A collision criterion for products of factorial divisors

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_proof: The bilinear bound for products modulo powers of two, its composition into L63's representatives, a character form of the hypothesis, and the geometric-progression example.

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This result is recorded by Plasma AI. Part (i) is a conditional criterion for Problem 18(c): its collision hypothesis is unproved, so it gives no new bound for $h(n!)$. Part (ii) is unconditional. The proof uses Cauchy–Schwarz, Parseval's identity, Fourier inversion modulo $2^k$ and the order of $3$ modulo powers of two.

Significance

The accepted construction for part (b) converts flattening into Fourier decay for products by a comparison that squares at every averaging step. After $R$ steps its decay exponent is divided by $2^{R+2}$, and the displayed witnesses therefore cannot have polynomial parameters, as L66 shows. That source already ends with the classical bilinear bound, applied to a sum-product average. Part (i) applies the same bound to the product law itself. Two independent factors whose collision probabilities are below $2^{-j/2}$ by a factor $2^{-\eta j}$ give a product with additive Fourier coefficients at most $2^{-\eta j}$ at conductor $2^j$. No power grows with the number of steps. If the hypothesis holds, a product of $2s$ odd divisors of $K!$ is one divisor of $(2sK)!$, and the proof reaches L63's local interface directly.

The cost is that the whole difficulty moves into one explicit law. The proof records an exact identity: the hypothesis asks that few Dirichlet characters modulo $2^j$ stay close to $1$ on most odd primes up to $K$, for every $j$ up to $k$, while $K$ is only $k(\log k)^{O(1)}$. Counting supports, it also forces $K\gg k\log k/(1+\log s)$, which L63 allows.

Part (ii) shows that a proof of the hypothesis needs more than the stated nonconcentration. The powers of $3$ in $X$ satisfy the nonconcentration hypothesis of the weighted mixing theorem at every level, yet their products stay on few residues. So no argument from that nonconcentration alone can flatten every product law below $2^{-J/2}$ with subexponentially many factors. The source's theorem still gives Fourier decay after a sufficiently large number of factors depending only on $g$; its proof uses additive structure. The bilinear route needs information about the factorial law beyond this nonconcentration.

Dependency roles. [[theory/number_theory/L63_polylogarithmic_dyadic_representatives/_index|L63 turns the residue representatives built in the proof of (i) into the bound for h(n!)]], with its exponents $a=2A$, $b=A$ and $d=2A+1$.

Standing and remaining obligation

Tier 0, author-recorded; no independent review or formalization here. There is no computational evidence entry point: verification consists of reading the proof, in particular the composition count in (i) and the residue count in (ii). The collision hypothesis of (i) remains open. Its [[research/erdos_18/dyadic_representative_route|bounded test on the editable route]] records what was tried, why the elementary pairing bound of L66 cannot reach it, and what would reopen the line.