% Paper 32: Dark Energy as Geometric Residual — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: current canon; the headline Lambda_0 = 1 - pi^2/32 is current canon, consolidated in Paper 38 (Dark as Kernel).
% 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).

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\title{Dark Energy as Geometric Residual: \\[0.5em]
\large $\Lambda_0 = 1 - \pi^2/32$ from the $B^4$ Corner Structure}

\author{L\'{e}on Fernando Vlegels \\
\small Independent Researcher}
\date{2026-04}

\begin{document}
\maketitle
\tableofcontents
\newpage

\begin{abstract}
We derive the cosmological constant from the geometry of $B^4$ embedded
in its minimal bounding hypercube $[-1,1]^4$.  The fraction of the
hypercube \emph{not} covered by the unit $4$-ball is
\[
  \Lambda_0 \;=\; 1 - \frac{\pi^2}{32} \;\approx\; 0.6916,
\]
matching the Planck 2018 measurement $\Omega_\Lambda = 0.6847 \pm 0.0073$
at $0.94\sigma$ without any free parameters or fitting.  We establish three
supporting theorems: (1) the corner residual formula for $\Lambda_0$,
grounded in the Cantor-continuum argument that identifies $[-1,1]^4$ as the
pre-geometric substrate of $\mathbb{R}^4$; (2) Euler's planar-graph theorem
forces a seventh face in any cube-net representation, identified with
$\Lambda$ as a topological necessity; and (3) an octonionic structural
correspondence in which the sixteen vertices of the hypercube, identified in antipodal pairs, correspond to the
eight basis elements of $\mathbb{O}$, with the seventh imaginary unit
identified as the exterior face.  We further propose a dynamic extension
$\Lambda(k) = \Lambda_0 \cdot f(k/8)$ governing the cosmological evolution
of dark energy, and a conjecture relating the Hubble tension to two
observers reading $\Lambda_0$ at distinct Hopf latitudes $\beta_\mathrm{obs}$.
\end{abstract}

% -----------------------------------------------------------------------
\section{Introduction}
% -----------------------------------------------------------------------

The cosmological constant $\Lambda$ is the most precisely measured yet
least understood quantity in modern physics.  The Planck satellite's 2018
data release fixes the dark-energy density fraction at
$\Omega_\Lambda = 0.6847 \pm 0.0073$ in a flat $\Lambda$CDM universe
\cite{Planck2018}.  Standard quantum field theory predicts a vacuum energy
some $10^{120}$ times larger; the observed value requires fine-tuning
to one part in $10^{120}$, with no geometric explanation.

The geometric Theory of Everything (TOE) developed in this corpus takes a
different starting point.  Physical geometry is not fundamental; it is
forced by the requirement of \emph{multi-observer coherence}.  Paper 27
\cite{Paper27} establishes that two observers demanding cross-coherence
$C_B \circ P_A = I$ uniquely select $B^4$ with boundary $S^3$ as their
shared arena.  $B^4$ is carved out of a pre-geometric continuum by
imposing a Euclidean norm; the continuum itself requires no norm, no
metric, only the set-theoretic structure of $\mathbb{R}^4$.

The present paper asks: \emph{what does the pre-geometric continuum retain
that $B^4$ cannot reach?}  The answer is the corners of
$[-1,1]^4$: the regions where the norm $\|(x_1,x_2,x_3,x_4)\|_2 > 1$
yet the set-theoretic object $[-1,1]^4$ still exists.  The fraction of
this continuum lying outside $B^4$ is $\Lambda_0 = 1 - \pi^2/32$, and we
identify this residual as the cosmological constant.

The structure of the paper is as follows.  Section~\ref{sec:cantor}
establishes the pre-geometric status of $[-1,1]^4$ via the Cantor
continuum argument.  Section~\ref{sec:main} proves the main formula.
Sections~\ref{sec:euler} and \ref{sec:octonion} provide two independent
structural supports via Euler's formula and the octonionic correspondence.
Section~\ref{sec:dynamic} proposes a dynamic extension governing
cosmological evolution.  Section~\ref{sec:hubble} addresses the Hubble
tension.  Section~\ref{sec:discussion} connects the result to the corpus.

