\section{Introduction}
\emph{This paper is a companion to three other papers, of which only the first is published at present
\cite{RT-ICFA}\cite{RT-Positron-q}\cite{RT-LHC2}. Quite long sections appear, modified slightly at most, 
in these papers.}

A series of previous papers has shown how a ``prototype ring'' (PTR)\cite{PTR-Bad-Honnef}  
can be used to model the performance
of precision experiments of electric (EDM) and magnetic (MDM) dipole moments, with the aim of 
detecting beyond the standard model (BSM) physics.  Until now these proposals have required the use 
of a ``beam accumulator'' ring (BA) to produce the required $10^{10}$ minimum total number of stored 
counter-circulating particles to perform the operations needed to measure, for example, $EDM_p$ for the
proton or $EDM_h$ for the helion individually.  More importantly, it is what is needed to measure 
the $EDM_p-EDM_h$ difference, a combination from which the dominant systematic EDM error cancels. 

This paper emphasizes the dramatic improvement in EDM and MDM measurement capability made possible
by the huge increase in isotope source currents the light ion linac (LIL) component of the proposed 
Center for Linac Isotope Project) CLIP project promises.  The paper describes how this increase in source 
current will render bunch accumulation in a BA ring unnecessary, reducing both cost and complexity.

Using (admittedly optimistic) assumptions, the counting statistcs EDM error after continuous running for 
one year is expected to be $\pm\,10^{-30}$\,e-cm. This is four orders of magnitude less than the previous 
upper BSM physics limit derived from measurement of the neutron EDM (as quoted in the abstract)
which includes both systematic and counting statistics errors).  

As originally intended, a major goal for PTR was to measure the proton $EDM_p$, in order to guide the estimation 
of and minimization of systematics of a full scale, all-electric EDM ring.  To reduce the PTR cost
(to k\texteuro 16,600 in 2020 euros) many cost saving design measures were taken; led by room temperature 
(non-cryogenic) operaion and the use of inexpensive magnetic shielding.  

Using counter-circulating beams of different particle type was first suggested by 
Koop\cite{Koop-different-particles}.  Parameters for the baryons Koop discussed are given in 
Fig.~\ref{fig:KoopParameters}.
\begin{figure}[hbt]
\centering
\includegraphics[scale=0.35]{pdf/KoopParameters.pdf}
\caption{\label{fig:KoopParameters}Parameters for particle types considered by Koop as possible candidates
for EDM determination. Koop's symbol ``a'' for anomalous moment is replaced by ``G'' in this paper.
A necessary, but not sufficient condition, for a pair of entries to be doubly-magic, is for their 
$G$-values (here $a$-values) to have opposite sign, as is true for $p$ and $He3$.  Though true also 
for $p$ and $d$ no doubly-magic pairing has, as yet, been found.  The same comment applies to the 
Li6,Li7 and He3,C13 pairings.  Seeming exceptions to this ``opposite sign'' rule are the all-electric
233\,MeV proton and 14.5\,MeV electron EDM rings.  This is discussed in detail in 
Appendix~\ref{sec:FixedPoint}.}
\end{figure}
Subsequent studies by the CPEDM group\cite{APS-FSU} have shown that many systematic errors can be reduced 
by phase-locked loop, frequency domain exploitation of ``mutual co-magnetometry''. This is epitomized by 
the doubly-magic proton-helion combination emphasized in the present paper. 

Significant uncertainties remain concerning the systematic errors associated with storage ring operation employing 
superimposed electric and manetic bending.  Though investigation of these systematic errors will be aided greatly 
by the above mentioned four order of magnitude imrovement in counting statistics, the systematic 
${\rm EDM}_p-{\rm EDM}_h$ difference can only be estimated.  Conservatively the systematic error in this 
difference will initially be less than $\pm\,10^{-27}$\,e-cm; and subject to continuing reduction from this
value, exploiting the available statistical accuracy.

Doubly-frozen kinematic conditions can always be found for simultaneous counter-circulation of
arbitrarily different pairs of charged particle types in a storage ring having superimposed electric
and magnetic bending.  It is always possible for at least one or the other of the beams to be said to 
be ``magic'' by virtue of having its spins ``globally frozen''---spins all pointing exactly forward, for example.  
In general, though, spins of the other beam can at most be ``pseudo-frozen''---meaning that, viewed stroboscopically 
at any location the spins ``appear'' to be frozen because their spin tunes are rational fractions, such that the spins
always return to the same orientaion at the observation point. For certain spin control procedures (those involving 
spin-phase locking capability) the pseudo-frozen feature is sufficient, but ``magic'' freezing is necessary for EDM 
precession to increase monotonically for at least several minutes to enable accurate EDM measurement. 

The validity of statements in the previous paragraph (many of which are far from obvious \emph{a priori,})
and similar statements to follow, is based on carefully finding all real roots of the quartic 
Eq.~(\ref{eq:AbbrevFieldStrengths.3-rev}) and quadratic Eqs.~(\ref{eq:Wollnik-TM-params}).  

Because of technical limitations, for example the electric or magnetic fields may be unachievably strong,
or polarimetry may not be efficient enough, or have insufficient analyzing power, conditions that are achievable 
in principle may not be  experimentally practical.

Almost without exception a ring with superimposed electric and magnetic bending is required to achieve the frozen
spins needed for co-magnetometry.  With one of the beam polarizations frozen the magnetic moment of the particles
of the other beam particles can be determined by measuring the spin tune of that beam.

Two leading exceptions that do not require superimposed electric and magnetic fields are 14.5 MeV electrons 
(of either sign) or 233\,MeV protons (of either sign) in purely electric rings\footnote{Even these exceptions are 
disputed in Appendix~C.}.

For rings with superimposed electric and magnetic bending only two ``doubly magic'' cases are known, but there may be 
others involving higher Z nuclei.  The most promising of the known cases has doubly magic protons and He3 nuclei 
(of curiously close kinetic energies) both globally frozen; the doubled helion charge compensates for the
approximately doubled momentum.  Furthermore, counter-circulating in a ring of convenient and
inexpensive size, excellent polarimetry is also available.  A less promising doubly magic example has 
counter-circulating protons and positrons.  This case is less promising because the momenta are large enough to 
require an expensive ring, and because positron (and electron) polarimetry is not at all robust.

Doubly frozen pairings (with one or the other, but not both,  magic) of proton and deuteron beams, $pp$, $pd$, 
$dd$, and $dp$, are of special interest, both because their EDMs are of pre-eminent physical importance, and 
because these pairings can be investigated using well-established equipment and technology, most of which is 
currently available.  

Along with reproducible accelerating cavity frequency, doubly-frozen, the spin tunes (whose parameters other than 
$\beta$, $E_0$ and $B_0$  are constants of nature) can be stabilized to precision (inversely) proportional to run 
length.  \emph{This will permit precise electric and/or magnetic field reversal resettability to be performed.  The
reason this is important is that systematic errors can then be reduced by averaging over reversed configurations.}

In this way ``co-magnetometry'' can exploit the ``perfect'' gyroscopic stabilization provided by the circulating 
MDMs themselves.  A conceptual design for a storage ring complex capable of exploiting this technology is presented 
below.  It is based on a reconfiguration of the COSY storage ring complex in Juelich, Germany, which has been the 
storage ring at which most of the required spin control technology has been developed. 

Electric/magnetic frozen spin storage rings will also be capable of measuring isotope MDM 
values with unprecedented precision. Though such rings can be built inexpensively, they are not very flexible.
For this reason they need to be developed using existing facilities that match their requirements\cite{Kolya-Snowmass}.

Because the site for such an accelerator complex has not been formally selected, the ring circumference $\mathcal{C}$, 
nominally 100\,m, has been left left open in the computer codes producing the present report.  To maintain this 
flexibility the lattice design described can be expressed in ``scale invariant'' form.  But, to produce the 
numerical evaluations needed for concrete visualization, the PTR circumference is here taken to be 
$\mathcal{C}=102.2$\,m, as required to fit PTR, along with a bunch accumulator'' (BA) (required for rebunching 
the Juelich cyclotron proton or deuteron 25\,MHz cw-bunched cyclotron beams) into the COSY, Juelich beam hall.
A layout diagram for PTR is shown in Fig.~\ref{fig:PTR-layout-Toroidal8_102p2-mod}.   A perspective view of one
octant is shown in FIG~\ref{fig:SectorPerspective}.   

\begin{figure}[hbt]
\centering
\includegraphics[scale=0.28]{pdf/PTR-layout-Toroidal8_102p2-mod.pdf}
\caption{\label{fig:PTR-layout-Toroidal8_102p2-mod}Lattice layouts for PTR, the proposed 
``EDM prototype'' storage ring.  The single quadrant ``model space'' figure on the left
can be conformally mapped to one quadrant of the ``physical space'' figure on the right.
Straight lines on the left map to perfect circles on the right.  A ``compromise quadrupole
triplet'' is shown inset on the right.  The magnetic quad doublet and electric quad singlet
provide very weak ``optical trim correction''.  Their strengths are adjusted individually in 
the same $\eta_M,\eta_E$ ratio as the ring bend strengths.  Ideally these corrections would be
exactly superimposed, which would be mechanically challenging.  To thin element approximation 
this superposition is equivalent.  See Appendix~\ref{sec:Compromise}.}
\end{figure}

\begin{figure}[hbt]
\centering
\includegraphics[scale=0.4]{pdf/Fig-5-bender.pdf}
\caption{\label{fig:SectorPerspective}Perspective mock-up of one sector of PTR, the superimposed 
E/M prototype ring. Two ``short-circuited ends''cos\,$\theta$-dipoles surround the beam tube, within 
which the capacitor plates are accommodated.}
\end{figure}
\begin{figure}[hbt]
\centering
\includegraphics[scale=0.45]{pdf/COSY-hall-mod7.pdf}
\caption{\label{fig:COSY-hall-mod7}Pre-CLIP implemention of PTR in the COSY Hall. The bunch accumulator, BA,
consists of an  ``arcs-only COSY'', rebuilt with existing electron cooling, 
stripper-foil injection and electrostatic-magnetic extraction.  Both of these requirements would be met
by simply retaining existing equipment. }
\end{figure}

\section{Testing time reversal symmetry}
\subsection{Electric dipole moment (EDM) measurement} 
The excess of matter over anti-matter in the present-day universe suggests 
the existence of violations of P and T symmetries substantially greater than are required by the 
``standard model'' of elementary particle physics. This provides the leading motivation for 
measuring EDMs, which must vanish if P and T symmetries are respected. 

Ultimate proton EDM measurement has been thought to require simultaneously counter-circulating, 
frozen-spin 233\,MeV protons in a ``large'' (800\,m circumference) ``holy grail'' all-electric 
storage ring.  A step toward this goal is the ``small'' ($\sim$100\,m circumference) PTR ring  with 
superimposed electric and magnetic bending fields. Simultaneously counter-circulating frozen spin 
proton beams is one of the  ultimate ``holy grail'' features that can be demonstrated in PTR. 

The need for storage rings with predominantly electric bending is based on the
requirement to ``freeze the spins'' of stored particle beams. This is the first step
toward the measurement of electric dipole moments (EDMs) of the circulating
particles, as well as for other symmetry violation investigations.
Electrons (of either sign) along with the proton and triton baryons, are 
fundamental particles whose electric EDM's are potentially measureable in 
all-electric frozen spin storage rings. With superimposed magnetic bending
other particles, especially deuterons and helions (He3 nuclei), can have 
frozen spins.

