arxiv	2602.00134

Six Birds: Foundations of Emergence Calculus

Six Birds: Foundations of Emergence Calculus

Tsiokos, Ioannis

arXiv: 2602.00134

Zenodo: https://zenodo.org/records/18365949

DOI: 10.5281/zenodo.18365949

This repository is the full research workspace for the math-only preprint. It includes the manuscript, formal scaffolding, reproducibility checks, and supporting registries/audits.

What this repository provides

A math-only framework—Six Birds Theory—for reasoning about hierarchical description, emergence, and open-ended theory growth in multiscale systems. The central construct is a theory package with three components:

• Lens / definability structure — what distinctions are expressible at a given layer
• Completion (packaging) rule — modeled as an idempotent (or approximately idempotent) endomap capturing which descriptions stabilize and become objects
• Audit / certificate — quantities that remain monotone (or provably well-behaved) under observation and coarse-graining

Within this framework, the project formalizes:

• Object formation as fixed points of completion: stable objects arise as the fixed points of the completion map
• Saturation under repeated closure: iterating a fixed completion rule does not yield indefinite novelty; it tends to stabilize and saturate
• Open-endedness via strict extension: genuine open-ended growth requires strict theory extension—changes in definability/closure—rather than repeated application of a single closure rule

The project also introduces:

• An induced empirical completion operator built from a Markov kernel, a lens, and a timescale
• A computable total-variation idempotence defect that quantifies packaging stability
• Arrow-of-time defined as path-space KL divergence with a data processing inequality proof
• A finite forcing–style counting lemma for anti-saturation and novelty
• Six primitives (P1–P6) as a minimal vocabulary of closure-changing moves

Scope and limitations

This is a math-only foundations paper. It provides:

• A reusable emergence calculus that cleanly separates stability (idempotence/fixed points), novelty (non-definability/extension), and directionality (audits monotone under lenses)—including explicit non-implications
• A lightweight reproducibility backbone: a small mechanized proof core (Lean) for closure/idempotent structures and a deterministic Python harness for sanity checks
• Appendix-level theorem templates for capacity control and bounded interface complexity

It does not provide domain instantiations—this is the foundational math layer only.

Build

Build the paper PDF (requires latexmk, pdflatex, or tectonic):

bash scripts/build_paper.sh

Build the Lean scaffold:

bash scripts/check_lean.sh

Test

Run the lightweight consistency checks:

python3 scripts/check_tex_refs.py manuscript/paper.tex
python3 scripts/check_paper_contract.py
python3 scripts/check_deps_dag.py
python3 scripts/check_kb_pointers.py

Regenerate indices

python3 scripts/extract_kb_index.py

Repository layout

• manuscript/paper.tex — main LaTeX source.
• manuscript/ — build helpers and manuscript notes.
• docs/spec/ — scope/claim inventory and spec references.
• docs/registries/ — generated registries and indices.
• docs/audits/ — audit notes and edge cases.
• formal/ — Lean4 + mathlib scaffold (minimal formal map).
• scripts/ — build/check tooling for reproducibility.
• reports/ — check logs and review reports.

Notes

• The LaTeX toolchain is optional unless you want the PDF.
• The checks above are designed to keep references, scopes, and dependency claims consistent with the manuscript.