To Plot a Stone with Six Birds: Constructing and Auditing Emergent Geometry from Markov Dynamics

Abstract

Can points, distances and curvature be built from nothing more than a Markov chain and a decision about what to ignore? This paper studies that question for finite Markov dynamics, within the Six Birds emergence calculus. A lens groups microstates into macro states, each represented by a prototype distribution; lifting, evolving for τ steps and repackaging gives the macro dynamics. The distance between two macro states is the cost of the cheapest chain of macro transitions, each step costing the negative logarithm of its symmetrized, floored transition probability. We prove, and formalize in Lean 4, that this always yields an extended pseudometric, that each macro state's persistence defect equals its escape probability, and that the worst escape probability bounds the closure defect on every microstate. A layer is coherent when its closure, persistence, route and normalized distortion defects all vanish along a declared refinement family. Main results Grid. Block lenses on a lazy grid walk (stage τ = 5) give coherent layers whose normalized metrics, in declared distance units, converge to the L1 unit square. Fractal. Recursive cells on a Sierpiński gasket walk give coherent layers whose normalized metrics, in declared distance units, converge to a self-similar compact space of Hausdorff dimension log 3/log 2, in which arbitrarily small balls about every point resist Hilbert approximation to within one sixteenth of their radius. Curved. Reservoir chains on the sphere give coherent layers whose normalized metrics, in declared distance units, converge to the round unit sphere; a landmark transport estimator recovers its curvature from the Markov metrics. Pythagoras as accounting. For the isotropic walk on the lattice, centred negative-log probabilities converge uniformly to a quadratic form on diffusive windows, so the normalized readout √(Fn(z)/2), in diffusive units z/√n, converges to Euclidean distance and the Pythagorean identity becomes a law of accounting. A matched correlated walk keeps a quadratic cost but breaks orthogonality of the coordinate axes. Limits. Spectrally learned lenses give valid metrics but fail persistence; for the canonical grid and gasket partitions this holds for every choice of prototype. A local-MDS holonomy estimator can overestimate curvature by more than 20% in the refinement limit, even on exact spherical distances. Every result states its basis: Lean mechanization, written proof, exact computation or floating computation. The coherent lenses are supplied using the structure of each substrate; the constructions establish existence, not automatic discovery. Version 3 follows a mathematical review of the constructions, code and experiments and is a full rewrite: corrected cost rule and numerical pipeline, a Lean formalization of the metric and closure foundations, the three coherent constructions, the uniform quadratic limit theorem, the holonomy-estimator analysis, and new figures. The version history is in Appendix D of the paper. Code, Lean sources, configurations, certificates and audit records: https://github.com/ioannist/six-birds-space