On the metaphysics of $\mathbb F_1$

TL;DR

This paper explores the conceptual foundations of the field with one element ($\\mathbb{F}_1$) by examining rings of polynomials over the sphere spectrum and relating them to number systems, offering a new perspective on its metaphysics.

Contribution

It introduces a novel approach to understanding $\\mathbb{F}_1$ through rings of polynomials over the sphere spectrum and connects this to the Riemann-Roch theorem for integers.

Findings

The sphere spectrum $\\mathbb S$ is a strong candidate for the absolute base of $\\mathbb{F}_1$.

Riemann-Roch theorem for $\\mathbb Z$ is interpreted via polynomial rings over $\\mathbb S$.

New links between number systems and $\\mathbb{F}_1$ metaphysics are established.

Abstract

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of $\mathbb S[\mu_{n,+}]$-polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring $\mathbb Z$ of the integers that we obtained recently uses the understanding of $\mathbb Z$ as a ring of polynomials $\mathbb S[X]$ in one variable over the absolute base $\mathbb S$, where $1+1=X+X^2$. The absolute base $\mathbb S$ (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious $\mathbb F_1$.