In all cases the EDM values are so small that suppression of systematic errors
makes it essential for the storage ring to support simultaneous counter-circulating
beams.  With superimposed electric and magnetic bending, constructive bending superposition 
in one direction implies destructive superposition in the other direction. By itself, this 
does not preclude the possibility of simultaneously counter-circulating beams---but it does 
require major difference between the beams, either in particle type or momentum.

In contrast to a ``single particle table-top trap'', the essential feature of a storage ring 
``trap'' is that some $10^{10}$ particles can have their spins aligned in a polarized stored beam.
As demonstrated by Hempelmann et al. at the COSY lab\cite{Hempelmann},
this is a nunber of  particles large enough for the beam polarization to be detected externally, 
and fed back to permit external control of the beam polarization. 
Though the table large enough for any such a ``storage ring trap'' would be quite large, the 
level of achievable spin control, though classical, not quantum mechanical, can be comparable
to the control of one or a small number of polarized particles in a low energy trap.
Precision spin tune control  at COSY has been pioneered by  Eversmann et al.\cite{Eversmann}.

In true ``globally frozen spin'' operation the bunch polarizations remain globally frozen, for example parallel 
to the circulating beam momentum vector. For all-electric bending 
this fixes all kinematic quantities uniquely. But with superimposed electric and magnetic bending, 
the frozen spin condition can be met over bands of E/B ratios, specific to particle type. Furthermore, 
some experiments can be performed with ``pseudo-frozen spins'' that appear frozen locally, but 
not globally.

The leading observable effect of a static particle EDM would be an ``out-of-plane'' 
spin precession, orthogonal to MDM-induced ``in-plane'' horizontal spin precession caused by 
whatever magnetic and/or electric fields cause the particle orbit to follow a sequence
of horizontal circular arcs. With standard model EDM predictions being much smaller than 
current experimental sensitivities, detection of any particle's non-zero EDM would 
signal discovery of New Physics. 

A  ``nominal experimental proton EDM detectability target'' has conventionally been 
defined to be $10^{-29}$e-cm.  An EDM of this magnitude could help to account for the 
observed matter/antimatter asymmetry of our universe while, at the same time being 
plausibly (only one or two orders of magnitude) larger than existing standard model 
calculations. 

Currently the proton EDM upper limit (as inferred indirectly by measuring the
Hg atom EDM) is roughly $2.1\times10^{-25}e \cdot\,$cm.\cite{CYR}
The original goal for PTR was to serve as prototype for the nominal, full-scale, 
all-electric proton EDM ring.  It has been possible, however to make the PTR design flexible
enough to support quite accurate EDM determinations and to achieve other goals, such
as precision measurement of anomalous magnetic moments of light nuclear isotopes.

\subsection{Spin tune comparator measurement of light isotope MDMs and EDMs}
(Except for synchrotron radiation) first generation electron/positron colliders such 
as CESR have demonstrated CPT-symmetry, and robust backward/forward (bf)-symmetry.
\emph{But the present paper considers only storage rings in which particle spins can be globally frozen}. 
This rules out rings with pure magnetic bending---there are no candidates. One turns, therefore, 
to \emph{electric} rings.   Of special interest are 
the 232.8\,MeV ``Holy Grail'' all-electric proton EDM ring, which has been under discussion for more than a decade.
Also of interest (especially at Wilson Lab) eould be a (small) all-electric 14.50\,MeV e+/e- storage ring in which 
globally-frozen spin positrons and electrons can counter-circulate simultaneously. 

For particles at rest ``co-magnetometry'' in low energy ``table-top particle traps'' has been essential. 
For example, Gabrielse\cite{Gabrielse-eEDM} has (with excellent justification) described the measurement of the 
electron magnetic moment (with 13 decimal point accuracy) as ``the standard model's greatest triumph'', based 
on the combination of its measurement to such high accuracy and on its agreement with theory to almost the 
same accuracy.  

Especially for the \emph{direct} measurement of EDMs, storage ring technology with beam pairs that can counter-circulate 
simultaneously in a storage ring with superimposed electric and magnetic bending is required.  In this context the term 
``mutual co-magnetometry'' can be used to apply to ``beam type pairings'' for which both beams have frozen spins.
Usually only one of the beams can be ``globally'' frozen or ``magic'', however.  

Surprisingly, \emph{counter-circulating beams can  follow  accurately identical orbits, even when the bending 
is performed by superimposed electric and magnetic fields.} Here we show how this can enable the precise comparison of 
the spin tunes, and hence MDMs, of  different elementary particle types---hence the name 
``spin tune comparator''.  Both beams are assumed polarized and one of the beam type's MDM may be much 
better known than the other; for example e- better than p, or e- better than e+. 

To the extent both MDM's are already known, the known ratio can be used to adjust a compensating element 
to match this ratio.  Though EDM measuremnt is scarcely mentioned in this spin tune comparator section, 
this remains the goal and \emph{spin tune comparator operation will permit Koop-wheel rotation reversal with 
almost arbitrarily high, frequency domain, accuracy. Within each run a precision EDM measurement (including systematic 
error contribution) can then be obtained.} Koop\cite{Koop-SpinWheel} argues (persuasively) that intentional roll of 
the Koop wheel can improve spin coherence duration.  Extraction of the EDM value then relies on linear
interpolation of data taken at negative and positive roll rates to cancel the torque applied to drive the ``roll''.

All of the most important measurement methods for ultimate EDM precision can be developed in the PTR ring. 
Furthermore, one can be more relaxed about parameter tolerance specifications in a small ``inexpensive'' 
prototype than in an eight times larger eventual ``ideal'' ring.  One of the tasks for PTR will be to 
investigate how tight the tolerances have to be, and how easily they can be met.
In this spirit, the present section derives quite tight tolerances to be aimed for (but not necessarily achieved) 
for measuring EDM's in a small ring with accuracy rivalling what can be achieved in the full scale ring.
This is work in progress. Unresolved technical issues are discussed in other sections.

Especially for the \emph{direct} measurement of EDMs, storage ring technology with beam pairs that can counter-circulate 
simultaneously in a storage ring with superimposed electric and magnetic bending is required.  In this context the term 
``mutual co-magnetometry'' can be used to apply to ``beam type pairings'' for which both beams have frozen spins.
Usually only one of the beams can be ``globally'' frozen or ``magic'', however.  

Compared to the electron, MDM accuracies of all other particles
are inferior by three orders of magnitude or more. 
A task for a spin-tune-comparator can be to
``transfer'' some of the electron's precision to the
measurement of other magnetic dipole moments. 
For example, the
positron EDM could perhaps be determined to almost the current 
accuracy of the electron's, and the (presumed) equality 
(in magnitude) of electron and positron MDM's could be improved
by several orders of magnitude.

Another motivation for developing spin tune comparison capability is to study 
beam-based self-compensation and self-calibration.  Accurately known counter-circulating 
MDM's can be used to cancel unknown differences between electric and
magnetic bending field shapes. ``Introspectron'' might be an apt task for such a 
ring---the purpose then being to investigate its own self-consistency. 
This is not without pure physics interest. With spins always a bit mysterious, 
confirmation (or not) to high accuracy, that CW and CCW spin tunes are identical could 
be informative. This possibility is pursued briefly in Appendix~E, which suggests 
investigation of the Aharonov-Anandan phase.  One supposes that any gravitational 
phase shift will be opposite for CW and CCW circulation.

The final section contains currently unresolved technical issues, the most important of which
concerns the achievability of strong electric fields surrounded by weak magnetic fields. 
This is only an issue for the small prototype ring PTR.  For an eight times larger ring, for
example situated in the AGS or Wilson lab tunnel, electric field strengths much in excess of 
a million volts per meter are not required (at least for the $p,h$ EDM difference measurement). 

\subsection{General relativistic influence on spin precession}
Various authors\cite{Obukhov:2016vvk}\cite{Silenko:2004ad}\cite{OrlovFlanaganSemertzidis}
have pointed out that general relativity (GR) introduces
effects that could be measureable at the precision level promised by proposed EDM rings. A
paper by L\' aszl\'o and Zimbor\' as\cite{Laszlo-Zimboras} has 
calculated the GR influence on storage rings designed for
MDM measurement, such as the PTR ring under discussion here.

The nominal EDM value can be compared to a general relativistic (GR) 
out-of-plane precession effect, mimicking an EDM of the order of  $10^{-28}$\,e-cm,
associated with the storage ring's orientation in the earth's magnetic field. 
Depending on storage
ring details, this reliably calculable ``background precession'' wiil provide a 
``standard candle of convenient magnitude'' calibration of any EDM 
measurement.  

Other GR references include
\cite{Silenko:2006er}\cite{Silenko:2015jqa}\cite{Khriplovich:1997ni}\cite{Pomeransky:2000pb}.
These authors have calculated the GR effect of the earth's gravitaional 
field on the experiment referred to above as the ``holy grail'' of EDM
measurement. They show that the GR effect mimics the EDM effect with a
magnitude such that, if mistakenly ascribed to the proton EDM, would 
produce a spurious proton EDM value of $3\times10^{-28}$\,e-cm. 
This is about thirty times greater than the precision anticipated for
the holy grail ring and not inconsistent with the Orlov, Flanagan,
and Semertzidis estimate\cite{OrlovFlanaganSemertzidis}.

\clearpage

\section{Co-magnetometry provided by common orbits in the same ring}
\subsection{Fractional bending coefficients}
To aid the discussion of rings with superimposed electric and magnetic bending, (to be abbreviated as 
``EM-superposion'') fractional bending coefficients $\eta_E$ and $\eta_m$ can be defined by
\begin{equation}
\eta_E = \frac{q}{pc/e}\,\frac{E_0r_0}{\beta},\ 
\eta_M = \frac{q}{pc/e}\,cB_0r_0,
\label{eq:BendFrac.2}
\end{equation}
neither of which is necessarily positive.  These bending fractions satisfy
\begin{equation}
\eta_E + \eta_M = 1\quad\hbox{and}\quad
\frac{\eta_E}{\eta_M} = \frac{E_0/\beta}{cB_0}.
\label{eq:BendFrac.2p}
\end{equation}
The ``potencies'' of magnetic and electric bending are in the ratio $cB_0/(E_0/\beta)$ because  the electric
centripetal force in electric field $E_0$ is stronger than the centripetal force produced by magnetic field 
$cB_0=E_0$ by the factor $1/\beta$, as regards bending charge $q$ onto an orbit with the given radius of curvature 
$r_0$. Also, when expressed in term of spin tunes, the  ``potencies'' of magnetic and electrically induced MDM 
precessions are in the same ratio as the bending potencies.  Defining a ``normalized momentum'' 
\begin{equation}
\widetilde{p} = \frac{pc/e}{q},
\label{eq:BrhoErhro.1}
\end{equation}
the fractional bending coefficients can be expressed in terms of ``Brho'' and ``Erho'' factors;
\begin{equation}
\eta_E = \frac{1}{\widetilde{p}}\,E_0r_0/\beta
       = \frac{E_0r_0/\beta}{E_0r_0/\beta + cB_0r_0}, \quad
\eta_M = \frac{1}{\widetilde{p}}\,cB_0r_0       
       = \frac{cB_0r_0}{E_0r_0/\beta + cB_0r_0}.
\label{eq:BrhoErhro.2M}
\end{equation}
With superimposed electric and magnetic fields, the combination $E_0r_0/\beta + cB_0r_0$ plays the role
that $cBr_0$ plays for purely magnetic lattices.  This imports into EM-superposition the ``geometrization
of transverse optics''---which is to say ``Newtonian optics with lens strengths quantified by focal lengths''. 
\emph{An essential inference from Eqs.~(\ref{eq:BendFrac.2}) through (\ref{eq:BrhoErhro.2M}) is that, if
forward and backward focusing strengths are in the same ratio as E/M bending strengths then 
the forward and backward focal lengths are in the same ratio as the normalized $\tilde p$ values}.
This is strictly true for vertical focusing (irrespective of electric and magnetic bending fractions) but
true only to a first approximation (i.e. vanishing in the $\eta_M=0$ limit) for horizontal focusing.
With this limitation, forward and backward focal lengths will be the same, element by elemeent and, 
for a lattice that is forward/backward symmetric the forward and backward $\beta_y$-functions will be
identical and the $\beta_x$-functions approximately the same.  

