Geometry without a background space
Builds points, distances, and curvature from scratch, without assuming any background space. Audits detect when the emergent geometry is coherent and when it breaks down.
This paper treats geometry as an emergent property rather than a background container. "Points" are groups of fine-grained states that a chosen lens cannot tell apart. "Distance" is the minimum cost of transitioning from one group to another. "Curvature" is detected by moving a reference frame around a loop and checking whether it comes back unchanged. A reproducible pipeline builds a coarse geometry from a substrate and tests its coherence across different levels of detail, separating flat, curved, constrained, and fractal regimes.
Geometry is what remains when packaging, protocols, and costs stabilize together.
- Ioannis Tsiokos
Points from packaging
A "point" is a group of fine-grained states that the lens cannot distinguish. Different lenses produce different points.
Distance from accounting
Distance is the cheapest way to get from one point to another, measured in transition cost. Nearby = cheap to reach; far = expensive.
Coherence audits
Four checks guard the geometry: stability under re-packaging, consistency across scales, bounded distortion, and connectivity (no isolated islands).
Curvature as holonomy
Move a reference frame around a small loop. If it comes back rotated, there is curvature. No coordinate system is needed.
Core lens
What is a point
Points
Groups of fine-grained states that the lens labels identically become the "locations" of the emergent geometry.
What is a distance
Distance
The minimum transition cost between two points defines a metric. This cost comes from the dynamics, not from an assumed coordinate system.
What counts as geometry
Audits
Coherence is checked by looking more closely (refinement) and by transporting around loops. If the geometry is real, it survives both tests.
Highlighted results
Flat grids stay coherent
On uniform grids, the emergent metric is connected and stable across different levels of detail.
Loop residue separates flat from curved
On flat substrates, transporting around loops leaves near-zero residue. On sphere-like substrates, the residue is large.
Constraints deform geometry
Restricting which directions are traversable warps the emergent distances, producing anisotropic (direction-dependent) geometry.
Pythagoras emerges under diffusion
Under uniform spreading dynamics, the stable distance law becomes approximately Pythagorean (d^2 = x^2 + y^2). A Manhattan-distance control does not show this.
Methods and reproducibility
Sanity checks
Media-ready
The repository provides deterministic scripts that regenerate every figure and summary table from scratch.
- Loop-transport sweeps and curvature diagnostics
- Distortion and connectivity checks
- Transition cost and Pythagorean residual plots
Limitations and scope
Resources
Read the paper (DOI)
Zenodo DOI record
Code and reproducibility
Emergent geometry pipeline
Framework paper landing page
Six Birds: Foundations of Emergence Calculus
Access
Open-access preprint with reproducible geometry audits.
Citation
Ioannis Tsiokos (2026). To Plot a Stone with Six Birds: A Geometry is A Theory. Zenodo. https://doi.org/10.5281/zenodo.18494975
BibTeX
@misc{tsiokos2026plot,
title = {To Plot a Stone with Six Birds: A Geometry is A Theory},
author = {Tsiokos, Ioannis},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.18494975},
url = {https://doi.org/10.5281/zenodo.18494975}
}Press and contact
For media inquiries, figures, or walkthroughs of the artifacts, reach out directly.
Ioannis Tsiokos
ioannis@automorph.io
Corresponding author - Press contact