Finite congruence levels for sums of factorials

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Fix an integer $a\ge1$ and a prime $p$, as in Problem 404. The following reductions are author-recorded arguments of Plasma AI, without independent review. They give no pair $(a,p)$ with unbounded valuation.

Put $t_K=\min\{t:v_p(t!)\ge K\}$. If $K>v_p(a!)$, then $a<t_K\le pK$. All factorials of index at least $t_K$ vanish modulo $p^K$. Hence a sum of distinct factorials beginning with $a!$ can be divisible by $p^K$ if and only if a subset of $\{a,\ldots,t_K-1\}$ containing $a$ has factorial sum zero modulo $p^K$. This is an exhaustive finite decision. If $K\le v_p(a!)$, the singleton $a!$ already works.

Unboundedness at this fixed pair is equivalent to an infinite increasing sequence beginning with $a$ whose partial sums have valuations tending to infinity. Indeed, represent a choice by $(x_i)_{i\ge a}\in\{0,1\}^{\mathbb N}$ with $x_a=1$. The congruence at precision $K$ constrains only finitely many coordinates. Unboundedness makes each such closed set $C_K$ nonempty, and $C_{K+1}\subseteq C_K$. Compactness, or a diagonal subsequence of finite choices, supplies a choice in every $C_K$. Its support is infinite: a finite positive factorial sum cannot be divisible by every power of $p$. For each fixed $K$, once the selected indices have passed $t_K$, every subsequent partial sum is zero modulo $p^K$. Thus its valuations tend to infinity. Conversely, the finite partial sums of any such sequence give unbounded valuations at its first index.

If $p\mid a+1$, every factorial of index greater than $a$ has valuation strictly above $v_p(a!)$. Every allowed sum then has valuation exactly $v_p(a!)$, so $f(a,p)=v_p(a!)$.

For another exact example, modulo $128$ the only potentially nonzero terms of index at least $4$ are

$$4!\equiv24,\qquad5!\equiv120,\qquad 6!\equiv80,\qquad7!\equiv48.$$

Adding $24$ to the eight subset sums of $\{120,80,48\}$ gives $24,16,104,72,96,64,24,16$ modulo $128$. None is zero, whereas $4!+5!+7!=5184=2^6\cdot81$. Thus $f(4,2)=6$.

The general remaining obligation is a lifting or survival argument at every precision for one fixed pair, or a finite level with no admissible choice. Larger individual lower witnesses do not establish infinite survival. The separate reported tables and lower bounds on the problem's claim pages retain their stated pending standing; this argument does not replay them.