Teleodynamic Learning a new Paradigm For Interpretable AI

Enrique ter Horst; Juan Diego Zambrano

cs.LGMar 11, 2026

Teleodynamic Learning a new Paradigm For Interpretable AI

Authors: Enrique ter Horst, Juan Diego Zambrano

Abstract

We introduce Teleodynamic Learning, a new paradigm for machine learning in which learning is not the minimization of a fixed objective, but the emergence and stabilization of functional organization under constraint. Inspired by living systems, this framework treats intelligence as the coupled evolution of three quantities: what a system can represent, how it adapts its parameters, and which changes its internal resources can sustain. We formalize learning as a constrained dynamical process with two interacting timescales: inner dynamics for continuous parameter adaptation and outer dynamics for discrete structural change, linked by an endogenous resource variable that both shapes and is shaped by the trajectory. This perspective reveals three phenomena that standard optimization does not naturally capture: self-stabilization without externally imposed stopping rules, phase-structured learning dynamics that move from under-structuring through teleodynamic growth to over-structuring, and convergence guarantees grounded in information geometry rather than convexity. We instantiate the framework in the Distinction Engine (DE11), a teleodynamic learner grounded in Spencer-Brown's Laws of Form, information geometry, and tropical optimization. On standard benchmarks, DE11 achieves 93.3 percent test accuracy on IRIS, 92.6 percent on WINE, and 94.7 percent on Breast Cancer, while producing interpretable logical rules that arise endogenously from the learning dynamics rather than being imposed by hand. More broadly, Teleodynamic Learning unifies regularization, architecture search, and resource-bounded inference within a single principle: learning as the co-evolution of structure, parameters, and resources under constraint. This opens a thermodynamically grounded route to adaptive, interpretable, and self-organizing AI.

Figures & tables

Figure 1: Teleodynamic system architecture. The system state

s\in\mathcal{S}

comprises structure (hypotheses

\mathcal{H}

), parameters (

\theta\in\mathcal{M}

), energy

E

, and history

\tau

. Observations

(x,y)

trigger the coalgebraic step function

\gamma

, producing an observation

o\in\mathcal{O}

and a successor state

s^{\prime}

. Structural actions (genesis, wedge) and parametric updates (natural gradient on the manifold) compete; the local teleodynamic objective

J

gates action selection. The resource

E

couples both dynamics: structural actions cost energy, while predictive success replenishes it.

Figure 2: Manifold carving on 2D data. Left: Input space partitioned by atomic halfspaces. Each colored region corresponds to a hypothesis that “owns” that region. Center: The parameter manifold

\mathcal{M}

showing atom orientations (arrows) and positions (dots). Right: The decision boundary after 20 epochs, showing how compositional forms (conjunctions and disjunctions of atoms) create complex, interpretable regions.

Figure 3: Decomposition of the teleodynamic objective. The three components of

J

—loss (blue), complexity penalty (orange), and energy cost (green)—evolve over training on IRIS. Early phases show high loss but low complexity; structural growth increases complexity while reducing loss; equilibrium balances all three. The total

J

(black, dashed) decreases monotonically until structural freeze (vertical line), then fluctuates around a stable value as parameters fine-tune.

Figure 4: Convergence of inner dynamics. Left: Loss trajectory showing rapid initial descent followed by oscillation around a minimum. Center: Gradient norm decreasing over time, indicating approach to a stationary point. Right: Fisher information diagonal (selected atoms), showing adaptation of the metric geometry. The shaded region marks the structural phase; the unshaded region is pure parametric optimization.

Method	IRIS	WINE	BC
DE11 (Regime B)	93.3%	7 rules	82	Yes
Logistic Regression	91.1%	—	15	Partial
Decision Tree	93.3%	8–15 leaves	—	Yes
Random Forest	95.6%	100 trees	$\sim$ 1000	No

Table 1: Classification accuracy (%) on test sets. Mean

\pm

std over 10 seeds. DE11 Regime B achieves competitive performance with interpretable rules.

