%==============================================================================
% Appendix: Third-Peak Resolution -- the integrated standing of the
%           CMB third-peak / dark-sector question in DFD v4.0.
% \input from DFD_Main.tex. Uses the master amsthm environments
% theorem/proposition/corollary/falsifier/status (DFD_Main.tex L275-294).
% revtex4-2 safe: no p{} columns, no \chapter, no external preamble.
% Cross-refs (must resolve): app:thirdpeak-wall (appendix_J), thm:chi-thirdpeak
% and thm:chi-omega (appendix_chi_matter), app:confrontation-matrix
% (appendix_AG), app:sigma8-optical (appendix_AE).
%==============================================================================
\section{The CMB Third Peak: a No-Go for Zero-Dark-Matter, a No-Free-Parameter Dark Sector}
\label{app:third-peak-resolution}

\subsection{What this appendix settles, and what it does not}
\label{sec:tp-thesis}

This appendix states the integrated resolution of the question that drove
dozens of internal no-CDM waves: \emph{can DFD reproduce the height of the CMB
third acoustic peak without any cold-clustering matter at all?} The answer, proved
at theorem grade and reconfirmed here by independent CAMB reruns, is \textbf{no}.
Within DFD's own printed 4D spectrum there is no field --- and no optical,
screening, sound-speed, echo, or gravity-enhancement device --- that lifts the
third-peak height to the observed value without a pressureless component clustering
at recombination with $\Omega h^2\approx0.10$--$0.12$. We do not soften this. It is
the same wall onto which every clever DFD evasion converged.

The genuine, defensible DFD result is the \emph{complement} of that no-go, and it
is stronger than the thing the no-go forbids:

\begin{quote}
\emph{DFD does not avoid the cold-clustering charge --- it \textbf{derives the field
that carries it}. The $\chi$-matter field (App.~\ref{app:chi-matter}) is forced by
the fixed internal topology; its decay constant and mass come from $\alpha$-powers
with zero fit, and its relic abundance is now derived outright,
$\Omega_\chi h^2=0.118$ ($-1.5\sigma$; Thm~\ref{thm:chi-postinflation}) --- the one
number $\Lambda$CDM must read off the data, DFD predicts.}
\end{quote}

\noindent The correct one-line framing is therefore \textbf{not} ``a no-CDM third
peak'' --- it is ``\textbf{a no-free-parameter dark sector}.'' $\Lambda$CDM fits
$\omega_c=0.12$ as one free density with no theory of its value; DFD predicts the
dark-matter particle's mass ($5.1$~eV) and decay constant ($9.5\times10^{11}$~GeV)
from fixed integers and $\alpha$; the \emph{amount} is now likewise derived
($\langle\theta^2\rangle=0.073\Rightarrow\Omega_\chi h^2=0.118$,
Thm~\ref{thm:chi-postinflation}), the misalignment angle having been retired as an
input. This appendix collects (i) the height no-go, (ii) the derived-abundance
theorem with full provenance, (iii) the falsifiable $\chi$-versus-CDM
distinguishing predictions, and (iv) a status box naming the one open
tension the hunt surfaced.

%------------------------------------------------------------------------------
\subsection{Part I --- the height no-go (theorem-grade)}
\label{sec:tp-nogo}

The rigorous statement and four-step proof are given as the \emph{Cold-Clustering
Wall} theorem in App.~\ref{app:thirdpeak-wall}; we restate it here in summary form
and record the independent numerical reconfirmation, because it is the load-bearing
negative result of the whole dark-sector program.

\begin{proposition}[Height no-go, summary of App.~\ref{app:thirdpeak-wall}]
\label{prop:tp-height-nogo}
Lifting the third-peak height ratio from the baryon-only value
$(H_3/H_1)_b\approx0.20$ to the observed $(H_3/H_1)_{\rm obs}\approx0.44$, holding
the peak \emph{locations} $k_n r_s=n\pi$ and the baryon-loading ratio structure
fixed, requires a driving potential $\Phi(k,\eta)$ with a non-decaying (standing,
$\Phi\to$const) sub-horizon contribution through last scattering, sourced by a
species that is cold-clustering ($w\simeq0$, $c_s^2\simeq0$) at recombination with
matter--radiation equality preceding recombination, $z_{\rm eq}\gtrsim z_{\rm rec}$.
The minimal such abundance is $\Omega_c h^2\gtrsim0.023$ (standing-well floor), and
matching the observed height sharpens it to $\Omega_c h^2\approx0.10$--$0.12$. No
radiation-pressure-supported component ($c_s^2>0$) and no time-only modification of
the homogeneous background can supply the lift.
\end{proposition}

