A Threshold Model for Quantum-Classical Duality: Wave and Particle State as Binary Expressions of a Unified Fundamental Field

--- VERSION 3. 0 MAJOR REVISION UPDATE (JUNE 2026) --- This manuscript has been comprehensively updated to Version 3. 0. This major revision transitions the framework from a static dimensional scaling model into a dynamic, non-linear field theory. Key Additions: The Hambardzumian Substrate Equation: A formalized, non-linear Schrödinger-Newton system that scales gravito-electromagnetic self-attraction by 1/ using exclusively fundamental constants. Analytical Soliton Convergence: Direct mathematical derivation showing the collapsed wave substrate naturally stabilizes at the exact spatial radius of the classical electron (rₑ). Temporal Collapse Constant: Derived an instantaneous wave-to-particle localization timeframe (1. 76 10^-19 s) that perfectly matches the intrinsic Zitterbewegung frequency of the electron field. Falsifiable Laboratory Target: Establishes a definitive, low-energy vertical boundary condition for wave function collapse at exactly 3, 730 eV (soft X-rays), offering a concrete path for terrestrial experimental verification. The original Version 1. 0 abstract follows below We propose a reframing of the relationship between quantum mechanics and general relativity, and present a parameter-free analytical expression for the hierarchy problem as a consequenceof that reframing. Rather than treating quantum mechanics and general relativity as twoincompatible theories requiring reconciliation, we argue they are two categorical expressions ofa single underlying field — the same fundamental substrate viewed from opposite sides of athreshold. Wave and particle states are not two behaviors of the same object but two entirelydifferent modes of existence separated by a hard binary boundary. A particle is not a wave thathas been measured — it is a concentration of the wave field that has crossed a critical threshold, analogous to the way sufficient spacetime curvature produces a black hole event horizon. Weshow that both thresholds — the quantum-classical boundary and the event horizon — arestructurally identical: both are the unique dimensionless threshold condition available in theirrespective theories, both produce hard binary phase transitions, and both are expressions of thesame substrate at different scales. From the scale separation between the two thresholds wederive the expression: Hierarchy = alpha x (mP / mₑ) ², which produces the correctelectromagnetic-to-gravitational force ratio of 10⁴2 with no free parameters, to within 0. 015%precision. This result is consistent with the framework being correct and motivates theexperimental test that would constitute proof: a discrete, reproducible collapse of wave behaviorat approximately 3, 730 eV of electromagnetic field energy — derived from existing constantswith no free parameters — which standard quantum mechanics does not predict and thethreshold model specifically does.