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Computational Constraints on Quantum State Transitions in a Finite-Resource Substrate: A Reinterpretation of the Uncertainty Principle and the Derivation of the Operational Density Zquant

Yuhei Miyazaki

Published Jan 6, 2026• 0 citations
Publisher

Abstract

Within the framework of the Computational Resource Finiteness Hypothesis (CRFH), this work presents a computational reinterpretation of the quantum mechanical uncertainty principle as a constraint arising from finite operational resources on a Universal Substrate. Uncertainty is shown to emerge from unavoidable competition between information writing and state-transition buffering, rather than from intrinsic randomness or measurement disturbance. By imposing resource conservation expressed through $L^2$-norm preservation and minimizing operational dissipation, we derive an effective operational density that converges to a stable rated point,\[Z_{\mathrm{quant}} = \frac{1}{\sqrt{2}} \approx 0.707 .\]This value is interpreted not as an exact instantaneous bound, but as an engineering-rated operating point ensuring long-term stability under finite computational capacity. The result suggests that probabilistic structures in quantum mechanics may reflect an underlying optimization principle governing resource allocation on the Universal Substrate. This formulation provides a scale-consistent bridge between microscopic uncertainty relations and macroscopic operational density limits observed in larger physical systems.

DOI
10.5281/zenodo.18162659

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Within the framework of the Computational Resource Finiteness Hypothesis (CRFH), this work presents a computational reinterpretation of the quantum mechanical uncertainty principle as a constraint arising from finite operational resources on a Universal Substrate. Uncertainty is shown to emerge from unavoidable competition between information writing and state-transition buffering, rather than from intrinsic randomness or measurement disturbance. By imposing resource conservation expressed through L2L^2L2-norm preservation and minimizing operational dissipation, we derive an effective operational density that converges to a stable rated point,[Z_{\mathrm{quant}} = \frac{1}{\sqrt{2}} \approx 0.707 .]This value is interpreted not as an exact instantaneous bound, but as an engineering-rated operating point ensuring long-term stability under finite computational capacity. The result suggests that probabilistic structures in quantum mechanics may reflect an underlying optimization principle governing resource allocation on the Universal Substrate. This formulation provides a scale-consistent bridge between microscopic uncertainty relations and macroscopic operational density limits observed in larger physical systems.
The uncertainty principle is among the most widely cited results of quantum mechanics, yet it is also one of the most persistently misunderstood. It is frequently explained as a limitation of measurement, experimental disturbance, or technological precision, despite being fundamentally independent of observation. This paper reframes the uncertainty principle as a structural constraint arising from the geometry of admissible physical paths under finite temporal resolution. We argue that tradeoffs between localization and momentum coherence do not reflect epistemic ignorance or measurement back-action, but instead express an unavoidable incompatibility between competing structural requirements. Within a structure-first framework, uncertainty emerges naturally from the need to maintain global coherence while restricting admissible continuation. This interpretation preserves the standard formalism of quantum mechanics while clarifying the conceptual origin of uncertainty and preparing a foundation for later discussions of quantum fields as pre-nodal harmonic structure.
The uncertainty principle is fundamentally rooted in the algebraic asymmetry between observables. We introduce a new class of uncertainty relations grounded in the resource theory of asymmetry, where incompatibility is quantified by an observable's intrinsic, state-independent capacity to break the symmetry associated with another. This ``operator asymmetry,'' formalized as the incompatibility norm, leads to a variance-based uncertainty relation for pure states that can be tighter than the standard Robertson bound. Most significantly, this framework resolves a long-standing open problem in quantum information theory: the formulation of a universally valid, product-form uncertainty relation for the Wigner-Yanase skew information. We demonstrate the practical power of our framework by deriving tighter quantum speed limits for the dynamics of nearly conserved quantities, which are crucial for understanding non-equilibrium phenomena such as prethermalization and many-body localization. This work provides both a new conceptual lens for understanding quantum uncertainty and a powerful, versatile toolkit for its application.