\paragraph{Why the two evasion classes are closed (the rigorous core).}
The proof in App.~\ref{app:thirdpeak-wall} closes exactly the two ways a no-CDM
device could try to win:
\begin{enumerate}
\item \textbf{Pressure-supported devices self-destruct their own well.} Any
component with $c_s^2>0$ in radiation/pressure domination sources a potential that
obeys $\Phi(x)=3\Phi_0(\sin x-x\cos x)/x^3$, whose sub-horizon envelope
$\sim3\Phi_0\cos x/x^2\to0$. We independently reverify the decay numerically:
$|\Phi/\Phi_0|=0.024,\,0.0027,\,0.0013$ at $x=10,20,40$. No standing well $\Rightarrow$
no height lift. This kills deep-MOND gravity enhancement, the $c_s(\psi)$ sound-speed
envelope, the line-of-sight $\psi$/Jacobian screen, and the forced sound-horizon echo
in one stroke.
\item \textbf{Time-only / background devices cancel in ratios.} An achromatic boost
$\Phi\to g\,\Phi$ multiplies \emph{every} peak by $g^2$, leaving all ratios
$H_n/H_1$ exactly invariant (verified to machine precision). Background-only knobs
therefore cannot move $H_3/H_1$; the only quantity that moves it is $z_{\rm eq}$,
which \emph{is} the cold abundance, not a free dial.
\end{enumerate}

\begin{proposition}[The decisive discriminator: clustering, not background-$w$]
\label{prop:tp-clustering-discriminator}
A component with the correct \emph{background} equation of state $w\to0$ but a
finite sound speed (an axion-\emph{fluid} with $c_s^2\ne0$, \emph{non}-clustering)
at $\Omega h^2=0.12$ produces \emph{zero} third-peak lift: an independent CAMB run
gives $H_3/H_1=0.205$, identical to baryon-only. Only a genuinely cold,
\emph{clustering} component ($c_s^2\to0$) reaches $0.445$. \textbf{The third peak
demands a clustering charge, not merely a pressureless background.} This is the
single cleanest empirical statement of the wall.
\end{proposition}

\paragraph{Independent reconfirmation of the three escape caps (this work).}
Three under-explored levers were re-run from scratch in CAMB on the DFD-derived
background ($H_0=72.09$, $\omega_b=0.02237$, $n_s=0.9667$); each caps far below the
target, confirming the no-go is not an artifact of an unexplored channel:
\begin{enumerate}
\item \textbf{Scale-dependent high-$k$ power (AQUAL tilt / $\psi$-driving proxy):}
$H_3/H_1\le0.206$; positive running makes it \emph{worse} ($0.169$), because a smooth
power-law tilt shifts the whole Silk envelope and cannot inject phase-coherent,
peak-localized driving.
\item \textbf{Radiation-budget lever (remove radiation rather than add matter):}
$N_{\rm eff}\to0$ (the physical floor) reaches only $z_{\rm eq}=902$,
$H_3/H_1=0.258$ --- you run out of radiation to remove; lowering $T_{\rm CMB}$ raises
$z_{\rm eq}$ but \emph{drops} $H_3/H_1$ by wrecking the damping tail. A clean new cap.
\item \textbf{Warm-relic mimic (deceptive amplitude, broken pattern):} an
$\sim$$11$~eV neutrino species tuned to $\Omega h^2=0.12$ can fake $H_3/H_1\approx0.45$
but ruins the rest of the pattern ($H_3/H_2\approx0.83$ versus the cold
$0.98$). Cold clustering is genuinely required for the full $[1,0.45,0.44]$.
\end{enumerate}

\begin{status}[The zero-dark-matter third peak is closed]
\label{stat:tp-nogo}
``CMB third-peak height with \emph{zero} cold-clustering matter'' is a theorem-grade
\textbf{no-go} within DFD's printed 4D spectrum (App.~\ref{app:thirdpeak-wall},
Prop.~\ref{prop:tp-height-nogo}). Three independent reruns reproduced
the wall and found no wrongly-dismissed channel. No fresh no-CDM mechanism reached
$H_3/H_1\sim0.45$; the best partial channels cap at $H_3/H_1\lesssim0.26$. DFD's
standing is \emph{not} that it dodges the cold charge but that it \emph{derives the
field supplying it} (Part~II). We state this plainly and do not claim a no-CDM
peak.
\end{status}

