S-4_Shadow Quantum Gravity Equation from Unified Spacetime Scale Structure

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Autore principale: Morita, Kuniyuki
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Pubblicazione: Zenodo 2026
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author Morita, Kuniyuki
author_facet Morita, Kuniyuki
contents <p>This work (S-4) unifies the time-scale structure developed in S-2 and the spatial scale structure developed in S-3, and derives the Shadow Quantum Gravity Equation. Under three minimal conditions—differentiability, local invertibility, and consistency—the spacetime scale transformation is uniquely generated by</p> <p>    K = K_t + K_x = k_t * partial_t + k_x * Delta,</p> <p>which yields a closed spacetime scale flow for any field Phi_ell:</p> <p>    d/dell Phi_ell = k_t * partial_t Phi_ell + k_x * Delta Phi_ell.</p> <p>The time direction produces a decay structure, while the spatial direction produces a coarse-graining structure, and both arise from a single unified scale principle.</p> <p>In the Kerr background, the spatial Laplacian is replaced by Delta_Kerr, leading to a scale-dependent radial equation that defines the Shadow Quantum Gravity Equation. This formulation clarifies the geometric origin of the QNM decay structure D(L), and naturally produces the scale-dependent deformation of the Kerr potential and the resulting QNM frequency shifts.</p> <p>This paper completes the mathematical foundation of spacetime scale hierarchies and establishes the framework of shadow quantum gravity. The S-4 equation provides a unified theoretical basis for scale-dependent QNM analysis and opens the path toward the source-side theory (S-5).</p>
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id zenodo_https___doi_org_10_5281_zenodo_19637085
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publishDate 2026
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spellingShingle S-4_Shadow Quantum Gravity Equation from Unified Spacetime Scale Structure
Morita, Kuniyuki
spacetime scale structure
shadow quantum gravity
scale-dependent geometry
time-scale generator
spatial-scale generator
Laplacian operator
Kerr spacetime
quasinormal modes
scale-dependent QNM
radial equation
coarse-graining
decay structure
operator semigroup
mathematical physics
quantum gravity foundations
<p>This work (S-4) unifies the time-scale structure developed in S-2 and the spatial scale structure developed in S-3, and derives the Shadow Quantum Gravity Equation. Under three minimal conditions—differentiability, local invertibility, and consistency—the spacetime scale transformation is uniquely generated by</p> <p>    K = K_t + K_x = k_t * partial_t + k_x * Delta,</p> <p>which yields a closed spacetime scale flow for any field Phi_ell:</p> <p>    d/dell Phi_ell = k_t * partial_t Phi_ell + k_x * Delta Phi_ell.</p> <p>The time direction produces a decay structure, while the spatial direction produces a coarse-graining structure, and both arise from a single unified scale principle.</p> <p>In the Kerr background, the spatial Laplacian is replaced by Delta_Kerr, leading to a scale-dependent radial equation that defines the Shadow Quantum Gravity Equation. This formulation clarifies the geometric origin of the QNM decay structure D(L), and naturally produces the scale-dependent deformation of the Kerr potential and the resulting QNM frequency shifts.</p> <p>This paper completes the mathematical foundation of spacetime scale hierarchies and establishes the framework of shadow quantum gravity. The S-4 equation provides a unified theoretical basis for scale-dependent QNM analysis and opens the path toward the source-side theory (S-5).</p>
title S-4_Shadow Quantum Gravity Equation from Unified Spacetime Scale Structure
topic spacetime scale structure
shadow quantum gravity
scale-dependent geometry
time-scale generator
spatial-scale generator
Laplacian operator
Kerr spacetime
quasinormal modes
scale-dependent QNM
radial equation
coarse-graining
decay structure
operator semigroup
mathematical physics
quantum gravity foundations
url https://doi.org/10.5281/zenodo.19637085