> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # WSFC Round 27 ## 2007 Conjecture Ledger / Authority Audit: Classical Closure, WSFC Conditional Proofs, and Formal Debt **English Title:** *The 2007 Conjecture Ledger: Classical Closures, WSFC Conditional Proofs, and Remaining Formal Debt* **Document ID:** `WSFC-R27-v0.1` **Version:** v0.1 **Date:** 2026-09-24 **Author:** Neo.K **AI Research Collaborator:** Aletheia / OpenAI GPT-5.6 Sol **Series:** Worm Sequence Freedom Closure **Prerequisites:** `WSFC-R21`–`WSFC-R26` **Status:** `AUTHORITY AUDIT / NO NEW GLOBAL OVERCLAIM / CONJECTURE-LEVEL LEDGER` **Canonical Source:** UTF-8 Markdown --- # Abstract In 2007, Panraksa–Wetzel–Wichiramala concluded *The broadest three-segment unit arc* with seven general-$n$ conjectures: 1. $\Gamma_n$ is unique up to rigid motions; 2. $\Gamma_n$ is convex; 3. $\Gamma_n$ is a drape; 4. bilateral symmetry for even $n$, no symmetries for odd $n>3$; 5. $t_n$ strictly increases to the broadworm breadth $t_0$; 6. the endpoint distance $b_n$ strictly decreases to the broadworm endpoint distance; 7. the arclength-parametrized $\Gamma_n$ uniformly converges to the broadworm. Round 27 is about authority normalization, not adding a new theorem. The most important correction is: $$ \boxed{(2)\text{ convexity and }(3)\text{ drape are classical-source consequences}.} $$ Adhikari–Pitman 1989 already proved that for any fixed segment cap $M$, a minimum-length polygonal arc of width at least one must be convex. Scale the broadest $n$-segment unit arc to breadth $1$, then embed a lower segment count in an exact-$n$ representation by collinear subdivision to obtain conjecture (2). The same paper also states, for a convex minimizer, that if the arc leaves the perpendicular strip of its endpoints, perpendicular truncation gives a shorter arc with greater breadth. The drape definition of Maki–Wetzel–Wichiramala 2005 is precisely a convex arc lying in this endpoint strip. Thus conjecture (3) is also a consequence of an existing published theorem chain. The correct authority classification of the seven conjectures is therefore not “WSFC solves everything,” but: $$ \boxed{2\text{ classical source closures}+5\text{ WSFC provisional research closures}.} $$ Conjecture (6) contains a decisive computer-assisted interval certificate; no new WSFC-specific claim is currently closed in a proof assistant or externally independently reviewed. --- # 1. Source Ledger The original 2007 conjectures: $$ \begin{array}{ll} (1)&\Gamma_n\text{ unique up to rigid motions},\\ (2)&\Gamma_n\text{ convex},\\ (3)&\Gamma_n\text{ a drape},\\ (4)&\text{even bilateral symmetry / odd no symmetry},\\ (5)&t_n\uparrow t_0\text{ strictly},\\ (6)&b_n\downarrow b_0\text{ strictly},\\ (7)&\Gamma_n\to\Gamma_{\rm BW}\text{ uniformly}. \end{array} $$ The source also establishes uniqueness for $n=2,3$, and: $$ t_n\le t_{n+1} convex arc lying in the closed semi-infinite endpoint strip. Adhikari–Pitman's strip reduction for convex minimizers gives this stronger condition, so: $$ \boxed{\Gamma_n\text{ is a drape}.} $$ This is not merely generic “drapeability.” --- # 4. Conjecture (1): Uniqueness R24's two strict-convexity engines control integer allocation and continuous geometry; strict Jensen then makes the tangent support partition unique. R23 excludes lower-stage optimizers. Conditional on G0: $$ \boxed{\operatorname{Argmax}(\mathcal F_n)/\operatorname{Isom}(\mathbb R^2)\text{ is a singleton}.