This work reinterprets the Uncertainty Principle within an adynamical, constraint-based framework, showing that uncertainty relations arise from global consistency requirements rather than measurement disturbance or intrinsic indeterminacy. By treating conjugate quantities as projections of a single, closure-constrained relational structure, the formal inequalities emerge as necessary conditions for admissibility under No Preferred Reference Frame (NPRF) and boundary-based closure. The framework removes reliance on dynamical narratives and instead derives uncertainty as a structural feature of representation—an invariant limitation imposed by the geometry of admissible configurations. This approach integrates naturally with Projection Residual Geometry and related constraint-based formalisms, providing a unified, minimal ontology for quantum constraints.
Gate-based quantum computing, as demonstrated in The Structural Limitsof Quantum Computing, exhibits a fundamental mismatch between continuousquantum dynamics and the discrete stability required for digital computation. Thisstructural tension motivates a reformulation of computation itself. In this work,we propose a geometric and physically grounded foundation in which computationis defined as convergence. Our contributions are threefold:• A convergence-based axiomatization of computation. We definecomputation as the action of a convergence operator C : X → X satisfying aLyapunov-type monotonicity condition V (xt+1) < V (xt). This situates digitalstability within the general framework of dissipative dynamical systems andprovides a unified mathematical description of stable information processing.• A physical interpretation grounded in geometric stability. We showthat convergence operators arise naturally from physical principles: gradientflows, energy minimization, and topological invariants all realize stabilitythrough contraction toward fixed points. In this view, intrinsically topological architectures exemplify physical realizations of convergence rather thanspecial-purpose corrections to quantum instability.• A geometric reformulation of computational complexity. We reinterpret algorithmic difficulty in terms of the geometry of convergence: the length of convergent trajectories, the structure of attraction basins, and thecurvature of effective potential landscapes. This perspective separates complexity from the step-count assumptions of Turing-style models and insteadties it to physically meaningful stability mechanisms. Taken together, these results establish a physically motivated and geometricallycoherent foundation for computation. Convergence—rather than discrete logicaloperations—emerges as the fundamental mechanism enabling reliable informationprocessing in natural and engineered systems alike. This framework provides aprincipled path toward scalable, stable computation beyond gate-based quantumarchitectures and sets the stage for unified models of physical and algorithmicreasoning.
We derive the structural modifications and physical implications of the Heisenberg Uncertainty Principle under the framework of the Three-Dimensional Complex Necessity (3DCN) principle. In standard quantum mechanics, the uncertainty relation is bounded by a constant scalar value derived from the axiomatic commutation relation [x, p] = i*hbar. By replacing the static imaginary unit i with the dynamically locked universal complex structure operator J_U, we prove that the commutator deforms into a state-dependent operator equation [x, p] = J_U*hbar. We demonstrate that the lower bound of quantum uncertainty is no longer invariant, but fluctuates based on the local geometry of the real phase space substrate and the energy density of the flavor-locking mechanism. This deformation naturally regularizes sub-Planckian spatial measurements, establishing a fundamental physical cutoff that prevents infinite information density localized in real space, thereby avoiding gravitational collapses at the quantum level.
Author: Marco Lindenbeck Title: Deterministic Quantum Mechanics: Replacing the Copenhagen Pillars of Probability and Uncertainty with Discrete Substrate Logistics Description: For a century, the Copenhagen Interpretation has forced theoretical physics to accept a subatomic universe governed by random chance, probabilistic wave functions, and fundamental fuzziness. This paper demonstrates that these principles are not foundational truths of nature, but mathematical artifacts born from a single categorization error: treating the vacuum of space as a continuous, empty void. By applying the Ω\OmegaΩGLR framework—defining the vacuum as a discrete, pre-tensioned geometric network (the Topological Substrate) governed by a strict Pythagorean processing limit (c2=u2+v2c^2 = u^2 + v^2c2=u2+v2)—this paper entirely eliminates probability and uncertainty from the atomic scale. By mapping the discrete hardware that underlies quantum software, local realism, determinism, and absolute mechanical causality are fully restored to atomic physics. Key Mechanical Derivations Included: Replacing Schrödinger: The Topological Resonance Equation redefines electron energy levels as deterministic network routing paths. Replacing Heisenberg: The Uncertainty Principle (ℏ/2\hbar/2ℏ/2) is mathematically mapped as the Substrate Resolution Limit—an asymptotic hardware buffer overrun (Bandwidth Saturation). The Bohr Radius & Rydberg Protocol: Re-derived as strict geometric hardware constraints and topological data exhaust. The Photoelectric Effect: Re-derived as localized structural bandwidth saturation. The Casimir Effect: The 111 Atmosphere measurement discrepancy is resolved mathematically via the discrete Bilateral Elasticity Factor (2ϕ−22\phi^{-2}2ϕ−2), eliminating the need for virtual zero-point particles.