FBGT Axiomatization of Superstring Theory

Author: Zhang Xun | Version: Preprint, October 2026 | Base system: FBGT (Function–Branch Growth Theory) over ZFC + supacg Overview This monograph reconstructs superstring theory as a strictly axiomatic, background-independent system. Continuous spacetime is discarded as first-order ontology: physics is identified with global sections of a locally free sheaf over the four-dimensional compact simply-connected Poincaré manifold M4M4, observed through local truncation functors. Core Results Sheaf-theoretic ontology (§1). Strings are not objects moving in a manifold; they are sheaf-theoretic projections of higher-dimensional algebraic structures onto low-dimensional observation sections. The string's geometric identity is the Kakeya set. Truncation and entropy loss. Every finite observation is a local truncation functor; singularities and wave-function collapse are section-extension obstructions with strictly positive KL / von Neumann entropy loss. Existence criterion. A string is physically real iff its intrinsic section measure HS(Σ)=T⋅C>0, where T is BIBD topological rigidity and C algorithmic rigidity. Spectral identity of four forces (§2). Gravity, electromagnetic, weak and strong fields are graded ideals of one SAG super-algebra; their spectra {3,−1,−1,−1} and {0,0,0}{0,0,0} are isomorphic under the dimension-penetration operator Φ4Φ4. End of the landscape (§3). Compactification is discrete BIBD encoding, not Calabi–Yau geometry. The physical vacuum is uniquely locked to the (4,4,3,3,2) design; all 10^500pseudo-vacua are eliminated by spectral-gap degeneration. Dynamics and singularities (§4–5). The Schrödinger equation is derived (not postulated) from Stone's theorem on a complex line bundle; curvature singularities dissolve into essential-spectrum continuization and loss of L2L2 integrability. Bekenstein–Hawking entropy follows from BIBD state counting; UV divergences are cut off algebraically. Four-dimensional Kakeya conjecture (Appendix A). Within FBGT we prove the Kakeya conjecture in dimension four: any Kakeya set in a compact four-dimensional cognitive manifold satisfying full directional coverage must have semantic Hausdorff dimension exactly 44. The proof replaces tube-neighborhood estimates by Fisher-matrix non-degeneracy plus a non-zero topological obstruction class; lower dimensions ≤2 and the critical case 33 are both excluded. Physical picture (Appendix B). Five geometric models (right conoid, conic sections, Gosper curve, Tarski boundary, parallelogram sliding) visualize the results, each explicitly tagged with its formal status and open gaps A–E. Open Problems Minimal non-Lie structure; categorical correspondence between filtered geometry and the Lie skeleton; conservative extension back to plain ZFC (O6); closing gaps A–E. Licenses Creative Commons Attribution 4.0 International 中文: FBGT;超弦公理化;挂谷集;四维挂谷猜想;BIBD 编码;谱同一性;算子拓扑;真空唯一性 English: FBGT; superstring axiomatization; Kakeya set; four-dimensional Kakeya conjecture; BIBD encoding; spectral identity; operator topology; vacuum uniqueness Knowledge Belongs to All Humanity.