This issue is discussed in numerical detail in Section~\ref{sec:PTRLatticeOptics}.
It is shown that forward/backward symmetry can be satisfied closely but not quite exactly in the 
PTR ring.  Though it is possible to superimpose electric and magnetic bending with high precision, it is much
more difficult to superimpose electric and magnetic lumped quadrupole fields.  For this reason, a significant 
fraction of the present paper is devoted to designing a PTR ring that is stable and robust, using
toroidal field shaping electrodes and magnet coils along with ``triplet, compromise quadrupoles'' (described in detail
in Appendix~\ref{sec:Compromise}) that approximately 
superimpose electric and magnetic lumped quadrupoles in the same ratio as the bending fields. To linearized
 order, the lattice properties will then be almost identical for forward and backward traveling beams. 

\subsection{Frozen spin co-magnetometry}
In the idealized storage ring to be discussed, the electromagnetic
fields are ``cylindrical'' electric ${\bf E}=E_0{\bf\hat x}r_0/r$ (with $E_0<0$) and, 
superimposed, uniform magnetic field ${\bf B}=B_0{\bf\hat y}$ (with $B_0>0$).
The bend radius is $r_0>0$. Terminology is useful to specify the relative
polarities of electric and magnetic bending:
Cases in which both forces cause bending in the same sense will be called
``constructive'' or ``frugal'';  Cases in which the electric and magnetic
forces subtract will be referred to as ``destructive'' or ``extravagant''.

There is justification for the ``frugal/extravagant'' terminology. 
Electric bending is notoriously weak (compared to magnetic bending) and
iron-free (required to avoid hysteretic effects) magnetic bending is also 
necessarily weak. As a result, an otherwise-satisfactory configuration can be
too ``extravagant'' to be experimentally feasible.

For a particle with spin circulating in a (horizontal) planar magnetic storage ring,
its spin axis precesses around a vertical axis at a rate proportional to the particle's 
anomalous magnetic dipole moment, $G$.  For an ``ideal Dirac particle'' (meaning $G=0$) 
\emph{in a purely magnetic field} the spin precesses at the same rate as the momentum---pointing always 
forward, for example.
Conventionally the spin vector's orientation is specified by the angle $\alpha$ between 
the spin vector ${\bf S}$ and the particle's momentum vector ${\bf p}$ (which is tangential, by definition). 
For such a ``not-anomalous'' particle the spin-tune $Q_M$ 
(defined to be the number of $2\pi$ spin revolutions per particle revolution) 
therefore vanishes, in spite of the fact that, in the laboratory, the spin axis has actually precessed 
by close to $2\pi$ each turn.  

In general, spin 1/2 particles are not ideal Dirac particles; the directions of their spin axes deviate 
at a rate proportional to their anomalous magnetic moments, $G$, and their spin tunes differ from 
zero even in a purely magnetic field.  Note also, that a laboratory electric field produces a magnetic 
field in the particle rest frame, so a particle in an all-electric storage ring also has, in general, a 
non-vanishing spin tune $Q_E$. Along with $G$ and $Q$, sll of these comments apply equally to the 
polarization vector of an entire bunch of polarized circulating particles.  

By convention, in the BMT-formalism\cite{BMT}\cite{Wolski}, the orientation of the spin vector 
${\bf S'}$ is defined and tracked in the rest frame of the circulating particle, while the 
electric and magnetic field vectors are expressed in the lab. The spin equation of motion 
with angular velocity $\omega$ is 
\begin{equation}
\frac{d{\bf S'}}{dt} = \omega\times{\bf S'},
\label{eq:BMT.1} 
\end{equation}
with orbit in the horizontal $(x,z)$ plane assumed, where
\begin{align}
\omega
 &=
-\frac{qe}{\gamma mc}\,
\bigg(\Big(G\gamma\Big)cB_0 + \Big(\big(G - \frac{1}{\gamma^2-1}\big)\gamma\beta^2\Big) \frac{E_0}{\beta}\bigg)\,{\bf\hat y} \notag\\
 &\equiv
-\frac{qe}{\gamma mc}\,\bigg((Q_{M})cB_0 + (Q_E)\,E_0/\beta\bigg)\,{\bf\hat y},
\label{eq:BMT.2} 
\end{align}
With $\alpha$ being the angle between the in-plane component of
beam polarization and the beam direction, the ``spin tune'' is defined to
be the variation rate per turn of $\alpha$, expressed as a fraction of 
$2\pi$. Spin tunes in purely electric and purely magnetic rings are given by
\begin{equation}
Q_E = G\gamma - \frac{G+1}{\gamma},
\quad
Q_M = G\gamma,
\label{eq:BendFrac.7}
\end{equation} 
where $\gamma$ is the usual relativistic factor.
Note that the sign of $Q_M$ is the same as the sign of $G$, which is positive for
protons---proton spins precess more rapidly than their momenta in magnetic fields. 
Deuteron spins, with $G$ negative, lag their momenta in 
magnetic fields.  With $G$ positive, $Q_E$ increases from -1 at zero velocity, eventually switching sign
at the ``magic'' velocity where the spins in an all-electric ring are ``frozen'' (relative 
to the beam direction).  When a particle spin
has precessed through 2$\pi$ in the rest frame it has also completed one full revolution
cycle from a laboratory point of view; so the spin-tune is a frame invariant quantity. 

\subsection{Forward-backward beam pairings}
For brevity one can discuss just electrons (of either sign), 
protons($p$), deuterons($d$), tritons($t$), and helions($h$); or even just $p$,
$d$, and $h$, based on the consideration that (with the exception of
helion beam) most of the apparatus, and all of the 
technology, needed for their EDM  measurement is presently available at COSY
laboratory in Juelich, Germany.  Polarized helion beams of appropriate
energy will soon be available at BNL.

The circulation direction of a so-called 
``master beam'' (of whatever charge $q_1$) is assumed to be CW or, equivalently,
$p_1>0$. A secondary beam charge $q_2$ is allowed to have either 
sign, and either CW or CCW circulation direction.

Ideally both beam polarizations would be frozen ``globally'' (meaning spin tune $Q_S$ is zero and the angle $\alpha$ 
between polarization vector and momentum is constant everywhere around the ring).  (Somewhat weaker) ``doubly-frozen'' 
can (and will) be taken to mean that a ``primary beam'' locked to $Q_S=0$, circulates concurrently with a ``secondary'' 
beam that is ``pseudo-frozen'', meaning the spin tune is locked to an unambiguous, exact, rational fraction (other 
than zero).  Only if the secondary beam rational fraction is exactly zero, would the terminology ``doubly-magic'' 
be legitimate.

Such particle pairings are expected to make direct MDM or EDM difference measurements of unprecedented 
precision possible.  For any  arbitrary pairing of particle types $(p,p),\ (p,d),\ (p,e+)$, etc., continua of such 
doubly-frozen pairings are guaranteed.  This is taken to imply that both beam polarizations can be phase-locked,
which is one of the requirements for mutual co-magnetometry.

A design particle has mass $m>0$ and charge $qe$, with electron charge 
$e>0$ and $q=\pm 1$ (or some other integer). These values produce circular 
motion with radius $r_0>0$, and velocity ${\bf v}=v{\bf\hat z}$, where the motion
is CW (clockwise) for $v>0$ or CCW for $v<0$. With $0<\theta<2\pi$ being 
the cylindrical particle position coordinate around the ring, the angular 
velocity is $d\theta/dt=v/r_0$. 

(In MKS units) $qeE_0$ and $qe\beta c B_0$ are commensurate forces, 
with the magnetic force relatively weakened by a factor $\beta=v/c$ 
because the magnetic Lorentz force is $qe{\bf v}\times{\bf B}$. 
By convention $e$ is the absolute value of the electron charge.  When $e$
appears explicitly, usually as a denominator factor, its purpose in 
MKS formulas is to allow energy factors to be expressed in electron volts (eV)
in formulas for which the MKS unit of energy is the joule. 
Newton's formula for radius $r_0$ circular motion, expressed in terms of 
momentum and velocity (rather than just velocity, in order to be relativistically valid)
can be expressed using the total force per unit charge in the form
\begin{equation}
\beta\frac{pc}{e} = \Big(E_0 + c\beta B_0\Big)\,qr_0,
\label{eq:CounterCirc.1} 
\end{equation}
Coming from the cross-product Lorentz magnetic force, the factor $q\beta cB_0$
is negative for backward-traveling orbits because the $\beta$ factor 
is negative.

A ``master'' or primary beam travels in the ``forward'', CW direction. 
For the secondary beam, the $\beta$ factor can have either sign.
For $q=1$ and $E_0=0$, formula~(\ref{eq:CounterCirc.1}) reduces to a standard 
accelerator physics ``cB-rho=pc/e'' formula.  For $E_0\ne 0$ the formula 
incorporates the relative ``bending effectiveness'' of $E_0/\beta$ 
compared to $cB_0$.  As well as fixing the bend radius $r_0$,
this fixes the magnitudes of the electric and magnetic bend field values 
$E_0$ and $B_0$. To begin, we assume the parameters of a frozen spin ``master'',
charge $qe$, particle beam have already been established, including the signs
of the electric and magnetic fields consistent with $\beta_1>0$ and $p_1>0$.  
In general, beams can be traveling either CW or CCW.  For a CCW beam both $p$ and 
$\beta$ have reversed signs, with the effect that the electric force is unchanged, but the 
magnetic force is reversed. The $\beta$ velocity factor can be expressed as
\begin{equation}
\beta = \frac{pc/e}{\sqrt{(pc/e)^2 + (mc^2/e)^2}}.
\label{eq:CounterCirc.2} 
\end{equation}
Introducing the abbreviations
\begin{equation}
\mathcal{E} = qE_0r_0,
\quad\hbox{and}\quad
\mathcal{B} = qcB_0r_0, 
\label{eq:Alterations.2}
\end{equation}
it has been shown in the companion paper\cite{RT-ICFA} that all circular orbits
satisfy the equation 
\begin{equation}
p_m^4 - 2cB_0(qr_0)p_m^3 + (\mathcal{B}^2-\mathcal{E}^2)p_m^2 - \mathcal{E}^2m^2 = 0,
\label{eq:AbbrevFieldStrengths.3-rev} 
\end{equation}

Curiously, expressed as quartic equation for $p_m$, this equation is ``independent of $r_0$'' 
(or rather of the product ``$qr_0$'') in the sense that the product factor $(qr_0)$ can be chosen
arbitrarily without influencing any essential implications derivable from the equation.  This and other 
properties can be confirmed by pure reasoning, based on the structure of the equation, or by explicit 
partially-numerical factorization of the left hand side into the product of four factors linear in $p_m$, 

These considerations have removed some, but not all of the sign ambiguities introduced by the quadratic 
substitutions used in the derivation of Eq.~(\ref{eq:AbbrevFieldStrengths.3-rev}). 
The electric field can still be reversed, thereby doubling the set of solutions of the equation. 
Note that this change cannot be compensated by switching the sign of $q$, which also reverses the 
magnetic bending.

