The Preservation Tradeoff: A Thermodynamic Bound in the Diminishing-Returns Regime

TL;DR

This paper reframes preservation of information against thermal fluctuations as a distinct thermodynamic problem, introducing the preservation stiffness $\mathcal{S}_\kappa$ and the Stiffness-Odds Identity $\mathcal{S}_\kappa(\kappa^*) = \frac{1-\kappa^*}{\kappa^*}$. It derives the diminishing-returns regime through two independent routes—Shannon channel capacity and Landauer dissipation—showing convergence on an exponential-saturation reliability form $R(\kappa) = 1 - e^{-g(\kappa)}$ with concave $g$, and defines the Thermodynamic Efficiency Frontier with $a = g'(0) \in [2,3]$. In this regime, the optimal maintenance fraction $\kappa^*$ typically lies in the 30–50% band, a result supported by empirical data from E. coli proofreading and TCP/IP protocol overhead, and it provides a substrate-agnostic diagnostic via stiffness to detect inefficiency or regime departures. The framework clarifies why diverse systems cluster in this band, attributes the convergence to fundamental limits on information preservation and dissipation, and offers falsifiable predictions and a practical diagnostic for assessing preservation efficiency in biological and engineered contexts.

Abstract

Thermodynamic systems that preserve information against thermal fluctuations face a tradeoff distinct from transmission (Shannon) or erasure (Landauer). We establish a feasibility condition for this preservation problem within a broad class of systems exhibiting diminishing returns in error suppression. By defining the preservation stiffness S_kappa, a response function analogous to magnetic susceptibility, we derive an optimality condition linking S_kappa to the resource odds. This identity provides a substrate-agnostic diagnostic: deviations reveal thermodynamic inefficiency or operation outside the smooth reliability regime. For systems in the diminishing-returns regime -- a class we argue is typical of physically realistic error-correction -- the optimal maintenance allocation is bounded above by 50%; for the physically significant subclass exhibiting smooth saturation, it is further constrained to a 30-50% band. We derive this regime from two independent physical principles -- Shannon's channel capacity and Landauer's erasure bound -- whose convergence on the same functional form constitutes our central theoretical contribution. We validate this framework against kinetic proofreading data in E. coli and protocol overhead in TCP/IP networks, and specify conditions under which the prediction is falsifiable.

Figures (3)

• Figure 1: Equilibrium condition for preservation in the diminishing-returns regime. The dashed black curve shows the resource odds $(1-\kappa)/\kappa$. Colored curves show the preservation stiffness $\mathcal{S}_\kappa = (e^{a\kappa}-1)/(a\kappa)$ for rate parameters $a = 2, 3, 5$ (Eq. \ref{['eq:chi_linear']}). Intersections (dots) define the optimal maintenance fraction $\kappa^*$. The gray band marks the predicted 30--50% regime. For $a = 3$ (typical of biological proofreading), the intersection occurs at $\kappa^* \approx 0.36$.
• Figure 2: Preservation phase diagram in the $(\kappa, \mathcal{S}_\kappa)$ plane. The dashed curve shows the resource-odds relation $\mathcal{S}_\kappa = (1-\kappa)/\kappa$ (Eq. \ref{['eq:identity']}), which forms the equilibrium locus where stiffness matches the leverage of maintenance resources. The region above the curve (shaded red) corresponds to a stiff mode, $\mathcal{S}_\kappa > (1-\kappa)/\kappa$, in which error suppression is saturated and additional maintenance yields small reliability gains relative to the payload capacity sacrificed. The region below (shaded blue) corresponds to a soft mode, $\mathcal{S}_\kappa < (1-\kappa)/\kappa$, where the system is under-protected and additional maintenance would improve effective throughput. A gray vertical band marks the predicted maintenance range $\kappa \in [0.30, 0.50]$ for systems in the diminishing-returns regime. Markers indicate illustrative operating points: empirical $\kappa$ estimates paired with stiffness values computed from the linear reference model (Eq. \ref{['eq:chi_linear']}). E. coli kinetic proofreading ($\kappa \approx 0.37$, $\mathcal{S}_\kappa \sim 1.7$) and TCP bulk transfer ($\kappa \approx 0.35$, $\tilde{\mathcal{S}}_\kappa \sim 1.5$; normalized stiffness, see Section \ref{['sec:validation']}) both lie near the equilibrium locus; hypothetical over-engineered and under-protected configurations are shown for comparison.
• Figure 3: The Stability Landscape ($a=3$). The penalty for deviating from the optimal maintenance band is asymmetric. To the left ($\kappa < 0.30$), systems face an "Error Cliff" where reliability collapses exponentially, leading to catastrophic failure. To the right ($\kappa > 0.50$), systems enter a "Stagnation Slope" where throughput decays only linearly due to capacity waste. This asymmetry creates strong selection pressure favoring the preservation band as a "safe harbor," trapping optimized systems in the 30--50% regime.

Theorems & Definitions (7)

• Definition 1: Diminishing-Returns Regime
• Definition 2: Smooth Saturation
• Theorem 1: Upper Bound
• proof
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:upper']}