Here’s a Zenodo-ready description you can paste directly (Zenodo supports Markdown). To Plot a Stone with Six Birds applies Six Birds Theory (SBT) to the emergence of space-like geometry from micro-dynamics—without assuming an ambient coordinate system or background “container space.” In the SBT lens, geometry is a higher-layer description that becomes available when repeated compression makes three notions coherent under refinement: Indistinguishability (packaged macro-states that function as “points”), Composition of moves (protocols), and Cost of moving (accounting). Operationally, we instantiate an SBT-native pipeline: Substrate (finite microstate space + Markov micro-dynamics)→ Lens ladder (multi-scale packaging maps + refinement relations)→ Closed macro dynamics (induced macro kernel)→ Accounting-derived cost (negative log transition likelihood)→ Emergent distance (minimal protocol cost via shortest paths) We introduce and use a suite of falsification-first coherence audits—including closure idempotence defect, prototype stability, route mismatch, inter-scale distortion (with rescaling), connectivity checks, and a curvature-like diagnostic based on loop holonomy (transport noncommutativity measured through local neighborhood alignment). Headline exhibits (controlled, reproducible) Plane-like regime: isotropic grid dynamics yields a coherent, connected induced metric across refinement. Curvature as protocol residue: under a deterministic holonomy protocol, loop residue concentrates near zero for plane-like substrates but shifts strongly upward for a sphere-like substrate, separating flat vs curved regimes under matched diagnostics. Fractal candidate: Sierpiński-like substrates support coherent closure without smoothing toward Euclidean neighborhoods (scale-stable structure rather than smooth local tangent behavior). Constraints deform geometry: directional feasibility constraints induce systematic anisotropic deformation of the emergent metric. Pythagoras as emergent accounting law: for staged isotropic diffusion, the negative-log transition cost becomes approximately quadratic and separable, collapsing a Pythagorean residual across stages; a Manhattan (L1) negative control fails in the expected way (diamond contours). What’s included This record accompanies the manuscript with reproducible artifacts (as available in the accompanying repository release), typically including: Experiment harness + configuration files Canonical “run packs” with plots and summary metrics referenced in the paper Paper-ready comparison figures and quotable tables Minimal Lean4 anchors supporting the metric construction narrative (triangle inequality via path concatenation; separation quotient for pseudometrics; Euclidean Pythagoras sanity anchor) Scope and guardrails This work provides diagnostic, auditable evidence for emergent geometry under controlled substrates and documented failure modes. It does not claim continuum-limit convergence to smooth manifolds or infer exact curvature tensors from holonomy estimates; those are framed as future upgrades.
Ioannis Tsiokos (Thu,) studied this question.