% Addendum to Paper 32: the Cantor Bridge — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Formalises Phase 0.1 — the Cantor Bridge argument.
% Status: current canon; Paper 32's headline Lambda_0 = 1 - pi^2/32 is current canon, consolidated in Paper 38.
% Registry: no registry items — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P032.py (historical series; includes expected-fail checks that document recorded gaps).

\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}

\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{remark}{Remark}

\title{Addendum to Paper 32:\\The Cantor Bridge: $\mathbb{R}^4$ as the Pre-Geometric Hypercube}
\author{L\'{e}on Fernando Vlegels}
\date{2026-04}

\begin{document}
\maketitle

\begin{abstract}
Paper 32 derives $\Lambda_0 = 1 - \pi^2/32$ as the corner residual of $B^4$ inscribed in its minimal bounding $4$-cube, but does not justify why the hypercube is the correct container. This addendum provides that justification via the \emph{Cantor bridge}: the hypercube $[-1,1]^4$ is not a modelling choice but the pre-geometric structure of $\mathbb{R}^4$ itself, requiring no metric or inner product beyond the product topology. The ball $B^4$ is carved from this container by imposing the Euclidean norm. $\Lambda_0$ measures the fraction of the raw continuum that the norm cannot reach.
\end{abstract}

\section{Motivation}

Paper 32 \S3 Theorem 1 establishes:
\[
\Lambda_0 = 1 - \frac{V_4(1)}{2^4} = 1 - \frac{\pi^2/2}{16} = 1 - \frac{\pi^2}{32} \approx 0.6916,
\]
where $V_4(1) = \pi^2/2$ is the volume of the unit $4$-ball and $2^4 = 16$ is the volume of $[-1,1]^4$. The theorem is geometrically correct but relies on the choice of $[-1,1]^4$ as the bounding container. Without a principled argument for this choice, $\Lambda_0$ could be dismissed as parameter-dependent. The Cantor bridge eliminates this objection.

\section{The Cantor Bridge}

\begin{proposition}[Cantor Bridge]
The unit hypercube $[-1,1]^4$ is the canonical pre-geometric container of $\mathbb{R}^4$. It requires no metric, no inner product, and no geometric structure beyond the product topology. The unit ball
\[
B^4 = \{ x \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 + x_4^2 \leq 1 \}
\]
is the image of $[-1,1]^4$ under the Euclidean norm constraint. The corner residual
\[
\Lambda_0 = 1 - \frac{\mathrm{Vol}(B^4)}{\mathrm{Vol}([-1,1]^4)}
\]
measures the fraction of the raw continuum that the norm cannot reach.
\end{proposition}

\begin{proof}
\textbf{Step 1. $\mathbb{R}^4$ is the hypercube before physics.}
$\mathbb{R}^4 = \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}$ is the Cartesian product of four copies of the real continuum. No metric or inner product is assumed at this stage, only the product topology. The unit container in this product space is $[-1,1]^4$: the Cartesian product of four copies of $[-1,1]$. This is not a choice; it is the minimal closed unit region in the product topology, forced by the independence of the four real axes.

\textbf{Step 2. The ball is carved from the cube by adding structure.}
$B^4$ does not exist in $\mathbb{R}^4$ without a norm. Imposing $\|x\|_2 \leq 1$ is an additional constraint: it introduces geometry (inner product structure) on top of the pre-geometric product space. Structurally and logically: \emph{cube first, ball second}. The ball is what physics does to $\mathbb{R}^4$; the cube is what $\mathbb{R}^4$ is before physics.

\textbf{Step 3. The corners are excluded by the norm.}
The $2^4 = 16$ corners of $[-1,1]^4$ are the points $(\pm 1, \pm 1, \pm 1, \pm 1)$. Each has $\|({\pm}1,{\pm}1,{\pm}1,{\pm}1)\|_2 = 2 > 1$, so all $16$ corners lie strictly outside $B^4$. These are precisely the points where all four real axes simultaneously attain their unit extremes; there the raw continuum structure is maximally concentrated.

