% 328_Addendum_SubIFrontier.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (i), Heart 2 grounding probe: the three OPEN items of
% per-category operator uniqueness. Five sectors are machine-verified forced
% (A304 T_cycle, A312 renormalization incl. Delta_psi, A312 fiber Laplacian,
% A314 boundary Laplacian UP TO SCALE, A314 bulk Dirac^2 Lichnerowicz constant
% on the round S^3). This addendum re-grounds and gates the three remaining
% items, and takes the first concrete step on the crispest one (scale-fixing).
% ITEM 1 (scale/shift of the Laplacians) CLOSES: the unit ball fixes the metric
% scale a=1 (a corpus metric FACT, not a fiat), and annihilating-constants
% fixes the shift b=0, so Delta_S^3 and Delta_S^1 go from unique-up-to-scale to
% FULLY FORCED. ITEM 2 (full bulk B^4 Dirac^2) is PARTIAL: Clifford-uniqueness
% of D (hence D^2) is machine-verified, but the full B^4 boundary-value-problem
% spectrum is not reduced to a single spectral identity the way the round S^3
% was. ITEM 3 (V_self/M formula-uniqueness) is a DEFINITIONAL FLOOR: re-grounded
% against P18, V_self is the saturating exponential (E_self/m_0^2)(1-e^{-r/r_0}),
% NOT the quartic the probe brief carried (that is the OI-287-1 production op);
% the formulas are P18 posits, parallel to rho/G1 (A327). Fan-out: ONE probe
% (ITEM 2, the full B^4 Dirac^2 BVP); ITEM 1 closed here, ITEM 3 a floor.
% Verifier: verify_P328.py.

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\newtheorem{proposition}{Proposition}[section]
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\title{Addendum 328: The Sub-result (i) Frontier\\
\large Scale-fixing closes the Laplacians; the bulk Dirac$^2$ is partial; the
multiplication-sector formulas are a definitional floor}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} GROUNDED + STEP (ITEM 1 closed; ITEM 2 partial; ITEM 3
floor; fan-out: one probe) \quad|\quad
\textbf{Addresses:} P18-T2 sub-result (i); A314; A312; A304; A327 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P328.py} --- all PASS

\begin{abstract}
P18-T2 sub-result (i) asks whether each sector operator of $\hat O$ is the only
operator in its category. Five sectors are machine-verified forced: the layer
cycle $T_{\rm cycle}$ (A304), the renormalization generator
$R=r\,d/dr+\Delta_\psi$ including the additive shift (A312), the fiber Laplacian
$\Delta_{S^1}$ (A312), the boundary Laplacian $\Delta_{S^3}$ up to scale (A314),
and the bulk Dirac$^2$ Lichnerowicz constant $R/4=3/2$ on the round $S^3$
(A314). Three items remain open. This addendum, the Heart 2 grounding probe,
re-grounds each against its canonical verifier and gates it, then takes the
first concrete step on the crispest. \textbf{Item 1}, the scale-fixing of the
Laplacians, closes: the unit ball $|x|\le 1$ (P18) fixes the unit-radius round
metric, whose Laplace--Beltrami operator has the spectrum $-L(L+2)$ with
multiplier exactly one, so $a=1$ is a corpus metric fact, not a convention; and
the corpus condition that the boundary operator is the pure Laplace--Beltrami
operator of the round metric, equivalently that it annihilates constants, fixes
$b=0$. So $\Delta_{S^3}$ and, by the identical argument, $\Delta_{S^1}$ go from
unique up to $a\,\mathrm{Lap}+b\,I$ to fully forced. \textbf{Item 2}, the full
bulk $B^4$ Dirac$^2$ operator, is partial: the Dirac operator is the unique
first-order operator with principal symbol Clifford multiplication, machine-
verified through the dimension-four Clifford relation, so $D^2$ is forced once
$D$ is; but the full $B^4$ boundary-value-problem spectrum is not reduced to a
single spectral identity the way the round $S^3$ slice was. \textbf{Item 3}, the
multiplication-sector formula-uniqueness of $V_{\rm self}$ and $\mathcal{M}$, is
a definitional floor: re-grounded against P18, $V_{\rm self}$ is the saturating
exponential $(E_{\rm self}/m_0^2)(1-e^{-r/r_0})$, not the quartic the probe
brief carried (that is the OI-287-1 numerical production operator); given the
formulas both are forced as multiplication and projection operators, but the
formulas themselves are P18 posits, parallel to $\rho$ and $G1$ (A327). The
pre-registered fan-out is a single probe on Item 2; Item 1 closes here, Item 3
is a floor. Verifier \texttt{verify\_P328.py}: 14 PASS / 0 FAIL.
\end{abstract}

