Physics Programme
A Representational Research Agenda in Mathematical Physics
Stefaan Vossen
Dot Theory Research Programme
Purpose
This page outlines the physics-facing research programme associated with Dot Theory.
Dot Theory is not presented as a completed physical theory. Instead, it proposes a modelling question concerning state representation and prediction in complex dynamical systems.
The programme investigates whether predictive models become limited when contextual information influencing system dynamics is excluded from the state representation used in computation. A minimal implementation of this benchmark is available as a public repository:
https://github.com/stefaanvossen/dot-theory
The central aim is therefore modest but precise:
to determine whether extending a model’s state representation to include contextual structure improves predictive accuracy in regimes where such structure influences observable outcomes.
The programme proceeds through computational benchmarks, modelling experiments and formal analysis rather than through immediate modification of established physical theory.
Representational Hypothesis
Many physical and computational models operate on a defined state space.
Let
ψ ∈ ℋ
represent a conventional state description.
Dot Theory proposes examining an augmented representation
Ψ = (ψ, μ) ∈ ℋ × ℳ
where μ represents contextual variables associated with the system’s regime, measurement conditions or environmental structure.
The projection
π(Ψ) = ψ
recovers the conventional model.
The research question becomes:
Under what circumstances does the projection π discard information required to predict observable outcomes?
If two augmented states
Ψ₁ ≠ Ψ₂
project to the same ψ but produce different observable distributions, then modelling exclusively in ℋ may be representationally incomplete in that regime.
This hypothesis is domain-specific and must be tested empirically.
Relation to Existing Physics
This programme does not claim that established physical theories are incorrect.
Quantum Mechanics, Quantum Field Theory and General Relativity remain extraordinarily successful predictive frameworks.
Any representational extension must therefore satisfy strict conditions.
1. Limit Recovery
When contextual variables μ are irrelevant or averaged out, the augmented model must reduce to the standard formulation.
2. Symmetry Preservation
Extensions must preserve or explicitly justify any modification to established symmetries such as:
Lorentz invariance
gauge invariance
unitarity
3. Mathematical Coherence
The extended state space ℋ × ℳ must support well-defined dynamics and observables.
4. Empirical Discriminability
The extension must produce at least one measurable difference from the reduced model in a defined regime.
Without these conditions the programme has no standing as mathematical physics.
Reproducible Test
Representational Incompleteness in a Regime-Switching Dynamical System
To make the representational hypothesis testable, a minimal computational benchmark is used.
The benchmark models a dynamical system whose behaviour depends on an unobserved regime variable.
The system alternates between two regimes producing different dynamics.
If a predictive model observes only the system state ψ, it cannot always distinguish between regimes when they produce identical instantaneous observations.
However, if a contextual variable μ representing the regime is included, predictive accuracy may improve.
The benchmark therefore tests whether augmenting the state representation improves prediction when regime switching occurs.
System Description
Consider a discrete-time dynamical system.
Hidden system state
sₜ
Observed variable
oₜ = sₜ + noise
Regime variable
μₜ ∈ {0,1}
The regime determines the system’s evolution.
Example dynamics:
If μₜ = 0
sₜ₊₁ = 0.9 sₜ + ε
If μₜ = 1
sₜ₊₁ = −0.5 sₜ + ε
where ε represents small stochastic noise.
The observation alone does not always uniquely determine the governing regime.
Competing Predictive Models
Two predictive models are evaluated.
Baseline Model
The baseline predictor receives only the observation
Input
oₜ
Prediction
ŝₜ₊₁ = f(oₜ)
This corresponds to modelling with state ψ alone.
Augmented Model
The augmented predictor receives both observation and contextual information
Input
(oₜ, μₜ)
Prediction
ŝₜ₊₁ = f(oₜ, μₜ)
This corresponds to modelling with the extended state
Ψ = (ψ, μ).
Evaluation
Prediction accuracy is measured across the full simulation.
A typical metric is mean squared prediction error.
If contextual variables influence system dynamics, the augmented model should achieve lower error.
If contextual variables are irrelevant, both models should perform similarly.
Implementation
A minimal implementation of this benchmark is available as a public repository:
https://github.com/stefaanvossen/dot-theory
The repository contains a compact Python script that
generates regime-switching data
trains baseline and augmented predictors
evaluates prediction error
visualises the results.
The experiment can be executed with a single command:
python dot_benchmark.py
Expected Output
The benchmark produces a simple plot comparing prediction error over time.
Horizontal axis
time step
Vertical axis
prediction error
Two curves are shown
baseline model error
augmented model error
When regime switching occurs, the baseline model error increases because the model cannot distinguish between governing dynamics.
The augmented model tracks the system more accurately because regime information is available.
Scope
This benchmark is intentionally minimal.
It does not claim to modify existing physical theory.
Instead it demonstrates a modelling principle:
predictive performance may depend on whether the state representation used by a model captures the contextual structure governing system dynamics.
If the representational hypothesis proves useful in simple computational systems, future work may explore more structured domains including
experimental measurement modelling
simulation environments
complex dynamical systems.
Programme Outlook
The long-term aim of this research programme is to examine how state representation influences predictive modelling across physical and computational systems.
The work proceeds incrementally through:
minimal computational benchmarks
formal modelling frameworks
empirical evaluation in increasingly structured domains.
The present benchmark serves only as a first test of the representational hypothesis.
Closing Remark
This programme proposes a modest but testable question.
If modelling performance depends on the completeness of the state representation used to describe a system, then identifying regimes of representational incompleteness may become a useful tool in the study of complex dynamical systems.
The benchmark provided here is intended simply as a transparent starting point for investigating that possibility.
Thank you for your time, attention and consideration,
Stefaan