The Pentachoric Tower and the Riemann Hypothesis: Unconditional Results and a Conditional Reduction
Abstract
From the prime sequence, a free scale constant, and a selection predicate, we prove three things. The selected polytope is the 4-simplex Σ4Σ4. The Hessian of its squared volume at the regular point is (l6/2304)(−5I+2A)(l6/2304)(−5I+2A), with spectrum +7,−3,−9+7,−3,−9 and signature (+1,−9)(+1,−9). Every S5S5-invariant operator on its edge space has at most three distinct eigenvalues. Under an SU(2) edge postulate, the edge spectral zeta continues meromorphically to CC with simple poles at s=1,−1,−3,…s=1,−1,−3,…, and we compute every residue. None of these results carries prime data beyond the selection step, and none bears on the zeros of ζζ. We prove that RH follows from one hypothesis: a self-adjoint operator built from the tower whose point spectrum contains the nontrivial zero ordinates. The converse is trivial, so the content lies entirely in constructing that operator. RH also follows from the trace-identity form of the hypothesis, by a uniqueness lemma proved here. The most direct prime-weighted candidate is a Berry–Keating operator along the supermetric direction of a prime functional. It reduces to one-dimensional pieces, and with a constant boundary phase its point spectrum grows linearly, too sparse to contain the zeros. With a varying phase it can contain any countable set, which is exactly where the zeros would have to be inserted by hand. The uniform-gap premise of the tower limit is ill-posed beyond Layer 0, because the volume of the rectified simplex is not differentiable at the regular point. No trace whose period support is finitely generated can match the Weil explicit formula, and the commuting family that Postulate E does supply fails it by sign. Positivity of periodic weights survives any distributional limit, so the tower limit cannot repair the sign. As a control, the proved finite-field case and Ramanujan's ττ show what the tower lacks: a commuting prime-indexed family forced by the geometry, and a trace formula for the zeros proved as a theorem