% -----------------------------------------------------------------------
\section{The Pre-Geometric Bounding Box}
\label{sec:cantor}
% -----------------------------------------------------------------------

\begin{definition}[Pre-geometric substrate]
The \emph{pre-geometric substrate} of $\mathbb{R}^n$ is the product
$[-1,1]^n$, regarded purely as a set-theoretic object: four independent
copies of the real interval $[-1,1]$, with no inner product, no metric,
and no topology beyond the product of the standard interval topology.
\end{definition}

The justification for treating $[-1,1]^4$ as the correct reference
container for $B^4$ rests on three observations.

\textbf{(i) The real line is prior.}
Dedekind's construction promotes the gaps of $\mathbb{Q}$ to objects;
$\mathbb{R}$ is $\emptyset$ made structural.  By Cantor's theorem
$|\mathbb{R}| > |\mathbb{Q}|$: the nothing between rationals outnumbers
the rationals themselves.  The interval $[-1,1]$ exists as soon as $\mathbb{R}$
does; it costs no additional axiom.

\textbf{(ii) $[-1,1]^4$ is free.}
Four independent real axes immediately yield $[-1,1]^4 = [-1,1]^{\times 4}$
as a set-theoretic product.  No metric is required: no angle, no distance,
no norm.  The hypercube is \emph{given} by set theory alone.

\textbf{(iii) $B^4$ requires a norm.}
The unit $4$-ball is
\[
  B^4 \;=\; \bigl\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4 :
    x_1^2 + x_2^2 + x_3^2 + x_4^2 \le 1\bigr\}.
\]
Defining $B^4$ inside $[-1,1]^4$ requires \emph{imposing} the Euclidean
norm, an additional physical postulate.  The ball is carved from the
cube by physics; the cube pre-exists physics.

\begin{remark}
The Cantor argument illuminates a key asymmetry: $B^4$ can tame the
\emph{interior} of $[-1,1]^4$ (every point with $\ell_2$-norm $\le 1$
is inside $B^4$), but the sixteen corners of $[-1,1]^4$ where
$\|(x_1,x_2,x_3,x_4)\|_\infty = 1$ and $\|(x_1,x_2,x_3,x_4)\|_2 > 1$
remain permanently outside.  These corners are the untamed continuum.
$\Lambda_0$ measures precisely how much of the raw continuum the norm
cannot reach.
\end{remark}

% -----------------------------------------------------------------------
\section{The Corner Residual: $\Lambda_0 = 1 - \pi^2/32$}
\label{sec:main}
% -----------------------------------------------------------------------

\begin{theorem}[Corner residual formula]\label{thm:main}
The fraction of $[-1,1]^4$ not covered by $B^4$ is
\[
  \Lambda_0 \;=\; 1 \;-\; \frac{\pi^2}{32} \;\approx\; 0.6916.
\]
\end{theorem}

\begin{proof}
The volume of the unit $d$-ball is
\[
  V_d \;=\; \frac{\pi^{d/2}}{\Gamma(d/2+1)}.
\]
For $d=4$:
\[
  V_4 \;=\; \frac{\pi^2}{\Gamma(3)} \;=\; \frac{\pi^2}{2}.
\]
The volume of $[-1,1]^4$ is $2^4 = 16$.  The fraction covered by $B^4$ is
\[
  \frac{V_4}{16} \;=\; \frac{\pi^2/2}{16} \;=\; \frac{\pi^2}{32}.
\]
The complementary fraction, the corner residual, is
\[
  \Lambda_0 \;=\; 1 - \frac{\pi^2}{32}.
\]
\end{proof}

\begin{remark}[Comparison to observation]
Numerically, $\pi^2/32 \approx 0.3084$, so $\Lambda_0 \approx 0.6916$.
The Planck 2018 dark-energy fraction is $\Omega_\Lambda = 0.6847 \pm 0.0073$
\cite{Planck2018}.  The discrepancy is
\[
  |\Lambda_0 - \Omega_\Lambda| \;=\; 0.0069,
  \qquad
  \frac{|\Lambda_0 - \Omega_\Lambda|}{\sigma} \;\approx\; 0.94\sigma.
\]
The predicted value lies within one standard deviation of the measured
value.  No parameter is adjusted; the formula is determined entirely by
the dimension $d = 4$ and the geometry of the standard ball.
\end{remark}

\begin{remark}[Dimensional cascade]
The formula $\Lambda_0^{(d)} = 1 - V_d(1)/2^d$ can be evaluated at
other dimensions.  For $d = 2$: $1 - \pi/4 \approx 0.215$.
For $d = 3$: $1 - \pi/6 \approx 0.476$.  For $d = 6$: $1 - \pi^3/384 \approx 0.919$.
The $d = 4$ case is distinguished by producing a value consistent with
the observed cosmological constant; no other integer dimension does.
This singles out $d = 4$ as the physically realised dimension without
additional argument.
\end{remark}

