% 312_Addendum_RenormSectorForced.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2, sub-result (i), a second sector made concrete. A304 forced the
% layer-cycle sector T_cycle and reported the renormalization generator
% R = r d/dr + Delta_psi as open: the dilation part unique up to scale, the
% additive Delta_psi a free c-number. This addendum closes that freedom. The
% additive constant is fixed by self-adjointness in the geometric measure:
% the dilation generator skew-symmetric in the boundary-field radial measure
% r^{d-2} dr is r d/dr + (d-1)/2, so Delta_psi = 3/2 for d=4. The remaining
% sectors are surveyed and classified. Sub-result (i) is advanced, not
% closed; A294's re-typing of the whole P18-T2 stands. Verifier: verify_P312.py.

\documentclass[11pt]{article}
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\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 312: The Renormalization Operator is Forced\\
\large A verified result on P18-T2 sub-result (i) for a second sector,
with a survey of the rest}
\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): renormalization sector forced incl.
$\Delta_\psi$; remaining sectors surveyed) \quad|\quad
\textbf{Addresses:} P18-T2 sub-result (i); A304; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P312.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 sub-result a first-order local result; the A306--A310 chain
established the second; and A304 gave the first a verified result for one
sector, the $\mathbb{Z}_3$ layer-cycle generator $T_{\rm cycle}$. A304 also
examined the renormalization generator $\mathcal{R} = r\,\partial_r +
\Delta_\psi$ of P18 \S3.6 and reported it open: the dilation part
$r\,\partial_r$ is the unique scale-transformation generator up to scale,
but the additive $\Delta_\psi$ was flagged as a free $c$-number. This
addendum closes that freedom. The dilation generator is unique up to the
affine family $a\,(r\,\partial_r) + b\,I$, with eigenfunctions $r^p$ and
eigenvalue the scaling dimension $p$. The additive shift $b$ is not free: a
proper unitary scale generator must be skew-symmetric in the geometric
inner product. Integration by parts in the radial measure $r^k\,dr$ gives
the adjoint $D^\ast = -D - (k+1)\,I$, so the skew-symmetric shift is
$c = (k+1)/2$. For the boundary-field radial measure $r^{d-2}\,dr$
($d = 4$, weight $r^2$) this is $c = 3/2 = (d-1)/2$, which is exactly
$\Delta_\psi$. The additive constant is therefore forced by self-adjointness
in the geometric measure, and the renormalization operator is a second
forced sector of sub-result (i). The remaining sectors are surveyed and
classified. The result is an honest partial: sub-result (i) is advanced (two
sectors now forced), not closed; the multiplication and projection sectors
($V_{\rm self}$, $\rho$, $\mathcal{M}$) are forced as operators given their
formulas but the formula-uniqueness is open, and A294's re-typing of the
whole P18-T2 stands.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Forcing}

\begin{proposition}[The renormalization operator, including $\Delta_\psi$,
is forced by self-adjointness in the geometric measure]
Let $D = r\,\partial_r$ on the radial half-line, the generator of dilations
$r \mapsto e^t r$. Then:
\begin{enumerate}
\item[\textnormal{(i)}] $D$ has eigenfunctions $r^p$ with eigenvalue $p$,
the scaling dimension: $D\,r^p = p\,r^p$. Any scale-covariant first-order
operator is $a\,D + b\,I$; the operator part is unique up to the affine pair
$(a, b)$.
\item[\textnormal{(ii)}] In the radial inner product $\langle f, g\rangle =
\int_0^\infty f\,g\,r^k\,dr$, integration by parts gives
$D^\ast = -D - (k+1)\,I$. The unique additive constant making $D + c$
skew-symmetric, $\,(D+c) + (D+c)^\ast = 0$, is
\[
c = \tfrac{k+1}{2}.
\]
\item[\textnormal{(iii)}] For the boundary-field radial measure $k = d-2$
(the radial part of the cone measure over the $(d-1)$-sphere boundary;
$d = 4$ gives $k = 2$), the skew-symmetric shift is
\[
c = \frac{d-1}{2} = \frac{3}{2},
\]
which is the scaling dimension $\Delta_\psi$ of P18 \S3.6.
\end{enumerate}
Hence the renormalization operator $\mathcal{R} = r\,\partial_r + \Delta_\psi$
with $\Delta_\psi = 3/2$ is the unique generator of dilations that is a
proper unitary scale transformation on $L^2(r^{d-2}\,dr)$. The additive
constant A304 flagged as a free $c$-number is fixed; the operator of the
renormalization sector is forced.
\end{proposition}

