The Scale Geometry Framework: Topology, Projection, and Static Universe

Based on the noncommutative geometry of the quantum-deformed 3-sphere Sq³, this work establishes a self-consistent unified physics framework named Scale Geometry. Taking the Scale Self-Duality Axiom as the core and integrating the Spectral Action Principle and quantum group representation theory, it realizes the geometric unification of quantum mechanics, general relativity, and the Standard Model of particle physics. A rigorous mathematical foundation is laid through the complete real spectral triple G= (A, H, D, J, ) (the absolute geometric ontology of the universe) and the Inverse Limit Isomorphism Theorem, resolving long-standing fundamental physics puzzles such as the origin of free parameters in the Standard Model, the cosmological constant fine-tuning problem, and the quantum measurement problem. Key achievements include: deriving the full gauge group SU (3) c SU (2) L U (1) Y and chiral fermion representations of the Standard Model from geometric structures; rigorously proving the Representation Splitting Theorem to explain the geometric origin of three generations of fermions; locking the fine-structure constant (^-1 136. 9137. 1) and fermion mass power-law formula (₀ 0. 63) from first principles; naturally deriving discrete flavor symmetries S₃ (quarks) and A₄ (leptons), as well as the CKM and PMNS mixing matrices consistent with experiments; deriving the Einstein-Hilbert action and solving the cosmological constant fine-tuning problem; and interpreting quantum measurement as scale projection to resolve the measurement problem. Nine falsifiable experimental predictions are proposed, classified by verification time windows: 1. Neutrino Physics (3-10 years): Strict normal ordering of neutrino masses, ₁₃ 8. 5^; constraints on Majorana phases via the A₄ flavor symmetry and Seesaw mechanism (tested by JUNO, DUNE, Hyper-K). 2. Cosmology (5-10 years): Closed S³ spacetime topology with ₖ < 0 (magnitude 10^-310^-4) (tested by CMB-S4, Euclid, LSST). 3. Quantum Gravity (3-10 years): Noncommutative correction to photon dispersion relation, with a time delay of t 10^-2 s for 1 EeV photons (O (1) ) (tested by LHAASO, CTA). 4. Particle Physics (10-20 years): Proton lifetime ₚ 10^34 years, with p e^+ ⁰ as the dominant decay channel (tested by Hyper-K, DUNE, future large-scale detectors). 5. Higgs Physics (10-20 years): Higgs self-coupling constant deviating by O (0. 1) from the Standard Model (tested by HL-LHC, CEPC, FCC-ee). 6. Black Hole Physics (10-20 years): Quantum gravity corrections to the shadow radius of intermediate-mass black holes (order of 10^-5 relative deviation) (tested by ngEHT, next-generation radio interferometer arrays). 7. Early Universe Physics (Over 20 years): Primordial gravitational wave spectral index deviating from the slow-roll consistency relation (nₜ 0. 007) (tested by next-generation CMB polarization satellites, LISA, DECIGO). 8. Gravitational Physics (Over 20 years): Violation of the weak equivalence principle with Eötvös parameter 10^-122 (tested by high-precision space equivalence principle experiments). 9. Cosmic Spacetime Topology (Over 20 years): Closed S³ universe confirmed by topological signatures and all-sky imaging effects (tested by JWST, CSST, future high-precision survey facilities). This framework achieves unification with minimal assumptions, no free fitting parameters, and strict mathematical consistency, providing a falsifiable new scheme for fundamental physics unification. The author would like to express sincere gratitude to Alain Connes for laying the foundational framework of noncommutative geometry, and to Albert Einstein for his profound insights into geometric unification and spacetime nature, which have inspired the development of this work Paper 1DOI: https: //doi. org/10. 5281/zenodo. 19389313内容: 尺度几何框架基础奠基, 搭建核心公理与S³量子形变几何算子, 是全系列的理论总起点。 2. Paper 2DOI: https: //doi. org/10. 5281/zenodo. 19389392内容: 精细结构常数的几何锁定, 从第一性原理推导出≈1/137, 证明其为时空几何不变量。 3. Paper 3DOI: https: //doi. org/10. 5281/zenodo. 19389478内容: 彻底解决量子场论紫外发散问题, 用尺度自对偶性自动抵消所有无穷大, 构建全阶有限量子场论。 4. Paper 4DOI: https: //doi. org/10. 5281/zenodo. 19389527内容: 三代费米子的几何起源, 通过表示分裂定理证明三代费米子是量子群表示分解的必然结果。 5. Paper 5DOI: https: //doi. org/10. 5281/zenodo. 19389584内容: 宇宙膨胀是观测者幻觉, 将暗能量、加速膨胀解释为观测者尺度分辨率提升的几何效应, 彻底抛弃暗能量假设。 6. Paper 6DOI: https: //doi. org/10. 5281/zenodo. 19389686内容: 静态块宇宙模型, 用循环尺度投影替代大爆炸, 构建无暴涨、无奇点的静态宇宙, 解决大爆炸核心危机。 7. Paper 7DOI: https: //doi. org/10. 5281/zenodo. 19389835内容: 强决定论与循环时间复发, 完成理论闭环, 解释时间箭头、自由意志幻觉与量子测量问题, 统一粒子物理、宇宙学与观测者体验。 wxsq1638@outlook. com ORCID;0009-0000-3175-1681 Name;Xinyu Zheng