It is important also to realize, in cases where efficient polarimetry is available, that \emph{the sign of 
$B$ (or $E)$ can be reversed or reset to high accuracy without the need even to measure $B$ (or $E$).} 
This feature, exploiting the constancy of electron MDMs, has enabled the masses of all resonances discovered 
in electron/positron colliders to be determined with such high accuracy.

\subsection{Electric and magnetic contributions to spin tune}
The combined field spin tune $Q_s$ can be expressed in terms of the fractional bending coefficients;
\begin{equation}
Q_S = \eta_EQ_E + \eta_MQ_M.
\label{eq:BendFrac.6} 
\end{equation}
Superimposed electric and magnetic bending permits beam spins to be frozen 
``frugally''; i.e. with a ring smaller than would be required for all-electric 
bending; for spin tune $Q_S$ to vanish requires
\begin{equation}
Q_S = \eta_{_E}Q_E + (1-\eta_{_E})Q_M = 0.
\label{eq:SpinPrecess.5m}
\end{equation}
Solving for $\eta_{_E}$ and  $\eta_{_M}$, 
\begin{equation}
\eta_{_E} = \frac{G\gamma^2}{G+1}, \quad
\eta_{_M} = \frac{1+G(1-\gamma^2)}{G+1} = \frac{1-G\beta^2\gamma^2}{G+1}.
\label{eq:SpinPrecess.5}
\end{equation}
For example, with proton anomalous magnetic moment $G_p=1.7928474$, trying $\gamma=1.25$, 
we obtain $\eta_{_E} = 1.000$ which agrees with the known proton 
233\,Mev kinetic energy value in an all-electric ring. 
For protons in the non-relativistic limit, 
$\gamma\approx1$ and $\eta^{\rm NR}_E \approx2/3$.

The electric/magnetic field ratio for the primary beam to be magic is
\begin{equation}
\frac{\eta_E}{\eta_M} = 
\frac{E_0/\beta}{cB_0} = \frac{G_1\gamma_1^2}{1-G_1\beta_1^2\gamma_1^2}.
\label{eq:SpinPrecess.5pp}
\end{equation}
For given $\beta_1$, along with this equation and the required bend radius $r_0$, this fixes the 
electric and magnetic fields to the unique values that globally freeze the primary beam spins.
With $1\rightarrow2$ subscript replacement, the same frozen beam formulas apply to the secondary 
beam; note, though, that the $\beta$ factor has opposite sign. To be ``doubly-magic'' both beams must 
satisfy this relation.  At this time the only known dual solutions are $(p,p)$, $(p,h)$, and $(p,e+)$.

\section{Prototype ring (PTR) EDM measurement capability}
\subsection{Toroidal focusing PTR lattice design}
\emph{Responsible planning} for an eventual ``{Holy Grail''} all-electric EDM storage ring with simultaneously 
counter-circulating $\mathcal{E}=232.8$\,MeV frozen spin proton beams \emph{requires the construction of a prototype 
ring (PTR).} To maintain flexibility the PTR lattice design initially is ``scale-invariant'' and site independent.
But, for numerical examples: $\mathcal{C}$=102.5\,m, roughly 1/8 scale relative to the Holy Grail ring. Except for 
construction itself, PTR needs only present day technology and apparatus available in the COSY facility.

Though planned initially as a prototype, and constrained by cost minimization compromised performance, PTR
will, itself, be capable of the performance promised in the title of the present paper.  \emph{This is largely
the result of improved understanding of the {\rm EDM} measurement capability improvement produced by spin
tune phase-locking, understanding the {\rm BSM} power of measuring the difference of proton and 
helion {\rm EDM}s, and recognizing that ``quadrupole free'' toroidal focusing will assure long spin coherence 
time, especially with betatron phase space ``mixing'' provided by running on the ``difference resonance''.}

There is a substantial literature describing the physical importance of measuring EDMs of 
elementary particle and the all-electric storage rings needed for this purpose.  The most recent
comprehensive report is due to the CPEDM collaboration, based mainly on experience of the JEDI group using the COSY 
ring in Juelich, Germany, and described in a CERN Yellow Report\cite{CYR} (referred to here as ``CYR'') 
feasibility study for a storage ring to search for EDMs of charged particles.

Measurement of the proton EDM relies on the development of a storage ring with predominantly electric bending, 
capable of storing simultaneously counter-circulating frozen spin proton beams.  
The CYR analyses the feasibility and states that a low energy (as economical as practical) prototype EDM ring,
here referred to as ``PTR'', is needed to investigate problematical issues. The Phase~I goals for PTR are, 
first, to store an intense (not necessarily polarized) proton beam and, second, to store a simultaneous 
counter-circulating beam. Phase~II will superimpose magnetic bending in order to store and freeze an intense 
frozen-spin 45\,MeV proton beam.  

There is considerable overlap between Chapter~7 of the CYR report and the present paper.  
FIG~\ref{fig:SectorPerspective} is a perspective mock-up of one sector of PTR copied
from that report.  Though the PTR superperiodicity is $N_{\rm super}=4$, it is sufficient 
for most purposes to display just one octant. The ring is also forward/backward symmetric. 
There have been substantial lattice design changes subsequent to the CYR manuscript commit  date.  
The currently revised design is exhibited in Fig.~\ref{fig:PTR-layout-Toroidal8_102p2-mod}.

Quadrupoles are shown but, as mentioned previously, they are intended to be as weak as possible,
consistent with their main roles, which are to enable operational procedures requiring large
changes in ring optics, and to compensate for any unexpected systematic optical deficiencies
associated with the fabrication of the toroidal electrodes.

Based on much the same optics that has enabled scanning transmission electron microscopy (TEM and STEM) of
such spectacularly acute spatial resolution, toroidal focusing can be expected (for the same reason) to provide
exceptionally long spin coherence times (SCT) for polarized beams.  This can be expected to enable experimentation 
into the role played by particle spin in time-reversed situations.

The decision to rely entirely on toroidal focusing, carries with it a significant operational 
penalty resulting from the fact that, once fabricated, no smooth way of tuning the toroidal 
focusing $m$ value has been established. Theoretical discussion of these issues in detail is
contained in companion paper\cite{RT-Positron-q}.  Cancellation of spin polarization decoherence
is greatly enhanced by toroidal focusing.  As mentioned at the start of this paper there is
considerable overlap of that paper with this paper. Here we list only essential references from 
that paper: 
\cite{ConformalHandbook},
\cite{AlbrechtToroidal},
\cite{ToffolettoLeckeyRiley},
\cite{CourantSnyder},
\cite{CourantDirichlet},
\cite{Erni-TEM},
\cite{Good},
\cite{Magiera},
\cite{Nehari}.
In combination, along with other technical details, these papers explain how the requirement 
of conformal invariance is satiisfied by the PTR design in the present paper. 

The dominant bending and focusing is performed by the red and blue toroidal sectors shown
in Fig.~\ref{fig:PTR-layout-Toroidal8_102p2-mod}.  The $m$-values are displaced slightly, with 
$m_{blue}<=m_{red}$. The purpose for this is to preserve 
flexibility while the focusing design is being refined.  At the time of final construction it 
will be preferable for red and blue sectors to be identical.  Otherwise, every sector bend has 
identical superimposed electric and magnetic bending.  As fabricated, all magnet bends 
will probably be powered in series (with possible small shunt compensation) and all electrode
voltages only slightly adjustable (for closed orbit steering). 

In spite of the small number of adjustable parameters, there is sufficient flexibility to tune PTR
to some of the operational MODES mentioned in previous reports.  Computationally the optical design
has been performed using a MAPLE-BSM code, where ``BSM'' is an acronym for ``Belt and Suspenders, 
Modifiable''.  The lumped quadrupoles can provide the somewhat stronger (separated function) focusing 
needed for robust ring commissioning operations.  However, a COLLIDING BEAM mode described in previous 
reports cannot be achieved.  This is because backward/forward symmetric ``low beta squeeze''
(for increased luminosity) is impractical with superimposed electric and magnetic bending.  

As well as violating forward/backward lattice function symmetry, lumped quadrupoles inevitably
introduce chromaticity into ring optics.  Customarily in storage rings this chromaticity is
compensated by one, or typically, at least two, families of sextupoles.  Such sextupoles prove to
be acceptable for unpolarized beams, but their presence impairs the ring performance for polarized
beams. Their presence is the leading cause of spin precession decoherence, which limits the
spin coherence time (SCT).  For ultimate EDM measurement precision, it will therefore be essential
for the lumped quadrupoles to be as weak as possible.

As mentioned previously, the various lattice optical designs are scale invariant and site-independent.
But, to make lattice parameters and plots more intuitive, the ring circumference is taken to be 
102.5\,m, matched exacly to the ring circumferences of the proposed bunch accumulator (BA) described 
below.  As fixed by this circumference, in all subsequent lattice function plots, the horizontal axis 
coordinate is the longitudinal position coordinate $s$. 

\begin{figure}[hbt]
\centering
\includegraphics[scale=0.5]{pdf/ToroidalShape.pdf}
\caption{\label{fig:ToroidalShape}Copied from Albrecht\cite{AlbrechtToroidal} who provided the original
toroidal focusing formulation, this figure shows field lines and and equipotential contours in the central
horizontal ring plane for the toroidal coordinate system used in developing the Wollnik transfer matrices 
used in the present paper. Electrodes cut from the full toroid surface are shown by bolder curve sections.
Electrically isolated, they are held at equal but opposite electric potentials, with electric field lines
joining exactly normal at both electrodes.  Electrodes, and normal entry particle trajectories 
are perfect circles.  The Wollnik linear transfer matrices used in the present paper correspond to 
propagation through the good field regions between these electrodes.  }
\end{figure}

\subsection{EDM measurement strategy}
As regards the orientation of the beam polarization, it is essential to distinguish between ``in-plane'' and 
``out-of-plane'' orientations, where ``the plane'' refers to the ring beam plane, which is presumed to be 
horizontal.  \emph{In-plane precession}, is routinely induced by ideal (rest frame) magnetic fields acting 
on beam particle magnetic dipole
moments (MDMs).  Ideally, \emph{out-of-plane precession} can be induced only by P- and/or T-violating symmetry violating
torques due to electric fields acting on particle EDMs.  But, in practice, the inevitable existence of unintentional
magnetic fields acting on particle MDMs can also induce out-of-plane precession.  The average radial magnetic field
$\langle B_r\rangle$ is expected to be the dominant source of systematic error in the determination of particle EDMs.

The leading strategy for reducing this source of systematic error is to produce simultaneously counter-circulating 
frozen spin beams on ``identical'' orbits.  This condition is to be achieved by adjusting local vertical beam deflection
corrections to cause the counter-circulting orbits to be identical.  This cancels the dominant systematic error source 
$\langle B_r\rangle$ by canceling the average out-of-plane (vertical) orbit separation. We refer to this capability as  
``self-magnetometry''. The precision with which the orbits can be matched vertically depends on the precisions of the 
beam position monitors (BPMs) that measure the vertical beam positions, and on the ring lattice sensitivity to the 
magnetic field errors causing the orbits to be vertically imperfect. 

Assuming both beam spins are frozen, at least the ``primary'' beam-1 will, by convention,  
be globally frozen; spin tune $Q_{s1}=0$.  Ideally 
both beams would have $Q_s=0$ but, ordinarily, the ``secondary'' beam-2 will be only locally frozen; $Q_{s2}$ exactly equal 
to a rational fraction other than $0/1$.  In this condition both beam polarizations can be phase-locked,  allowing both
beam spin tunes to be set and re-set with frequency domain precision, meaning that synchronism can be maintained for runs 
of arbitrary duration.  Since the RF frequency can also be restored to arbitrarily high precision, conditions 
can be set 
and  re-set repeatedly, without depending upon high precision measurement to the electric and magnetic bend fields.
This allows, for example, the magnetic bending field to be reversed with high precision, as would be required to 
interchange CW and CCW beams. 
This capability can be referred to as \emph{stabilizing all fields  by phase locking both revolution frequencies
and both beam polarizations, using their own MDMs as ``magnetometric gyroscopes''}.

