Fractal Non-Closure: Lean 4 Formal Core

Lean 4 formal core for the paper Fractal Non-Closure (doi:10.5281/zenodo.21272282).

This package formalizes a small grammar for level-indexed closure. A base dynamical system may remain open under its own operation while a lawful factor reading closes one level up to an invariant law state. The formal point is not that every such reading is substantively informative, but that lawfulness, closure, and retention of distinctions are separate requirements.

The core definitions are RenormSystem (a type of states with one distinguished operation), ElementClosed and OrbitClosed (finite terminal and periodic closure of an orbit), InstanceOpen (absence of both at the base level), Factor (a lawful reading into a second system, expressed by the commuting square), LawClosure (factor-level closure to an invariant, nondegenerate law state), and OpenWithLawClosure (the target shape: base-level openness with law-level closure).

The principal calibration results bracket the abstract layer from both sides: the one-point reading commutes with every dynamics and law-closes every orbit while retaining no distinction (TerminalReading.onePointCalibration); a universal closure relation makes law closure automatic (Factor.lawClosed_of_universal_closesTo); and the target shape is nonetheless satisfiable by a non-terminal reading that retains a base distinction (LatchReading.openWithLawClosure_without_terminal_reading). Together these isolate the structural claim: a commuting square proves lawfulness of a reading, not by itself the substantive adequacy of the reading for a domain.

Release v0.1.0, commit ffa9cc3. Builds with lake build --wfail on Lean v4.31.0; imports Std only, no Mathlib dependency; no sorry and no added axioms. This version formalizes the abstract scaffolding used by the paper, not Hausdorff dimension, scenery flow, or a specific measure-theoretic example; those are targets for later extensions.

Source repository: https://github.com/LarsenClose/fractal_non_closure