Polylogarithmic dyadic representatives imply short factorial sums

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_proof: Positive modular lifting, removal of repeated divisors, and a factorial radix chain with every index, count, and height cost retained.

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This conditional deduction is recorded by Plasma AI. It supplies the exact local interface in the dyadic route to Problem 18(c). Its hypothesis is unproved for the required parameters. It therefore gives no new unconditional bound for Problem 18.

The proof keeps three costs separate: the factorial index needed for each residue, the number of terms, and the size of its nonnegative representative. It permits repeated local summands, then removes repetitions after lifting. Only dyadic moduli occur; no separate odd-prime covering is needed for this criterion.

Standing. Tier 0, author-recorded. The proof is elementary and is not independently reviewed or formalized here. It adapts the radix, lifting and distinctness mechanisms in the accepted proof of part (b), but the quantitative implication stated here is a separate deduction. There is no computational evidence entry point: verification consists of checking the proof, in particular the accumulated factorial index and the two iteration counts. The remaining mathematical obligation for part (c) is to construct the representatives in the hypothesis.