Six Birds Verified VII: Formation and Explanation

In plain words

Building a theory means three jobs: forming its records, explaining its conclusions, and running its dynamics. Each tempts a shortcut. One is to read a small count of generators as a small count of roles. Another is to treat why as a single yes or no status. A third is to treat the running program, or its public behavior, as the object itself. This paper replaces each shortcut with exact mathematics for finite systems.

For formation, records are produced from seeds and admitted providers by finite rules. On three finite examples with 10, 16 and 121 records, the fewest providers needed to generate everything is 0, 1 and 2. Yet all six role meanings of Six Birds Theory stay separately representable. For explanation, a claim over a finite family of models is forced, impossible or contingent. A constructed normalizer returns a receipt exactly when the evidence submitted is valid, and the receipt meets the requested grade exactly when the evidence covers every requested obligation.

For running, a finite system is induced on a quotient, stored as tables and run. When every gate holds, the run agrees with the original on every command sequence and every formula. The sharpest result is a limit. Two four state systems, one that keeps a hidden bit and one that erases it, show identical public behavior forever, yet are not the same dynamics. A complete public theory does not determine what runs underneath. Most results are formalized in Lean 4, with exceptions listed.

What it shows

• A generation calculus with exact minimum provider counts of 0, 1 and 2 on three instances.
• Explanation as a typed receipt whose ingredients are defined independently.
• A stored, table driven run that matches the original when every gate holds.
• Two systems with identical public behavior at every horizon that are not conjugate.

What it does not claim

The provider counts concern generators in fixed examples. They do not answer the question of whether exactly six roles are needed, and they do not reduce the six roles. A receipt that meets its grade discharges the requested obligations; it does not prove its target true.

Cite

Tsiokos, I. (2026). Six Birds Verified VII: Formation and Explanation. Zenodo. https://doi.org/10.5281/zenodo.23097928