Fixed Point Hierarchy in Higher-Dimensional Geometry: Mathematical Foundations and String Theory Connections
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2025
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| _version_ | 1866901312005734400 |
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| author | Pound, Richard Thomas |
| author_facet | Pound, Richard Thomas |
| contents | <p>We report the discovery of a universal fixed point hierarchy in higher-dimensional geometric optimization. Analysis of dimensions 6D through 100D reveals that all dimensions share the same geometric exponent (2.020924) with zero error, producing a mathematical sequence of fixed points that converges to unity. We prove three fundamental theorems: (1) convergence of the fixed point sequence xD satisfying xD = x + 1 to xD = 1 + 1/(D − 1) + O(1/D2) → 1 as D → ∞ with explicit error bounds, (2) uniqueness of fixed points in each dimension, and (3) strict monotonicity of the sequence. We derive a closed-form generating function with 0.0068% accuracy and establish connections to the Padovan sequence and plastic constant. The fixed point hierarchy appears in all string theory dimensions (6D Calabi-Yau, 10D superstring, 11D M-theory, 26D bosonic), suggesting a geometric foundation for string compactifications. The plastic constant (1.324718) appears as a universal fixed point in every dimension, connecting crystal structures (3D), spacetime geometry (5D), and higher-dimensional physics. This work provides rigorous mathematical foundations for the unified geometric optimization principle discovered in ϕ-Geometry Paper V.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17647706 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Fixed Point Hierarchy in Higher-Dimensional Geometry: Mathematical Foundations and String Theory Connections Pound, Richard Thomas fixed point hierarchy higher-dimensional geometry plastic constant Padovan sequence String theory geometric optimization convergence theorem uniqueness theorem monotonicity conjecture asymptotic analysis mathematical foundations x^D = x + 1 numerical verification Calabi-Yau manifolds M-theory bosonic string theory <p>We report the discovery of a universal fixed point hierarchy in higher-dimensional geometric optimization. Analysis of dimensions 6D through 100D reveals that all dimensions share the same geometric exponent (2.020924) with zero error, producing a mathematical sequence of fixed points that converges to unity. We prove three fundamental theorems: (1) convergence of the fixed point sequence xD satisfying xD = x + 1 to xD = 1 + 1/(D − 1) + O(1/D2) → 1 as D → ∞ with explicit error bounds, (2) uniqueness of fixed points in each dimension, and (3) strict monotonicity of the sequence. We derive a closed-form generating function with 0.0068% accuracy and establish connections to the Padovan sequence and plastic constant. The fixed point hierarchy appears in all string theory dimensions (6D Calabi-Yau, 10D superstring, 11D M-theory, 26D bosonic), suggesting a geometric foundation for string compactifications. The plastic constant (1.324718) appears as a universal fixed point in every dimension, connecting crystal structures (3D), spacetime geometry (5D), and higher-dimensional physics. This work provides rigorous mathematical foundations for the unified geometric optimization principle discovered in ϕ-Geometry Paper V.</p> |
| title | Fixed Point Hierarchy in Higher-Dimensional Geometry: Mathematical Foundations and String Theory Connections |
| topic | fixed point hierarchy higher-dimensional geometry plastic constant Padovan sequence String theory geometric optimization convergence theorem uniqueness theorem monotonicity conjecture asymptotic analysis mathematical foundations x^D = x + 1 numerical verification Calabi-Yau manifolds M-theory bosonic string theory |
| url | https://doi.org/10.5281/zenodo.17647706 |