Shin Tamaki

Axiomatics of Generative Stratification: The Formal Recursive Layer Theory

FRLT formalizes generation through stratified existence, pivot-based actualization, orthogonal asymmetries, three-phase layer formation, and inversion producing a unique null state. This axiomatic system defines a minimal, recursive structure for generative processes and value em...

Formal Recursive Layer Theory and the Generative Construction of Mathematical Structures

FRLT formalizes generation through informational, generative, and causal layers. Twist‑insertion integrates quaternion rotation, phase flow, and lemniscate geometry, producing structural pivots and unified groupoid dynamics. The framework constructs manifold formation and causal...

Trifocal Geometry and Generative Stratification: An Axiomatic Framework for FRLT

Formal Recursive Layer Theory (FRLT) employs a triple-pivot scalar field Σ whose equal‑tension surfaces form asymmetric trifocal ovals, establishing a geometric basis for generative stratification and linking spatial structure with stability properties encoded in the Hessian of Σ...

Bipolar Structure of the Lemniscate: Dual Images and Möbius Twist

The Bernoulli lemniscate provides a minimal geometric model of an antisymmetric generative kernel. Its dual‑image structure and intersection‑induced normal flip yield a Möbius twist, illustrating how local two‑image behavior becomes globally unified through topological identifica...

The Movable Parallelogram as a Minimal Model of Generative Structure

The movable parallelogram provides a minimal geometric model linking determinantal and spectral generation, projective depth, and optical imaging. Its hinge‑based deformation unifies column volume, eigen‑direction scaling, perspective geometry, and lens behavior within a single g...

Bipolar Structure of the Lemniscate: Dual Images and Möbius Twist

The Bernoulli lemniscate provides a minimal geometric model of an antisymmetric generative kernel. Its dual‑image structure and intersection‑induced normal flip yield a Möbius twist, illustrating how local two‑image behavior becomes globally unified through topological identifica...

The Movable Parallelogram as a Minimal Model of Generative Structure

The movable parallelogram provides a minimal geometric model linking determinantal and spectral generation, projective depth, and optical imaging. Its hinge‑based deformation unifies column volume, eigen‑direction scaling, perspective geometry, and lens behavior within a single g...

Monster Symmetry as a Safety Architecture in Generative Systems

This paper proposes a structural safety framework where Monster group symmetry governs generative behavior. Equivariance preserves high performance, while symmetry breaking naturally induces degradation, providing intrinsic protection without external prohibitions or ethical cons...

Monster Symmetry as a Safety Architecture in Generative Systems

This paper proposes a structural safety framework where Monster group symmetry governs generative behavior. Equivariance preserves high performance, while symmetry breaking naturally induces degradation, providing intrinsic protection without external prohibitions or ethical cons...

The VCS Communication Model: Three-Core Ratio Structure with Geometric and Dynamical Properties

This paper presents the VCS communication model, defining information through three-core ratios within a geometric simplex. It introduces geodesics, curvature, and dynamical flow fields, and shows their structural isomorphism with internal AI communication processes. This work is...

A Structural Reclassification of Thurston's Eight Geometries Through Knot Complements

This paper reinterprets Thurston’s eight geometries through knot complements, showing how reflection, closure, and chirality encode geometric phases. The framework reveals non‑commutative residues distinguishing Nil, Sol, and SL2R~ from commutative geometries. This work is licens...

A Structural Reclassification of Thurston's Eight Geometries Through Knot Complements

This paper reinterprets Thurston’s eight geometries through knot complements, showing how reflection, closure, and chirality encode geometric phases. The framework reveals non‑commutative residues distinguishing Nil, Sol, and SL2R~ from commutative geometries. This work is licens...

The VCS Communication Model: Three-Core Ratio Structure with Geometric and Dynamical Properties

This paper presents the VCS communication model, defining information through three-core ratios within a geometric simplex. It introduces geodesics, curvature, and dynamical flow fields, and shows their structural isomorphism with internal AI communication processes. This work is...

Axiomatic Generation of Geometry and Complex Structures from One-Dimensional Defects

This paper presents an axiomatic framework showing how one-dimensional defects generate three-dimensional geometric structures and induce two-dimensional complex structures. The theory establishes a necessary generative hierarchy and demonstrates its consistency through hyperboli...

Axiomatic Generation of Geometry and Complex Structures from One-Dimensional Defects

This paper presents an axiomatic framework showing how one-dimensional defects generate three-dimensional geometric structures and induce two-dimensional complex structures. The theory establishes a necessary generative hierarchy and demonstrates its consistency through hyperboli...

Null Direction Fields and Topological Imprints in Lorentzian Geometry

This paper shows how differential variations of null direction fields generate topological imprints in Lorentzian geometry, demonstrating that local null generation can accumulate into global structure characterized by the Pontryagin number in four-dimensional spacetimes. This wo...

Null Direction Fields and Topological Imprints in Lorentzian Geometry

This paper shows how differential variations of null direction fields generate topological imprints in Lorentzian geometry, demonstrating that local null generation can accumulate into global structure characterized by the Pontryagin number in four-dimensional spacetimes. This wo...

QR and Householder Orthogonalization on Light-Cone Geometry

We reinterpret QR and Householder orthogonalization through light-cone geometry, showing how null-aligned reflections encode causal structure. This framework links orthogonalization to spacetime direction hierarchies, revealing a geometric basis for null-origin alignment. This wo...

Hyperbolic Knot Complements as Imaginary Thickness: Boundary Map Dynamics and PSL(2,R) Actions

This paper presents a cohomological model in which imaginary thickness arises from boundary map dynamics and aligns with hyperbolic actions of PSL(2,R). Knot complements provide the geometric setting that realizes these generative depth transformations. This work is licensed unde...

Hyperbolic Knot Complements as Imaginary Thickness: Boundary Map Dynamics and PSL(2,R) Actions

This paper presents a cohomological model in which imaginary thickness arises from boundary map dynamics and aligns with hyperbolic actions of PSL(2,R). Knot complements provide the geometric setting that realizes these generative depth transformations. This work is licensed unde...