\section{The frontier and the discipline}

A294 re-typed P18-T2 into three sub-results: (i) each sector operator is the
only operator in its category; (ii) additive composition is the only admissible
composition; (iii) the coefficients are the only spectrum-matching assignment.
Sub-result (i) has advanced through A304, A312, and A314 to five
machine-verified forced sectors. Three items remain, and the binding solution-bar
discipline carried over from Heart 1 governs this probe: do not force closure, do
not accept-as-primitive prematurely, and do not manufacture a faithful object to
justify a fan-out. This addendum establishes the framing and gates the three
items; it is not required to close anything, and it reports each gate honestly.

The geometry is P18's unit ball: $M=B^4=\{x\in\mathbb{R}^4:|x|\le 1\}$ with
boundary $\partial M=S^3$, the metric $ds^2=dr^2+r^2\,d\Omega_3^2$ where
$d\Omega_3^2$ is the round metric on $S^3$ (P18 \S2.1). The bulk operator is
$D_{B^4}^2=-\Delta_{B^4}+R/4$ (P18 \S3.1, the Lichnerowicz form).

\section{Item 1: scale-fixing of the Laplacians (CLOSED)}

\subsection{Re-grounding}

A314 forced $\Delta_{S^3}$ only up to the affine family $a\,\mathrm{Lap}+b\,I$:
the $\mathrm{SO}(4)$-invariant second-order operator space is two-dimensional,
spanned by $\{I,C_2\}$ with $C_2$ the $\mathfrak{so}(4)$ Casimir, and A314
identified $C_2$ with $-\Delta_{S^3}$ through $4j(j+1)=L(L+2)$ on the symmetric
$(j,j)$ irreps. A312 forced $\Delta_{S^1}$ only up to scale by the same
isometry-group argument. The open question is whether $a$ and $b$ are fixed by a
corpus normalization or remain a convention the corpus sets by fiat. This is the
framing-audit question for the item: is the normalization a genuine corpus metric
fact or a convenient fiat?

\subsection{Gate: machine-derivable. The scale is a corpus metric fact}

The answer is that the normalization is a metric fact, and the item closes. The
Laplace--Beltrami operator of a fixed Riemannian metric is unique; there is no
free multiplier once the metric is fixed. The metric is fixed by the unit ball:
$|x|\le 1$ forces the boundary to be the unit-radius round $S^3$, whose
Laplace--Beltrami spectrum is exactly $-L(L+2)$, with multiplier one. We verify
(checks 1--4) that the $\mathfrak{so}(4)$ Casimir reproduces $L(L+2)$ for
$L=0,\dots,4$ as $0,3,8,15,24$ with multiplier exactly one, that the
metric-rescaling law $g\mapsto c^2 g$ sends the spectrum to $L(L+2)/c^2$ so that
the unit ball $c=1$ fixes $a=1$, that a free scale is observable (a value $a=2$
gives a spectrum $2\,L(L+2)$ distinguishable from $L(L+2)$ for every $L\ge 1$,
so the scale is not a hidden convention), and that the identical argument fixes
$a=1$ for $\Delta_{S^1}$ on the unit Hopf fiber.

The shift $b$ is fixed by the corpus condition that the boundary operator is the
pure Laplace--Beltrami operator of the round metric (P18 lines 113--114), which
is equivalent to the condition that it annihilates constants. We verify (checks
5--7) that the Laplacian annihilates the constant $L=0$ mode exactly, that an
affine $a\,\mathrm{Lap}+b\,I$ with $b\neq 0$ leaves $b$ on the constant mode and
so fails the annihilation condition, forcing $b=0$, and that $b=0$ is equivalent
to the operator having no zeroth-order piece, i.e.\ to its being the pure
Laplace--Beltrami operator.

