A minimal analysis exploring whether gene expression recovery after hypoxia remains uniformly accessible across all trajectories.

While biological systems are often described as robust, this work examines whether recovery within finite time windows may be heterogeneous.

We introduce a simple observable:

delta-recovery, which quantifies whether gene expression trajectories return toward baseline within a fixed observation window.

Results suggest that most genes return or remain near baseline, while a non-negligible subset remains displaced at 24h.

This pattern is consistent across thresholds and robust to replicate variability.

These observations are consistent with the possibility that some regions of state space may remain structurally present but become dynamically inaccessible within finite recovery timescales.

This analysis does not demonstrate inaccessibility directly, but provides an observable pattern consistent with time-constrained recoverability.

The framework is intentionally minimal, open, and designed for falsifiability.

A minimal framework exploring whether system failure may emerge from loss of recoverability rather than loss of state space.

The ERC models investigate how accessibility to stable states may depend not only on total perturbation magnitude, but on the relationship between input rate and internal recovery processes.

Numerical results suggest that systems may retain valid states while losing the ability to reach them when driven faster than they can resolve within finite temporal windows.

A temporal extension of the ERC framework showing that loss of recoverability may depend not only on the magnitude of perturbations, but on the rate at which they are applied.

Simulations reveal that even with identical total input (AUC), faster forcing can drive the system beyond its recoverable regime, while slower inputs remain accessible.

This rate dependence appears as a shift in the effective recoverability threshold: the attractor persists, but becomes dynamically unreachable when the system is driven faster than it can recover.

A minimal extension exploring how loss of recoverability emerges at the level of initial condition distributions.

While the minimal model shows that a stable attractor can become dynamically inaccessible, this work examines how that transition appears across an ensemble of initial states.

As internal constraint increases, the fraction of initial conditions that return to the base attractor progressively contracts, providing an operational measure of accessibility.

This reveals that loss of function may first appear statistically, as a shrinking basin of recovery, before becoming absolute at the level of individual trajectories.

The attractor remains present in the state space, but becomes reachable only from a diminishing subset of initial conditions.

This suggests that system robustness depends not only on structure, but on how accessibility is distributed across state space under constraint.

This ensemble perspective extends the ERC framework by introducing a measurable indicator of recoverability loss.

A minimal dynamical system exploring whether loss of recoverability may emerge without removal of underlying states.

The model shows that a stable attractor can persist while becoming dynamically inaccessible due to an internal constraint that reshapes the effective accessibility boundary.

As trajectories evolve, a moving boundary in phase space determines whether recovery remains possible. Crossing this boundary results in loss of access to the attractor, despite its continued existence.

This suggests a separation between structural persistence and functional recoverability, governed by constraint-dependent accessibility within the state space.

A minimal network-based model suggesting that recoverability may depend not only on the existence of return pathways, but on their accessibility under constraint.

By progressively restricting allowed transitions, the system may lose the ability to return to baseline configurations, even when the underlying structure remains unchanged.

This suggests a possible separation between structural integrity and functional recoverability, governed by constraint-dependent accessibility within the state space.

A minimal descriptor-based projection suggesting that oxygen species may separate into distinct regimes rather than forming a continuous space.

O–O and O–H dominated species appear to occupy different regions, with consistent placement of held-out species.

Status: Minimal model + preliminary separation in descriptor space

An operational framework suggesting that system failure may be defined by finite-time recovery constraints rather than asymptotic stability loss.

A boundary appears when recovery time exceeds a finite coherence window, rendering return dynamically inaccessible.

A structural hypothesis suggesting that oxygen-related species may organize as a harmonic system, with ratios appearing to align with simple vibrational proportions.

Observed ratios appear to cluster near simple values (4/3, 7/6, √2, φ).

Status: Hypothesis-generating + reproducible framework