Figure 5: Learning dynamics on IRIS (Regime B). Top-left: Test accuracy rises rapidly in early epochs, stabilizes after structural freeze. Top-right: Number of hypotheses grows during structural phase, then freezes at 22-24 rules. Bottom-left: Energy accumulates with correct predictions, with visible drops at structural actions. Bottom-right: Train accuracy (solid) vs test accuracy (dashed), showing mild overfitting stabilized by freeze.

Figure 6: Trade-offs at training endpoint. Each point is one run (10 seeds

\times

4 datasets). Left: Accuracy vs. structural complexity. Middle: Accuracy vs. final energy. Right: Complexity vs. energy. The Pareto frontier (dashed) shows optimal trade-offs. High-accuracy solutions require moderate complexity and sufficient energy investment.

Figure 7: Extracted rules on IRIS (seed 0). Each hypothesis is a logical form over atomic halfspaces. Forms are displayed in Laws of Form notation; atoms reference learned parameters. Reliability indicates historical accuracy. The rules partition input space into class-specific regions.

Method	Accuracy	Rules/Nodes	Parameters	Interpretable
DE11 (Regime B)	93.3%	7 rules	82	Yes
Logistic Regression	91.1%	—	15	Partial
Decision Tree	93.3%	8–15 leaves	—	Yes
Random Forest	95.6%	100 trees	$\sim$ 1000	No

Table 2: Detailed comparison on IRIS. DE11 achieves accuracy comparable to decision trees with fewer, more coherent rules.

Figure 8: Pareto frontier in (Energy, Accuracy) space. Each point is one training run. The frontier (dashed) represents runs where higher accuracy cannot be achieved without more energy investment (or vice versa). Regime B runs cluster near the frontier; Regime C runs are Pareto-dominated.

Configuration	Test Accuracy
Full (Regime B)	$93.3\pm 4.2$
No natural gradient (vanilla SGD)	$87.2\pm 6.1$
No complexity penalty ( $\lambda_{c}=0$ )	$89.5\pm 5.8$
No energy cost ( $\lambda_{e}=0$ )	$91.1\pm 4.9$
No structural freeze (Regime A)	$86.7\pm 8.4$
No structure (Regime C)	$66.7\pm 12.3$

Table 3: Ablation study on IRIS. Removing any component degrades performance.

Figure 9: Phase diagram in (Complexity, Energy) space. Trajectories for 10 seeds on IRIS (Regime B). Arrows indicate time direction. Three phases are visible: (I) rapid complexity growth at low energy (under-structuring), (II) diagonal ascent with coupled growth (teleodynamic), (III) vertical ascent with fixed complexity (equilibrium). The transition boundary (dashed) separates regimes.

Figure 10: Teleodynamic trajectory in 3D state space. A single run on IRIS showing the coupled evolution of complexity, energy, and accuracy. Color encodes time (blue

\to

yellow). The trajectory spirals through three phases before settling at a high-accuracy, moderate-complexity, high-energy attractor.

Figure 11: Structural transition rate over training. The rate starts high (frequent genesis), peaks during refinement (wedge actions), then collapses to zero at structural freeze. The transition is sharp, characteristic of a phase transition.

Figure 12: Energy budget over training. Left: Cumulative energy trajectory. Middle: Energy change per step (rewards minus costs). Right: Structural costs (stepped) vs. predictive rewards (continuous). Early phases show net energy drain (costs exceed rewards); later phases show net accumulation.

Panel of Figure 13 — Figure 13: Teleodynamics on (Regime B) vs. off (Regime A). Regime B exhibits controlled structural growth with stable accuracy. Regime A (unconstrained) shows unstable oscillation and over-structuring.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Parameter	Description	Default
$E_{0}$	Initial energy	10,000
$\eta$	Learning rate	0.02
$\beta$	Fisher decay	0.95
$\lambda_{c}$	Complexity coefficient	0.001
$\lambda_{e}$	Energy coefficient	0.001
$c_{\text{gen}}$	Genesis cost	5.0

Appendix

Table 4: DE11 Configuration Parameters

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