\begin{proposition}[$\Delta_{S^3}$ and $\Delta_{S^1}$ fully forced]
On the unit ball geometry of P18, the boundary Laplacian $\Delta_{S^3}$ and the
fiber Laplacian $\Delta_{S^1}$ are fully forced: $a=1$ by the unit-radius round
metric, $b=0$ by the pure-Laplace--Beltrami / annihilates-constants condition.
The affine freedom A314 and A312 left open is removed. This is a derivation, not
a fiat: the metric is corpus-fixed, and the Laplace--Beltrami operator of a fixed
metric is unique.
\end{proposition}

This advances sub-result (i) from five sectors forced (two Laplacians only up to
scale) to seven sector facts forced, with the two Laplacian scales now closed.

\section{Item 2: full bulk $B^4$ Dirac$^2$ operator (PARTIAL)}

\subsection{Re-grounding}

A314 checked only the round-$S^3$ spectral identity, the Lichnerowicz constant
$R/4=3/2$. The full claim is that $D_{B^4}^2=-\Delta_{B^4}+R/4$ (P18 line 151,
the Weitzenb\"ock/Lichnerowicz formula) is the unique second-order operator in
the bulk Dirac category on the four-ball with its spin structure.

\subsection{Gate: partial. Clifford-uniqueness is machine-checkable; the full
BVP spectrum is not}

The Dirac operator is the unique first-order operator with principal symbol
equal to Clifford multiplication; this is what forces $D$, and $D^2$ is forced
once $D$ is. We verify (checks 8--9) the dimension-four Euclidean Clifford
relation $\{\gamma_i,\gamma_j\}=2\delta_{ij}I$ on an explicit chiral-basis spin
representation, and that each $\gamma_i$ is Hermitian and squares to the
identity, confirming $D$ is the genuine spin-Dirac operator of the metric and
spin connection rather than an ad hoc first-order operator. The Weitzenb\"ock
identity $D^2=\nabla^*\nabla+R/4$ then follows structurally.

We further verify (check 10) that the four-ball adds a genuinely new datum over
the round-$S^3$ slice A314 already did: the flat interior has $R=0$ so the
curvature term $R/4$ vanishes in the bulk, while the round $S^3$ boundary has
$R=6$ so $R/4=3/2$ there. The bulk curvature term is carried entirely by the
boundary, consistent with the radial form (P18 line 154) having no constant
interior potential beyond $R/4$.

\begin{remark}[Gate: PARTIAL]
The Clifford-uniqueness of $D$, hence of $D^2$, is machine-verified
representation theory, and the Weitzenb\"ock form is a structural identity. What
is not reduced to a single spectral identity, the way the round $S^3$ was, is the
full $B^4$ boundary-value-problem spectrum. The structural content is forced; the
full BVP spectrum is the residual (check 11).
\end{remark}

\section{Item 3: multiplication-sector formula-uniqueness (DEFINITIONAL FLOOR)}

\subsection{Re-grounding, and a correction}

A312 noted that $V_{\rm self}$, $\rho$, and $\mathcal{M}$ are forced as
multiplication or projection operators given their explicit formulas, but that
the formula-uniqueness is open. $\rho$'s formula-uniqueness is OI-287-1 (Heart
1), which A327 showed bottoms out in a flagged posit. We must re-ground
$V_{\rm self}$ and $\mathcal{M}$ against P18 directly, and here the canonical
source corrects a memo claim: P18 line 216 gives
\[
V_{\rm self}^{\rm can}(r)=\frac{E_{\rm self}}{m_0^2}\bigl(1-e^{-r/r_0}\bigr),
\qquad E_{\rm self}=13.177,
\]
a saturating exponential with $V_{\rm self}(0)=0$ (regularity at the origin, P18
line 228), \emph{not} the quartic $E_{\rm self}\,x^2(1-x)^2$ that the probe brief
carried. That quartic is the numerical production operator of the OI-287-1
thread, a different object; the two forms differ (check 12). The moment operator
is $\mathcal{M}=\sum_{n\ge 0}(\mu_n/\mu_0)P_n$ (P18 line 315), a spectral
projector sum weighted by the corpus moments.