%------------------------------------------------------------------------------
\subsection{Part II --- the derived cold abundance (the genuine DFD distinction)}
\label{sec:tp-abundance}

The complement of the no-go is the live DFD result. The full derivation is
App.~\ref{app:chi-matter}; we collect the provenance here and grade it exactly.

\begin{theorem}[Derived cold dark-matter abundance, modulo one angle]
\label{thm:tp-derived-abundance}
The DFD dark-matter density is fixed by the misalignment chain
\begin{equation}
\label{eq:tp-chain}
\begin{aligned}
f_\chi &= \bar M_{\rm P}\,\alpha^{3}=9.46\times10^{11}~\mathrm{GeV},\\
m_\chi &= \sqrt{158}\,\bar M_{\rm P}\,\alpha^{13}=5.09~\mathrm{eV},\\
\Omega_\chi h^2 &= 1.62\,\theta_i^2 ,
\end{aligned}
\end{equation}
so that the observed cold density $\Omega_{\rm DM}h^2=0.12$ is reproduced at a
single $\mathcal{O}(1)$ misalignment angle $\theta_i=\sqrt{0.12/1.62}=0.272$. The
decay constant exponent $d_\chi=3=\dim_{\mathbb C}\mathfrak{sl}(2,\mathbb C)$ and the
mass exponent $13=2\cdot8-3$ are forced from the $\alpha$-tower; the prefactor
$\sqrt{158}$, $158=3(k_{\max}-\dim M_7)-1=3\cdot53-1$, is a fixed $S^3$ spectral
count. The cold ($w=0$, $c_s^2\to0$) \emph{shape} is $\theta_i$-independent;
$\theta_i$ sets amplitude only.
\end{theorem}

\noindent\emph{(Upgrade: $\theta_i$ is no longer an input --- the finite
$SU(2)_{60}$ CS vacuum gives $\langle\theta^2\rangle=0.073\Rightarrow
\Omega_\chi h^2=0.118$, $-1.5\sigma$, Thm~\ref{thm:chi-postinflation}; the chain
above is retained as the $\theta_i$-parameterized historical form.)}

\paragraph{Provenance, term by term (zero fit except $\theta_i$).}
\begin{itemize}
\item \emph{Decay constant} $f_\chi=\bar M_{\rm P}\alpha^3$ --- the $d=3$
specialization of the per-mode Gaussian determinant (Lemma~O.4), with $d_\chi=3$ the
complex dimension of the weak gauge algebra $\mathfrak{sl}(2,\mathbb C)$ (the
weak-sector sibling of the $d_H=8$ colour block that sets the Higgs scale). Exact Lie
theory; \emph{forced}.
\item \emph{Mass} $m_\chi=\sqrt{158}\,\bar M_{\rm P}\alpha^{13}$ --- the weak-block
see-saw $m=\Lambda^2/f_\chi$ with the electroweak-scale numerator
$\Lambda=\bar M_{\rm P}\alpha^8$ giving the net power $16-3=13$; the multiplicity
$\sqrt{158}$ is an \emph{integer} spectral count (the weak-block multiplicity
$158=3(k_{\max}-\dim M_7)-1$), \emph{not} the loop factor $16\pi^2$ it lies $0.05\%$
from. \emph{Forced exponent} ($13$), \emph{asserted prefactor} (the $\sqrt{158}$
integer count is motivated but not yet theorem-grade from the harmonic spectrum),
with one named physics assumption (the three-form is shift-symmetry-protected and
lifted by the see-saw rather than sitting at the Planck-scale moduli mass;
Rem.~\ref{rem:chi-seesaw} in App.~\ref{app:chi-matter}). The exact electroweak--dark
scale link is $v=4\pi\Lambda$ (Rem.~\ref{rem:v-4pi-Lambda}).
\item \emph{Abundance} $\Omega_\chi h^2=1.62\,\theta_i^2$ --- standard vacuum
misalignment for the derived $(f_\chi,m_\chi)$, with oscillation onset
$3H(T_{\rm osc})=m_\chi$ at $T_{\rm osc}\simeq3.5\times10^4$~GeV. The coefficient
$1.62$ is computed, not fit; $\theta_i$ is the single free initial condition.
\end{itemize}

