Deriving Gravitational Geometry and Schrodinger Dynamics from Finite Accessibility - From Physical Constraints to Mathematical Necessity: Derivation Chain and Experimental Programs

This work demonstrates that the mathematical structures of general relativity and quantum mechanics are not independent postulates but necessary consequences of operational constraints on what observers can certify. Beginning from three axioms—(i) finite refinement cost (No-Free-Refinement), (ii) finite observational accessibility, and (iii) intersubjective stability—we derive gravitational geometry and Schrödinger dynamics as the unique stable descriptive regimes. The gravitational coupling G and inertial mass coefficient m are identified via Quantum Fisher Information (QFI) density rather than derived, providing the sole constitutive input beyond operational axioms. Key results include: The data processing inequality forces the stationary vacuum profile to satisfy Laplace's equation, whose unique point-source solution is the 1/r Newtonian form Intersubjective stability requires exactly three spatial dimensions: only d=3 permits radius-independent total distinguishability Conservation of distinguishability volume yields the PPN parameters γ=1 and β=1 without invoking Einstein's equations Hard exhaustion and landmark rigidity select the Schwarzschild exterior as the unique scale-invariant fixed point When flip-rate saturation renders pointwise descriptions inadmissible, admissibility conservation, composable updates, and multi-agent stability force linear norm-preserving dynamics, yielding the Schrödinger equation The framework reveals a shared operational backbone between gravity and quantum mechanics: the harmonic 1/r profile governing gravitational accessibility reappears as the scalar coordination load in the relational regime. Experimental programs are proposed to test for residual non-metric vacuum effects, frame dissolution decoherence, entanglement bandwidth limits, and gravitational wave dispersion. Detailed protocols and sensitivity estimates are provided in companion documents on operational holonomy and gravitational wave dispersion. This work inverts the standard presentation of fundamental physics: rather than postulating geometric or quantum structures, we ask what mathematical bookkeeping is forced upon observers facing fundamental certification limits. The metric, the Newtonian potential, and Schrödinger dynamics emerge as necessary structural consequences required to maintain intersubjective comparability under finite accessibility. Keywords General Relativity Quantum Mechanics Foundations of Physics Quantum Fisher Information Information Geometry Operational Physics Intersubjective Stability Schwarzschild Metric Schrödinger Equation Data Processing Inequality Finite Accessibility Observer-Dependent Physics Dimensional Analysis PPN Formalism Frame Dissolution Purification Quantum Reference Frames Certification Limits Harmonic Profiles Hard Exhaustion Landmark Rigidity ISCO Photon Sphere Gravitational Time Dilation Entanglement Decoherence Fisher Distinguishability Inference Theory Multi-Agent Systems Relational Quantum Mechanics No-Free-Refinement Flip-Rate Saturation Bandwidth Constraints Measurement Theory Experimental Signatures Gravitational Wave Dispersion Optical Clock Tests Subject Categories (Zenodo) Primary: Physics > Theoretical Physics Secondary: Physics > General Relativity and Quantum Cosmology Physics > Quantum Physics Physics > Mathematical Physics Related Identifiers Is supplement to: 10.5281/zenodo.18225913 (Intersubjective Stability) Is supplement to: 10.5281/zenodo.18225876 (Frame Failure) Is supplement to: 10.5281/zenodo.18009076 (Rigidity vs Simultaneity) Is supplement to: 10.5281/zenodo.18600688 (Operational Now) Is supplement to: 10.5281/zenodo.18226009 (Purification) Cites: 10.5281/zenodo.18180824 (Operational Holonomy) Cites: 10.5281/zenodo.18226034 (Gravitational Wave Dispersion)