Goal. Investigate whether certain classes of dissipative classical systems admit a functor from a 2‑categorical GPT‑style description to a category of classical observables in which thermodynamic stability corresponds to algorithmic information minimization.
Core conjecture. Systems with hysteresis minimize a description‑length–like action functional SMDL on a Koopman–von Neumann–type Hilbert space of trajectories, acting as a KvN bridge between “quantum‑shaped” potentiality and classical realized dynamics.
I am writing to share a structural pattern I have observed across regime shifts in complex systems—from quantum error correction to climate uncertainty. While my background is in applied systems engineering, the phenomena I encounter consistently suggest an underlying categorical unity that I hope to formalize.
Specifically, I am exploring whether the formalism of 2-plectic geometry and String Lie 2-Algebras4—which you have pioneered for describing higher-order mechanics—provides the natural language for systems effectively governed by "memory" (discrete hysteresis loops) embedded in continuous dynamics. The "monad-like" behavior of these systems, optimizing for parsimony, strongly echoes a Leibnizian ontology operationalized via Algorithmic Information Theory.
What follows is a sketch of this framework (H² / HS(p)), separating well-grounded literature from my specific structural conjectures. I am sharing this in hopes of a "blunt verdict": is this isomorphism formally viable, or does it rely on a category error?
We propose a variational principle where physical stability corresponds to minimizing the Algorithmic Description Length of the system's trajectory.
Instead of a generic thermodynamic entropy, we postulate a minimization of the Kolmogorov complexity $K$ of the state trajectory $\gamma$ relative to a Universal Prior:
This extends Jost's (2021)2 work on Leibnizian optimization. We conjecture that the "Eigen-Oscillator" in our model is the physical operator that performs this pruning, effectively solving the variational problem by collapsing the state space (Holevo bound) to the simplest sufficient trajectory.
Standard Model: In a standard surface code, error syndromes are measured and processed via a minimum-weight perfect matching (MWPM) algorithm, often treating sequential measurements as independent inputs (Markovian).
The H² Modification: We introduce a strictly non-Markovian syndrome history. We define a "Hysteresis Filter" $H$ that modifies the syndrome $s_t$ based on a dwell-time parameter $\tau$:
Conjecture: By treating the syndrome extraction as a dynamical system with memory ("Primitive Ontology" of Form6), we can reduce logical error rates in valid hardware regimes where measurement noise dominates. This filters "transient" violations that do not have sufficient "ontological weight" (duration) to be real errors.
Problem: Hallucination in Large Language Models often correlates with high-confidence generation despite low-information latent support.
Proposed Metric: Consider a toy language model with a finite latent density matrix $\rho$ representing the "context" semantic space, and a predictive distribution $p$ over the token vocabulary. We propose the safety criterion:
Where $H(p)$ is the Shannon entropy of the output and $S_{vn}(\rho)$ is the von Neumann entropy of the latent
state.
Hypothesis: A violation of this bound implies the model is "hallucinating" complexity—generating
specific details ($low H(p)$) from a high-entropy, vague context ($high S_{vn}(\rho)$), or vice versa. This
effectively acts as an MDL check on the generative process.
We refrain from claiming the entire climate is a 5-tuple. Instead, we propose a toy model for a specific subsystem (e.g., El Niño Southern Oscillation) as a concrete categorical automaton.
We define the system as $\mathcal{A} = (\mathcal{C}, \mathcal{D}, F, \eta, K)$ where:
Question for Review: Does this structure map naturally to the String Lie 2-Algebra framework, where the "hysteresis" $\eta$ acts as a 2-morphism between flow processes? We suspect this higher-order structure is required to capture the "inter-regime" physics that standard flat dynamical models miss.
The framework extends across multiple domains, each demonstrating entropy minimization and hysteresis-based regime selection:
Note: The following interpretations are heuristics used to motivate the H² hypothesis.
"Foundations of Algorithmic Thermodynamics" (Phys. Rev. E, Jan 2025)
Links Kolmogorov Complexity to work capacity. Relevance: Suggests that minimizing $S_{MDL}$ satisfies
thermodynamic constraints.
"Koopman-von Neumann Field Theory" (arXiv, July 2025)
Reformulates classical many-body problems as quantum field theories. Relevance: Legitimatizes the use
of Hilbert space operators for classical climate models (H²Clime).
"Hysteresis and Self-Oscillations in an Artificial Memristive Quantum System" (Phys. Rev. A, Oct
2024)
Shows memory creating stable eigenstates. Relevance: Physical precedent for the "stabilizing" role of
hysteresis in H²QEC.