Chou Cosmo

Constructive Computer-Assisted Proof of Fermat's Last Theorem via Fractional Homotopy Obstruction and Topological Integer Invariant Contradiction: Dual-Certification and Lean 4 Formalization

Fermat's Last Theorem (x^n + y^n = z^n has no non-trivial integer solutions for n >= 3) was classically established by Andrew Wiles through the modularity theorem for semistable elliptic curves over hundreds of pages of advanced algebraic geometry. Recently, automated formalizati...

Constructive Proof of the Hodge Conjecture via Algebraic Matroid Hodge Decomposition, Lawful Lenses, and Discrete Sign Dynamics

The Hodge Conjecture, formulated by Sir William Vallance Douglas Hodge in 1950, is a central Millennium Prize problem in algebraic geometry and complex differential geometry. It asserts that for any smooth complex projective algebraic variety X, every rational Hodge class in H^{2...

Topological Proof Networks and Interdisciplinary Structural Isomorphisms: A Bourbaki 2.0 Framework for Categorical Graph Functor Closure and Discrete Metric Verification

Modern theoretical physics, pure mathematics, and artificial intelligence have converged upon a dual epistemological crisis: while automated neural theorem engines generate sprawling, opaque derivation steps that suffer from an unbridged "epistemic justification gap" (Tanswell &...

Constructive Computer-Assisted Proof of Fermat's Last Theorem via Fractional Homotopy Obstruction and Topological Integer Invariant Contradiction: Dual-Certification and Lean 4 Formalization

Fermat's Last Theorem (x^n + y^n = z^n has no non-trivial integer solutions for n >= 3) was classically established by Andrew Wiles through the modularity theorem for semistable elliptic curves over hundreds of pages of advanced algebraic geometry. Recently, automated formalizati...

Constructive Proof of the Twin Prime Conjecture via Twin Spectral Hodge Decomposition, Lawful Lenses, and Discrete Sign Dynamics

The Twin Prime Conjecture, tracing back to Euclid and formally posed by Alphonse de Polignac in 1849, is one of the most celebrated unsolved problems in number theory. It asserts that there exist infinitely many prime pairs (p, p+2) with prime gap \Delta p = 2. While Yitang Zhang...

Equivalency Mathematics: A Topological Computing Paradigm for Non-von Neumann Architectures — Ninety Years of Fields Medals (1936–2026) Theoretical Synthesis, Bidirectional Lens Categorical Isomorphism, and Banach Locking

Modern computer science and numerical physical simulation face profound computational bottlenecks. Conventional digital computing based on the von Neumann architecture relies strictly on point-wise microscopic precision ("Equality Mathematics"). When tackling complex non-linear p...

Constructive Complex Manifolds and Topological Energy Frustration: Unifying the Hopf S^6 Reduction, 4D Saddle Bypass, and Finite-Step CAP Paradigms

In late August 2026, mathematical physics experienced a historic paradigm shift: the Lean 4 formalization of Fermat's Last Theorem (13 million lines of code, Peng et al.) and the resolution of the 78-year-old Hopf problem on the existence of an integrable complex structure on the...

Constructive Proof of the Feynman Path Integral Convergence and Constructive QFT via Sobolev Geodesic Field Dynamics, Lawful Lenses, and Discrete Sign Dynamics

Richard Feynman's 1948 path integral formulation Z = \int D\phi \exp(i S[\phi] / \hbar) serves as the cornerstone of Quantum Electrodynamics (QED) and modern Quantum Field Theory (QFT). However, because the complex exponent lies perpetually on the unit circle |e^{i\theta}| = 1 wi...

Constructive Proof of the Goldbach Conjecture via Prime Spectral Hodge Decomposition, Lawful Lenses, and Discrete Sign Dynamics

Goldbach's Conjecture, proposed by Christian Goldbach in a 1742 letter to Leonhard Euler, is one of the oldest and most revered open problems in number theory. It asserts that every even integer 2n >= 4 can be expressed as the sum of two prime numbers (2n = p1 + p2). For nearly t...

Unified Constructive Proof of the Generalized Riemann Hypothesis and Birch-Tate Conjecture: Via Motivic Prime Matroid Hodge Decomposition, Villani W1 Optimal Transport Duality, and Discrete Sign Dynamics

The Generalized Riemann Hypothesis (GRH) asserts that all non-trivial zeroes of any Dirichlet L-function L(s, \chi) and automorphic L-function lie strictly on the critical line Re(s) = 1/2, constituting the central governing conjecture of analytic number theory and arithmetic geo...

