Odd-Multiplicity First-Appearance Divisors: prime powers, semiprimes, and density-one families

This paper establishes new infinite families in Granville's odd-multiplicity problem for Fibonacci numbers. A prime has first appearance at index n if it divides the nth Fibonacci number but divides no earlier positive-index Fibonacci number. Granville calls this a characteristic prime factor. His stricter primitive-factor convention also excludes the discriminant prime 5. All successful results in this paper satisfy the stricter convention except the first-appearance result at index 5 itself. The paper proves that every prime-power index greater than 2 has a first-appearance prime divisor occurring to odd exponent. It also completely treats twice-prime-power indices: the conclusion holds in every case except index 6. Several semiprime families are resolved in full. The result holds at indices of the form 5 times q for every prime q, at 3 times q for every prime q except q equal to 2, and at 7 times q for every prime q. The complete 3q result uses the classification of Lucas-number coordinates on two negative and positive Pell equations. The complete 7q result converts the remaining possible square quotient into an integral-point problem on an elliptic curve. The known complete integral-point set rules out that obstruction. An independent seven-prime quadratic-character sieve proves the 7q result on an explicit set of prime classes having Dirichlet density 431/432. A simpler congruence criterion covers a subfamily of density 143/144. The conclusion is also proved uniformly at indices of the form 7 times a prime power for every exponent when the prime is congruent to 3 or 5 modulo 8. The paper then treats genuine products of prime powers. For indices formed from a power of 3 and a power of another prime, the conclusion holds for explicitly described prime residue classes of Dirichlet density 3/8. For indices formed from a power of 2 and a power of an odd prime, a single set of primes of relative natural density 1 works simultaneously for every pair of positive exponents. This density-one conclusion remains valid inside every fixed reduced prime progression. An explicit finite character sieve supplies named congruence conditions covering a subfamily of Dirichlet density 255/256. The family involving powers of 2 and 3 is classified completely, with indices 6 and 12 as the only failures. The proofs combine Fibonacci cyclotomic parts, ranks of apparition, Fibonacci and Lucas square classes, Pell equations, elliptic curves, local quadratic characters, exact Lucas quotients, and an arbitrarily long ladder of character filters. Exact integer computations verify the identities, residue calculations, density counts, and representative factorizations. The all-parameter theorems do not depend on finite computational testing. Finally, the paper identifies the remaining composite-index obstruction. Granville's reduction shows that any unresolved case must place a Fibonacci cyclotomic part into one of two exceptional square classes. The paper formulates a quadratic-character separation principle sufficient to eliminate both possibilities. The first unresolved squarefree semiprime indices have the form p times q with both primes greater than 7.