Factorial exponents and occupied prime intervals

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Let $h(n)$ count the distinct exponents in the prime factorization of $n!$, as in Problem 912. Plasma AI's following argument gives an exact representation and a conditional asymptotic. The asserted prime-occupancy law is an unproved premise, not a theorem or a numerically fitted conclusion.

Legendre's formula gives $v_p(n!)=\sum_{j\ge1}\lfloor n/p^j\rfloor$. For $p>\sqrt n$ this equals $\lfloor n/p\rfloor$. Put $m=\lfloor\sqrt n\rfloor$. The values $\lfloor n/j\rfloor$ decrease strictly for $1\le j\le m+1$: for $j<m$, use $n\ge j(j+1)$; for $j=m$, split at $n=m(m+1)$ to obtain a drop from at least $m$ to at most $m-1$, or from at least $m+1$ to at most $m$. The higher-power summands decrease weakly with $p$, so small-prime valuations are distinct and exceed all large-prime valuations. The same separation holds for the floor quotients. Consequently

$$h(n)=\#\{\lfloor n/p\rfloor:p\le n,\ p\text{ prime}\}.$$

Equivalently, this counts small primes separately and the occupied bins $(n/(j+1),n/j]$ containing a prime above $\sqrt n$. Write $H=\sqrt{n/\log n}$. The small-prime count is $O(\sqrt n/\log n)=o(H)$. At $j=tH$ for fixed $t>0$, the bin length is asymptotic to $\log n/t^2$ and its location to $\sqrt{n\log n}/t$. A Poisson heuristic assigns occupancy $1-e^{-2/t^2}$, but that heuristic alone proves nothing about these particular intervals.

The precise sufficient premise is that, for every fixed $0<a<b<\infty$,

$$\frac1H\sum_{aH<j\le bH} \mathbf1_{\{(n/(j+1),n/j]\text{ contains a prime}\}} \longrightarrow\int_a^b(1-e^{-2/t^2})\,dt.$$

Here $n$ tends to infinity through every integer. Under this premise the remaining tails are controlled: $j\le aH$ contributes at most $aH$; occupied bins with $j>bH$ inject into primes at most $n/(bH)$, whose number is $(2/b+o(1))H$ by the prime number theorem. Letting $a\downarrow0$ and $b\to\infty$, and integrating by parts, gives

$$\frac{h(n)}H\longrightarrow \int_0^\infty(1-e^{-2/t^2})\,dt=\sqrt{2\pi}.$$

This is a complete conditional deduction at author standing. The held Erdős order bound gives $h(n)\asymp H$, not the occupancy premise. Its statement and proof sketch were inspected; no independent source-proof audit is asserted. A usable advance must control averaged occupancy in this reciprocal partition at logarithmic interval lengths, with errors summable over the bins. A usual prime-number-theorem estimate on much longer intervals does not decide their emptiness.