# Information Bottlenecks and Bounded Rational Decision Making: A Strict Proof of Fitness Beats Truth

**Target Venue:** *Journal of Theoretical Biology*

## Abstract
Donald Hoffman's "Fitness Beats Truth" (FBT) theorem argues that perception is tuned to utility, not reality. We provide a mathematically rigorous proof of FBT using Bounded Rational Decision Making and the Information Bottleneck method. Previous models failed by using standard Rate-Distortion Theory, which requires a fixed distortion matrix. We rectify this by defining biological distortion directly as the utility loss: $D(x, y) = -F(x, \arg\max_a \mathbb{E}_{X' \mid y}[F(X', a)])$. Because the optimal action $a^*(y)$ relies on the perceptual channel $p(y|x)$ via Bayesian inference, the optimization is non-linear. By explicitly formulating a joint optimization over the perceptual encoder $p(y|x)$ and the actor policy $a(y)$, we mathematically prove that minimizing expected distortion under a channel capacity bound $C$ forces the organism to completely obliterate structural isomorphism.

## 1. Introduction
Standard Rate-Distortion theory assumes an objective distortion metric independent of the channel. Biological perception, however, is a joint policy optimization where subjective inference directly defines the biological cost.

## 2. Joint Optimization of Perception and Action
The agent possesses a bounded channel capacity $I(X;Y) \le C$. 
Let $p(y|x)$ be the perceptual encoder and $a(y)$ be the actor policy. The true biological cost is the negative expected fitness: $\mathbb{E}[-F(x, a(y))]$.
We formulate the biological survival problem as an Information Bottleneck applied to decision theory:
$$
\min_{p(y|x), a(y)} \left( \mathbb{E}[-F(x, a(y))] + \frac{1}{\beta} I(X;Y) \right)
$$
where $\beta$ is a Lagrange multiplier enforcing the strict channel capacity bound $C$.

## 3. Minimizing Distortion Destroys Isomorphism
Because this is a joint optimization, the optimal actor policy $a^*(y)$ depends on the posterior $\mathbb{P}(X|y)$, which is determined by the encoder $p(y|x)$. 
The fitness landscape $F(x, a)$ generically possesses symmetries and gradients completely orthogonal to the metric topology of the true state $x$. To minimize the functional under a strict capacity bound, the optimal encoder $p(y|x)$ will aggressively cluster topologically distant points in $X$ that share identical optimal actions $a^*$. 
Any bits of the strictly limited capacity $C$ spent on distinguishing points with identical fitness payoffs strictly increase the Lagrangian cost. Therefore, the joint optimization mathematically forbids veridical structural isomorphism.

## 4. Conclusion
By correctly classifying perception as Bounded Rational Decision Making, we prove that bounded capacity organisms must abandon truth to jointly optimize their sensory-motor policies for survival.

## References
1. Hoffman, D. D., Singh, M., & Prakash, C. (2015). *The interface theory of perception*. Psychonomic Bulletin & Review.
2. Ortega, P. A., & Braun, D. A. (2013). *Thermodynamics as a theory of decision-making with information-processing costs*. Proc. R. Soc. A.