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.
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.
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:
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.
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.
