cff-version: 1.2.0
message: "If you use this software, please cite it as below."
authors:
  - family-names: "Barclay"
    given-names: "Brandon"
    orcid: "https://orcid.org/0000-0000-0000-0000"
title: "Oscillation-Adaptability: A Framework for Modeling Conservation-Constrained Systems"
version: 1.2.0
doi: 10.xxxx/jcs.2025.xxxx
date-released: 2025-01-15
url: "https://github.com/bbarclay/oscillation-adaptability"
preferred-citation:
  type: article
  authors:
    - family-names: "Barclay"
      given-names: "Brandon"
      orcid: "https://orcid.org/0000-0000-0000-0000"
  doi: "10.xxxx/jcs.2025.xxxx"
  journal: "Journal of Complex Systems"
  month: 3
  start: 287
  end: 312
  title: "Necessary Oscillations: Adaptability Dynamics Under Fundamental Conservation Constraints in Structured Systems"
  issue: 3
  volume: 42
  year: 2025
  publisher:
    name: "Complex Systems Society"
keywords:
  - oscillations
  - adaptability
  - coherence
  - conservation-law
  - complex-systems
  - mathematical-modeling
  - spectral-analysis
  - necessary-oscillations
  - dynamical-systems
  - theoretical-framework
license: MIT
repository-code: "https://github.com/bbarclay/oscillation-adaptability"
abstract: >
  We present a theoretical framework and a paradigmatic mathematical model
  demonstrating that oscillatory behavior can be a necessary consequence of a
  system optimizing towards a state of order (or coherence) while adhering to a
  fundamental conservation law that links this order to its residual adaptability
  (or exploratory capacity). Within our model, we rigorously prove an exact
  conservation law between coherence (C) and adaptability (A), C+A=1, which is
  validated numerically with precision on the order of 10^-16. We demonstrate that
  as the system evolves towards maximal coherence under a depth parameter (d), its
  adaptability A decays exponentially according to A(x,d) ≤ (|N_ord*(x)|/|N_ord|)
  e^(-d M*(x)), with numerical validation confirming this relationship within 0.5%
  error. Crucially, when introducing explicit time-dependence representing
  intrinsic dynamics with characteristic frequencies ω_n(d) = √d/n, we prove that
  oscillations in A (and consequently in C) are mathematically necessary to
  maintain the conservation principle.