Formal Derivation: Schrödinger's Equation from Information Dynamics

This paper presents a novel derivation of Schrödinger's equation from first principles based on quantum information dynamics rather than traditional quantization procedures. Working within the Pentagonal Quantum Information Substrate (PQIS) framework, we demonstrate that the wave function and its evolution equation emerge naturally from the dynamics of information processing in a discrete network with pentagonal (D₅) symmetry. Through rigorous mathematical analysis, we establish that information depth distributions project onto configuration space as wave functions, while their evolution through the network yields precisely the Schrödinger equation. We formally derive the Planck constant ℏ as J·χ₍ₐᵥₖ₎·ℓ₍ₚ₎, connecting quantum behavior to fundamental network parameters, and show how the Hamiltonian operator emerges from information exchange terms in the QIS action. Our derivation accounts for all essential quantum features including superposition, interference, probability interpretation, and measurement collapse through the meltdown mechanism. Quantum corrections for finite-sized systems and relativistic extensions are also presented. This work suggests that quantum mechanics is not a fundamental description but rather an emergent approximation of more basic information dynamics, with significant implications for our understanding of quantum foundations and potential paths toward quantum gravity.