Six Birds Verified XI: Self-Carried Closure

In plain words

A cell rebuilds its enzymes with enzymes. A compiler compiles its own next version. A repair crew fixes its own tools. Each keeps itself going, and each was started by something else: the first enzymes, the first binary, the first tools. This paper makes that exact for finite rule systems in which every rule names the capability that runs it, and capabilities wear out after a set number of rounds. With that one change, nothing is generated from an empty start. Rules that look self originating on paper need a first member present to actually run.

Four ideas are kept apart: a set that supports itself, a set generated from a seed, a set some run keeps present forever, and a set whose records are audited. Whatever is carried must lie in the largest self renewing part of what the seed can build. With no limit on how many rules fire at once, that is enough once the seed outlasts the building by one round, and a single run that fires every enabled rule settles the question. The smallest starting provisions are the minimal sets that meet every trap touching the target, where a trap is a set that can never be refilled once empty.

With one firing per round, keeping a fully seeded self renewing set alive is a known scheduling puzzle, pinwheel scheduling. So, using a cited theorem of Kawamura, five sixths is the best density threshold that works in every case. Completing the records of what the rules could produce is not producing it. When every auditor must rank above what it audits, the top records are never audited from within. The general question is PSPACE complete.

What it shows

• Nothing is realized from the empty seed; minimal seeds are the minimal transversals of the siphons meeting the target.
• With unbounded capacity, a seed lasting one round beyond the build time suffices, and this bound is attained.
• With capacity one, the question becomes pinwheel scheduling, and density five sixths is optimal.
• The same public output can come from a self carried system or a fed one; only the log tells them apart.
• Self carriage is decidable and PSPACE complete.

What it does not claim

It models no organism or chemistry and offers no criterion of life. The seed lifetime bound is sufficient, not necessary, and under finite capacity density alone is not a criterion. Everything else is checked in Lean 4, but the PSPACE completeness proof and the use of Kawamura's theorem are written proofs, not formalized.

Cite

Tsiokos, I. (2026). Six Birds Verified XI: Self-Carried Closure. Zenodo. https://doi.org/10.5281/zenodo.23097936