\subsection{Gate: definitional floor}

Given the formulas, both sectors are forced as multiplication and projection
operators: $V_{\rm self}$ is diagonal in the position basis, hence commutes with
$\rho$ (check 13), and $\mathcal{M}$ is the projector sum $\sum_n w_n P_n$ in the
eigenbasis (check 14). That much A312 already had. The formulas themselves,
however, are P18 posits: $V_{\rm self}$'s constants $E_{\rm self}$, $r_0$, $m_0$
are cited values, and $\mathcal{M}$'s weights $\mu_n/\mu_0$ are corpus moments
(P02, P17). Neither is produced by a uniqueness argument within P18.

\begin{remark}[Gate: DEFINITIONAL FLOOR]
$V_{\rm self}$ and $\mathcal{M}$ bottom out in ``P18 states the formula'',
exactly as $\rho$ does (OI-287-1; A327 showed its magnitude is an irreducible
flagged posit). This is a definitional floor, not a derivation, and naming it
honestly is the result. We do not force these to look closable; they are floors,
parallel to $\rho$ and $G1$.
\end{remark}

\section{The gate ledger and the pre-registered fan-out}

\begin{center}
\begin{tabular}{lll}
\toprule
Item & Gate & First step / status \\
\midrule
1. Laplacian scale/shift & MACHINE-DERIVABLE & CLOSED here ($a=1$, $b=0$) \\
2. Full bulk $B^4$ Dirac$^2$ & PARTIAL & Clifford-uniqueness verified; BVP residual \\
3. $V_{\rm self}$ / $\mathcal{M}$ formulas & DEFINITIONAL FLOOR & P18 posits, parallel to $\rho$/$G1$ \\
\bottomrule
\end{tabular}
\end{center}

Item 1 closes in this probe: the metric fixes the scale and the
pure-Laplace--Beltrami condition fixes the shift, a derivation not a fiat. Item 3
is a definitional floor; there is no derivation to fan out to, only a posit to
name, which we have done. Item 2 carries genuine machine-derivable content beyond
what was checked: the Clifford-uniqueness of $D$ is verified here, but the full
$B^4$ boundary-value-problem spectrum, with the spin structure and the boundary
condition at $S^3$, is not reduced to a single spectral identity the way the
round $S^3$ slice was.

\begin{proposition}[Pre-registered fan-out: one probe]
The fan-out is a single probe on Item 2: construct the radial $B^4$ Dirac$^2$
boundary-value problem (P18 line 154, $-\partial_r^2-(3/r)\partial_r+
L_{S^3}^2/r^2+R/4$) with the spin structure and a self-adjoint boundary
condition at $r=1$, and test machine-side whether its spectrum is forced by the
Weitzenb\"ock decomposition together with the boundary Laplacian (now fully
forced by Item 1). If the BVP spectrum reduces to the bulk $\nabla^*\nabla$
spectrum plus the (vanishing interior) curvature term, the operator is forced
and Item 2 closes; if a boundary-condition freedom survives, that freedom is the
residual to gate. Item 1 needs no probe (closed here); Item 3 needs no probe (a
floor).
\end{proposition}

This is a deliberately small fan-out, consistent with the no-manufacture
discipline: only Item 2 has machine-derivable content distinct from what is
already established, so only Item 2 is fanned out.

\section{Conclusion}

Sub-result (i) of P18-T2 advances. Item 1, the scale-fixing of the Laplacians,
closes: the unit ball fixes the metric scale $a=1$ as a corpus fact and the
pure-Laplace--Beltrami condition fixes the shift $b=0$, so $\Delta_{S^3}$ and
$\Delta_{S^1}$ are fully forced. Item 2, the full bulk $B^4$ Dirac$^2$, is
partial: Clifford-uniqueness of $D$ (hence $D^2$) is machine-verified, the full
$B^4$ BVP spectrum is the residual, and a single probe is pre-registered. Item 3,
the multiplication-sector formulas $V_{\rm self}$ and $\mathcal{M}$, is a
definitional floor: P18 posits the formulas, parallel to $\rho$ and $G1$, and the
honest result is to name the floor. A294's re-typing of the whole P18-T2 stands;
sub-result (ii) (the A310 chain) and sub-result (iii) (A302) are unchanged. No
canon source files are edited; this is a standalone addendum. Verifier
\texttt{verify\_P328.py} reaches 14 PASS / 0 FAIL and exits zero.

\end{document}
