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 strict Rate-Distortion Theory. Previous models failed by embedding the Data Processing Inequality over a causal collider, destroying the dependency on the true state of the world. We rectify this by defining the distortion function directly as the actual fitness penalty incurred when the true world state is $x$, but the agent acts optimally based only on its perception $y$: $D(x, y) = -F(x, \arg\max_a \mathbb{E}_{X' \mid y}[F(X', a)])$. We mathematically prove that minimizing this distortion under a strict channel capacity bound $C$ forces the optimal perceptual mapping $p(y|x)$ to completely obliterate structural isomorphism.
To prove FBT using Information Theory, the distortion metric cannot integrate out the true state of the world. It must compare the true state to the subjective optimal action.
The agent possesses a bounded channel capacity $C$ for the mapping $X \to Y$.
The perceptual distortion when true state $X=x$ is mapped to $Y=y$ is defined as the loss of actual utility when the agent takes the optimal action dictated by $y$.
Let $a^*(y) = \arg\max_a \mathbb{E}_{X' \mid y}[F(X', a)]$ be the subjectively optimal action given $y$.
The organism must find the mapping $p(y|x)$ that minimizes the expected distortion $\mathbb{E}_{x,y}[D(x,y)]$ subject to $I(X;Y) \le C$.
Because the fitness landscape $F(x, a)$ generically possesses symmetries and gradients completely orthogonal to the metric topology of $x$, the optimal reconstruction $Y$ 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 expected distortion. Therefore, the optimal perceptual channel mathematically forbids veridical structural isomorphism.
By correctly defining biological distortion as actual utility loss based on subjective optimal action, standard Rate-Distortion theory proves that bounded capacity organisms must abandon truth to optimize survival.