\noindent
{\bf Encapsulation of the EDM measurement strategy: }
\begin{itemize}
\item 
 Use threefold phase locking:
     \begin{enumerate}
     \item Primary beam~1, CW,  \emph{globally frozen}
     \item Secondary beam~2, CCW, in general, locally ($Q_s=$ non-zero rational fraction) frozen
     \item Synchronous bunch capture onto almost identical orbits with different ``best matched'' 
           beam~1 and beam~2 harmonic numbers in a common RF cavity,
     \end{enumerate}
\item
which accurately matches beam~1 and beam~2 orbits.
\item
and allows conditions to be reset precisely, including magnetic or electric field reversal, 
without the need for unachievably high precision direct electric and magnetic field measurement.
This supports averaging over interchange of conditions to reduce systematic error. 
\end{itemize}

\subsection{High intensity beam injection from CLIP to replace bunch accumulation using BA}
Fig.~\ref{fig:CLIP-p-accelerator-pic} displays a conjectured placement of PTR in the polarized beam 
injection complex proposed for the BNL CLIP project. Fig.~\ref{fig:CCL_Comparison} provides parameters and 
performance information of the proposed LIL linac. The anticipated polarized injected beam currents from LIL 
are a factor of roughly 6000 time larger than from the COSY cyclotron, which is comparable with the 10,000 
times increase in bunch current promised by the BA bunch accumulator shown in Fig.~\ref{fig:COSY-hall-mod7}. 

Furthermore CLIP promises a gourmet menu of polarized nuclear isotopes, including most, if not all, shown
on Koop's menu, Fig.~\ref{fig:KoopParameters}.  Most exciting of all the choices is the combination
of protons, $p$, and helions, $h$.  It is this pairing that can provide the most precise test of BSM physics
that is likely to be achievable in the quite near future. Furthermore the range of available beam energies
is luxurious enough to require no detailed discussion.

As mentioned previously, co-magnetometry with PTR can be expected to permit convenient and precise MDM 
measurements for any nuclear isotope having lifetime in excess of a minute or so.  Because of its
concentration on the proton/helion EDM difference measurement, this capability is not
discuused further in the present paper.
\begin{figure}[hbt]
\centering
\includegraphics[scale=0.6]{pdf/CLIP-p-accelerator-pic.pdf}
\includegraphics[scale=0.18]{pdf/PTR-in-CLIP.pdf}
\caption{\label{fig:CLIP-p-accelerator-pic}{\bf Above:\ }CLIP sector layout; 
{\bf Below:\ }PTR situated in vicinity of CLIP Building 1.}
\end{figure}
\begin{figure}[hbt]
\centering
\includegraphics[scale=0.55]{pdf/200_MeV_LINAC.pdf}
\includegraphics[scale=0.55]{pdf/CCL_Comparison.pdf}
\caption{\label{fig:CCL_Comparison}200\_MeV\_LINAC parameters.}
\end{figure}

\clearpage

\section{Lattice design and performance}
\subsection{Ideal thick lens toroidal focusing\label{sec:ThickToroidalFocusing}}
Transfer matrices from one symmetry point to the next of a storage ring with superperiodicty $N_s$ 
have the form
\begin{equation}
{\bf M_1} = 
\begin{pmatrix} \cos\mu        & \beta\sin\mu     \\
         -\frac{\sin\mu}{\beta} &  \cos\mu   
\end{pmatrix}
 \equiv
\begin{pmatrix} C         & \beta S     \\
         -\frac{S}{\beta} &       C 
\end{pmatrix},    
\label{eq:ThickToroidal.1}
\end{equation}
where $\mu$ can stand for $\mu_x$ or $\mu_y$, $\beta$ can stand for $\beta_x$ or $\beta_y$
and similarly for $C\equiv\cos\mu$ and $S\equiv\sin\mu$. Squaring ${\bf M_1}$ and
applying trigonometric identities produces
\begin{equation}
{\bf M_2}\equiv {\bf M_1^2}  
 =\begin{pmatrix} \cos2\mu      & \beta\sin2\mu \\
         -\frac{\sin2\mu}{\beta} &  \cos2\mu   
\end{pmatrix}.
\label{eq:ThickToroidal.2}
\end{equation}
Squaring again and applying trigonometric identities produces
\begin{equation}
{\bf M_4}\equiv {\bf M_1^4}  
 =\begin{pmatrix} \cos4\mu      & \beta\sin4\mu \\
         -\frac{\sin4\mu}{\beta} &  \cos4\mu   
\end{pmatrix}.
\label{eq:ThickToroidal.3}
\end{equation}
By applying these formulas separately to horizontal, $x$, and vertical,$y$ transverse coordinates produces
\begin{equation}
{\bf M^x_4}\equiv {{\bf M^x_1}^4}  
 =\begin{pmatrix} \cos4\mu_x      & \beta_x\sin4\mu_x \\
          \frac{\sin4\mu_x}{\beta_x} &  \cos4\mu_x   
\end{pmatrix}
\quad\hbox{and}\quad
{\bf M^y_4}\equiv {{\bf M^y_1}^4}  
 =\begin{pmatrix} \cos4\mu_y      & \beta_y\sin4\mu_y \\
         -\frac{\sin4\mu_y}{\beta_y} &  \cos4\mu_y   
\end{pmatrix}.
\label{eq:ThickToroidal.4}
\end{equation}
\emph{What makes these formulas fascinating is that the parameters $\beta_x$ and $\beta_y$ can be 
chosen arbitrarily, and not necessarily equal.}  This is important for maintaining conformal mapping validity with 
a substantial fraction of the horizontal focusing being ``geometric'' which does not contribute to vertical focusing.
Then, by constraining the betatron phase advances $\mu_x$ and $\mu_y$, for example as
\begin{equation}
\mu_x = \mu_y + \frac{\pi}{2},
\label{eq:ThickToroidal.5}
\end{equation}
one obtains simpler relations.  In this case 
\begin{equation}
4\mu_x = 4\mu_y + 2\pi  \overset{eff.}{\ =\ } 4\mu_y.
\label{eq:ThickToroidal.6}
\end{equation}
In this paper lattice design meeting this constraint is referred to as
``ideal, thick lens toroidal focusing''.   
Even simpler for tuning the PTR ring would be the choice $\mu_x\approx\pi/2$, in which case
\begin{equation}
{\bf M^x_4}
 \approx\begin{pmatrix} 1      & \beta_x\sin4\mu_x \\
       -\frac{\sin4\mu_x}{\beta_x} &  1   
\end{pmatrix}
\quad\hbox{and}\quad
{\bf M^y_4}
 =\begin{pmatrix} 1      & \beta_y\sin4\mu_y \\
       -\frac{\sin4\mu_y}{\beta_y} &  1   
\end{pmatrix},
\label{eq:ThickToroidal.7}
\end{equation}
which are close to identity matrices which, in linear approximation, describe drifts of length 
$\beta_x\sin4\mu_x$ and $\beta_y\sin4\mu_y$ with weak focusing components) 
where $\beta_x$ and $\beta_y$ are independently adjustable, and all matrix elements are simply 
expressible in terms of the two parameters, $C$ and $S$, defined in Eq.~(\ref{eq:ThickToroidal.1})
and satisfying the condition $S^2+C^2=1$.  

Currently the PTR lattice design shown in Fig.~\ref{fig:PTR-layout-Toroidal8_102p2-mod} 
applies these formulas only up to ${\bf M^x_2}$ and ${\bf M^y_2}$.  For operational flexability, the ring
optics is then controllable by the triplets (located at symmetry points) and masquarading as superimposed 
electric/magnetic quadrupoles.  In this sense the ring behavior will resemble a separated function
FODO lattice with free, arbitrarily strong,  $F$ and $D$ focusing strength parameters. (All of this has 
to be consistent with $F$ and $D$ being turned off as nearly as possible during actual EDM measurement.)

\subsection{Transfer matrix for beam evolution through toroidal bending elements\label{sec:Wollnik}}
The content of this paper have been limited to this point, mainly to resolving issues identified in the
(pre-Covid) CERN Yellow Report.  ``Covid-enforced leisure'' has resulted in the revolutionary promotion
of PTR from ``EDM prototype ring'' to ``ambitious research project'' as explained so far and in the rest of
the paper.

All-electric bending fields exist in the tall slender gaps 
between inner and outer, almost-plane-vertically, curved-horizontally, 
electrodes. The radial electric field dependence is 
\begin{equation}
E= E_r\sim\frac{-1}{r^{1+m}},
\label{eq:WW-AG-CF.1}
\end{equation}
Our convention has the electric field $E_0$ being negative since the nominal beam charge
is positive and the force is centripetal. 
The corresponding electric potential parameterization specializes to inverse power law 
dependence on the bend radius $r$ with 
the electric potential defined to vanish on the design orbit;
expressed as power series in $\xi=x/r$, it is given by
\begin{align}
V(r)
 &=
\frac{E_0r_0}{ m }\,
\bigg(
(1-\xi)^m-1
\bigg)  \label{eq:WeakFoc-mod.2}\\
 &=
-E_0r_0
\bigg(
\xi + \frac{1-m}{2}\xi^2 + \frac{(1-m)(2-m)}{6}\xi^3\dots
\bigg).
\notag
\end{align}
This formula simplifies spectacularly for the Kepler $m$=1 case.  As it happens, for relativistic particles, 
$m=1$ also marks the edge of a region to which the term ``toroidal optics'' applies. Beyond this edge 
orbits spiral in toward the center.  The other edge is marked by the $m=0$ ``cylindrical edge''; beyond this edge 
vertical orbits diverge to infinity.  These edges are defined more accurately below by Eqs.~(\ref{eq:Wollnik-TM-params}).

