetzel–Wichiramala 2010's Simple Approximation Theorem proves that for any rectifiable arc: $$ \gamma $$ and: $$ \varepsilon>0, $$ there exists a simple polygonal path: $$ \pi $$ such that: $$ L-\varepsilon < \ell(\pi) \le L $$ and it is close to $\gamma$ in the Hausdorff/neighbourhood sense. The main consequence is that simple polygonal unit arcs suffice to approximate all unit arcs for covering problems. Thus: $$ \boxed{ \text{qualitative polygonal density} } $$ already has classical/source support. Round 22's new work is a fixed-$n$ uniform quantitative bound. --- # 2. Why Not Use the Inscribed Chord Length Deficit? A traditional inscribed polygon: $$ A_0A_1\cdots A_N $$ has only: $$ \ell(P_N)\le1. $$ For an arbitrary rectifiable curve on a fixed equal mesh, the deficit: $$ 1-\ell(P_N) $$ has no obvious uniformly small bound independent of smoothness. Directly rescaling the inscribed polygon is therefore not the cleanest route. The tent construction bypasses this problem entirely: it enforces, during the construction of each block: $$ \boxed{ \ell(P_i)=\ell(\gamma_i)=\Delta. } $$ Exact global unit length is therefore a local invariant. --- # 3. Local Tent Lemma If: $$ |A-B|\le\Delta, $$ there exists a two-edge path: $$ A\rightarrow M\rightarrow B $$ with: $$ |AM|=|MB|=\Delta/2. $$ The height from the midpoint to the apex is: $$ h = \frac12 \sqrt{ \Delta^2-|A-B|^2 }. $$ Hence: $$ |AM|^2 = \frac{|A-B|^2}{4} + h^2 = \frac{\Delta^2}{4}. $$ --- # 4. Exact Segment Count $N$ blocks produce: $$ \boxed{ 2N } $$ nonzero segments, each of length: $$ \Delta/2. $$ There is therefore no zero-segment problem. For arbitrary: $$ n\ge2, $$ take: $$ m= \left\lfloor \frac n2 \right\rfloor. $$ First construct $2m$ segments. If: $$ n=2m+1, $$ perform one collinear split of any segment. The trace, length and Hausdorff error remain unchanged. A unit polygonal path with **exactly $n$ segments** therefore exists: $$ P_n. $$ --- # 5. Full-Domain Hausdorff Cover Radius We obtain: $$ \boxed{ r_n^H := \sup_{\gamma\in\Gamma} \inf_{P\in\mathcal F_n} d_H(\gamma,P) \le \frac{ 1 }{ 2\lfloor n/2\rfloor }. } $$ For even: $$ n=2m, $$ $$ \boxed{ r_n^H\le\frac1n. } $$ For odd: $$ n=2m+1, $$ $$ \boxed{ r_n^H\le\frac1{n-1}. } $$ Consequently: $$ \boxed{ r_n^H\rightarrow0. } $$ G1 is therefore not merely qualitative closure, but quantitative exhaustion. --- # 6. Parameterized Uniform Cover Traverse every half-edge synchronously at unit speed. On the first half of a block: $$ t\in[t_i,t_i+\Delta/2], $$ we have: $$ |\gamma(t)-A_i| \le t-t_i, $$ and: $$ |P(t)-A_i| = t-t_i. $$ Therefore: $$ |\gamma(t)-P(t)| \le 2(t-t_i) \le \Delta. $$ The second half is symmetric with respect to $B_i$. Hence: $$ \boxed{ \|\gamma-P_{2N}\|_\infty \le \frac1N. } $$ For general $n$: $$ \boxed{ d_\infty \le \frac1{\lfloor n/2\rfloor}. } $$ --- # 7. Simple vs. General Polygonal Semantics Semantic discipline must be retained here. Round 22's tent construction: ```text DOES NOT guarantee global simplicity. ``` The 2007 paper on the broadest $n$-segment arcs, however, itself permits crossing polygonal-chain candidates; its three-segment proof specifically handles an intersection of nonadjacent segments. Wetzel–Wichiramala 2010 state even more explicitly: ```text curve / path / arc are regarded as synonymous ``` They impose `simple polygonal` as an additional stronger restriction. For the 2007 polygonal-chain $F_n$ optimization semantics, this round's quantitative construction is therefore appropriate. If a future theorem requires: ```text simple polygonal exact-n uniform modulus ``` this round does not claim to have proved it; the 2010 source supplies qualitative near-density. --- # 8. G2 Effective Approximation Round 20 defines: $$ r_n = \sup_{\gamma\in\Gamma} \inf_{\eta\in\Gamma_n} d(\gamma,\eta). $$ We now have, in the Hausdorff metric: $$ \boxed{ r_n \le \frac{ 1 }{ 2\lfloor n/2\rfloor }. } $$ Therefore: $$ \boxed{ \text{G2 = CLOSED}. } $$ This is not the tail of a chosen history. It is a full-domain uniform cover. --- # 9. Breadth Error Round 20 has proved: $$ |b(K)-b(L)| \le 2d_H(K,L). $$ Substituting: $$ r_n^H \le \frac1{ 2\lfloor n/2\rfloor }, $$ gives: $$ \boxed{ |b(\gamma)-b(P_n)| \le \frac1{ \lfloor n/2\rfloor }. } $$ Thus the global task residual satisfies: $$ \boxed{ r_n^Q \le \frac1{ \lfloor n/2\rfloor }. } $$ --- # 10. Squeezing the True Broadest Values Let: $$ t_0 = \max_{\gamma\in\Gamma} b(\gamma) $$ be the unit broadworm's breadth. Let: $$ t_n = \max_{P\in\mathcal F_n} b(P). $$ Take the broadworm: $$ \Gamma_0. $$ By the coverage theorem, there exists: $$ P_n\in\mathcal F_n $$ such that: $$ b(P_n) \ge t_0 - \frac1{\lfloor n/2\rfloor}. $$ We have: $$ t_n\ge b(P_n). $$ Also, since: $$ \mathcal F_n\subset\Gamma, $$ $$ t_n\le t_0. $$ Therefore: $$ \boxed{ 0 \le t_0-t_n \le \frac1{ \lfloor n/2\rfloor }. } $$ Consequently: $$ \boxed{ t_n\rightarrow t_0. } $$ This value convergence does not require G0 reduced-form exactness. --- # 11. 