Does a covering system exist using only moduli of the form p-1 (p prime >= 5)? Search for a witness (Erdos #273)

A covering system is a finite set of congruences x = a_i (mod m_i), i=1,...,k, with distinct moduli m_i > 1, such that every integer satisfies at least one congruence. Erdos #273 asks: is there a covering system all of whose moduli have the form p-1 for a prime p >= 5? The admissible moduli are {p-1 : p prime, p>=5} = {4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, ...}. The parent question is not finitely decidable in full generality (moduli could be arbitrarily large), but it has two finite, computational faces: (a) WITNESS SEARCH. Search for a covering system using admissible moduli up to a bound M -- a finite constraint-satisfaction / SAT / ILP problem. Exhibiting one resolves the problem affirmatively. (b) BOUNDED NON-EXISTENCE. Prove that no covering system exists using only admissible moduli <= M, for an explicit M, closing off that region of the search.

Acceptance. FULLY RESOLVES (affirmative): an explicit covering system with every modulus of the form p-1 (p prime >= 5), given as a list of pairs (a_i, m_i), with a machine-checkable proof that it covers Z (verify it covers 0,1,...,lcm(m_i)-1). PARTIAL PROGRESS: a verified proof that no covering system exists using only admissible moduli <= M for an explicit M (with the SAT/ILP unsatisfiability certificate); OR an admissible-moduli system covering all integers outside a set of density < delta for an explicit small delta. Provide the encoding and solver artifacts so the result can be re-checked.