For focusing index in the (approximate) range $0 < m < 1$ linearized horizontal evolution of initial displacement 
vector ${\bf V_{x0}}$ through a thick bending and focusing element element of length $w$ is given by 
\begin{equation}
 {\bf V_{x1}} = {\bf M_4}{\bf V_{x0}} = 
\begin{pmatrix} c_x     &  s_x    & 0 &  d_xN_t    \\
              -k_x^2s_x &  c_x    & 0 & s_xN_t/R_0 \\
            -s_xN_t/R_0 & -d_xN_t & 1 &  -N_t^2    \\
                0      &    0    &   0   &    1    
\end{pmatrix}{\bf V_{x0}},
\quad\hbox{where}\quad 
{\bf V_{x0}} = \begin{pmatrix} x_0 \\ \theta_{x0} \\ s \\ \delta p /p \end{pmatrix}
\label{eq:Wollnik-TM-x}
\end{equation}
\begin{equation}
{\bf V_{y1}} = {\bf M_2}{\bf V_{y0}} = 
\begin{pmatrix}   c_y    &  s_y       \\
               -k_y^2s_y &  c_y       
\end{pmatrix}{\bf V_{y0}},
\quad\hbox{where}\quad 
{\bf V_{y0}} = \begin{pmatrix} y_0 \\ \theta_{y0} \end{pmatrix}.
 \label{eq:Wollnik-TM-y}
\end{equation}
the parameters are given by \footnote{(Just in this section) Lebedev's $R_0$ replaces 
$r_0$ as bend radius.} 
\begin{equation}
k_x = \frac{1}{R_0}\sqrt{1 - m + \frac{\eta_E}{\gamma^2}},\quad
k_y = \frac{1}{R_0}\sqrt{m},\quad
N_t = 1 + \frac{\eta_E}{\gamma^2}, 
\label{eq:Wollnik-TM-params}
\end{equation}
and the transfer matrix components for elements of length $L$ are expressed in terms of expressions
\begin{equation}
c_x = \cos(k_xL),\ s_x = \sin(k_xL),\ d_x = \frac{1-c_x}{R_0k_x^2},\ t_d=\frac{k_xL-s_x}{R_0^2k_x^2},\\
c_y = \cos(k_yL),\ s_y = \sin(k_yL).
\label{eq:MissingEqName}
\end{equation}
Due originally to Wollnik\cite{Wollnik}, but here expressed in more conventional modern notation, these
formulas have been copied from Lebedev\cite{Lebedev}, except for having been further generalized to allow superimposed
electric and magnetic bending.
\footnote{Subtle detail: ring tunes estimated from transfer matrices based on accepting 
large values of $L$ in advance in Eqs.~\ref{eq:MissingEqName} may be seriously incorrect due to incorrect aliasing with 
thick element focusing. Rearranging Eqs.(4.6) from reference\cite{RT-ICFA}, and solving for $\tan\psi$ 
produces an expression for phase $\psi(s)$ in terms of matrix elements evaluated locally, for example at arbitrary origin;
\begin{equation}
\psi(s)
 = 
\tan^{-1}\,
\frac{{\bf M}_{1,2}(0, s)}
     {\beta_0{\bf M}_{1,1}(0, s) - \alpha_0{\bf M}_{1,2}(0, s)}.
\label{TM.17}
\end{equation}
Like all inverse trigonometric formulas, this equation has multiple solutions.
But, with the {\tt .sxf} granularity being required to be fine enough, one
can (in principle) sequentially obtain unique phases. With
$\psi(s)$ starting from $\psi(0)=0$, as $s$ increases
from $s_i$ to $s_{i+1}$ there is a unique solution of Eq.~(\ref{TM.17}),
$\psi(s_i)\ge\psi(s_{i-1})$ such that the function $\psi(s)$ increases 
monotonically, as required.
Because the interval from $s_{i=1}$ to $s_i$ is non-zero, $\psi$ can,
superficially, advance discontinuously; the correct solution is the least
discontinuous. The same calculation has to be done for both the $x$ 
and $y$ betatron sectors. 
Even though, theoretically, the phase advances 
monotonically, numerical errors can cause Eq.~(\ref{TM.17}) to
give local phase decrease. Requiring betatron phase $\psi(s)$ to increase monotonically
(as shown most satisfyingly in Fig.~\ref{fig:FDDF-ToroidalQuadrant-betas}) can cause $\beta(s)$ shape
variation to seem curious.  This behavior was encountered especially while producing 
Fig.~\ref{fig:0p32349-0p99990-0p00909}.  This issue is pursued further in 
Appendix~\ref{sec:Aharonov-Anandan}.}

This accounts for the appearance of $\eta_E$ in
Eqs.~(\ref{eq:Wollnik-TM-params}). (Recall that $\eta_E+\eta_M=1$, noting that one or the other of $\eta_E$ and 
$\eta_M$ can be negative, in which case the centripetal forces combine ``destructively''. 
But the fractions must sum to 1 to produce the correct $E/B$ bend ratio.)

Operating on orbit displacement vector $V$, $MV(L)$ describes, to linear approximation, 
how $V$ has evolved after propogation through bend element of length $L$.
For pure magnetic bending $\eta_E=0$.  For partial electric bending the $1+\eta_E/\gamma_0^2$ combination, a time
dilation correction factor to the ``geometric focusing'' supplied by the curvature, approaches 1+$\eta_E$  in the 
nonrelativistic limit. 

Notice that there is dependence, in Eqns.~(\ref{eq:Wollnik-TM-params}), 
on bending radius $R_0$ in both $k_x$ and $k_y$ and dependence on $\eta_E$ in $k_x$, though not in $k_y$.  
In the present paper the absence of dependence of $k_y$  on $\eta_E$ is referred to as ``vertical only, geometricization''. 
This means that vertical focusing, and therefore $\beta_y$, is independent of beam momentum (provided the
bending force matches the design ring radius). 
\emph{But horizontal focusing depends on beam momentum (or, rather,  momentum/charge) through its 
dependence of $k_x$ on $\eta_E$}.  

In the case of doubly-frozen or doubly-magic 
counter-circulating beams, the vertical focusing is the same for both beams, independent of beam 
circulation direction.  But the horizontal lattice optics depends on beam momentum and is different for
counter-rotating beams.  Fortunately, for motion in a purely horizontal plane, both spin tunes 
$Q_x$ and $Q_y$, as defined in Eqs.~(\ref{eq:BendFrac.7}), are ``constants of the motion'',
independent of horizontal betatron oscillation amplitude to all orders.  By symmetry, the spin tunes are 
independent of vertical betatron oscillation amplitude, but only to linear order.  This makes it 
important, therefore, for $\beta_y(s)$ to be exactly independent of beam circulation direction, as it is. 
(This comment is pursued in Appendix~\ref{sec:Hybrid} which discuusses the so-called
``hybrid option'' for ring focusing.) 

The matrix ${\bf M_2}$ has determinant 1, which is the only requirement for a 2x2 matrix to be 
symplectic.   Block diagonalized into 2x2 matrices, the blocks 
on the main diagonal of  ${\bf M_4}$ are individually symplectic, and the determinant of the blocks on the 
other diagonal vanish. Taken together, these conditions show that ${\bf M_4}$ is also ``symplectic''.

The quotation marks here are intended to serve as a warning that, iterated indefinately, 
 ${\bf M_4}$ is actually divergent.  The lower two rows of ${\bf M_4}$ include ``chromatic'' 
i.e. momentum-dependent effects---primarily dispersion and time-of-flight. Another idiosyncrasy of 
accelerator formalism is that (based on a fast-slow approximation) these chromatic effects are counted 
as ``second'' rather than ``first'' order effects.  This is related to the fact that it is only the orbit 
shape that can be said to be stable for all time.  (No meaning whatsoever is attached to orbit position
as a function of time.) 

The imporatant feature of conformally-mapped toroidal optics is that ``everything of importance'' electrodes, 
equipotentials, on and off-momentum closed orbits remain perfect circles.  The divergent behavior of iterating 
${\bf M_4}$, represents the longitudinal beam spreading associated with the spread of particle velocities.
So a bunched beam will gradually ``de-bunch'', eventually becoming a uniform ``coasting beam.  As established by 
McMillen and Veksler, this debunching can be prevented by using an RF cavity to capture all (or most) of the
particles into stable longitudinal ``buckets''.

\section{PTR lattice optics\label{sec:PTRLatticeOptics}}
\subsection{Parameters for PTR situated in the COSY beam hall}
Parameters and lattice function values for the lattice design illustrated in 
Fig.~\ref{fig:PTR-layout-Toroidal8_102p2-mod} are given in Table~\ref{tbl:MODEparameters2}. Included in this
table are parameters appropriate for Fig.~\ref{fig:COSY-hall-mod7}, which shows the
positioning of PTR and ``BA'' which is a ``bunch accumulater ring'' needed to prepare polarized beam
bunches for injection into PTR.  

\setlength{\tabcolsep}{2pt}
\begin{table}[h]\small
\caption{PTR and COSY-arcs-only bunch accumulater (BA) parameters. \label{tbl:MODEparameters2}}
\centering
\begin{tabular}{ccccc}             \hline
file name             & variable         & unit &    BA             &     PTR        \\ 
                      &  name            &      &  COSY-arcs-only   &               \\ \hline
circumference         & {\tt circum}     &  m   &   102.250         &    102.250     \\ \hline
bend radius           &  {\tt r0}        &  m   &                   &     11.0       \\
E fld., 30 MeV prot.  &  E               & MV/m &                   &    5.370       \\ \hline 
long strt. length     &  {\tt llsnom}    &  m   &                   &    4.142        \\
tot. avail. strt. sec. & $16\times${\tt llsh} & m &                 &     32          \\
electrodes/quadrant   &                  &      &                   &      4         \\
bend/electrode        & {\tt Thetah}     & radian &                 &  $2\pi/16$     \\
electrode length      & {\tt Leh}        &  m   &                   &   4.32       \\ \hline 
\rowcolor[gray]{.9}
PTR stored p's no BA  &                  &      &                   & $0.6\times10^7$ \\ 
\rowcolor[gray]{.9}
COSY-arcs-only BA     &                  &      & $0.6\times10^{11}$ & $0.6\times10^{11}$ \\ \hline
min/max horz. beta    &   $\beta_x$      &  m   &                   &  9.60/10.83   \\
min/max vert. beta    &   $\beta_y$      &  m   &                   &  16.6/24.8         \\ \hline
horizontal tune       &    $Q_x$         &      &                   &  1.726        \\
vertical tune         &    $Q_y$         &      &                   &  0.673       \\ \hline 
\end{tabular}
\end{table}


\subsection{``No lumped quadrupole'' toroidal optics lattice functions}
Figure~\ref{fig:FDDF-ToroidalQuadrant-betas} illustrates the result of lattice tuning in a storage ring with
``no lumped quadrupoles''.  The quotation marks here acknowledge that PTR will, in fact, have lumped quadrupoles.
But their srengths will be adjustable such that (under unrealistically ideal conditions) their strengths
will be tuned to exactly zero. The tuned-up situation is shown in the middle row of the figure; 
the upper/lower rows represent varying $m_{\rm nom}$ (where $m_{\rm nom}$ is another name for the $m$ parameter 
appearing in Eqs.~(\ref{eq:Wollnik-TM-params})) below/above the tuned-up condition.  For fixed ring geometry, 
quadrupole free, toroidal focusing required for long term stability in both horizontal and vertical planes 
depends on the value of this single parameter $m_{\rm nom}$.  

From the point of view of operational practiality, this behavior is problematical in two ways; once constructed,
neither the ring geometry nor the value of $m_{\rm nom}$ can be tuned conveniently.  One can contemplate altering  
$m_{\rm nom}$ by using a smooth mechanical way of adjusting the vertical curvature of the electrodes producing
the electric field---the horizontal curvature is fixed by the constant bending radius constraint---but no satisfactory
design exists for adjusting the electrode shape. This is because an electrode supple enough to be easily adjustable, 
would almost certainly be insufficiently rigid to guarantee the long term mechanical and thermal shape 
constancy required for precision co-magnetometry.  The absence of lumped quadrupoles would mean that the entire ring 
design has to be ``dead-reckoned''.
\begin{figure}[hbt]
\centering
\includegraphics[scale=0.60]{pdf/FDDF-ToroidalQuadrant-betas-0p19947.pdf}
\includegraphics[scale=0.60]{pdf/FDDF-ToroidalQuadrant-tune-advances-0p19447.pdf}
\includegraphics[scale=0.60]{pdf/FDDF-ToroidalQuadrant-betas-0p29947.pdf}
\includegraphics[scale=0.60]{pdf/FDDF-ToroidalQuadrant-tune-advances-0p29447.pdf}
\includegraphics[scale=0.60]{pdf/FDDF-ToroidalQuadrant-betas-0p39947.pdf}
\includegraphics[scale=0.60]{pdf/FDDF-ToroidalQuadrant-tune-advances-0p39447.pdf}
\caption{\label{fig:FDDF-ToroidalQuadrant-betas} PTR optical functions for one quadrant centered
at a D-focusing quadrupole straight section, with all quadrupole strengths set to zero. 
{\bf Left:\ }beta functions 
plotted aginst longitudinal coordinate $s$; {\bf Right:\ }betatron tune advances plotted aginst 
longitudinal coordinate $s$; full ring tunes are four times the right hand intercepts. Regarded 
as a spectrometer, the upper case quadrant is ``under-focused'' the bottom case ``overfocused''.  
The central row is approximately matched to $\alpha_{\rm D}=0$ at the crossing point with all 
quadrupoles turned off. From this condition the full ring would be stablized with very weak 
quadrupole strengths.}
\end{figure}


For this reason the quadrupoles shown in Figure~\ref{fig:FDDF-ToroidalQuadrant-betas} are critical to the design
but need to be accurately tuneable to small vlues, consistent with achieving adequate ring stability.  
One might reasonably expect at least one iteration in which, having first established the quadrupole settings 
needed for stability, the electrodes could be mechanically deformed during a period of ring shutdown to enable 
the quadrupoles to be weaker.

