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\title{Cost-Penalized Interface Games: Thermodynamic Limits and Replicator Dynamics in the Fitness-Beats-Truth Theorem}
\author{Mark Randall Havens}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
Hoffman's ``Fitness Beats Truth'' (FBT) theorem posits that evolutionary processes drive veridical perception to extinction. We formalize this by mapping perceptual strategies to an Information Bottleneck framework, penalizing the ``Truth'' strategy with the metabolic cost of information processing via Landauer's limit. We define the explicit evolutionary payoff integral and derive the optimal perceptual encoder as a Gibbs distribution. Through formal replicator dynamics and trajectory analysis, we prove that the population frequency of Truth asymptotically approaches zero ($\lim_{t \to \infty} x_T(t) = 0$). Furthermore, we establish the explicit Evolutionarily Stable Strategy (ESS) conditions, demonstrating that a heuristic fitness-tuned population strictly resists invasion by veridical mutants due to the thermodynamic cost of representation.
\end{abstract}

\section{The Payoff Integral and the Gibbs Encoder}
Let $\mathcal{M}$ be the continuous objective world manifold, and $\mathcal{Y}$ be a finite set of discrete perceptual states. The expected evolutionary payoff $f_i$ for a strategy $i$ is defined by taking the expectation over both the world states and perceptual mapping:
\begin{equation}
f_i = \int_{\mathcal{M}} \sum_{y \in \mathcal{Y}} W(x, a_i(y)) p_i(y|x) p(x) \, d\mu(x) - C(i)
\end{equation}
where $W(x, a)$ is the fitness utility of taking action $a$ in state $x$, $a_i(y)$ is the action policy, $p_i(y|x)$ is the perceptual encoder, and $C(i)$ is the metabolic penalty.

Following Ortega and Braun \cite{Ortega2013}, the metabolic cost of maintaining a high-fidelity homomorphic representation $T$ (Truth) is bounded by Landauer's principle: $C(T) = \beta^{-1} \int_{\mathcal{M}} D_{KL}(p_T(y|x) \parallel p_0(y)) p(x) \, d\mu(x)$, where $\beta^{-1} \propto \eta_{\text{bio}} k_B T \ln 2$ and $p_0(y)$ is the marginal prior distribution over perceptual states.  

Optimizing the free-energy functional yields the optimal perceptual encoder as a Gibbs distribution:
\begin{equation}
p^*(y|x) = \frac{p_0(y) e^{\beta W(x, a_i(y))}}{Z(x)}
\end{equation}
This establishes that the optimal evolutionary encoder is tuned strictly to the utility function $W$, not the structural homomorphism of $x$, explicitly decoupling perception from objective reality.

\section{Replicator Extinction and ESS Analysis}
Let $x_T$ and $x_F$ be the population frequencies of the Truth ($T$) and Fitness ($F$) strategies. The continuous-time replicator equation is:
\begin{equation}
\frac{dx_T}{dt} = x_T(f_T - \bar{f})
\end{equation}
where $\bar{f} = x_T f_T + x_F f_F$. Because the heuristic strategy $F$ operates with $C(F) \ll C(T)$ while achieving comparable or superior utility via the Gibbs encoder, we have $f_F > f_T$. 

To prove extinction, we analyze the population trajectory directly. Since $f_T < \bar{f}$ for all $x_T \in (0,1)$, we find $\frac{dx_T}{dt} < 0$. Therefore, the system is asymptotically stable at $x_T = 0$, proving $\lim_{t \to \infty} x_T(t) = 0$.

Furthermore, evaluating the invasion fitness, a monomorphic population of $F$ resists invasion by $T$ because the frequency-independent condition $f_F > f_T$ strictly holds. Since the metabolic tax strictly reduces the payoff of the mutant $T$ without providing a commensurable increase in $W$, the strict inequality holds. Thus, Fitness is a formal Evolutionarily Stable Strategy (ESS).

\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{Hoffman2015} D. D. Hoffman, M. Singh, C. Prakash, \textit{Psychon. Bull. Rev.} \textbf{22}, 1480 (2015).
\bibitem{Ortega2013} P. A. Ortega, D. A. Braun, \textit{Proc. R. Soc. A} \textbf{469}, 20120683 (2013).
\end{thebibliography}
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