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Six Birds Verified V: The Cost of Access
Knowing a quantity is determined by what you see is not the same as computing, checking or sourcing it.
In plain words
Saying a quantity can be read off from what a layer observes is often taken to mean much more: that it is cheap to compute, easy to check, and that its sources are known. Each of these can fail while the others hold. This paper pulls them apart with exact statements. Part I works with finite systems seen through a public lens. Part II works with countable systems of rules that each use finitely many premises.
In Part I, a target has a public factor exactly when no two states that look the same have different targets. But a factor is not a program. AND on two bits is determined by its inputs, yet no program in the simple affine language computes it. Costs are vectors on ten axes, so correct programs can be incomparable. Checking and searching also differ: a yes answer for OR on n bits is checked with one query, while every exact deterministic decider needs all n on some input.
In Part II, the set of everything derivable can be indexed uniformly by computable means, and every witnessed derivation lifts to a receipt that records its sources and erases back to the derivation. But the sources a derivation used are not the sources its conclusion needs. The smallest sufficient source sets can be infinitely many, and no algorithm lists them exactly for every effective rule system. Lean 4 formalizations accompany both parts.
What it shows
- Being determined by observations does not give a program in a chosen language.
- Cost is a ten axis vector, and some correct programs cannot be ranked against each other.
- Checking an answer can take one query while deciding it takes n.
- The minimal sources behind a conclusion cannot be listed by any uniform algorithm.
What it does not claim
The query bounds concern deterministic query trees only and say nothing about P versus NP. An append only history preserves the origins it records; it does not authenticate them.
Cite
Tsiokos, I. (2026). Six Birds Verified V: The Cost of Access. Zenodo. https://doi.org/10.5281/zenodo.23097919