Expanded running-lcm series has no lattice product decomposition

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_proof: A product decomposition would split the recession cone along rational subspaces; its irrational facets admit no such split for three primes.

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Statement

Let $P$ be a finite set of primes with $\lvert P\rvert\ge3$, and put $L_u=\log u$ for $u\in P$. Let $\Omega$ be the set of pairs $(m,k)$ of vectors $m=(m_u)_{u\in P}$ and $k=(k_u)_{u\in P}$ in $\mathbb Z_{\ge0}^P$ such that

$$\sum_{v\in P}m_vL_v<(k_u+1)L_u\qquad\text{for every }u\in P.$$

There are no integers $d_1,d_2\ge1$, nonempty sets $\Omega_1\subset\mathbb Z^{d_1}$ and $\Omega_2\subset\mathbb Z^{d_2}$, and group isomorphism $T:\mathbb Z^{d_1}\times\mathbb Z^{d_2}\to\mathbb Z^P\times\mathbb Z^P$ with $\Omega=T(\Omega_1\times\Omega_2)$.

Significance and scope

Let $S_P=\sum_m 1/D(m)$ be the series of Problem 269, summed over all positive integers $m$ whose prime factors lie in $P$, where $D(m)$ is the running lcm at the cutoff $m$. Since $D(m)=\prod_{u\in P}u^{\lfloor\log_u m\rfloor}$, and since $u^{-\lfloor x\rfloor}=(1-u^{-1})\sum_{k\ge\lfloor x\rfloor}u^{-k}$ with $k\ge\lfloor x\rfloor$ exactly when $k+1>x$, one has

$$S_P=\prod_{u\in P}\bigl(1-u^{-1}\bigr) \sum_{(m,k)\in\Omega}\ \prod_{u\in P}u^{-k_u}.$$

The weight is a character of the lattice. A decomposition $\Omega=T(\Omega_1\times\Omega_2)$ would therefore factor the series into two weighted lattice-point sums of lower dimension.

For two primes $P=\{p,q\}$ such a decomposition exists: the unimodular change of variables $k'_p=k_p-m_p$, $k'_q=k_q-m_q$ turns the conditions into $m_qL_q<(k'_p+1)L_p$ and $m_pL_p<(k'_q+1)L_q$, which involve disjoint sets of variables. These strict inequalities force $k'_p,k'_q\ge0$, so the original nonnegativity conditions introduce no further coupling. The two factors are the one-slope sums of the two-prime factorization, and their product is the factorization recorded there. This claim shows that the mechanism has no analogue for three or more primes in this representation. For three primes, after the same change of variables, each $m_v$ occurs in two of the three irrational conditions; the proof shows that no unimodular change of variables separates them.

The claim concerns this lattice-point representation and product decompositions of it. It does not exclude identities of other kinds, and it says nothing about the irrationality or transcendence of $S_P$. The three-prime research page uses it, together with [[research/smooth_lcm_three_prime/finite_separation|the known infinite-rank obstruction to finite termwise separation]], to delimit the separation route. The [[theory/irrationality/L190_three_prime_no_finite_planar_partition/_index|finite halfspace partition obstruction]] additionally excludes signed partitions into one-slope polyhedral pieces in the same six coordinates.

Current standing and verification

The complete proof, by Plasma AI, is author-recorded at tier 0. It has no identified gap and no independent review. It uses the linear independence of $\log u$ ($u\in P$) over $\mathbb Q$, which follows from unique factorization, and elementary facts about recession cones and faces of polyhedral cones. No native claim is consumed and no source theorem is imported. No computation is required; the expansion identity above is a rearrangement of nonnegative terms.