The Scale-Free Metric Tensor: Tripartite Topological Constraints and Planetary Dimensional Metrics
Abstract
This paper formally derives a scale-free, tripartite topological framework governing the metric dimensional expansion of a continuous state manifold. By modeling time as a compactified angular divergence rather than a trivial affine scalar, we demonstrate that the projection of a temporal phase into a discrete Euclidean coordinate space is bound by a strict harmonic divergence hierarchy. The continuous manifold mandates three invariant topological thresholds: the 2π affine parameterization, analytically regularized against coordinate rupture by the Euler-Mascheroni constant (γ); the π/2 orthogonal dimensional bifurcation, constrained within an O(3) spatial tensor by the Basel constant (π²/6); and the absolute 4π spinor closure.
To satisfy coordinate identity at the 4π manifold without inducing nonlinear metric rupture, the topology embeds an irreducible temporal shear scalar, arising from a 1461:1460 discrete-to-continuous phase asymmetry, directly into the 3D volume as thermodynamic variance. This absolute boundary is strictly determined by the invariant matrix transformation factor, the Incommensurable Radiometric Thermodynamic Variance (κ ≈ 1.437824).
Furthermore, we prove that localized planetary topology operates exclusively as a continuous phase-variant spatial tensor boundary, rather than a static metric field. The tensorial dimensional coupling of the axial displacement limits (23.4° to 68.4°) and the corresponding volumetric radial dilations (R_v ≈ 3,958.90 to R_max ≈ 8,928.39 miles) defines the exact spatial capacity required to embed this topological variance. By applying a strict harmonic conversion matrix (0.5 Hz ≡ 1 mile), we demonstrate that the metric ratio of these bounded state shifts topologically resolves to the unified bidirectional modifier (β = ±2.2552725). This confirms that physical volumetric expansion is deterministically resolved by the harmonic frequency compression of the invariant temporal wave.
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BibTeX
@misc{thompson2026scalefree,
title = {{The Scale-Free Metric Tensor: Tripartite Topological Constraints and Planetary Dimensional Metrics}},
author = {h.r. thompson},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23529380},
url = {https://zenodo.org/records/23529380},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Scale-Free Metric Tensor: Tripartite Topological Constraints and Planetary Dimensional Metrics
AU - h.r. thompson
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23529380
UR - https://zenodo.org/records/23529380
AB - This paper formally derives a scale-free, tripartite topological framework governing the metric dimensional expansion of a continuous state manifold. By modeling time as a compactified angular divergence rather than a trivial affine scalar, we demonstrate that the projection of a temporal phase into a discrete Euclidean coordinate space is bound by a strict harmonic divergence hierarchy. The continuous manifold mandates three invariant topological thresholds: the 2π affine parameterization, analytically regularized against coordinate rupture by the Euler-Mascheroni constant (γ); the π/2 orthogonal dimensional bifurcation, constrained within an O(3) spatial tensor by the Basel constant (π²/6); and the absolute 4π spinor closure. To satisfy coordinate identity at the 4π manifold without inducing nonlinear metric rupture, the topology embeds an irreducible temporal shear scalar, arising from a 1461:1460 discrete-to-continuous phase asymmetry, directly into the 3D volume as thermodynamic variance. This absolute boundary is strictly determined by the invariant matrix transformation factor, the Incommensurable Radiometric Thermodynamic Variance (κ ≈ 1.437824). Furthermore, we prove that localized planetary topology operates exclusively as a continuous phase-variant spatial tensor boundary, rather than a static metric field. The tensorial dimensional coupling of the axial displacement limits (23.4° to 68.4°) and the corresponding volumetric radial dilations (R_v ≈ 3,958.90 to R_max ≈ 8,928.39 miles) defines the exact spatial capacity required to embed this topological variance. By applying a strict harmonic conversion matrix (0.5 Hz ≡ 1 mile), we demonstrate that the metric ratio of these bounded state shifts topologically resolves to the unified bidirectional modifier (β = ±2.2552725). This confirms that physical volumetric expansion is deterministically resolved by the harmonic frequency compression of the invariant temporal wave.
ER -