\textbf{Step 4. Cantor's theorem.}
By Cantor's diagonal argument, $|\mathbb{R}| > |\mathbb{Q}|$: the gaps in the rationals (the $\emptyset$-content of the real line) outnumber the rationals themselves. The real continuum is \emph{more void than content}. In $\mathbb{R}^4$, the corners are the points of maximal void concentration: all four axes simultaneously at their extreme, where the continuum's gap-structure is unmediated by the norm. The ball excludes them; the cube preserves them. $\Lambda_0$ is the geometric measure of this preserved void.

\textbf{Step 5. The corner residual.}
$\mathrm{Vol}(B^4) = \pi^2/2$, $\mathrm{Vol}([-1,1]^4) = 16$. Therefore
\[
\Lambda_0 = 1 - \frac{\pi^2/2}{16} = 1 - \frac{\pi^2}{32}.
\]
No step required a choice of bounding geometry. The container was given by $\mathbb{R}^4$ itself. \qed
\end{proof}

\begin{corollary}
$\Lambda_0 = 1 - \pi^2/32$ is observer-independent and scale-invariant. It is not fitted to observation; it is forced by the corner geometry of $\mathbb{R}^4$ once a Euclidean norm is imposed.
\end{corollary}

\begin{proof}
Scale invariance follows because $\Lambda_0$ is a ratio of volumes. Observer-independence follows because the product topology and the Euclidean norm are defined independently of any observer position on $S^3$. The local deviation $\Lambda_\mathrm{local} = \Lambda_0(1 + \varepsilon \cos 2\beta_\mathrm{obs})$ from Paper 32 \S6 introduces observer-dependence only through the projection geometry, not through $\Lambda_0$ itself.
\end{proof}

\section{The Graph-Paper Intuition}

The Cantor bridge was first observed visually. On a Cartesian grid (itself an approximation to $\mathbb{R}^2$), inscribing a circle in a unit square makes the same structure explicit in two dimensions: the square is given (the grid exists before any drawing act), and the circle is imposed by choosing a distance function. The corner residual $1 - \pi/4 \approx 0.215$ in $d=2$ is the $2$-dimensional analogue of $\Lambda_0$.

Growth from the initial circle to a stable scale is governed by the boundary capacity (24 slots in $d=4$, corresponding to the 24 elements of $S_4$ acting on the quaternion axes) and the accumulated phase $\Theta(x) = 4\pi^3 x^4 + \pi^2 x^3 + \pi x^2$. At $x=1$, $\Theta(1) = \alpha^{-1}$. This is not a coincidence: the cube was always there, the circle was imposed, and $\alpha^{-1}$ is the total phase budget when the imposition is complete.

\section{Relationship to Euler Forcing}

Paper 32 \S4 Theorem 2 proves that the exterior face of the cube net is Euler-forced ($V - E + F = 2$ for the planar embedding of the cube net, with $F=7$ faces including the exterior). The Cantor bridge and the Euler argument are logically independent but structurally consistent:

\begin{itemize}
\item \textbf{Cantor bridge}: explains why $[-1,1]^4$ is the correct container for $B^4$ (pre-geometric necessity).
\item \textbf{Euler forcing}: explains why one face of the net is always exterior (topological necessity).
\item \textbf{Together}: the bounding hypercube is given, and within it, Euler guarantees that exactly one degree of freedom ($\Lambda$) is always withheld from the physical interior.
\end{itemize}

\section{Relation to Prior Work}

\begin{itemize}
\item Paper 00 (\S2): $\emptyset \equiv^* \infty$ at the limit. The corners are where $\emptyset$ is geometrically present inside the something, consistent with the limit equivalence.
\item Paper 08 Theorem 3.1: self-intersection counts $c_1 = 2, c_2 = 3, c_3 = 16$ from prime structure $(2^1, 3^1, 2^4)$. The factor $2^4 = 16 = \mathrm{Vol}([-1,1]^4)$ is the volume of the Cantor container; the Cantor bridge connects Paper 08's counting argument to Paper 32's geometric argument.
\item Paper 32 \S3 Theorem 1: the main result, now justified by Proposition 1 of this addendum.
\end{itemize}

\end{document}
