# Verification of the exact finite recursion

The square identity, potential bounds, two-candidate theorem and compressed
algorithm have complete mathematical proofs in the manuscript. Acceptance
rests on the originating researcher's detailed self-audit, not a separate
reviewer or proof assistant.

The standard-library checker passes 30,039 exact conditions in both normal
and optimized modes. It verifies the polynomial identity coefficientwise
over Q(sqrt(3)), compares all splits through n=3000, and reconstructs
optimal paths for inputs of up to 4097 bits. Exhaustive dynamic programming
examines 4,501,500 split scores; the sparse implementation follows a
separate control flow.

The original single-C5 density conjecture is not resolved. The finite
recursive construction, recurrence, cubic leading coefficient and
directed-star extremal interpretation retain their original attribution.
The pair-forbidden asymptotic corollary uses Bodnar-Leon-Liu-Pikhurko's
existing O(1)-error structural theorem.

The all-input proof is distinct from its finite regression tests. The
algorithm outputs a compressed block path, not all hypergraph edges.
Its arithmetic-operation bound is logarithmic; bit complexity is
polynomial in the binary input length, not unit cost for huge integers.