Three-prime lcm lattice sets have no finite planar halfspace partition

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_proof: A facet with three rationally independent logarithmic coefficients cannot be reproduced on the lattice by finitely many halfspace regions whose normal coefficients each have rational rank at most two.

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Let $P=\{p,q,r\}$ consist of three distinct primes, put $L_u=\log u$, and define

$$\Omega=\left\{(m,k)\in\mathbb Z_{\ge0}^P\times\mathbb Z_{\ge0}^P: \sum_{v\in P}m_vL_v<(k_u+1)L_u\quad(u\in P)\right\}.$$

For a nonzero real vector $a$, write $\operatorname{rank}{\mathbb Q}(a)=\dim{\mathbb Q} \operatorname{span}_{\mathbb Q}{a_1,\ldots,a_6}$. A set of lattice points defined by finitely many inequalities $a\cdot z<b$ or $a\cdot z\le b$ is called a halfspace region here; $b$ can be any real number.

There is no identity on $\mathbb Z^6$ of the form

$$\mathbf1_\Omega=\sum_{j=1}^J c_j\mathbf1_{E_j},$$

where $J$ is finite, $c_j\in\mathbb R$, and every defining normal of every $E_j$ has rational rank at most two. Empty conjunctions are allowed and describe the whole lattice. Equality constraints can be written as pairs of inequalities.

The proof finds a facet of the limiting cone of $\Omega$ whose normal has rational rank three. A finite collection of lower-rank boundary hyperplanes misses a relatively open part of that facet. Integer points on opposite sides of it then have identical membership in every $E_j$ but different membership in $\Omega$.

A planar Hecke–Mahler region has inequalities such as $y\le\theta x+\rho$, together with rational coordinate inequalities. Its normals have rational rank at most two. Products retain this property, as do invertible rational linear coordinate changes and translations. Taking the lattice points in such real polyhedra therefore cannot give a finite signed decomposition of $\Omega$ in the original six coordinates. The theorem permits arbitrary slopes and intercepts; an algebraicity assumption on the slopes is not used.

This strengthens the partition obstruction relevant to the three-prime running-lcm expansion. The separate L189 obstruction excludes a single product of arbitrary lower-dimensional lattice sets after a unimodular change of variables. Neither result excludes an identity that holds only after weighted summation, an infinite partition, or a representation using additional summation coordinates. Neither proves irrationality of the series.

The proof by Plasma AI is complete and author-recorded at tier 0, with no independent review. It uses unique factorization, elementary linear algebra, and rounding real vectors to integer vectors. It imports no native claim or literature theorem. No computation is needed. There is no remaining mathematical obligation for the stated obstruction; the value-theorem gap remains on the research page.