} $$ Status: ```text RESEARCH-PROOF-LEVEL PROVISIONALLY CLOSED FORMAL OPEN INDEPENDENT REVIEW OPEN ``` --- # 5. Conjecture (4): Symmetry Even stage: $$ q_L=q_R,\qquad x=d/2, $$ The left/right support lists are mirror images, giving bilateral symmetry. Odd stage: $$ |q_L-q_R|=1. $$ Every symmetry of the trace must preserve the endpoint set. If it fixes both endpoints, the nonidentity candidate is reflection in the endpoint line, but this moves a one-sided convex arc to the other side; if it exchanges the endpoints, it exchanges the left/right support chains, forcing $q_L=q_R$, a contradiction. Thus odd $n>3$ has no nontrivial Euclidean symmetry. Status: provisional at research-proof level, conditional on G0 + balancing. --- # 6. Conjecture (5): Breadth Sequence R22: $$ 0\le t_0-t_n\le\frac1{\lfloor n/2\rfloor}, $$ Hence $t_n\to t_0$. R23 proves strictness using tangent-gap splitting and exact base cases: $$ t_n0$ and uses an adaptive interval certificate to cover balanced continuous support-count interpolation: ```text processed boxes 28244 unresolved terminal boxes 0 minimum K lower bound 0.0006339178281571378 ``` Together with exact finite base cases and R22 convergence: $$ \boxed{b_n\text{ strictly decreases to }b_0.} $$ Status: provisional at research-proof level + a decisive computer-assisted component; formal verification remains open. --- # 8. Conjecture (7): Uniform Convergence Without fixing placement in the original statement, $\Gamma_n$ is defined only up to rigid motion, so literal coordinate convergence requires normalization. The safest formulation is: $$ \boxed{\Gamma_n\to\Gamma_{\rm BW}\quad\text{modulo Euclidean isometries}.} $$ Alternatively, fix the initial endpoint at the origin, the final endpoint on the positive $x$-axis, and a reflection convention. The arclength parametrizations are all 1-Lipschitz. Arzelà–Ascoli gives a subsequential uniform limit. R22 gives $t_n\to t_0$, and breadth continuity carries the limiting breadth to $t_0$. Lower semicontinuity of length gives $L\le1$; if $L<1$, scaling to unit length would yield breadth $>t_0$, a contradiction. Hence $L=1$, and the limit is a maximal-breadth unit arc. By broadworm uniqueness, all subsequential limits are the normalized broadworm, so the full normalized sequence converges. Thus (7) is closed at research-proof level in normalized/mod-isometry form. --- # 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 ``` Current 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 The targeted search in 2026-09 covered the original title, exact conjecture phrases, later broadest-$n$ phrases, the 2025 worm-problem review, and the authors' publication lists. No article was located that explicitly announces a later-literature solution of all seven general-$n$ conjectures as a whole. But: $$ \boxed{\text{targeted search}\neq\text{priority proof}.} $$ A formal novelty claim still requires review at the MathSciNet / zbMATH / citation-tree / author-contact level. --- # 11. Anti-Overclaim Rules Prohibited: 1. calling (2), convexity, a new WSFC theorem; 2. calling (3), drape, a new WSFC theorem; 3. calling (6) a purely analytic formal proof; 4. calling a targeted search a proof of novelty; 5. writing (7) as literal coordinate convergence without normalization; 6. conflating an odd mirror pair with uniqueness up to orientation-preserving rigid motions. --- # 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. Next Steps Round 28 should become: $$ \boxed{\textbf{Independent Verification Pack}.} $$ Separate it into: ```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 ``` Each packet should retain only its theorem statement, minimal assumptions, proof skeleton, machine certificate, dependencies, and attack checklist. --- # Conclusion Round 27 centers on authority normalization: $$ \boxed{2\text{ classical source closures}+5\text{ WSFC provisional research closures}.} $$ The next stage is not accumulating more theorems, but authority elevation: formalization, independent replay, independent mathematical review, and literature-priority review. ===END===