% 314_Addendum_LaplacianDiracForced.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2, sub-result (i), two argued items made machine-checked. A312 forced
% the layer-cycle and renormalization sectors by machine and reported the
% boundary Laplacian Delta_S^3 and the bulk Dirac^2 D_B^4 as forced but
% argued, not machine-verified. This addendum converts both, where they are
% machine-checkable. The boundary Laplacian: the so(4) = su(2)+su(2) Casimir
% on the symmetric (j,j) irrep equals 4 j(j+1) = L(L+2), the Laplace-Beltrami
% spectrum, and the space of degree-2 SO(4)-invariant operators is
% 2-dimensional, spanned by the identity and the Casimir, so the Laplacian is
% unique up to an affine transformation. The Dirac square: the Lichnerowicz
% constant R/4 = 3/2 on the round S^3 is recovered from the Dirac and
% connection-Laplacian spectra. The scale of the Laplacians and the full bulk
% B^4 Dirac square remain stated boundaries. Sub-result (i) is advanced, not
% closed; A294's re-typing of the whole P18-T2 stands. Verifier: verify_P314.py.

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\newtheorem{proposition}{Proposition}[section]
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\title{Addendum 314: The Boundary Laplacian and the Dirac Square\\
\large Two argued items of P18-T2 sub-result (i) made machine-checked}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (sub-result (i): boundary Laplacian + Dirac$^2$
Lichnerowicz constant machine-verified) \quad|\quad
\textbf{Addresses:} P18-T2 sub-result (i); A312; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P314.py} --- all PASS

\begin{abstract}
P18-T2 (assembly uniqueness) requires three independent sub-results. A294
re-typed the question as pure mathematics and offered no new tools; A302 gave
the third a first-order local result; the A306--A310 chain established the
second; and A304 with A312 gave the first verified results for two sectors,
the $\mathbb{Z}_3$ layer cycle $T_{\rm cycle}$ and the renormalization
operator $\mathcal{R}$. A312 surveyed the remaining sectors and recorded the
boundary Laplacian $\Delta_{S^3}$ and the bulk Dirac square $D^2_{B^4}$ as
forced but argued, not machine-checked. This addendum converts both, where
they are machine-checkable. For the boundary Laplacian: the eigenfunctions on
$S^3$ are organized by $\mathrm{SO}(4)$ irreps, with eigenvalue $-L(L+2)$ and
multiplicity $(L+1)^2$ on the degree-$L$ block, and the quadratic Casimir of
$\mathfrak{so}(4) = \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ on the symmetric
$(j,j)$ irrep ($L = 2j$) equals $4\,j(j+1) = L(L+2)$, so the Casimir is the
Laplace--Beltrami operator up to sign and scale. The space of degree-2
$\mathrm{SO}(4)$-invariant operators is two-dimensional, spanned by the
identity and the Casimir, so the only nonconstant second-order invariant is
the Laplacian up to an affine transformation $a\,\Delta + b\,I$. For the Dirac
square: the Lichnerowicz constant $R/4 = 3/2$ on the round $S^3$ is recovered
from the Dirac spectrum $\pm(k + 3/2)$ and the connection-Laplacian spectrum
$(k+3/2)^2 - 3/2$, independent of $k$. The result is an honest partial: this
machine-verifies the boundary Laplacian uniqueness up to scale and the
Dirac-square Lichnerowicz constant; the full bulk $B^4$ Dirac square and the
scale fixing remain stated boundaries. Sub-result (i) is advanced, not closed;
A294's re-typing of the whole P18-T2 stands.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Boundary Laplacian}

\begin{proposition}[The boundary Laplacian on $S^3$ is the Casimir, and is
the unique second-order $\mathrm{SO}(4)$-invariant operator up to scale]
Let $\mathfrak{so}(4) = \mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$ act on
functions on $S^3$, and let the spherical harmonics of degree $L$ carry the
symmetric irrep $(j,j)$ with $L = 2j$. Then:
\begin{enumerate}
\item[\textnormal{(i)}] The quadratic Casimir
$C_2 = \sum_i (A_i^2 + B_i^2)$, written through the self-dual and
anti-self-dual generators $A_i = J_L^i + J_R^i$, $B_i = J_L^i - J_R^i$,
equals $2\,[\,j(j+1) + j(j+1)\,] = 4\,j(j+1)$ on the $(j,j)$ irrep. With
$L = 2j$ this is
\[
C_2 = 4\,j(j+1) = L(L+2),
\]
exactly the eigenvalue $-\Delta_{S^3}$ of the Laplace--Beltrami operator on
the degree-$L$ harmonic block, with block dimension $(2j+1)^2 = (L+1)^2$ equal
to the harmonic multiplicity. The Casimir is the Laplace--Beltrami operator up
to sign and scale.
\item[\textnormal{(ii)}] The space of $\mathrm{SO}(4)$-invariant operators
that are degree two in the generators, restricted to a reducible space
carrying two distinct $(j,j)$ blocks, is two-dimensional and spanned by the
identity $I$ and the Casimir $C_2$. Hence the only nonconstant second-order
$\mathrm{SO}(4)$-invariant operator on $S^3$ is the Laplacian up to the affine
family
\[
a\,\Delta_{S^3} + b\,I.
\]
\end{enumerate}
The boundary Laplacian is therefore the unique second-order
$\mathrm{SO}(4)$-invariant operator up to scale, identified with the
$\mathfrak{so}(4)$ Casimir.
\end{proposition}

