Period-doubling prefix palindromic length is not 2-regular

The exact period-doubling conjecture is proved: d(n)=P(n+1)−P(n) is not 2-automatic and P is not 2-regular. The proof uses a marker C(n) recording a singleton 1 in some globally optimal factorization. A Gray-code independent-set bound gives nonnegative palindrome-edge slack. Uniform constructions and an all-path diagonal obstruction distinguish infinitely many binary prefixes for C. The sole computer-assisted lemma is an all-length local inclusion proved by complete reachable-state closure, independently implemented twice; finite value tests are not its proof. This does not provide a full closed formula for P or settle the separate Fibonacci conjecture. Classical palindrome structure, automatic-sequence closure and state distinguishability, and the source regularity/difference equivalence are credited. Worldwide priority remains unassessed.

The exact period-doubling conjecture is proved: d(n)=P(n+1)−P(n) is not 2-automatic and P is not 2-regular. The proof uses a marker C(n) recording a singleton 1 in some globally optimal factorization. A Gray-code independent-set bound gives nonnegative palindrome-edge slack. Uniform constructions and an all-path diagonal obstruction distinguish infinitely many binary prefixes for C. The sole computer-assisted lemma is an all-length local inclusion proved by complete reachable-state closure, independently implemented twice; finite value tests are not its proof. This does not provide a full closed formula for P or settle the separate Fibonacci conjecture. Classical palindrome structure, automatic-sequence closure and state distinguishability, and the source regularity/difference equivalence are credited. Worldwide priority remains unassessed.