TSP 1.0 — Theory of Structural Prevalence

The problem it addresses

Many systems evolve under constraints that do not depend solely on their instantaneous variables. Their prior organization may remain incorporated into the attained state and accessible trajectories. Through a minimal dynamical description, TSP studies when a structure maintains a regulated regime and when it crosses a threshold that changes its possible evolution.

The proposal does not yet identify a universal physical realization: it provides a formal language for posing that question and deriving testable consequences.

Principle of structural prevalence

The attained structure is not a passive support: by incorporating history, it modifies which subsequent trajectories remain accessible.

In TSP 1.0, this principle guides the model construction. Its interpretation as a property of physical systems is a hypothesis awaiting validation; it is not presented as a demonstrated universal law.

Formulation, domain and parameters

The dimensionless dynamical family of TSP 1.0 is:

In this realization, u ≥ 0 is a normalized structural magnitude and τ is dimensionless time. Initially, χ̂ > 0 and δ > −1 are adopted, keeping the higher-order nonlinear term above first order and the origin regular. The physically admissible range of δ must be fixed when each system is operationalized.

• κ̂ > 0: may produce a regulated regime followed by a separatrix and accelerated growth, depending on the parameters.
• κ̂ = 0: reduces the model to growth with quadratic regulation.
• κ̂ < 0: introduces regulating feedback and may lead to saturation; by itself it does not generate a symmetric pitchfork.

The sign of κ̂ and the value of δ are not graphical details: they determine the dynamical class and must be estimated or fixed through an explicit physical correspondence.

Relation to existing frameworks

TSP’s dynamical family is not proposed in isolation. The canonical model’s topological transition is a standard saddle-node bifurcation; in coupled populations of trajectories, the collective variable can be formalized with established mean-field tools — diffusive coupling, or, when the phenomenon involves phase synchronization, Kuramoto-type coupling. The incorporation of history into individual evolution relates to the literature on non-Markovian processes and memory kernels.

What TSP specifically explores is the feedback between both levels: how a threshold reached at the collective level can modify the thresholds of the individual trajectories that generated it. That downward feedback — not each piece in isolation — is the open question the program aims to formalize.

Canonical case and stability clarification

Canonical case of TSP 1.0: χ̂ = 2.5, κ̂ = 1.0, δ = 1.0.

Dynamical realizations of structural prevalence

TSP is formulated as a dynamical principle that can admit different realizations. The original equation is retained to study regulation, saturation, threshold and escape. To represent specifically the symmetric coexistence of two furrows, the pitchfork normal form is incorporated as a particular case, not as a replacement:

For μ > 0, the origin is unstable and two stable states appear, x*± = ±√(μ/β). Here x is a signed order parameter, distinct from the non-negative variable u in the original model. A physical asymmetry may be introduced through dx/dτ = h + μx − βx³.

The pitchfork formalizes two prevalent configurations and their basins; it does not yet establish that a specific physical system follows this form.

Established results and present scope

Candidate physical system

The first calibration candidate will be a bistable autocatalytic reaction in an open reactor. It is selected because concentrations, feed flow, transition times and perturbations can be controlled, and because its kinetics can be compared with low-dimensional nonlinear models.

Before fitting, the correspondence must state what physical magnitude u or x represents, which experimental parameters correspond to χ̂, κ̂, δ, μ and h, and which observable defines the regime change. If the dynamical reduction cannot be justified, the candidate will be rejected rather than forced to fit TSP.

Ferromagnetic systems with hysteresis and forced Belousov–Zhabotinsky reactions remain later candidates. One favourable case would demonstrate applicability, not universality.

Quantitative prediction prior to testing

For the asymptotic regime dominated by the higher-order term, TSP 1.0 proposes:

Before testing, the definitions of S₀ and t_c, the asymptotic interval, controlled variables, independent method for fixing δ, admissible uncertainty and rejection criterion will be published. The prediction will be challenged if the observed exponent is incompatible with −(1+δ), if no reproducible scaling interval exists, or if an alternative model explains the data more robustly.

Spatial extension

Once the homogeneous model has been closed and calibrated, the reaction–diffusion extension will be studied:

This extension will allow the analysis of domains, fronts, spatial selection, competition between furrows and geometrical dependence. Its results will remain numerical and exploratory until a calibrated physical correspondence exists.

Testing programme

1. Operationally define the variables and units.
2. Derive or justify reduction to the selected dynamical model.
3. Fix parameters and prediction before observing the test outcome.
4. Fit with one dataset and evaluate against independent data.
5. Compare alternative models using explicit criteria.
6. Publish positive or negative results and only then extend to the spatial domain.

TSP will become physically falsifiable when this correspondence is fixed. At present it can already receive mathematical and methodological criticism, but it is not presented as a validated physical law.

Version history

Every substantial revision of the preprint is logged here with its versioned Zenodo DOI. Mathematical corrections identified after publication are documented explicitly, not buried in the archive.

Publication, resources, and citation

Author

Jesús Guillermo Paz Benavides

Title

TSP — Theory of Structural Prevalence

Version

TSP 1.0

Year

2026

DOI

10.5281/zenodo.20124711

Publication status

Published · Scientific preprint

Suggested citation

Paz Benavides, Jesús Guillermo. (2026). TSP — Theory of Structural Prevalence (TSP 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.20124711

Zenodo is the permanent scientific archive. GitHub contains the complementary code and reproducible material.