Koma Density Geometry ── Connection Conditions between Density-Derived Informational Structure and Effective Geometry

Abstract

This paper organizes the mathematical scope for extracting informational structure from the Koma density kernel $\rho$ and connecting it to an effective geometry. We introduce a Fisher-information-type functional for the density kernel, and consider its local gradient structure $$g_{ij}^{(F)} = \frac{\partial_i\rho\,\partial_j\rho}{\rho}$$ This tensor is positive semidefinite, and its rank at each point is at most 1. Accordingly, in the general case of $n>1$, it does not form a non-degenerate metric, and an ordinary Levi-Civita connection or Riemann curvature cannot be constructed from it alone. In view of this, this paper positions $g_{ij}^{(F)}$ not as the physical spacetime metric itself, but as a local information tensor representing the directional information contained in the density gradient. On this basis, we make explicit the central task, within the Koma Density Geometry Framework, of connecting density-derived informational structure to a physical or effective spacetime geometry, as the problem of constructing the effective metric map $$\rho_\kappa \;\mapsto\; g_{\mu\nu}^{\mathrm{eff}}$$ The conformal, gradient, and Hessian types are retained as candidate forms (ansatz) for examining this map, but none of them is treated in this paper as an already-adopted physical spacetime metric. Accordingly, Paper A v3.01 is not a document claiming that a physical geometry has already been generated from the density kernel; it is a document that formulates the connection conditions, scope of validity, and undetermined regions existing between density-derived informational structure and an effective geometry. This version reexamines the connection scope between the Fisher informational structure and density geometry in v3.00, and reflects the necessary Scope Restriction. For the correction history and differences, see the separate Paper A v3.01 Correction Note. Japanese and English versions are included in this record. Note on Mathematical Formalization This paper is presented as a structural hypothesis. Mathematical formalization and rigorous proof are intentionally left open, and external verification is welcomed. For correspondence: laboratory@dualbind.comLicense: Creative Commons Attribution 4.0 International (CC BY 4.0)