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Adaptive Constraint Cascade (ACC)_Case Study ACC_1.1: Three Illustrative Examples

Thanh Minh Nguyen

Published May 10, 2026• 0 citations
Publisher

Abstract

We extend the Adaptive Constraint Cascade (ACC) framework, introduced in ACC\_1 for the double-well potential, to three additional physical systems: (1) a qubit in a thermal bath, (2) a quantum harmonic oscillator with damping, and (3) a particle in a generic single-well potential. In each case, the ACC\_1 framework equation – a Lindblad-type master equation with a $\lambda$-dependent dissipation rate – is applied, where $\lambda = k_B T / \Delta E$ (or an analogous parameter) controls the quantum–classical transition. We show how the same mathematical structure seamlessly interpolates between pure quantum dynamics (Schr\"odinger equation) and classical thermal behavior (master equation), demonstrating the universality of the ACC approach. This work serves as an extension of ACC\_1, providing concrete, solvable examples for readers interested in applying ACC to their own systems.

DOI
10.5281/zenodo.20102752

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We introduce the concept of an Adaptive Constraint Cascade (ACC) as a novel framework for describing the continuous transition between distinct physical theories via a deformation of their underlying constraint structures. In this first case study (ACC_1), we focus on the quantum–classical transition in a double-well potential. Using a single control parameter λ = k_B T / V_barrier, we propose an interpolation equation – a Lindblad-type master equation with a λ-dependent dissipation rate – whose limiting behaviors recover the pure quantum (Schrödinger) and the classical master equation regimes. The ACC framework offers a smooth, contextuality-aware description of emergence without requiring a fixed ontology. This manuscript is the first of three planned works on ACC, with subsequent studies addressing Newtonian–relativistic gravity (ACC_2) and quantum field theory (ACC_3).
We derive a testable prediction from the Adaptive Constraint Cascade (ACC) framework, specifically from ACC_1 (quantum–classical transition). Using the ACC_1 framework equation – a Lindblad master equation with a λ-dependent dissipation rate γ(λ) = γ₀ tanh(λ), where λ = k_B T / ħω₀ – we compute the emission spectrum S(ω) of a qubit (two-level system) as a function of temperature T. The theory predicts a continuous crossover from a narrow Lorentzian peak (coherent regime) at low T to a broad Lorentzian peak (incoherent regime) at high T, with the linewidth Γ following Γ(λ) = Γ₀ + β tanh(λ), where β is a constant determined by the system–bath coupling. Rather than introducing a fundamentally new dynamical law, ACC provides a unified structural interpretation of crossover behavior that is often modeled using separate approximations in different regimes. Experimental verification is feasible with current superconducting qubit or trapped-ion platforms. We outline a concrete experimental protocol.
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We propose a tentative working definition of an Adaptive Constraint Cascade (ACC) – a framework for describing continuous transitions between distinct physical theories via a deformation of their underlying effective structural relations. An ACC is defined as a tuple (P, T, {C_p}, Φ) satisfying a set of heuristic axioms. The parameter space P is a connected topological space (e.g., an interval in R or a product of intervals). For each p in P, T(p) is a physical theory, and C_p denotes its associated effective structural relations (e.g., commutation relations, master equations, or constraint algebras). The family {C_p} is required to vary continuously with p in an appropriate sense. We show that the three case studies – ACC_1 (quantum–classical transition), ACC_2 (Newton → SR → GR), and ACC_3 (NRQM → QFT) – are instances of this definition. This foundational paper provides a coherent structural language for future developments, including categorical formulations and testable predictions, while remaining deliberately open to refinement.
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