% ----------------------------------------------------------------------
\section{The Dirac Square}

\begin{remark}[The Lichnerowicz constant and what stays open]
The bulk Dirac square is forced by the Lichnerowicz identity
$D^2 = \nabla^\ast\nabla + R/4$, where $R$ is the scalar curvature. The
constant is machine-checked on the round $S^3$, which is the checkable model.
There $R = 6$, so $R/4 = 3/2$. The Dirac operator on $S^3$ has eigenvalues
$\pm(k + 3/2)$ for $k = 0, 1, 2, \dots$, so $D^2$ has eigenvalues
$(k + 3/2)^2$, and the connection Laplacian on spinors has eigenvalues
$(k + 3/2)^2 - 3/2$. The difference is
\[
D^2 - \nabla^\ast\nabla = (k+3/2)^2 - \big[(k+3/2)^2 - \tfrac{3}{2}\big]
= \tfrac{3}{2} = \frac{R}{4},
\]
independent of $k$. The Dirac square is the unique second-order
spin-covariant operator with scalar principal symbol $|\xi|^2$ and no
first-order term, fixed by the spin connection; the Lichnerowicz form is that
operator, and the addendum records its constant as machine-verified.

The boundary of the present result is explicit. What is machine-verified is
the boundary Laplacian uniqueness up to scale, through the $\mathrm{SO}(4)$
Casimir and the two-dimensional invariant space, and the Dirac-square
Lichnerowicz constant $R/4 = 3/2$ on the round $S^3$. What remains a stated
boundary is the full bulk $B^4$ Dirac square as an operator, of which only the
round-$S^3$ spectral identity that fixes the constant is checked here, and the
scale of the Laplacians, which the isometry argument forces only up to a
scalar, with the scalar set by the metric normalization. The
multiplication-sector formula-uniqueness ($V_{\rm self}$, $\rho$,
$\mathcal{M}$) is unchanged from A312 and stays open, with $\rho$ the subject
of the OI-287-1 thread. A294's re-typing of the whole P18-T2 stands; the
problem is not closed.
\end{remark}

\paragraph{Status.} P18-T2 remains open. Its first sub-result now has machine
checks for five items: the layer-cycle operator $T_{\rm cycle}$ (A304), the
renormalization operator $\mathcal{R}$ (A312), the fiber Laplacian
$\Delta_{S^1}$ (A312), and now the boundary Laplacian $\Delta_{S^3}$ (unique
up to scale, via the $\mathrm{SO}(4)$ Casimir $= L(L+2)$ and a two-dimensional
invariant operator space) and the bulk Dirac-square Lichnerowicz constant
$R/4 = 3/2$. The full bulk $B^4$ Dirac square as an operator, and the scale of
the Laplacians, remain stated boundaries. The multiplication sectors
$V_{\rm self}$, $\rho$, $\mathcal{M}$ are forced as operators given their
formulas, with formula-uniqueness open. Sub-results (ii), the A306--A310
chain, and (iii), A302's first-order local result, are unchanged. No published
number changes.

\paragraph{Verifier.} \texttt{verify\_P314.py}: recomputes from scratch and
asserts that the $\mathfrak{su}(2)$ Casimir is $j(j+1)$ on each spin-$j$ rep;
that the $\mathfrak{so}(4)$ Casimir on the $(j,j)$ irrep is the scalar
$4\,j(j+1) = L(L+2)$ with $L = 2j$, matching the $-\Delta_{S^3}$ spectrum
($0, 3, 8, 15$) with multiplicity $(L+1)^2$ ($1, 4, 9, 16$); that the space of
degree-2 $\mathrm{SO}(4)$-invariant operators on the $j = 1/2 \oplus j = 1$
blocks is two-dimensional, spanned by $\{I, C_2\}$, so the Laplacian is unique
up to $a\,\Delta + b\,I$; and that the Lichnerowicz constant $R/4 = 3/2$ on the
round $S^3$ is recovered, $k$-independent, from the Dirac and
connection-Laplacian spectra. Eight checks, all PASS.

\end{document}
