Papers · Preprint · v2
To Plot a Stone with Six Birds: Constructing and Auditing Emergent Geometry from Markov Dynamics
Distance and curvature from a Markov chain, with no coordinates assumed.
In plain words
Points are equivalence classes under a lens. Distance is the minimal cost of composed transitions. Nothing else is assumed.
A pipeline goes from substrate to lens ladder to macro kernel to metric, with audits at each step: idempotence, prototype stability, route mismatch, distortion and connectivity.
On a grid the metric stays coherent under refinement. Loop residue is near zero on a flat grid and large on a sphere like substrate. With diffusion the cost law becomes quadratic and separable; a Manhattan control does not.
What it shows
- Emergent metric coherent across refinement.
- A holonomy diagnostic that separates flat from curved.
- A Pythagorean cost law emerging with timescale.
What it does not claim
Geometry here is a closure artifact of finite dynamics, not a claim about physical space.
Cite
Tsiokos, I. (2026). To Plot a Stone with Six Birds: Constructing and Auditing Emergent Geometry from Markov Dynamics. Zenodo. https://doi.org/10.5281/zenodo.18595948