Hormesis as a Geometric Necessity of Bounded Adaptive Systems: Quantitative Predictions From First Principles

To determine whether hormesis can be derived from the geometry of bounded adaptive biological endpoints, and whether the derivation yields quantitative predictions independent of curve fitting.

Bounded endpoints were represented in rapidity coordinates using the arctanh linearisation of the adopted Möbius composition law. Repair activation and damage accumulation were modelled as opposing bounded rapidity increments with independently measurable thresholds. The molecular-to-functional attenuation factor was derived for linear pathways from the metabolic-control summation theorem and separated from a testable multi-target extension. Predictions were compared with published H₂O₂, CdCl₂, and heat-shock dose-response data.

The model predicts a biphasic response whenever repair activation precedes toxicity (D_a < D_t) and high-dose damage ultimately exceeds bounded repair capacity. The exact hyperbolic model is the primary model; the product form is used only as a conservative analytical approximation. Aggregate predictions - peak-amplitude range, mean amplitude, and hormetic-zone width - matched the Calabrese hormesis database of more than 10,000 responses without parameter fitting; across three mechanistically distinct agents, independently published data confirmed the structural prediction that adaptive activation precedes toxicity, and pathway-specific attenuation distinguished linear NRF2-mediated responses (130-160%) from larger multi-target heat-shock responses (200-300%). The peak-location law is presented as a falsifiable prediction.

Within the stated scope of independently composing bounded adaptive endpoints, hormesis follows from finite repair capacity plus adaptive upregulation. The framework does not claim that all biological history is axiomatic; it identifies the conditional geometry that constrains dose-response shape once a bounded adaptive endpoint is specified.