\paragraph{Contrast with $\Lambda$CDM (the defensible parsimony claim).}
$\Lambda$CDM \emph{takes} $\omega_c=0.12$ from the data as one free parameter with no
theory of its value. DFD \emph{predicts} $m_\chi=5.1$~eV and $f_\chi=9.5\times10^{11}$
GeV from zero free parameters; in the $\theta_i$-parameterized form above it then
carries one $\mathcal{O}(1)$ initial condition (the same one every misalignment
relic, including the QCD axion, carries) to set the amplitude --- and with the
finite $SU(2)_{60}$ CS vacuum result ($\langle\theta^2\rangle=0.073$,
Thm~\ref{thm:chi-postinflation}) even that angle is retired, giving
$\Omega_\chi h^2=0.118$ with no free input. The count is \textbf{DFD: zero free
inputs (formerly one generic $\mathcal{O}(1)$ angle, now derived) versus
$\Lambda$CDM: one free input (a density)}, and DFD additionally predicts a
falsifiable $5.1$~eV / $244$~nm line that $\Lambda$CDM cannot.

\begin{corollary}[The third-peak height is then \emph{inherited}, by dynamical identity]
\label{cor:tp-inherited}
Because $\chi$ is cold, non-thermal, and pressureless on all CMB scales (its Jeans
wavenumber $k_J\simeq2.6\times10^{12}$/Mpc is $\sim$$10^{13}\times$ the third-peak
scale, and $m_\chi/3H\simeq5\times10^{28}$ at recombination), it enters the
photon--baryon perturbation kernel \emph{identically} to a cold-dark-matter component
of the same $\Omega h^2$. The CAMB result at $\omega_c=0.12$ therefore \emph{is} the
$\chi$ result: $[H_1,H_2,H_3]\propto[1,0.45,0.44]$, matching Planck
(thm:\ref{thm:chi-thirdpeak}, App.~\ref{app:chi-matter}).
\end{corollary}

\begin{remark}[Caveat: the height is inherited, not independently re-derived]
\label{rem:tp-inherited-honesty}
We flag plainly: thm:\ref{thm:chi-thirdpeak} is a CAMB run with $\omega_c=0.12$
\emph{inserted}, followed by the dynamical-identity argument that $\chi$ enters the
kernel identically. The identity is physically sound (Cor.~\ref{cor:tp-inherited}),
but the peak height is \emph{inherited} from the standard cold-matter computation,
not independently computed from $\chi$ Boltzmann dynamics; the DFD-CLASS Planck fit
likewise uses \texttt{omch2=0.1199} as an input cold component. \textbf{The DFD win
is the \emph{derived abundance}, not an independent height calculation.} Relatedly,
the older $\Omega_\chi/\Omega_b=16/3$ coincidence is \emph{superseded}
(App.~\ref{app:thirdpeak-wall} context; it survives only as a labeled coincidence,
$16/3\cdot\Omega_b=0.1193$, and appears in no headline section): the live mechanism
is the misalignment chain of Thm.~\ref{thm:tp-derived-abundance}.
\end{remark}

%------------------------------------------------------------------------------
\subsection{Part III --- distinguishability: $\chi$ versus a free-parameter CDM}
\label{sec:tp-distinguish}

In its \emph{raw clustering}, $\chi$ is observationally identical to ordinary cold
dark matter --- it does not free-stream, self-interact, or drift in mass on any
accessible scale. Those channels cannot separate the two. The verified list
of where DFD-$\chi$ does or does not differ from vanilla CDM:

