Simultaneous prime divisibility with arbitrary main and error coefficients.
general_determinant_log_lower_cutoff_congruence multiplies the local prime-power divisors
of an integer D at the good primes up to x, together with prescribed divisors at a set
P of excluded primes, and bounds log D from below. determinant_log_lower_of_local_divisors
packages this for a local divisor oracle (for each good prime an exponent with the required
lower bound) and absorbs the square-root error above a cutoff L ≥ d ^ 4;
determinant_log_lower_of_local_divisors_degree_four is the case L = 27 d ^ 4.
Prescribed primes excluded from the global sum contribute their entire
local valuation, in addition to the global lower bound. A lower cutoff costs
log L; perturbing p by C √p costs O(C / √L).
A lower cutoff costs log(L); perturbing p by C sqrt(p) costs O(C/sqrt(L)). All selected prime-power divisors are multiplied into the integer D.
Prime aggregation from a local divisor oracle. Every prime p > L up to x outside
bad supplies an exponent e with A / mass p - E ≤ e and p ^ e ∣ D, where
mass p ≤ p + K d² √p and d ^ 4 ≤ L; every p ∈ P ⊆ bad contributes p ^ v p ∣ D.
The case L = 27 d ^ 4 of determinant_log_lower_of_local_divisors, with local mass
p + K d² √p: the cutoff costs 4 log d + log 27.