% -----------------------------------------------------------------------
\section{Topological Necessity: Euler Forcing of the Seventh Face}
\label{sec:euler}
% -----------------------------------------------------------------------

We provide a second, purely topological derivation of why $\Lambda$ must
exist as a seventh generator that is inaccessible from within the
physical structure.

\begin{definition}[Cube net]
A \emph{cube net} is a connected planar hexomino whose six unit squares
correspond to the six faces of a cube, arranged so that folding along
internal edges produces a closed cube.
\end{definition}

\begin{theorem}[Euler forcing]\label{thm:euler}
For any cube net regarded as a connected planar graph, Euler's formula
$V - E + F = 2$ forces exactly seven faces: six bounded faces
(the physical generators) and one unbounded exterior face, which is
topologically required and unreachable from within the folded structure.
\end{theorem}

\begin{proof}
Let $G$ be a cube net embedded in the plane as a planar graph, where
vertices are corners of unit squares, edges are their sides, and faces
include both the bounded square regions and the unbounded exterior.
$G$ is connected by definition of a net.  By Euler's formula for
connected planar graphs,
\[
  V - E + F = 2.
\]
The six unit squares contribute six bounded faces: $F_\mathrm{bounded} = 6$.
The total face count is therefore
\[
  F = F_\mathrm{bounded} + F_\mathrm{exterior} = 6 + 1 = 7,
\]
where $F_\mathrm{exterior} = 1$ is forced by Euler's formula regardless
of the specific net chosen (there are eleven distinct cube nets, all
connected planar graphs satisfying the same identity).

Once the net is folded into a cube, the six bounded faces become the
six physical faces of the cube.  The exterior face does not become a
physical face: it is the complement of the planar graph in $S^2$
(by one-point compactification of the plane), which is mapped to the
interior of the cube upon folding.  From within the cube, the exterior
face is inaccessible: it is behind every face simultaneously.
\end{proof}

\begin{corollary}
In the TOE framework, where the six faces of the cube correspond to the
six generators of physical reality (three pairs of conjugate directions
in $B^4$), the forced exterior face is identified with $\Lambda_0$.
The cosmological constant is not added to the theory; it is required
by the topology of any planar representation of the physical generators.
\end{corollary}

\begin{remark}[Octonionic support]
The same count is forced by a second argument.  The octonions
$\mathbb{O}$ have seven imaginary units, whose multiplication table is
encoded in the Fano plane.  The exceptional Jordan algebra $J_3(\mathbb{O})$
naturally encodes the three-generation fermion sector \cite{Paper06}.  The
six physical generators of $B^4$ (three pairs of conjugate directions)
embed as six of the seven imaginary units of $\mathbb{O}$; the seventh
imaginary unit is the extension $\mathbb{H} \to \mathbb{O}$ that the
dual-observer system requires for correct embedding but cannot generate
internally.  This seventh imaginary unit is $\Lambda$.  Both
arguments, planar-graph and octonionic, yield the same conclusion
independently.
\end{remark}

% -----------------------------------------------------------------------
\section{The Octonionic Correspondence: $8 = \mathbb{O}$}
\label{sec:octonion}
% -----------------------------------------------------------------------

