Subexponential support of short factorial divisor blocks

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_proof: Small primes cost a sublinear number of bits, while the total bit budget permits only O(K/log K) large prime factors across all the blocks.

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This counting result is recorded by Plasma AI. Its proof counts the prime factors across all blocks together and requires no prime-counting estimate.

It identifies a limitation of the separated-digit construction in the dyadic route to Problem 18(c): shrinking its factorial index to $K(\log K)^{O(1)}$ destroys any fixed positive nonconcentration exponent. This does not exclude short sums of arbitrary factorial divisors, a different construction, or an estimate with parameters varying with $K$.

Standing. Tier 0, author-recorded; no independent review or formalization. Verification consists of checking the prime-factor count and its uniform asymptotics. No computation is used. The claim is closed as a counting statement. Combined with the necessary factor count $rg\ge1$ in L66, it also rules out a polylogarithmic value of the source's displayed block-expansion bound with scale-dependent parameters. Different laws or a different representation argument remain open.