Symbols kept distinct

Four maps are written with the letter P: XII-A's $\mathcal{P}$ in $H_{\gamma \gamma }=-\mathcal{P}{h}_0$ XIIAhad, with ${\mathcal{P}}^2=\mathrm{id}$ stated there; XII-B's $\mathcal{P}$, defined explicitly as trace reversal $\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r$ XIIB; XII-D's $\mathfrak{P}$ in ${G^{\mathrm{R}}}_{\mathrm{gr}}=-\mathfrak{P}{E_0}^{\mathrm{R}}$ XIID; and XIV-I's constant map ${\mathsf{P}}^L{}_M$ in (eq:T) Iresponse. XII-B's trace reversal satisfies ${\mathcal{P}}^2=\mathrm{id}$ in four dimensions: ${\mathrm{tr}}_{\eta }(\mathcal{P}r)={\mathrm{tr}}_{\eta }r-2{\mathrm{tr}}_{\eta }r=-{\mathrm{tr}}_{\eta }r$, so $\mathcal{PP}r=\mathcal{P}r+\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r=r$. This is consistent with XII-A's statement and identifies nothing. Lemma lem:fiber adds that XII-D's $a_1$ equals $-\mathcal{P}$ on symmetric tensors, a relation among XII objects only.

Four symbols are written with the letter R: the kernel $R$ of (eq:R); the output map $R_1$ of (eq:Rconv); the admission coefficient ${R_1}^{(1)}$ of (eq:admission), whose relation to $R_1$ is not reproduced; and XII-B's $\mathcal{R}=\frac{1}{2}K$ XIIB. None is identified with another. Likewise XII-D's $a_1$, $X_1$, $f$, $E^{\mathrm{A}}$ and XIV-H's $a_1$, $X$, $f$, ${E^{\mathrm{A}}}_m$ are distinct symbols until a sentence identifies them.