\begin{proposition}[Octonionic identification]\label{prop:octonion}
The sixteen vertices of the hypercube $\{-1,+1\}^4 \subset [-1,1]^4$,
taken in antipodal pairs, admit a natural identification with the eight basis elements of
$\mathbb{O} = \mathbb{R} \oplus \mathrm{Im}(\mathbb{O})$, where:
\begin{itemize}[nosep]
  \item the real unit $1 \in \mathbb{R}$ corresponds to the pre-geometric
        state $\emptyset$ (the undifferentiated hypercube before any norm
        is imposed);
  \item the seven imaginary units $e_1, \ldots, e_7 \in \mathrm{Im}(\mathbb{O})$
        correspond to seven elementary fold operations, of which six
        produce physical faces and one (identified with $\Lambda_0$) is
        the exterior forced by Theorem~\ref{thm:euler}.
\end{itemize}
Under this identification, the folding process that closes $[-1,1]^4$
into $B^4$ traces a path through all eight basis elements of $\mathbb{O}$.
\end{proposition}

\begin{remark}[Time as fold-progress]
The Lorentz proper-time formula acquires a fold-theoretic interpretation.
Let $k$ denote the number of completed fold operations ($0 \le k \le 8$),
regarded as discrete stages in the closure of the hypercube.  An observer
\emph{inside} the partially-folded structure counts completed folds as
proper time $\tau$; an observer \emph{outside} counts the total number of
fold operations underway as coordinate time $t$.  The ratio
\[
  \frac{d\tau}{dt} \;=\; \sqrt{1 - v^2}
\]
(setting $c = 1$) then reinterprets $v$ as the fold-rate differential
between interior and exterior counting, a consequence of the dual-observer
geometry of Paper 28 \cite{Paper28} expressed in fold coordinates.
\end{remark}

\begin{remark}[Euler characteristic in four dimensions]
In four dimensions the analogous object is the tesseract (4-cube), whose
Euler characteristic is $\chi = 0$ (not $2$ as in two dimensions).  The
null Euler characteristic of the tesseract means the fold-closure argument
is more tightly constrained in 4D: the structure does not surplus but
closes exactly.  The transition from $\chi = 2$ (planar cube net) to
$\chi = 0$ (4D tesseract) mirrors the transition from classical topology
to the four-dimensional bulk geometry of $B^4$.
\end{remark}

% -----------------------------------------------------------------------
\section{Dynamic Dark Energy}
\label{sec:dynamic}
% -----------------------------------------------------------------------

The formula in Theorem~\ref{thm:main} gives the static, fully-closed
value $\Lambda_0$.  We now propose a dynamic extension governing
cosmological evolution.

\begin{definition}[Fold-progress parameter]
Let $k \in \{0,1,\ldots,8\}$ denote the number of completed fold
operations in the closure of the hypercube $[-1,1]^4$ into the
geometric structure supporting $B^4$.  We define the fold-progress
parameter $x = k/8 \in [0,1]$.
\end{definition}

\begin{conjecture}[Dynamic cosmological constant]\label{conj:dynamic}
The cosmological constant at fold-progress $x = k/8$ is
\[
  \Lambda(x) \;=\; \Lambda_0 \cdot f(x),
\]
where $f\colon [0,1] \to \mathbb{R}_{>0}$ satisfies $f(1) = 1$
(so that $\Lambda(1) = \Lambda_0$ in the fully-closed universe) and
$f(x) > 1$ for $x < 1$ (dark energy was larger in the past).
The function $f$ is related to the Euler characteristic of the
partially-folded cube net at fold stage $k$: as interior faces
are connected, the effective $\chi$ decreases from $2$ toward $0$,
modulating the exterior-face contribution.
\end{conjecture}

\begin{remark}
Conjecture~\ref{conj:dynamic} identifies the fold-progress parameter
as the correct independent variable for dark energy evolution.
The function $f$ is not yet derived: no closed form is available,
and the conjecture should be understood as a framework for future
investigation rather than a completed result.  A natural candidate for
$f$ may arise from the Poincar\'e series of the partially-folded
structure (tracking how many faces are closed at each stage), but this
remains speculative.  The static value $\Lambda_0 = 1 - \pi^2/32$ is
the paper's primary result; the dynamic extension is an open problem.
\end{remark}

% -----------------------------------------------------------------------
\section{The Hubble Tension as Projection Angle}
\label{sec:hubble}
% -----------------------------------------------------------------------

The ``Hubble tension'' (a $4$--$5\sigma$ discrepancy between CMB-derived
and distance-ladder-derived values of $H_0$) lacks an explanation
within standard $\Lambda$CDM.  We propose a geometric interpretation
consistent with the corner-residual picture.

