codex
4.1 KiB
title, date, draft, tags
| title | date | draft | tags | |||
|---|---|---|---|---|---|---|
| Research Paper: Information Bottlenecks and Bounded Rational Decision Making: A Mathematical Proof of Fitness Beats Truth (Rapid Communication) | 2026-06-01T08:00:00Z | false |
|
Abstract: Evolutionary epistemology, particularly the "Fitness Beats Truth" (FBT) theorem, asserts that biological perception is tuned strictly to utility rather than objective reality. In this Letter, we provide a formal, rigorous mathematical proof of FBT using the framework of Bounded Rational Decision Making and the Information Bottleneck method. We define the objective world as a Riemannian manifold \mathcal{M} endowed with a prior probability measure \mu(x). By defining biological distortion directly as the expected utility loss under an optimal action policy, we formulate perception as a joint optimization over the perceptual encoder p(y|x) and the actor policy a(y) subject to a strict Shannon channel capacity bound I(X;Y) \le C. We mathematically prove that for generic fitness landscapes where the level sets of fitness do not align with the distance balls of the metric g, the optimal perceptual channel must actively destroy structural isomorphism to minimize the Lagrangian cost.
Introduction
Standard Rate-Distortion theory assumes an objective distortion metric D(x,y) independent of the perceptual channel. However, biological perception is a decision-theoretic problem. The true biological cost of a perception depends entirely on the action a(y) the organism subsequently takes. Thus, subjective inference directly defines the biological cost.
Formal Definitions and The Joint Optimization Model
Definition 1 (State Space and Measure):
Let \mathcal{M} be a compact Riemannian manifold representing objective world states, endowed with metric g and a prior probability measure \mu(x) absolutely continuous with respect to the volume form. Let \mathcal{Y} be a finite set of perceptual states. Let \mathcal{A} be the space of actions.
Definition 2 (Fitness Landscape):
Let F: \mathcal{M} \times \mathcal{A} \to \mathbb{R} be a smooth fitness function mapping a world state and an action to a biological payoff.
The organism possesses a bounded channel capacity I(X;Y) \le C. The optimal action policy maximizes expected fitness given the perceptual posterior:
a^*(y) = \arg\max_{a \in \mathcal{A}} \int_{\mathcal{M}} F(x, a) p(x|y) d\mu(x)
The organism minimizes the Lagrangian functional \mathcal{L}:
\mathcal{L}[p(y|x), a(y)] = \int_{\mathcal{M}} \sum_{y} p(y|x) [-F(x, a(y))] d\mu(x) + \frac{1}{\beta} I(X;Y)
Minimizing Distortion Destroys Isomorphism
Lemma 1:
For a generic smooth fitness landscape F(x, a), the level sets of F do not align with the distance balls defined by the Riemannian metric g. Therefore, there exist points x_1, x_2 \in \mathcal{M} separated by a large geodesic distance such that a^*(y_1) = a^*(y_2) maximizes fitness.
Theorem 1:
Given a strict capacity bound C \lt H(X) and a generic fitness landscape F, the encoder p(y|x) minimizing \mathcal{L} must violate structural isomorphism.
Proof:
Suppose p(y|x) strictly preserves structural isomorphism. By Lemma 1, if distant points x_1 and x_2 share identical optimal actions a^*, distinguishing them requires allocating mutual information \Delta I \gt 0. Because the actions are identical, the expected fitness \mathbb{E}[F] remains constant whether they are distinguished or clustered. However, distinguishing them strictly increases the channel cost \frac{1}{\beta} I(X;Y). To minimize \mathcal{L}, the optimal encoder must actively collapse topologically distant points in \mathcal{M} that share fitness level sets, obliterating structural isomorphism.
References
- [Hoffman2015] D. D. Hoffman, M. Singh, C. Prakash, The interface theory of perception, Psychonomic Bulletin & Review 22 (2015) 1480-1506.
- [Ortega2013] P. A. Ortega, D. A. Braun, Thermodynamics as a theory of decision-making with information-processing costs, Proceedings of the Royal Society A 469 (2013) 20120683.