This loss of operational flexibility has to be regarded as a serious impediment to achieving pure toroidal
focusing.  Nevertheless, the (eventual) benefits of toroidal focusing are so great, especially long SCT, that this 
``inconvenence'' has to be accepted.   Fortunately it is not difficult to make the quadrupoles strong enough to 
overcome realistically large initial design, fabrication, and positioning errors.  

\section{Forward and backward traveling ring optical functions\label{sec:BackwardForward}}
Forward and backward traveling beam optics dependence is illustrated in Fig.~\ref{fig:etaE-dependence}.
If the focusing is frugal in the forward direction, for example, $\eta_E=0.7$, $\eta_M=0.3$, then it
is extravagant, approximately $\eta_E=1.3$, $\eta_M=-0.3$ in the backward direction.
\begin{figure}[htb]
\centering
\includegraphics[scale=0.57]{pdf/0p29447_0p70_0p00909.pdf}
\includegraphics[scale=0.57]{pdf/0p29447_0p99990_0p00909.pdf}
\includegraphics[scale=0.57]{pdf/0p29447_1p30_0p00909.pdf}
\caption{\label{fig:etaE-dependence}PTR beta functions with $m_{nom}=0.29447$, 
corresponding to the ``tuned-up'' value for the central case in Fig.~\ref{fig:FDDF-ToroidalQuadrant-betas}.
In this configuration, in spite of being ``thick lenses'' the intervening bends act like drift sections
(of length not equal to the arc length in general).
Optically the full ring then acts like a separated function FODO lattice with alternating gradient
lumped quadrupole with focusing strength $del_q=0.1/r_0=0.009091$/m.  Fractional electric field
bending fraction $\eta_E$ varies from 0.7 in the top graph to 1.3 in the bottom.  This variation leaves
$\beta_y$ unchanged, but causes $\beta_x$ to vary (approximately proportional to $1/\sqrt{\eta_E})$.
 }
\end{figure}
Bend parameters for ``doubly magic'' (both spin tunes equal zero) forward proton beam and backward 
traveling helion beams are plotted in FIG.~\ref{fig:DoublyMagic-p-helion}. Similarly, 
FIG.~\ref{fig:MagicPseudoMagic-p-p} shows parameters for a ``doubly frozen'' proton/proton combination, 
on the left, and a  ``doubly frozen''proton/deuteron combination on the right. In these graphs a 
forward traveling, polarized proton beam has momentum such that, with the electric and magnetic fields shown, 
the forward traveling proton spin tune $Q_s$ vanishes; in other words the ``magic'' condition for frozen spin 
is satisfied for the forward traveling protons.  Because the ring elements are backward/forward symmetric the 
lattice optics is satisfactory for both forward and backward beam propagation.  

\begin{figure}[h]
\centering
\includegraphics[scale=0.52]{pdf/DoublyMagic-p-helion.pdf}
\caption{\label{fig:DoublyMagic-p-helion}Doubly magic, simultaneously-circulating frozen spin  proton
and helion (He3 nucleus) beams. The axes register electric ``field'' $E0/v$ and magnetic field $B0$, both
measured in Tesla units. Both axes apply to the ``forward'', $\beta_p\approx0.28$ frozen spin proton beam
indicated by vertical broken line. The $\pm$ magnetic field sign indicates the magnetic force is 
constructive for protons, destructive for helions. The vertical broken line at $\beta\approx0.17$ 
indicates that a simultaneously backward traveling helion beam with that velocity would circulate stably,
also with zero spin tune.
}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.42]{pdf/MagicPseudoMagic-p-p.pdf}
\includegraphics[scale=0.42]{pdf/MagicPseudoMagic-p-d.pdf}
\caption{\label{fig:MagicPseudoMagic-p-p}
Simultaneously counter-circulating beam kinematics in PTR. Left-hand (coarse) scale is $E/v$, 
right-hand (fine) scale is $\pm B$. The $\pm$ factor distinguishes forward/backward. The horizontal scale 
is $\beta=v/c$.  A forward traveling, 49.65\,MeV polarized proton beam has momentum such that spin tune 
$Q_s$ vanishes. A backward traveling (slightly adjustable) 24.73\,MeV proton beam can be tuned, e.g. 
to be pseudo-frozen.}
\end{figure}

On further reflection, forward/backward compatability  may seem 
to be impossible; with superimposed electric and magnetic bending, if the bending is ``frugal'' 
(i.e. constructive) for the forward traveling beam, it is ``extravagant'' (i.e. destructive) for 
the backward traveling beam.  Does this not make it impossible for same particle type, but opposite 
direction beams to counter-circulate simultaneously; one or the other beams would surely be lost. 

No!  There is a ``loop-hole'' in this argument. If both beams are protons, then the particle 
momenta in the two beams can be sufficiently different for their ``$B\rho$ and $E\rho$ values'' 
(confusing accelerator jargon for beam momentum matched to the required
bending) to be appropriate for both beams; i.e. for both beams to have the same radius of curvature.  
The motivation for the $B\rho$, $E\rho$ usage is to ``geometricize'' the transverse optics, effectively 
converting accelerator focusing design into Newtonian optics. In the language of geometric optics, quadrupole 
strengths are expressed as inverse focal lengths.  It is this feature that allows the choice of 
PTR length scale to be deferred\footnote{There is no equivalent trick for ``geometricizing'' 
longitudinal dynamics---parameters such as RF frequency depend on times-of-flight along
particle orbits, a concept not present in geometric optics.}.

For counter-circulating beams having the same particle type, for example protons, one may
have $\beta_p= 0.31304$ for the forward beam, and $\beta_p=-0.31304\times41/47$ for the backward beam.
FIG.~\ref{fig:MagicPseudoMagic-p-p} contains graphical descriptions by which the parameters of
simultaneously counter-circulating beams can be established. This requires reduced momentum for
a CCW proton beam that can counter-circulate in a ring with the same accelerator settings. These 
settings could, for example, freeze the spins of the CW beam. This would still leave some freedom
concerning the spin tune of the CCW beam.  Though the CCW beam could not be arranged to have 
globally frozen spins it could, in some cases, be adjusted to be 
pseudo-frozen.\footnote{With counter-circulating beams of different particle type, PTR could
serve as an ``MDM comparator''. This could be used, for example, to make a highly precise measurement 
of the ratio of spin-tunes (and hence MDMs) of two simultaneosly circulating beams.}

Though both beams on the left graph in FIG.~\ref{fig:MagicPseudoMagic-p-p} are protons, the spins of only
one can be frozen---with frugal bending there is only one magic combination of electric and magnetic 
bending.  There remains one freedom, however---for example to pseudo-freeze the counter-circulating beam.
The graph on the right of FIG.~\ref{fig:MagicPseudoMagic-p-p} shows the kinematics of counter-circulting
protons and deuterons.

\section{Canceling decoherence by intentional operation ``on the coupling resonance''}
A powerful strategy for increasing the spin coherence time (SCT) (which is equivalent to 
``canceling spin decoherence'') is to intentionally mix degrees of freedom. Some such cancellation 
occurs automatically in longitudinal synchrotron oscillation, as particle energies systematically cycle
between minimum and maximum valus.  What is proposed in this section 
is to cancel decoherence associated with transverse (i.e betatron) oscillations, by intentionally
mixing horizontal and vertical degrees of freedom.  

For the $\eta_E$=1 optics shown in Section~\ref{sec:BackwardForward} the \emph{fractional} betatron tunes
$Q_x/Q_y=1.726/0.673$ are quite close, meaning close to a difference resonance.  The effect of any skew quadrupole, 
present in the ring either intentionally or unintentionally, is to excite ``out of plane'' betatron oscillations.
As the tunes are brought closer together the amplitude of cross plane excitation increases until, exactly on 
resonance, the beams become round. 

Though $x,y$ coupled sum resonance is fatal, coupled difference resonances are especially stable for dual beam 
stability (though disappointingly not helpfil for high luminosity beam-beam operation, as first investigated 
and established at the Orsay storage e+/e- storage ring ACO\cite{Haissinski} as well as at 
CESR\cite{ChromaticitySharing}\cite{RoundBeams}.)  The optimal ACO tune ratio was $Q_x/Q_y=1.86/2.86$.

With careful tuning the fractional PTR tune equality can be made exact (as well as shifted away from third 
integer resonance, to which $Q_y$ has so far been unacceptably close.  The result of doing this tuning is shown in 
Fig.~\ref{fig:0p32349-0p99990-0p00909}.  Relative to the central case in Fig.~\ref{fig:etaE-dependence} this 
mainly entailed the change: $m_{nom}= 0.29447\longrightarrow0.32349.$ The result of this change is shown in
Fig.~\ref{fig:0p32349-0p99990-0p00909}.  The altered tunes have become equal, as shown at the bottom of the plot.
Otherwise lattice focusing functions have been little changed.
\begin{figure}[hbt]
\centering
\includegraphics[scale=1.0]{pdf/0p32349-0p99990-0p00909.pdf}
\caption{\label{fig:0p32349-0p99990-0p00909}
Lattice functions for essentially all electric bending ($\eta_E=0.9999$), and quite weak lumped quadrupole tuning
with  ($del\_q=0.009091\,{\rm m}^{-1}$).
The horizontal and vertical fractional tunes have been made equal (1.7073). In this condition, by itself, the presence of 
betatron motion causes no beam depolarization.  Raw lattice description files for PTR tuned to this condition are 
available (in four different formats) at 
https://github.com/jtalman/ual1/tree/master/examples/lattices
   /con-xml   /nocon-xml   /adxf    /sxf
/PTR\_m\_nomEQ0p0.32349\_sl4.sxf .
}
\end{figure}

Fig.~\ref{fig:0p32349-0p99990-0p00909} represents ``fully mixed'' long coherence time operation in the
all-electric case but, for realism, with a small, one part in 10,000 magnetic bending component.
Replacing $\eta_E$ by 1.0001 and/or reversing the sign of $del_Q$ in any combination, produces 
essentially the same graph. Replacing $\eta_E$ by 1, however, produces no solution.  This is
of no physical significance---the code failure is caused by zero denominators in the calculation 
of magnetic field influence on the kinematics.

On the coupling resonance any ``cross-plane coupling'' causes the beam to become round (or rather
to have equal horizontal and vertical emittances, because their sum $\epsilon_x+\epsilon_y$ 
is conserved)\cite{CourantSnyder}).  For any particular particle $\epsilon_x$ and $\epsilon_y$ can be 
interpreted as ``Courant-Snyder invariants'' (whose sum is a constant of the motion).   
By virtue of cycling regularly between horizontal and vertical betatron motion, in this condition, betatron 
motion induces the same spin precession for every particle having the same $\epsilon_x+\epsilon_y$ sum.
As a result there is no spin decoherence among all particles lying on any centered ellipse in betatron phase
space---which is to say, every particle.  In this condition, by itself, the presence of betatron motion 
causes no beam depolarization. This has been confirmed by simulation.