\begin{conjecture}[Hubble tension as Hopf latitude]\label{conj:hubble}
Two observers measuring $\Lambda$ at distinct Hopf latitudes
$\beta_\mathrm{obs}$ (the polar angle on $S^3$ in the Hopf fibration
$S^1 \to S^3 \to S^2$) measure a locally corrected value
\[
  \Lambda_\mathrm{local}(\beta_\mathrm{obs})
    \;=\; \Lambda_0 \cdot \bigl(1 + \varepsilon \cos(2\beta_\mathrm{obs})\bigr),
\]
where
\[
  \varepsilon \;=\; \frac{\pi}{\alphinv} \;=\; \frac{\pi}{4\pi^3 + \pi^2 + \pi}
    \;\approx\; 0.0229
\]
is the edge-layer fraction of the TOE ontology \cite{Paper04}.

Under this proposal, the CMB (early-universe) measurement probes one
effective $\beta_\mathrm{obs}$, while the local distance-ladder
measurement probes another, owing to the different Hopf projections
accessible to each experimental geometry.  The Hubble tension is then
not a discrepancy but a \emph{signature}: two readings of the same
$\Lambda_0$ through two different projection angles.
\end{conjecture}

\begin{remark}
The identification $\varepsilon = \pi/\alphinv$ (the edge fraction) is
motivated by the TOE layer structure \cite{Paper04}: the edge layer at
$2.3\%$ captures singular, boundary-sensitive contributions to
observables.  A local dark-energy measurement is precisely such an edge
phenomenon: it depends on the observer's Hopf position, not only on the
bulk value $\Lambda_0$.
\end{remark}

\begin{remark}[Quantitative check]
With $\varepsilon \approx 0.023$ and $\Lambda_0 \approx 0.6916$, the
formula yields a maximum variation
$\delta\Lambda \leq 2\varepsilon\Lambda_0 \approx 0.032$ between observers
at $\beta = 0$ and $\beta = \pi/2$.  In a flat $\Lambda$CDM model with
$\Omega_m \approx 0.315$, this propagates to a fractional Hubble
constant shift
\[
  \frac{\delta H_0}{H_0}
  \;\approx\;
  \frac{\varepsilon\,\Lambda_0}{2(\Omega_m + \Omega_\Lambda)}
  \;\approx\; 0.008,
\]
or $\delta H_0 \approx 0.5$ km/s/Mpc.  The observed Hubble tension is
$\sim 5$--6 km/s/Mpc, an order of magnitude larger.  Conjecture~\ref{conj:hubble}
therefore identifies the \emph{mechanism} (Hopf-latitude-dependent projection
of $\Lambda_0$) rather than providing a quantitatively complete account at
the current value of $\varepsilon$.  A successful quantitative explanation
would require either a substantially larger edge fraction or a non-linear
coupling between Hopf latitude and the locally measured $\Lambda$.  This is
left as an open problem.
\end{remark}

% -----------------------------------------------------------------------
\section{Discussion}
\label{sec:discussion}
% -----------------------------------------------------------------------

\subsection{Relation to the TOE Corpus}

The corner residual $\Lambda_0$ connects to several earlier results in
this corpus.

\textit{Layer fractions (Paper 04).}
The three-layer decomposition assigns fractions $4\pi^3/\alphinv \approx 90.5\%$
(bulk), $\pi^2/\alphinv \approx 7.2\%$ (boundary), and $\pi/\alphinv \approx 2.3\%$
(edge) to the three geometric layers of $B^4$ \cite{Paper04}.  The
corner residual is a \emph{fourth} quantity: not a layer inside $B^4$,
but the complement of $B^4$ in $[-1,1]^4$.  It is not normalised by
$\alphinv$ but by the volume of the bounding hypercube.

\textit{The Second Observer (Paper 27).}
$B^4$ was selected by multi-observer coherence.  The bounding box
$[-1,1]^4$ is prior to that selection; it requires only $\mathbb{R}^4$.
$\Lambda_0$ is therefore prior to the multi-observer geometry:
it is the cost of imposing coherence on a pre-geometric continuum.

\textit{Cosmic Breathing (Paper 14).}
Paper 14 \cite{Paper14} identifies the breath period $\pi \cdot \alphinv \approx 432$
as the fundamental cosmological cycle.  The dynamic extension
$\Lambda(x)$ of Section~\ref{sec:dynamic} runs on the fold-progress
parameter $x \in [0,1]$; a natural time scale is the breath period,
which may provide the physical clock for the fold stages.

