The Adèlic β-Plane: A Unified Parameter Space for Classical, Statistical, Quantum, and p-adic Computation
Ian Richard C Buckley
Abstract
The Maslov-Gibbs Einsum (MGE), $\pi_k(\beta) = e^{-\beta E_k}/\sum_j e^{-\beta E_j}$, is usually treated with $\beta \in \mathbb{R}{>0}$. We show that allowing $\beta$ to range over the adèle ring $\mathbb{A}\mathbb{Q} = \mathbb{R} \times \prod_p \mathbb{Q}_p$ produces a unified parameter space — the adèlic β-plane — in which every known computational model occupies a precise address. The positive real axis ($\beta > 0$) is statistical mechanics and the Forge ISA; the limit $\beta \to +\infty$ is the tropical $(\max,+)$ semiring and the Origami ISA; the imaginary axis ($\beta = it$) is unitary quantum evolution and the Meld ISA; the negative real axis ($\beta < 0$) is population inversion and laser physics; the complex plane ($\beta = \sigma + it$) is PT-symmetric quantum mechanics, with exceptional points as β-plane branch points; the origin ($\beta = 0$) is the Ambient smooth manifold; and the p-adic completions $\mathbb{Q}_p$ — attached at the origin but totally disconnected from $\mathbb{C}$ by Ostrowski's theorem — are the p-adic ISA. The Clifford group is identified as the quantum tropical regime (adèlically integral phases); magic states are phases non-integral at the prime $p = 2$; and Palmer's invariant set $I_U$ is the adèlically integral slice of Hilbert space. An obstruction-dodging theorem is proved: every $H^k$ obstruction is trivialised at level $k+1$ or at an alternative completion. Shor's algorithm is the canonical example — it is adèlic, exploiting both the imaginary axis (quantum Fourier transform) and the p-adic completions (period recovery by the Chinese Remainder Theorem). Ostrowski's theorem closes the map: no further completions of $\mathbb{Q}$ exist, so no further computational regimes are possible. Keywords: beta-plane, adelic, Maslov-Gibbs Einsum, p-adic, PT-symmetry, population inversion, Ambient manifold, Origami ISA, Forge ISA, Meld ISA, Clifford group, magic states, Palmer invariant set, Ostrowski theorem, obstruction theory, tropical geometry