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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19156316 |
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Table of Contents:
- <p>This paper proves four core consequences of the capacity geometry developed in Volume Two, Part One. First, the capacity coordinate <code>S_c = -log(1-R^2)</code> is forced, up to scale, by the exact multiplicative transport law for the slack <code>epsilon := 1-R^2</code>. Second, uniform admissible transport acts on the intrinsic tension <code>REG</code>, the intrinsic flux, and the integrated curvature budget by an exact common scaling law; in particular, slack-increasing uniform processing is monotone non-increasing on these quantities when <code>n >= 2</code>. Third, in the calibrated flat gauge <code>\tilde{\gamma} = e^{S_c}\delta</code>, <code>REG</code> is exactly the Dirichlet energy of the Yamabe conformal factor <code>v := e^{((n-2)/4)S_c}</code>. Fourth, among local first-order isotropic quadratic scalar tensions in the flat capacity gauge, the universal curvature-plus-flux bridge characterises <code>REG</code> up to scale. A final theorem gives local well-posedness and stability for the associated reconstruction problem under the standard nondegeneracy hypothesis on the linearised Dirichlet operator.</p> <p>The paper proves the uniqueness of the logarithmic capacity coordinate as the only continuous additive lift of the multiplicative slack variable, with Gaussian calibration fixing the unit coefficient. It then shows that the intrinsic Dirichlet tension <code>REG</code>, the intrinsic flux, and the integrated curvature budget share an exact common scaling law under uniform admissible transport. In the flat capacity gauge, the same tension functional reduces to the Dirichlet energy of the Yamabe factor, and a characterisation theorem shows that <code>REG</code> is the unique local first-order isotropic quadratic scalar tension admitting the universal curvature-plus-flux bridge.</p> <p>The final section recasts the reconstruction problem as a Yamabe-type Dirichlet problem and proves local well-posedness and stability under the usual linearised nondegeneracy condition. A supplementary appendix records the Gaussian Fisher calibration and the reference-chart audit tension as a Fisher-gradient energy, making explicit the bridge to later companion work.</p> <p>This paper serves as the canonicality and uniqueness layer for a broader companion arc in which the same capacity scalar and quadratic tension structure are used to develop quasi-local mass diagnostics, weighted Fisher bridges, forced Schrödinger dynamics, spectral mass ladders, operator-validation schemes, phase-sensitive observables beyond the quadratic ceiling, and gauge-covariant lifts. It therefore stands both as a self-contained mathematical-physics note and as the canonical middle layer between the geometric foundation in Part One and the downstream SGOC companion set.</p> <p><strong>Scope discipline.</strong> This paper does not claim that the functional form of <code>REG</code> is a new analytic primitive, that every global reconstruction problem in the capacity gauge is automatically well posed, or that the quasi-local mass interpretation from Part One becomes calibration-free. Its scope is narrower and explicit: it isolates the canonical logarithmic coordinate, the exact uniform transport scaling, the flat-gauge Dirichlet normal form, the curvature-plus-flux characterisation of <code>REG</code>, and the corresponding local reconstruction theory.</p> <p><strong>Keywords:</strong> capacity scalar; canonical coordinate; Dirichlet tension; REG; curvature-flux bridge; Yamabe normal form; local reconstruction; conformal geometry; Fisher bridge; mathematical physics</p> <p><strong>DOI list:</strong></p> <p><strong>Part One, Capacity Geometry: REG, Dirichlet Tension, and Quasi-Local Mass, v2.0 </strong><a href="https://doi.org/10.5281/zenodo.19156067" target="_blank" rel="noopener">https://doi.org/10.5281/zenodo.19156067</a></p> <p><strong>Part Two (this paper), The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge </strong><a href="https://doi.org/10.5281/zenodo.19156067" target="_blank" rel="noopener">https://doi.org10.5281/zenodo.19156316</a></p> <p> </p>