Self-Verifying Geometry (SVG)

Corpus Description This corpus consists of 16 mathematical research documents presenting a unified geometric-analytic framework called Self-Verifying Geometry (SVG), aimed at proving the Riemann Hypothesis (RH) and relating it to a dozen classical conjectures in number theory, geometry, and mathematical physics. Key Elements of the Corpus: Geometrization of the Gamma function: An ideal tetrahedron in hyperbolic space H3 is constructed from the completed Gamma function ΓR(s), with dihedral angles fixed by six classical analytic principles. Quaternionic symmetry Q8: The symmetries of the Gamma function are shown to generate an action of the quaternion group on the fundamental tetrahedron. Geometric rigidity principle: It is proven that any zero of ζ(s) off the critical line induces an antisymmetric phase defect that displaces the hyperbolic barycenter, breaking the Q8 symmetry. SVG framework for multiple conjectures: A hypercubic structure of four fundamental tetrahedra is proposed, encoding 12 conjectures (including Goldbach, Twin Prime, GRH, and others), with conditional propagation of SVG minimality. Complete proof of the Riemann Hypothesis: Global rigidity, self-regulated completeness, and exact equivalence theorems are integrated to derive RH as a theorem within the SVG framework. Complete Document List with Descriptions Core Theoretical Foundations 1. Anomaly-Free Structures (SVG)Foundational paper introducing Self-Verifying GeometryIntroduces the SVG framework where mathematical truth is encoded as existence and stability. Defines the inconsistency functional Γ and establishes the complete category of SVG objects. Presents RH as a corollary of geometric rigidity. 2. Self-Verifying Geometry (SVG) - Refined VersionExtended framework with multiple critical linesDevelops universal SVG objects for conjectures, introduces the category SVGcrit, and establishes alignment functors. Shows how multiple conjectures can be unified through geometric consistency. 3. Tetrahedron Geometry and Quaternionic SymmetryPure geometric construction from the Gamma functionConstructs the canonical hyperbolic tetrahedron associated with Γℝ(s) without arithmetic assumptions. Shows quaternionic group Q₈ action organizing the geometry into four isometric copies. Analytic Foundations 4. Phase Currents, Convexity, and RigidityAnalytic mechanism for defect exclusionDevelops phase currents, proves transverse convexity for log|Γℝ(½+it)|, establishes coercivity of phase energy, and proves the antisymmetric defect exclusion principle (ACE). 5. Riemann Hypothesis via Geometric-Analytic RigidityComplete proof integrating all componentsPresents the unified proof combining hyperbolic geometry, entire function theory, quaternionic symmetry, and variational coercivity. Reduces RH to pointwise minimization of the Levin-Pfluger indicator. Geometric Rigidity Theorems 6. Principle 7: Volumetric Centrality TheoremComplete proof of the central geometric principleDemonstrates equivalence between RH, barycenter invariance under Q₈, closure conditions, and stiffness matrix properties. Provides geometric reformulation of RH. 7. Geometric Rigidity and RH EquivalenceFrom antisymmetric defects to global closureConnects analytic defect exclusion (Article B) with tetrahedral framework (Article A) to prove RH as geometric rigidity condition. 8. Spectral Coercivity and Defect ExclusionSpectral approach completing the proofEstablishes spectral coercivity theorem, constructs rigidity operator A, and uses Q₈ symmetry decomposition to forbid antisymmetric phase defects. Unification Frameworks 9. Geometric Rigidity OverviewComprehensive synthesis of the frameworkPresents RH as rigidity condition for Q₈-symmetric Gamma complex. Shows how tetrahedral assembly is possible only when RH holds. 10. Verification Status and Conditional ConsequencesMeta-analysis of the framework's logical structureClarifies proved results, conditional implications, and verification requirements. Shows how 13 conjectures reduce to one rigidity statement. Multi-Conjecture Systems 11. Four Fundamental TetrahedraFirst presentation of tetrahedral organizationOrganizes 12 classical conjectures into four tetrahedra with conditional propagation from RH to all conjectures. 12. Framework of Four Tetrahedra - Explicit VersionFalsifiable implementation with explicit objectsProvides concrete table of 12 conjectures with SVG objects, connection diagrams, and detailed Goldbach example with falsifiability discussion. 13. SVG and Hypercubic Propagation of ConjecturesFormal categorical frameworkIntroduces SVG hypercube structure with foundational classification theorem. Works through Goldbach tetrahedron as detailed example. 14. Classical Conjectures - Integrated VersionComplete hypercubic propagation systemImplements the four tetrahedra in 4D hypercube with alignment morphisms. Shows conditional propagation from RH to all 12 conjectures. Extended Geometric Systems 15. Self-Regulated Hyperbolic FrameworkExtended system with icosahedra and ABC vectorsIntegrates tetrahedra, hypercubic lattice, rotated icosahedra, and ABC alignment operator. Proves global rigidity, completeness, and equivalence theorems. 16. Completed Proof of RH in Self-Regulated FrameworkFinal synthesis and complete derivationPresents RH as theorem of self-regulated hyperbolic geometry. Integrates all previous results into three fundamental theorems and provides complete derivation. Key Innovations Theoretical Contributions: Self-Verifying Geometry (SVG) - New paradigm where truth equals geometric stability Gamma-tetrahedron correspondence - Canonical hyperbolic tetrahedron from Γℝ(s) Quaternionic symmetry Q₈ - Natural symmetry group organizing the geometry Volumetric centrality - Geometric principle equivalent to RH Methodological Advances: Multiple critical lines alignment - Framework for unifying conjectures Hypercubic propagation - Systematic method for conditional proof propagation Phase energy formalism - Analytic approach to defect exclusion ABC alignment vectors - Self-regulating geometric system Verification and Falsifiability The framework includes explicit verification pathways: Numerical dihedral angle verification of the fundamental tetrahedron Phase energy positivity tests for artificial off-critical zeros Tetrahedral gluing simulations in hyperbolic space Spectral operator computations for coercivity verification Interdisciplinary Relevance Mathematics: Number theory (RH, Goldbach, twin primes) Geometric analysis (hyperbolic geometry, PDEs) Complex analysis (entire functions, phase currents) Representation theory (quaternionic symmetries) Potential Applications: Physics (quantum chaos, spectral rigidity) Computer science (verification frameworks) Mathematical foundations (truth paradigms) License:Creative Commons Attribution 4.0 International (CC BY 4.0)