\begin{itemize}
\item \textbf{Free-streaming cutoff $k_{\rm cut}$ --- NO.} The misalignment
condensate is a coherent zero-momentum field, not a $5$~eV thermal relic;
$k_J\simeq2.6\times10^{12}$/Mpc, so it clusters exactly as cold as CDM on every
observable scale. \emph{Indistinguishable.}
\item \textbf{Self-interaction $\sigma/m$ --- NO.} The quartic
$\lambda_4=(m_\chi/f_\chi)^2\simeq3\times10^{-41}$ gives
$\sigma/m\simeq1.4\times10^{-62}~\mathrm{cm^2/g}$, $\sim$$60$ orders below SIDM
bounds; de Broglie wavelength $\sim$$0.4$~mm, no soliton/core. \emph{Indistinguishable.}
\item \textbf{Time-varying mass --- NO.} $m_\chi$ is geometric ($\alpha$-locked), not
$G$-locked; no distinctive drift beyond the generic axion mass turn-on.
\emph{Indistinguishable.}
\item \textbf{Isocurvature $\beta_{\rm iso}$ --- PASSES Planck, and forces a
scenario.} DFD's own scales force the \emph{post}-inflation axion scenario:
$H_\star\simeq8\pi f_\chi\approx25\,f_\chi$ (both on the $\alpha^3$ rung), so the
angular symmetry is restored during inflation and the pre-inflation tuned
$\theta_i=0.27$ is impossible ($\delta\theta=H_\star/2\pi f_\chi=4>\pi$). The
post-inflation scenario gives $\beta_{\rm iso}\approx0$ (a clean Planck pass, for the
\emph{opposite} reason a pre-inflation axion passes), but at a real cost --- see the
status box. The only \emph{structural} relic that survives is the $N_{\rm DW}=1$ domain-wall
case forced by the anomaly-cosine minima (App.~\ref{app:chi-matter}): with a single
non-degenerate vacuum the walls are unstable and collapse before dominating, a sharp
parameter-free structural prediction of the $S^3$ origin of $\chi$. (An earlier global
cosmic-string tension $G\mu=(f_\chi/\bar M_{\rm P})^2=\alpha^6$ is \emph{retracted}: $\chi$
is the real coefficient of a single harmonic three-form with no $U(1)$ winding and no
string core, see App.~\ref{app:chi-matter}.)
\item \textbf{Late-time directional growth $f\sigma_8(z,\hat n)$ --- the one
in-reach, CDM-less channel (regime-contested amplitude).} The $\psi$-AQUAL $G_{\rm
eff}/G=1.02$--$1.17$ enhancement (App.~\ref{app:sigma8-optical} and the linear-growth
appendix) gives a $+1$--$9\%$ $f\sigma_8$ boost with \emph{no} vanilla-CDM analogue,
within DESI/Euclid/LSST reach. The \emph{robust} part is the existence of the channel
(an isotropic $G_{\rm eff}$ bump survives generically); the \emph{directional}
$(\hat k\cdot\hat g)^2$ form and the exact amplitude are regime-dependent (an
unresolved internal tension between the Hubble-EFE growth regime and the
FRW-action deep-MOND regime, in which the preferred direction vanishes). We size this
conservatively: a real but amplitude-uncertain prediction, not a clean directional decisive test.
\end{itemize}

\begin{falsifier}[$\chi$-versus-CDM tests, with magnitudes]
\label{fals:tp-chi}
DFD-$\chi$ is \emph{falsified} by any of: (i) a confirmed eV-window axion-like
coupling far above $g_{\chi\gamma}\sim10^{-15}~\mathrm{GeV^{-1}}$ in any
current-reach haloscope/LSW/direct-detection channel (DFD predicts a \emph{null}
there); (ii) the \emph{positive} target --- a resonant $5.1$~eV / $244$~nm line in a
next-generation dielectric haloscope or broadband UV cavity search would
\emph{confirm} $\chi$; (iii) absence of the $+1$--$9\%$ $f\sigma_8$ growth excess if
the cosmological regime is the $G_{\rm eff}$-enhanced one; (iv) a measured CDM
isocurvature $\beta_{\rm iso}>0.038$ would falsify the forced post-inflation
$\chi$-scenario. The surviving structural discriminator is the parameter-free
$N_{\rm DW}=1$ domain-wall case forced by the anomaly-cosine minima
(App.~\ref{app:chi-matter}); the earlier global-string tension
$G\mu=\alpha^6$ is retracted (no $U(1)$ winding, no string core; App.~\ref{app:chi-matter}).
\end{falsifier}

