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To Lay a Stone with Six Birds: Finite-State Semantics for Packaging, Directionality, and Coarse-Graining
What a system must contain to carry all six birds itself, with nothing smuggled in from outside.
In plain words
Six Birds describes emergence with six moves: rewrite, constraints, autonomous protocols, staging, packaging and accounting. The theory is meant to work on any substrate. So what must a substrate itself provide for the six to happen inside it? Inside means honestly inside: no outside schedule the dynamics does not represent, no grouping declared by fiat, no audit the system's own dynamics does not produce. The candidate is the class of finite Markov chains, with fixed lenses, packaging that evolves, forgets and rebuilds, and audits that compare a path with its reverse.
The main theorem says this class is enough as a compilation target. Every finite Markov source, with any family of exact packages, compiles onto one shared machine. The compiled machine keeps the packaging, the refinements, the operation sequences and every observed history, together with its arrow of time audit. A second construction also keeps the full hidden audit, for a narrower family of packages. Both guarantees are sharp in stated senses: a single packaging stage cannot do the job, and in one specified case no faithful machine can keep both the observed paths and the full hidden audit with a positive time identity stage.
Instead of claiming the class is the weakest possible, the paper proves scoped separations. A finite clock can only produce schedules that eventually repeat. Putting the schedule's phase faithfully into the state cannot erase an arrow of time. A stationary arrow on links that run both ways needs a driven cycle, while an arrow from a nonstationary start does not. Coarse graining can hide an arrow but never create one. Route mismatch is independent of direction. The Lean library checks 841 theorems.
What it shows
- Every finite Markov source with exact packages compiles onto one shared autonomous machine.
- Observed histories and their arrow of time audits are preserved; full hidden audits for a narrower family.
- Sharp limits on what a single stage, or a faithful identity stage, can do.
- Coarse graining can hide an arrow of time but never create one.
- A practical audit checklist, backed by 841 Lean theorems.
What it does not claim
It does not claim the class is the weakest possible substrate, or that any natural system is such a machine. All results are finite; nothing is said about continuum limits, and Markov chains serve as a normal form, not as a claim about fundamental physics.
Cite
Tsiokos, I. (2026). To Lay a Stone with Six Birds: Finite-State Semantics for Packaging, Directionality, and Coarse-Graining. Zenodo. https://doi.org/10.5281/zenodo.23131875