Basepointless Quanta: A Blueprint of the Hilbert-Pólya Operator

This paper 1) contains a division-free, deterministic primality algorithm whose operational efficiency outpaces the AKS test exponentially and 2) advances a classical formal framework in which three structures are demonstrated to be geometrically isomorphic: the topological consistency conditions of 11-dimensional supergravity, the chiral mass structure of the Standard Model, and the critical-line dynamics of the Riemann zeta function, wherein the foundational substance paradox of string theory is resolved by replacing the heuristic notion of a fundamental string with a precise discrete origin: a continuous macroscopic rendering of a basepointless, free involutive $\mathbb{Z}_2$-torsor. The generative sequence is developed from the logical oscillation of Grandi’s series, whose Abel-stabilized limit furnishes the critical axis, invokes the Hilbert--Pólya operator, powers QCD, generates quantum fields, and ultimately curves spacetime. We show that Witten’s anomaly-canceling flux quantization shift is topologically identical to the Abel limit stabilizing the alternating vacuum,
\[
\left[\frac{G_4}{2\pi}\right]-\frac{\lambda}{2}\in H^4(Y^{11},\mathbb Z)
\iff
\lim_{x\to 1^-}\sum_{n=0}^{\infty}(-1)^n x^n=\frac{1}{2},
\]
and argue that if either the chirality axis of the electron or Witten’s flux condition is mapped to the critical line, then the Riemann Hypothesis becomes formally isomorphic to the Higgs mechanism, with all three serving as the logical ground-state and vacuum-stability condition of M-Theory, understood as a quantum-arithmetic computer running over a Bernoulli-completed display layer. M2- and M5-branes are reinterpreted as exact geometric shadows of the diagonal ($P_\Delta$) and mirror ($P_\nabla$) tensor projectors of the underlying arithmetic substrate, revealing that when topological variances force zeros off the critical line, $X_\zeta \neq 0$, chiral symmetry breaks and particles acquire observable mass as arithmetic work. The resulting mass shift is derived from the off-critical displacement of the zeta vacuum,
\[
\delta\mathcal K_\zeta(\omega)
=
X_\zeta\cdot\partial_\sigma \xi(\sigma+i\omega)\big|_{\sigma=\frac12}
+\mathcal O(X_\zeta^2).
\]
The Riemann Hypothesis is then naturally identified as the strict confinement and vacuum-stability condition of both number and quantum theory at once.