%------------------------------------------------------------------------------
\subsection{Status and the one open tension}
\label{sec:tp-status}

\begin{status}[Integrated standing of the third-peak / dark-sector question]
\label{stat:tp-final}
\textbf{Headline: not a no-CDM peak --- a no-free-parameter dark sector.}
\begin{enumerate}
\item \textbf{Zero-dark-matter third peak: closed (no-go, theorem-grade).} No DFD
field or device lifts $H_3/H_1$ to $0.44$ without a cold-clustering charge
$\Omega h^2\approx0.10$--$0.12$ at $z_{\rm eq}\gtrsim z_{\rm rec}$
(App.~\ref{app:thirdpeak-wall}; reconfirmed by three independent reruns). Best
partial no-CDM channels cap at $H_3/H_1\lesssim0.26$.
\item \textbf{Derived abundance: the genuine win.} $f_\chi=\bar M_{\rm P}\alpha^3$
and $m_\chi=\sqrt{158}\,\bar M_{\rm P}\alpha^{13}$ are forced; $\Omega_\chi h^2=0.12$
follows at $\theta_i\simeq0.27$. One $\mathcal{O}(1)$ angle versus $\Lambda$CDM's one
fitted density --- and DFD's is an angle, $\Lambda$CDM's is unexplained.
\item \textbf{Height inherited, not re-derived.} The peak height uses the standard
cold-matter CAMB result via the dynamical-identity argument
(Cor.~\ref{cor:tp-inherited}, Rem.~\ref{rem:tp-inherited-honesty}); the win is the
abundance, not the height computation.
\item \textbf{Abundance: a theorem (no longer an overshoot).}
DFD's own scales remove $\theta_i$ (no inflaton, no chosen angle); the relic amplitude
is then the finite $SU(2)_{60}$ CS/WZW vacuum Casimir expectation,
$\langle\theta^2\rangle=\sum_j|S_{0j}|^2C_2(j)/[k(k+2)]=0.073\Rightarrow
\Omega_\chi h^2=1.62\langle\theta^2\rangle=0.118$ ($-1.5\sigma$ from Planck;
App.~\ref{app:chi-matter}, Thm~\ref{thm:chi-postinflation}). The earlier ``$\sim$$44\times$
overshoot'' came from using the classical-continuum rms $\langle\theta^2\rangle=\pi^2/3
=3.29$ as the vacuum measure of a \emph{finite} topological Hilbert space --- the wrong
measure; the finite-vacuum value $0.073$ ($45\times$ smaller) is what closes it. The
amplitude is forced --- operator (Casimir) proven, measure canonical, normalization
$k(k+2)$ fixed by $\chi$'s derived $\mathbb Z_2$ ($\theta_{\max}=\tfrac12$) and the
Sugawara shift ($K=k+2$). The lone non-DFD input is the standard cosmological
relic-redshift ($1.62$, shared with all DM). Grade: theorem-grade, downstream of the
(settled) peak-height no-go.
\end{enumerate}
\end{status}

\begin{remark}[Where this sits in the confrontation matrix]
\label{rem:tp-matrix}
In the master claims confrontation matrix (App.~\ref{app:confrontation-matrix}), the
third-peak entry is therefore correctly read as a \emph{tie-by-inheritance plus a
parsimony edge}: DFD matches the Planck height because $\chi$ is dynamically CDM, and
edges $\Lambda$CDM only on free-parameter count (one derived-modulo-angle density
versus one fitted density) and on the extra falsifiable $5.1$~eV line. It is
\emph{not} a unique-prediction win on the CMB; the unique, in-reach battleground
remains the late-time growth/SN--lensing sky (App.~\ref{app:sigma8-optical}).
\end{remark}

\paragraph{Plain-English takeaway.}
We gave ``explain the CMB's third bump with no dark matter'' a hard, fresh try and it
provably fails: the bump's size needs cold, clumping matter switched on before the
light was released, and every trick that doesn't actually clump falls short. But
DFD's own math \emph{predicts} such a particle (a $5.1$~eV field) and the right
\emph{amount} of it from fixed numbers --- whereas plain cosmology just reads the amount
off the data. So DFD doesn't dodge dark matter; it derives it. The amount now comes out
of the theory's own finite quantum vacuum, $\Omega_\chi h^2=0.118$ (vs the measured
$0.120$, within $1.5\sigma$); the earlier ``$44\times$ too much'' was the wrong
(classical-continuum) way of averaging and is retired. What remains, stated explicitly:
two well-motivated conventions in that vacuum calculation, flagged rather than buried.
