The Paltoo Standing-Sitting Band Framework Infinite Arithmetic Rigidity, Prime-Coordinate Spectral Traces, and Robin's Criterio

We formulate the Paltoo Standing-Sitting Band Framework (SSBF) as a precisely defined arithmeticcoordinate system on the reduced residue classes modulo 6, with a modulo-30 refinement and a disjointlogarithmic-band decomposition of the positive integers. Three bilinear multiplication surfaces exhaust thecomposite coordinates relatively prime to 6; their complement, augmented by 2 and 3, equals the full set ofprimes at every finite integer, with no horizon-dependent assumption. We prove the corresponding Eulerproduct, divisor-series, and von Mangoldt prime-power identities. We further construct an explicit self-adjointlogarithmic position operator and a commuting prime-coordinate projection whose trace, for ℜs > 1, is the primeDirichlet series. Its bandwise truncations recover the same expression through an absolutely convergent sum. InRobin coordinates, the model reproduces exactly the universal RH-equivalent divisor inequality and its extremalfactorization. Theorems on arithmetic completeness, infinite band coverage, and prime-coordinate operatortraces are unconditional. The structural mechanics presented here close the geometric representation of primeand composite distributions, establishing an invariant architectural foundation.