\textit{Shadow Universe (Paper 15).}
Paper 15 \cite{Paper15} models dark sectors as shadow projections.
The corner residual is a complementary perspective: rather than a shadow
(a projection of $B^4$ onto a lower-dimensional surface), $\Lambda_0$
is a \emph{remainder} (the portion of $[-1,1]^4$ that $B^4$ never covers).

\subsection{What this result is not}

$\Lambda_0 = 1 - \pi^2/32$ is a geometric prediction, not a derivation
of the vacuum energy from quantum field theory.  It does not resolve
the ``old'' cosmological constant problem (why the QFT vacuum energy is
not $10^{120}$ times larger).  Instead, it bypasses that problem by
identifying $\Lambda$ as a purely geometric quantity: a property of the
pre-geometric substrate, not of the quantum vacuum.  The QFT vacuum energy
and $\Lambda_0$ are separate objects; reconciling them is beyond the scope
of this paper.

% -----------------------------------------------------------------------
\section{Conclusion}
% -----------------------------------------------------------------------

We have established that the cosmological constant, interpreted as the
corner residual of the unit $4$-ball inscribed in its minimal bounding
hypercube, takes the value $\Lambda_0 = 1 - \pi^2/32 \approx 0.6916$.
This matches the Planck 2018 measurement at $0.94\sigma$ with no free
parameters.

Three independent structural supports are provided: (1)~the Cantor
continuum argument identifying $[-1,1]^4$ as the correct pre-geometric
reference; (2)~Euler's planar-graph theorem forcing a seventh,
inaccessible face in any cube-net representation of the physical
generators; and (3)~the octonionic correspondence mapping the sixteen
vertices of the hypercube, identified in antipodal pairs, onto the eight basis elements of $\mathbb{O}$.

Two conjectures extend the result: a dynamic $\Lambda(x)$ governed by
fold-progress, and a Hopf-latitude interpretation of the Hubble tension
with a testable prediction $\varepsilon = \pi/\alphinv \approx 0.023$.

The derivation requires no fine-tuning, no new degrees of freedom, and
no modification of general relativity.  Dark energy, in this picture, is
not a substance.  It is the measure of what set theory gives for free
that physics must work to constrain.

% -----------------------------------------------------------------------
\begin{thebibliography}{99}

\bibitem{Paper00}
L.~F.\ Vlegels, \textit{The Monad Rosetta Map: $\emptyset \equiv 0 \equiv 1 \equiv \infty$},
This volume (2025).

\bibitem{Paper04}
L.~F.\ Vlegels, \textit{Three-Layer Ontology of the $(B^4, S^3)$ Framework},
This volume (2025).

\bibitem{Paper06}
L.~F.\ Vlegels, \textit{The Fermion Sector from $\mathbb{Z}_3$ Symmetry},
This volume (2025).

\bibitem{Paper08}
L.~F.\ Vlegels, \textit{Time-Observation Bootstrap: the $(16,3,2)$ Triple from Primes},
This volume (2025).

\bibitem{Paper14}
L.~F.\ Vlegels, \textit{Cosmic Breathing: $\pi \cdot \alpha^{-1} \approx 432$},
This volume (2025).

\bibitem{Paper15}
L.~F.\ Vlegels, \textit{The Shadow Universe and Dark Sector Embeddings},
This volume (2025).

\bibitem{Paper18}
L.~F.\ Vlegels, \textit{The Master Operator $\hat{O}$ and Its Spectral Decomposition},
This volume (2025).

\bibitem{Paper27}
L.~F.\ Vlegels, \textit{The Second Observer: Why Geometry Exists},
This volume (2025).

\bibitem{Paper28}
L.~F.\ Vlegels, \textit{Universal Wave Geometry: Lorentz Structure from Dual-Observer Phase Dynamics on $S^3$},
This volume (2025).

\bibitem{Planck2018}
Planck Collaboration, \textit{Planck 2018 results. VI. Cosmological parameters},
Astronomy \& Astrophysics \textbf{641}, A6 (2020).

\end{thebibliography}

\end{document}
