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Actual Bottom Sibling Resolution

Abstract

Actual reset prefixes satisfy the complete bottom-sibling identification criterion for globally attached successful words.

Theorem 1.1 (Actual prefixes are identifiable exactly by bottom sibling coverage).

Lean statement: D5/S3/Arith/FibonacciAtomic/BottomSiblingBlockResolution.result

Proof. Machine-checked in Lean as D5/S3/Arith/FibonacciAtomic/BottomSiblingBlockResolution.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every H at least two, window limit t including zero, and reachable actual modular next row (u,v), result inhabits all seven clauses of BottomSiblingBlockCriterion.Target. Every residue has an actual source at one shared past depth, and every actual source number is positive. Equal original raw gcd futures are precisely equal source residues; replacing gcd by H/gcd preserves the full future equivalence, including failed End queries. For any finite available family of successful literal words, a finite adaptive protocol identifies the residue exactly when the globally attached centers cover all but at most one leaf of every prime-power bottom sibling block. The proof derives the actual query equation, prime-depth factorization and CRT injectivity, then uses the passive transcript bound and a fixed finite scan. Bottom coverage for all bounded successful words forces the Fibonacci cardinality bound, while the actual H=4 family with centers {0,2} shows cardinality alone is insufficient. Local prime-power centers are only projections of the same available global words.

References