Productive Asymmetry: A Framework for Bounded Deviation and Adaptive Progress

OverviewThis preprint introduces Productive Asymmetry, a general framework showing how bounded deviation (imperfection) can accelerate adaptation across physical, biological, and artificial systems. Intuition: zero error halts discovery; excessive deviation causes collapse; between them lies a non‑zero optimum that maximizes the adaptation rate. Let δ≥0 denote measurable deviation (e.g., noise variance, mutation rate, decoding temperature). We formalize the adaptation rate asG(δ) = - κ1 · J(δ) + κ2 · ψ( φ(δ) )and prove the existence and uniqueness of an interior maximizer δ* > 0 under mild regularity and stability conditions. A curvature lemma (“qp lemma”) clarifies when the benefit channel is concave, linear, or convex, and a worked example yields a closed‑form δ* via the Lambert–W function. Figures include a unified adaptation curve, a reproducible synthetic demonstration of the inverted‑U response, and a phase diagram mapping how δ* shifts with risk curvature and benefit scale. We state explicit falsifiers and a minimal estimation path to enable independent confirmation or refutation. ContextThe framework addresses a recurring but unresolved pattern: domain‑specific peaks at non‑zero variability. Examples include stochastic resonance in physics, noise‑aided perception in neuroscience, mutation‑rate trade‑offs in evolutionary biology, and regularization/noise scheduling in machine learning. The companion prior‑art survey synthesizes these literatures and highlights the gap this note resolves: a general mathematical link from deviation to adaptation rate. Illustrative analogiesWe sketch intuitive analogies—the obedience–innovation paradox (perfect replication halts progress; bounded deviation drives discovery) and modern AI testbeds (e.g., LLM decoding temperature/top‑p)—both aligning with the framework’s predictions. Files in this record Main PDF (theorem, lemma, worked example, figures, predictions/falsifiers, estimation recipe) Figure data and parameter files for reproducibility (CSV/JSON) Companion resources Prior‑art survey (supplement): 10.5281/zenodo.17438423 (cross‑domain synthesis and references). Code & data bundle: 10.5281/zenodo.17439704 (Python scripts + CSV/JSON to regenerate figures) License & metadataCC‑BY 4.0 · Version v1.0.0 · Funding: None (self‑supported under Divitia LLC, Atlanta, GA, USA) · ORCID: 0009‑0005‑3757‑6956