Chronos Perceptual Field: Focus, Blur, Learning, and Adaptive Perception — A Formally Verified Mathematical Framework
Matthew Hall · Zenodo (CERN European Organization for Nuclear Research) · 2026
This work introduces the Chronos Perceptual Field (CPF), a mathematical framework for describing perception as an adaptive, time-dependent process involving sensory access, selective focus, blur, attention allocation, learning, expertise, and continued model revision when observed conditions change. The framework began from a simple physical analogy: perception can be treated like a photographer operating a camera through an interface. The sensory apparatus determines what information can be resolved, while the observer determines where attention is directed, what is brought into focus, what is intentionally suppressed or blurred, and how that configuration changes as knowledge develops. The resulting model separates external reality, sensed information, interpreted perception, learned internal models, and active attention control. It distinguishes physical resolution from semantic understanding and shows mathematically why greater knowledge can improve interpretation even when the underlying sensory information is unchanged. Several consequences were explored computationally. These include simultaneous resolution of structures at different spatial scales, selective focus under limited perceptual capacity, the consequences of misdirected attention, broad-to-narrow perceptual search, learning-driven improvement in focus selection, and adaptation when the structure of the observed environment changes. The formal development is divided into two Lean-certified stages. CPF1 establishes a scalar mathematical core covering dual-scale visibility, conservation of an attention budget, improvement from correctly aimed selective focus, degradation from misdirected focus, learning-error contraction, repeated learning, positive error after environmental change, adaptive error reduction, and broad-then-narrow target recovery. CPF2 extends the framework into finite-dimensional vector perception and aimed attention. It formalizes vector attention budgets, zero-sum attention redistribution, aimed perceptual overlap, self-alignment, opposed and neutral attention, vector learning, exact squared-error evolution, strict learning contraction, and adaptation after a target or environment change. Both CPF1 and CPF2 were compiled directly with Lean 4.34.0 and Mathlib v4.34.0. The source audit found no uses of sorry, admit, or sorryAx and no custom axiom or constant declarations. The audited theorems depend only on the standard Mathlib foundations propext, Classical.choice, and Quot.sound. The Lean verification establishes the mathematical consequences of the stated CPF assumptions. Empirical investigation of how closely particular biological, educational, medical, photographic, or artificial-intelligence systems instantiate these structures remains a separate experimental question. This Zenodo record includes the explanatory research paper together with the CPF1 and CPF2 Lean source and verification packages so that the mathematical results can be independently inspected, compiled, reproduced, and extended.