Six Birds: Foundations of Emergence Calculus

This project develops 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:

1. Lens / definability structure — what distinctions are expressible at a given layer.

2. Completion (packaging) rule — modeled as an idempotent (or approximately idempotent) endomap capturing which descriptions stabilize and become objects.

3. 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.

To make these ideas operational, the project introduces an induced empirical completion operator built from a Markov kernel, a lens, and a timescale. It defines a computable total-variation idempotence defect that quantifies packaging stability and supports a qualified “theory birth” criterion: refinement can help or hurt, depending on how the lens aligns with the underlying dynamics.

To prevent spurious directionality claims, the project defines arrow-of-time as a path-space KL divergence between forward and time-reversed path measures, and proves a data processing inequality showing that coarse-graining cannot create time asymmetry. It also formalizes a protocol-trap / clock-audit principle that separates genuine drive from artifacts of hidden schedules.

For anti-saturation and novelty, the project proves a finite forcing–style counting lemma: definable predicates relative to a partition-language are exponentially rare, so generic predicate extensions are overwhelmingly non-definable. This provides a clean mathematical mechanism for strict ladder climbing.

Finally, Six Birds presents six primitives (P1–P6) as a minimal vocabulary of closure-changing moves—rewrites, gating, route dependence, sectorization, packaging, and audits—and includes a meta-theorem showing these primitives arise canonically as unavoidable closure mechanics under composability, limited access, and bounded interfaces.

Expected outcomes

• A reusable emergence calculus that cleanly separates and relates three certificates—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 demonstrating key invariants and sanity checks.

• Appendix-level theorem templates that sharpen how capacity control and bounded interface complexity enforce limits on runaway refinement (e.g., “No-Zeno” style constraints) in general multiscale settings.

This record hosts the manuscript and accompanying reproducibility artifacts, and serves as a stable reference point for subsequent domain instantiations of the framework (physics, life, consciousness, and civilization).