# Channel Capacity and Fitness: An Information-Theoretic Proof of FBT

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

## Abstract
Donald Hoffman's "Fitness Beats Truth" (FBT) theorem proves that evolution selects for fitness payoffs rather than veridical structural homomorphisms. We formalize this theorem purely using Information Theory and Channel Capacity. By treating the perceptual process as a sequence of explicitly non-commutative information channels—the Objective Channel (World $\to$ Sensor) and the Payoff Channel (Sensor $\to$ Fitness)—we demonstrate that a veridical mapping requires maintaining strict structural isometry. Because the payoff landscape is generically orthogonal to the objective state space, any channel optimizing for the Payoff Channel must discard the isometric mapping of the Objective Channel. FBT is thus proven not merely by bounded rationality or metabolic constraints, but as a strict algebraic consequence of optimizing transmission across non-commutative channel topologies.

## 1. Introduction
Evolutionary game theory demonstrates that veridical perception goes extinct (Hoffman et al., 2015). We seek to prove this using Shannon Information Theory without relying on arbitrary metabolic constraints or "bounded rationality" satisficing.

## 2. Non-Commutative Channel Topologies
Let $X$ be the objective state space, $Y$ be the perceptual state space, and $F$ be the fitness payoff space.
Perception is the channel $P(Y|X)$. The evolutionary environment defines a fixed mapping $W(F|X)$.
An organism survives by optimizing its decision channel $D(A|Y)$ to maximize expected fitness. 
If $Y$ is a veridical representation, there must exist an isomorphism $f: X \to Y$. 

## 3. The Algebraic Proof of FBT
To optimize fitness, the system must maximize the mutual information $I(Y; F)$. 
However, the mapping $W(F|X)$ is generically a highly non-linear, many-to-one function that destroys the topological structure of $X$. 
Because $W(F|X)$ is orthogonal to the structural isometry $f$, any channel $P(Y|X)$ that attempts to maintain the isomorphism (truth) will fundamentally restrict the channel capacity available to maximize $I(Y; F)$ (fitness).
The channel $P(Y|X)$ that maximizes fitness is the one that directly mimics the topology of $W(F|X)$, abandoning the topology of $X$ entirely. 

## 4. Conclusion
Fitness beats truth because the fitness channel and the objective reality channel do not commute. An organism cannot optimize for both simultaneously. Evolution guarantees that the perceptual interface is a map of payoffs, not a map of reality.

## References
1. Hoffman, D. D., Singh, M., & Prakash, C. (2015). *The interface theory of perception*. Psychonomic Bulletin & Review.
2. Shannon, C. E. (1948). *A Mathematical Theory of Communication*. Bell System Technical Journal.