2007 Conjecture (5) The 2007 conjecture (5) states: > The breadths $t_n$ strictly increase to $t_0$. Round 22 obtains: $$ \boxed{ t_n\rightarrow t_0 } $$ The original paper already knows: $$ t_n\le t_{n+1}. $$ Conjecture (5) is therefore currently split into: ```text CONVERGES TO t0 CLOSED NONDECREASING SOURCE-KNOWN STRICTLY INCREASING OPEN ``` This round does not claim strictness. --- # 12. Optimizer Compactness Let: $$ \Gamma_n $$ be any broadest $n$-segment unit arc: $$ b(\Gamma_n)=t_n. $$ Parametrize it by arclength: $$ \Gamma_n:[0,1]\rightarrow\mathbb R^2. $$ Apply suitable Euclidean isometries to fix its starting point. Unit length gives: $$ |\Gamma_n(t)-\Gamma_n(s)| \le |t-s|. $$ The family is therefore: - uniformly bounded; - equi-Lipschitz. By Arzelà–Ascoli, every subsequence has a uniformly convergent subsubsequence: $$ \Gamma_{n_k} \rightarrow \Gamma_\infty. $$ --- # 13. Limit Breadth Uniform convergence implies Hausdorff convergence of the traces. Breadth is continuous, so: $$ b(\Gamma_\infty) = \lim_k t_{n_k} = t_0. $$ Lower semicontinuity of arc length gives: $$ \ell(\Gamma_\infty) \le1. $$ If: $$ \ell(\Gamma_\infty)=L<1, $$ scaling it by: $$ 1/L $$ produces a unit-length path with breadth: $$ t_0/L > t_0, $$ a contradiction. Therefore: $$ \boxed{ \ell(\Gamma_\infty)=1. } $$ The limit itself is thus a unit broadworm. --- # 14. Broadworm Uniqueness The literature identifies the unit broadworm as the unique unit arc of maximum breadth, up to Euclidean isometry. Every subsequential optimizer limit therefore belongs to the broadworm orbit. Consequently: $$ \boxed{ \operatorname{dist}_{\rm quotient} ( \Gamma_n, [\Gamma_{\rm BW}] ) \rightarrow0. } $$ In other words: > After suitable isometries/canonical normalization, the broadest polygonal arcs converge uniformly to the broadworm. This gives the convergence statement of the 2007 conjecture (7), at least in its precise formulation modulo Euclidean isometry. --- # 15. Endpoint Distances Let: $$ b_n = | \Gamma_n(1)-\Gamma_n(0) |. $$ Uniform convergence after isometry normalization directly gives convergence of the endpoints. Hence: $$ \boxed{ b_n\rightarrow b_0. } $$ The 2007 conjecture (6) therefore becomes: ```text CONVERGES TO broadworm endpoint distance CLOSED STRICTLY DECREASES OPEN ``` --- # 16. G1 / G2 / G4 Status Update Round 20: ```text G1 OPEN G2 OPEN G3 CLOSED G4 CONDITIONAL ``` Round 22: ```text G1 FULL-DOMAIN COVERAGE QUANTITATIVELY CLOSED in polygonal-chain semantics G2 EFFECTIVE APPROXIMATION CLOSED G3 BREADTH STABILITY CLOSED G4 TRUE VALUE SQUEEZE CLOSED ``` with: $$ \boxed{ 0 \le t_0-t_n \le \frac1{\lfloor n/2\rfloor}. } $$ --- # 17. Repositioning G0's Role G0 remains very important. But it no longer blocks the question: > Can we prove $t_n\to t_0$? Its question is now: > Can the true finite-stage optimum $t_n$ be expressed exactly by the Round 03 reduced formulas? Thus: ```text VALUE LIMIT: no longer depends on G0 FINITE-STAGE STRUCTURE: still depends on G0 STRICT MONOTONICITY: still tied to G0/reduced proof PARITY / SYMMETRY: tied to G0 UNIQUENESS: still open ``` This is an important dependency correction. --- # 18. Source Cross-Check Following their finite-segment lower-bound argument, Adhikari–Pitman 1989 directly state: > Approximating arbitrary arcs by arcs with finitely many straight segments allows the estimate to pass to every arc. Wetzel–Wichiramala 2010 prove more formally that: - simple polygonal arcs provide classical approximation of arbitrary arcs; - simple polygonal unit arcs suffice for the unit-arc covering problem; - the Hausdorff metric is their main approximation language. G1's qualitative core is therefore not an assumption introduced without a source. Round 22 provides a more direct exact-length quantitative chain construction. --- # 19. Novelty Discipline The targeted search has not yet found: ```text r_n^H <= 1/(2 floor(n/2)) ``` or: ```text 0 <= t0-t_n <= 1/floor(n/2) ``` in this form in the broadest-$n$ literature. However: $$ \boxed{ \text{NO PRIOR FORMULA FOUND} \neq \text{NOVELTY PROVED}. } $$ An exhaustive literature review is still needed before formal external publication. --- # 20. Remaining Global Problems What remains is no longer full-domain value closure. The remaining questions are sharper structural ones: 1. Strictness: $$ t_n