\begin{remark}[Sector survey and what stays open]
With A304 (the layer cycle $T_{\rm cycle}$) and this addendum (the
renormalization operator $\mathcal{R}$), two of the sectors named in P18
\S3 are forced. The remaining sectors classify as follows.
\begin{itemize}
\item \emph{Bulk Dirac square $D^2_{B^4}$: forced.} By the Lichnerowicz
identity $D^2 = \nabla^\ast\nabla + s/4$, the square of the Dirac operator
is the unique second-order spin Laplacian compatible with the spin
structure on $B^4$. Argued from the geometric theorem, not machine-checked
here.
\item \emph{Boundary Laplacian $\Delta_{S^3}$ and fiber Laplacian
$\Delta_{S^1}$: forced up to scale.} The Laplace--Beltrami operator is the
unique second-order operator invariant under the full isometry group,
$\mathrm{SO}(4)$ on $S^3$ and $\mathrm{SO}(2)$ on $S^1$, up to a scalar. The
$S^1$ case is machine-checked: the periodic Laplacian has eigenfunctions
$e^{ik\varphi}$ with eigenvalue $-k^2$ and commutes with rotation, so any
invariant second-order operator on the circle is $a\,\partial_\varphi^2 +
b\,I$. The $S^3$ case is argued by the same isometry-group reasoning.
\item \emph{Self-lensing $V_{\rm self}$, density $\rho$, moment hierarchy
$\mathcal{M}$: forced as operators given their formulas; the formulas
remain open.} Each is a multiplication or projection operator, fixed once
its explicit formula is fixed ($V_{\rm self}(r)$ explicit, $\rho$ explicit,
$\mathcal{M} = \sum_n (\mu_n/\mu_0)\,P_n$ explicit). Whether each formula is
itself categorically forced is the deeper question this addendum does not
settle; for $\rho$ it is the subject of the OI-287-1 thread.
\end{itemize}
The result establishes per-category operator uniqueness for the
renormalization sector and records a reasoned classification of the rest.
Two sectors are machine-verified forced ($T_{\rm cycle}$, $\mathcal{R}$);
$D^2_{B^4}$ and $\Delta_{S^3}$ are argued forced (up to scale for the
Laplacian); $\Delta_{S^1}$ is machine-verified forced up to scale; and the
multiplication sectors are forced only conditionally on their formulas. The
boundary of the present result is explicit: it forces the renormalization
operator, including $\Delta_\psi$, given the geometric measure; the deeper
question of whether each multiplication sector's defining formula is
categorically unique remains open. Sub-result (i) is advanced, not closed.
Sub-results (ii), the A306--A310 chain, and (iii), A302's first-order local
result, are unchanged. A294's re-typing of the whole P18-T2 stands.
\end{remark}

\paragraph{Status.} P18-T2 remains open. Its first sub-result now has
verified results for two sectors: the layer-cycle operator $T_{\rm cycle}$
(A304) and the renormalization operator $\mathcal{R} = r\,\partial_r +
\Delta_\psi$ with $\Delta_\psi = 3/2 = (d-1)/2$ forced by self-adjointness
in the boundary-field radial measure $r^{d-2}\,dr$. The freedom A304 flagged
in the additive $\Delta_\psi$ is closed. The remaining sectors are
classified: $D^2_{B^4}$ forced (Lichnerowicz, argued); the Laplacians forced
up to scale by their isometry groups ($S^1$ machine-checked, $S^3$ argued);
$V_{\rm self}$, $\rho$, $\mathcal{M}$ forced as operators given their
formulas, with formula-uniqueness open. Sub-results (ii) and (iii) are
unchanged. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P312.py}: recomputes from scratch and
asserts that $D = r\,\partial_r$ has eigenfunctions $r^p$ with eigenvalue
$p$ (integer and non-integer $p$), that any scale-covariant first-order
operator is $a\,D + b\,I$, that the skew-symmetric additive constant in the
measure $r^k\,dr$ is $(k+1)/2$ (so $3/2$ in the boundary-field measure $r^2$
and $2$ in the bulk measure $r^3$), that $\Delta_\psi = 3/2 = (d-1)/2$ with
$d = 4$, and (survey item) that the periodic Laplacian on $S^1$ has
eigenfunctions $e^{ik\varphi}$ with eigenvalue tending to $-k^2$ and
commutes with rotation. Eight checks, all PASS.

\end{document}