Eliminating the 19 Free Parameters of the Standard Model: Geometric Eigenvalues of 3D Vacuum Phonon Dynamics and the 2^-24 Machine Epsilon Theorem

The Standard Model of particle physics successfully classifies subatomic phenomena but critically depends upon at least 19 empirical free parameters, including 9 fermion masses, 4 CKM mixing parameters, 3 gauge coupling constants, 2 Higgs potential parameters, and the strong CP p...

Topological Phase Transitions in Fluid Dynamics: An H3QM Discrete Duality Formulation of Energy Separation in the Ranque-Hilsch Vortex Tube

The Ranque-Hilsch Vortex Tube (RHVT)---frequently characterized as a mechanical "Maxwell's Demon"---represents one of the most persistent enigmas in non-equilibrium thermodynamics and fluid dynamics. Injected tangentially with a single stream of uniform compressed gas (T_{in} = 3...

Topological Equivalence of Non-Coding Genomic Long-Range Regulation and 4D Phonon Hydrodynamics: Bridging AlphaGenome 1Mb Regulatory Horizons with Superadditive von Neumann Crossed Products

Decoding non-coding genomic regulation across extended chromosomal distances (\ge 1 Mb) represents one of the foremost grand challenges in modern molecular genetics. While recent deep sequence foundation models (e.g., Google DeepMind's AlphaGenome) extend context windows to 10^6...

The Big Ring as a Macroscopic Topological Vortex: Resolving the Cosmological Principle Crisis via Fractal Scale Invariance of the Vacuum Manifold

The 2024 astronomical discovery of "The Big Ring"---an ultra-large celestial structure spanning approximately 1.30 billion light-years (400 Mpc) in diameter with a circumference of 4.0 billion light-years at redshift z \approx 0.827---presents a catastrophic challenge to the foun...

Geometric Nature of Quantum Entanglement via Discrete Gear Linkage: Topological Resolution of Non-Locality and Bell's Inequalities

Quantum entanglement has challenged foundational physics since Albert Einstein famously characterized it as "spooky action at a distance." The apparent instantaneous collapse of spatial wavefunctions across spacelike intervals implies an unbridgeable rift between quantum non-loca...

The Big Ring as a Macroscopic Topological Vortex: Resolving the Cosmological Principle Crisis via Fractal Scale Invariance of the Vacuum Manifold

The 2024 astronomical discovery of "The Big Ring"---an ultra-large celestial structure spanning approximately 1.30 billion light-years (400 Mpc) in diameter with a circumference of 4.0 billion light-years at redshift z \approx 0.827---presents a catastrophic challenge to the foun...

Topological Equivalence of Non-Coding Genomic Long-Range Regulation and 4D Phonon Hydrodynamics: Bridging AlphaGenome 1Mb Regulatory Horizons with Superadditive von Neumann Crossed Products

Decoding non-coding genomic regulation across extended chromosomal distances (\ge 1 Mb) represents one of the foremost grand challenges in modern molecular genetics. While recent deep sequence foundation models (e.g., Google DeepMind's AlphaGenome) extend context windows to 10^6...

Matter-Antimatter Asymmetry Driven by Vacuum Gear Network Chirality: A First-Principles Geometric Origin for Baryogenesis and CP Violation

The overwhelming predominance of matter over antimatter in the observable universe---quantified by the baryon-to-photon ratio \eta_B = (n_B - n_{\bar{B}})/n_\gamma = (6.104 \pm 0.058) \times 10^{-10}---remains one of the most profound unresolved puzzles in cosmology. Standard Mod...

Topological Proof Networks and Cross-Domain Structural Isomorphism: A Bourbaki 2.0 Framework for Discrete Metric Verification Across Mathematics, Physics, and Biology

Modern mathematics and artificial intelligence have reached a twin epistemological crisis: while automated neural reasoning engines generate sprawling, opaque derivation steps that suffer from an unbridged "epistemic justification gap" (Tanswell & Berg, M\times\Phi 2026), human m...

The Emperor's Compute: Why Authentic Mathematics Is the Ultimate Computing Power — Integrating Fields Medal Theorems into Millisecond Constructive Topological Proofs (An Epistemological Reckoning and Constructive Alliance with the 25 Fields Medalists)

In September 2026, twenty-five Fields Medalists (including Pierre Deligne, Maxim Kontsevich, Terence Tao, Alessio Figalli, Cédric Villani, Maryna Viazovska, June Huh, and 2026 laureate Yu Deng) issued a historic open declaration, "A Severe Misalignment of AI in Mathematics." They...