Proof of Bala's totient-period conjectures: OEIS A258899 and A258903, COMMENTS (3 and 8 August 2025)

The conjectures in Peter Bala's COMMENTS of 3 August 2025 in OEIS A258899 and 8 August 2025 in A258903 are proved: modulo every positive integer m, each sequence is eventually periodic with least eventual period dividing Euler's totient phi(m). The explicit sufficient onset is 2*m+4. A weighted divisor-factorial tail identity also proves the constant-tail conjecture in A121860 (COMMENTS, 8 August 2025), and gives the Carmichael exponent as an eventual period for all integer weights.