Dynamical Criticality and Statistical Regime Transitions: A Unified Theory of Structural Stability Boundaries, Bifurcation Transitions, Observation Pushforwards, and Critical Scaling

Abstract

Problem. Statistical regime transitions are often represented directly as parameter changes in observed sequences or as discrete hidden states, but such a statistical description does not by itself explain why a regime should change under continuously varying external control. Starting from parameterized families of deterministic dynamical systems, this paper establishes a rigorous reasoning chain from “parameter-space geometry—local bifurcation—invariant measure—observation distribution—statistical functional,” and studies when a continuous parameter path crossing a dynamical critical structure can produce a genuine statistical regime transition, as well as the mathematical distinction between continuous critical variation and finite-amplitude statistical jumps. Methods. First, local orbital structural stability within a parameter family is defined on a fixed compact invariant region, distinguishing the structurally stable region, dynamical orbital equivalence class, local bifurcation set, and statistical regime. We prove that any continuous path between distinct structural-stability regions must cross the structural critical set; moreover, if the endpoints belong to distinct dynamical orbital-equivalence classes, then the path must contain a dynamical regime-transition boundary, and a transverse crossing of a nondegenerate codimension-one bifurcation manifold produces the corresponding local bifurcation. Second, the observation distribution is defined by pushforward of an invariant probability measure followed by noise convolution, P_θ^Y = (h_#ν_θ) * Q, and qualitative and quantitative conditions for observational visibility are established. For linear-regression parameters, the intercept and slope are further expressed as population statistical projections under the invariant measure, removing arbitrariness from the parameter map. Finally, a general critical-branch expansion is used to derive a unified transfer law for observable critical exponents, and analytic results for continuous critical scaling, finite jumps, and hysteresis widths are obtained for saddle-node, transcritical, symmetric pitchfork, and cusp-type folds. The stochastic part is strictly confined to the local Ornstein–Uhlenbeck effective regime, and an analytic counterexample based on the full nonlinear stochastic system at the critical point identifies the boundary of its validity. Results. Four groups of core results are obtained. First, continuous traversal between structurally stable regions has a strict topological necessity, but crossing the critical set alone does not guarantee that the endpoint dynamical regimes differ; a content condition of distinct orbital-equivalence classes is therefore required for a genuine dynamical transition. Second, under continuity of the selected state branch and invariant measure, the statistical observation map is continuous within each stable region, so statistical jumps are localized to structural criticality or discontinuities in the state-selection rule; if the observation map is bi-Lipschitz on the relevant support, differences between dynamical measures transfer quantitatively to the observation level through the Wasserstein distance. Third, if the dynamical branch satisfies |x_s − x_c| ≍ |μ − μ_c|^{β_dyn} and the lowest nonzero Taylor order of the observation map at the critical state is k, then the corresponding observable statistic satisfies |T(P_μ) − T(P_c)| ≍ |μ − μ_c|^{kβ_dyn}; this unifies the statistical scaling of saddle-node, transcritical, and pitchfork bifurcations and explains invisibility induced by symmetric observations. Fourth, in cusp-type multistable folds, quasistatic branch selection produces finite state jumps and hysteresis; the jump amplitude scales as b^{1/2} in the cusp parameter, while the hysteresis width between the forward and reverse fold thresholds scales as b^{3/2}. Critical slowing down in the stochastic setting yields enhanced variance and autocorrelation only within the local linearization and small-noise window; at fixed nonzero noise, the full nonlinear system need not exhibit variance divergence. Significance. This paper does not interpret all statistical regime switching as bifurcation, nor does it assert an unconditional one-to-one correspondence between a particular bifurcation type and a particular statistical-parameter change. Its core contribution is a cross-layer, conditional mechanism theorem: a dynamical critical structure can produce a regime transition at the statistical level only after state selection and observation pushforward; continuous criticality, observational degeneracy, finite jumps, hysteresis, and critical slowing down are thus distinguishable mathematical objects within the same theoretical interface. The work is purely theoretical and uses no empirical data, data fitting, numerical simulation, or numerical experiment; all core conclusions follow from analytic derivations, standard theorems, and explicit counterexamples. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently determined by the author. Multiple AI tools assisted with code implementation, data presentation, and text drafting. The author bears full academic responsibility for all research content.