The true situation is not quite as simple as has just been described.  With betatron oscillations ignored, 
the presence of dispersion causes the beam to be dispersed radially, with off-momentum bending radius proportional 
to momentum, which is different for every particle.  At the same time there is a similar mixing in the synchrotron 
phase space.  This complicates the previous discussion.  The longitudinal mixing itself is purely beneficial
from the point of view of maximizing SCT.  But the chaotic mingling of transverse and longitudinal mixing
has the potential for introducing unpredictable resonance effects. This possibility will have to be addressed
by simulation or by more sophisticated analysis than has been attempted here.

To the extent that the relativistic $\gamma$ factor is the same for every particle and the resulting motion 
remains purely horizontal, synchrotron oscillation by itself would also not induce depolarization via decoherence.  
Since neither of these conditions is exactly true there will be a systematic precession proportional
to $\gamma$ offset. Since both electric spin tune $Q_E$ and magnetic spin tune $Q_M$ depend
(only) on $\gamma$, there is a strong tendency for spin decoherence to be closely correlated with
energy offset.  To leading order in synchrotron oscillation amplitude, the decoherence assocated with 
synchrotron oscillation would therefore still cancel. Regrettably, witout care, synchrotron oscillations tend to 
be quite nonliear. However, by superimposing weak third harmonic RF cavities, the spin decoherence from
this source can also be cancelled to high accuracy. These effects can also be studied by simulation. 

Optimistically one concludes that polarization decay will be sufficiently cancelled to have become insignificant,
even with no powered sextupoles employed to increase polarization SCT.
This does not yet mean that day-long runs can be anticipated.  The phase-locking on which everything in this paper 
is predicated relies on efficient polarimetry.  Yet, at this time, non-destructive polarimetry does not exist. 
One can assume, therefore, that run durations will be dominated by particle loss associated with 
polarimetry. 

It should be understood that most of the tasks PTR has to perform in its role as EDM prototype do not depend on
subtle effects such as have been discussed in this section, nor on the no-quadrupole requirement for optimizing 
EDM measurement precision that led to the lattice design shown in Fig.~\ref{fig:PTR-layout-Toroidal8_102p2-mod}.
As already anticipated in the CYR, the commissioning is expected to proceed in successive 
phases each lasting as long as a year.  It is only in later phases, for optimal MDM and EDM measurements to proceed, 
that the delicate toroidal focusing requirements will need to be met.  

\section{``Quadrupole free'' PTR transfer matrix lattice design}
\subsection{Transfer matrix algebra}
For a $2\times2$ transfer matrix ${\bf M}$ to be symplectic it must satisfy the algebraic relation
\begin{equation}
{\bf M}^{-1} = -{\bf S}{\bf M}^T{\bf S},
\quad\hbox{where}\quad {\bf S} = \begin{pmatrix} 0 & -1 \\ 1 & 0  \end{pmatrix},
\label{eq:M01p}
\end{equation}
and the only imposed constraint for 2x2 transfer matrices requires them to have unit determinant. For any higher 
(necessarily even) dimensions the same equation is satisfied with ${\bf S}$ being replicated along the diagonal.
This introduces symplecticity conditions going beyond the unity determinant requirement.

Transfer matrices ${\bf M_x}$ and ${\bf M_y}$ for one PTR superperiod as well as powers up to ${\bf M_x^8}$ 
and ${\bf M_y^8}$, which describe two consecutive complete revolutions follow: 
\begin{math}
\\
{\bf Mx} =
\left[\begin{array}{cccc}
 0.33015 &  9.04118 &  0.0 &  8.68780 
\\
 - 0.09855 &  0.33015 &  0.0 &  1.27816 
\\
 - 1.27816 & - 8.68780 &  1.0 &  204.00230 
\\
  0.0 &  0.0 &  0.0 &  1.0 
\end{array}\right]
\end{math}

\begin{math}\label{(14)}
{\bf My} = \left[\begin{array}{cc}
 0.95513 &  12.95238 
\\
 - 0.02167 &  0.95513 
\end{array}\right]
\end{math}

\begin{math}
{\bf Mx^2} = \left[\begin{array}{cccc}
- 0.78201 &  5.96980 &  0.0 &  23.11206 
\\
 - 0.06507 & - 0.78201 &  0.0 &  0.84395 
\\
 - 0.84395 & - 23.11206 &  1.0 &  385.79588 
\\
  0.0 &  0.0 &  0.0 &  1.0 
\end{array}\right]
\end{math}

\begin{math}
{\bf My^2} = \left[\begin{array}{cc}
 0.63162 &  24.74233 
\\
 - 0.04139 &  0.63162 
\end{array}\right]
\end{math}



\begin{math}
{\bf Mx4} =  \left[\begin{array}{cccc}
 0.22307 & - 9.33687 &  0.0 &  10.07647 
\\
  0.10177 &  0.22307 &  0.0 & - 1.31996 
\\
  1.31996 & - 10.07647 &  1.0 &  732.58076 
\\
  0.0 &  0.0 &  0.0 &  1.0 
\end{array}\right]
\end{math}

\begin{math}
{\bf My^4} \left[\begin{array}{cc}
- 0.62514 &  31.25562 
\\
 - 0.05229 & - 0.62514 
\end{array}\right]
\end{math}

\begin{math}
{\bf Mx^8} = \left[\begin{array}{cccc}
- 0.90048 & - 4.16564 &  0.0 &  24.64854 
\\
  0.04541 & - 0.90048 &  0.0 & - 0.58890 
\\
  0.58890 & - 24.64854 &  1.0 &  1491.76252 
\\
  0.0 &  0.0 &  0.0 &  1.0 
\end{array}\right]
\end{math}

\begin{math}
{\bf My^8} = \left[\begin{array}{cc}
- 1.24343 & - 39.07858 
\\
  0.06537 & - 1.24343 
\end{array}\right]
\end{math}

\subsection{Interpretation of the transfer matrices}
Even for the 4x4 natrices (mentally-partitioned 2x2) one can check mentally (at least at the 10\% level) that 
all determinant values are close to 1.  Displayed only to 6 place accuracy, each of these matrices satisfies 
Eq.~(\ref{eq:M01p}) to machine precision. In fact, this too is an understatement; these matrices are, in fact, 
analytically exact.  They have been obtained purely algebraicly. 

This is all that is required for the transverse propagation to be Hamiltonian, and hence loss-free.
No paraxial approximation, nor small angle, nor small amplitude approximations have been made; no truncated 
power series have been employed, no linear, quadrupole, sextupole and higher orders introduced. 
Admittedly these statements are true primarily because the starting transfer matrix was assumed exact.
This is an excellent (but not perfect) assumption for the toroidal focusing design, which has been
adopted in modern day (Scanning) Transmission Electron Microscopy (STEM), as described most recently
in a textbook by P. Erni\cite{Erni-TEM}, with clear elementary physical discussion, especially in Section~5.

\subsection{Longitudinal optics; beam capture into stable buckets}
In spite of this splendid transverse evolution description one cannot fail to notice that the horizontal 4x4 
matrices are, in some sense, divergent. Concentrating on the $M_{34}$ elements, referring to 
Eq.~(\ref{eq:Wollnik-TM-x}), and 
visualizing $\delta_p/p=10^{-5}$ as a possible fractional momentum offset, one finds that, after a single turn, the 
longitudinal arc length parameter $s$ will have advanced by 0.75\,cm, relative to a central particle. 

For a few turns in a ring of circumference 100\,m this advance would be scarcely noticeable, but circulating at
roughly a MHz rate for, say, two minutes, the beam spread will be of order $10^6\,$m. In other words the beam will
have by then spread around the ring into a uniformly distributed coasting beam.  
To keep the beam bunched, which is required for phase-locking, the PTR beam has to be quickly captured into stable 
RF buckets.  This should be straightforward.

\section{Practical simultaneous frozen spin beam combinations\label{sec:PracticalCombinations}}
A selection of practical frozen spin combinations are given in Tables~\ref{tbl:p-d2} and \ref{tbl:p-He3}.
Phase-locked parameters are given as rational nembers (i.e. ratios of integers). 
Table~\ref{tbl:p-d2} shows ``doubly frozen'' proton, deuteron combinations, for which the primary beam spin 
tune vanishes, while the secondary beam spin tune, though a rational fraction, is not zero.  
Table~\ref{tbl:p-He3} shows the ``doubly magic'' proton, helion case in which both spin tunes vanish.  
In all of these cases the RF cavity frequencies are also given as rational fractions.  In this condition the 
RF frequency can also be phase-locked, but it is only the primary beam for which the EDM-induced, out of 
plane precession accumulates monotonically. 

In all cases the frequencies are ``over-determined'' in the sense that the circumferences of primary and 
secondary beam, though close, are not quite identical.  The defects are spelled out in the figure captions.  
With orbits not quite identical, the precession-inducing error fields to which the beams are subject are not 
quite equal.  To leading order, the EDM measurement error will be canceled by averaging over runs with primary 
and secondary beams interchanged.

Initially this defect can be analyzed and compensated for by theoretical calculation of corrections to the 
measured EDM values.  The issue can also be faced experimentally be introducing a \emph{very weak} secondary 
RF cavity, synchronized to the secondary beam and strong enough to alter the secondary beam radius to match 
the primary beam radius.  To avoid affecting the primary beam, this longitudinal RF signal, synchronized to 
the secondary beam  would have to be gated to affect only the secondary beam. This would require gate opening 
and closing switching times at the 10\,ns level. 
\setlength{\tabcolsep}{1pt}
\begin{table}[htb]\footnotesize
\centering
\begin{tabular}{|c|ccccc|cc|cccccc|c|} \hline 
 bm  &     m1 & G1      &q &$\beta_1$&  K1  & E0*  &  cB0* & m2     &  G2   & q  &$-\beta_2/\beta_1$&  KE2 & Qs2  & bm \\ 
  1  &    GeV &         &  &         & MeV  & MV/m & cT/1e3& GeV    &       &    &  $*\dagger$      &  MeV &      &  2 \\ \hline
     & $r_0*=$ & 11.0\,m &  & Qs1=0  &&  {\bf PRESENT}    & {\bf DAY}  &        &       &    &     PTR         &      &      &    \\ \hline
   p & 0.9383 & 1.7928  & 1& 0.31304 & 49.7 & 6.26 & 8.11  & 0.9383 & 1.79  &  1 & 41/57           & 24.7 &-2/1  &  p \\ 
   p & 0.9383 & 1.7928  & 1& 0.29175 & 42.7 & 5.33 & 7.76  & 1.8756 &-0.57  &  1 & 53/100          & 22.8 &-9/8  &  d \\ 
   d & 1.8756 & -0.5713 & 1& 0.18000 & 31.1 &-7.73 & 74.12 & 1.8756 &-0.57  &  1 & 83/172          & 7.12 &-25/31&  d \\ 
   d & 1.8756 & -0.5713 & 1& 0.17760 & 30.3 &-7.52 & 73.11 & 0.9383 & 1.79  &  1 & 35/67           & 4.06 &43/127&  p \\  \hline
\end{tabular}
\caption{\label{tbl:p-d2}
Quantities expressed as rational fractions are exact and phase locked to arbitrarily high accuracy.
Except for PTR construction itself, these pairings use only \emph{present day technology and apparatus}.
Beam~1: is globally frozen in every case ; $Qs1=0/1$; so \emph{the beam~1 EDM signal accunulates monotonically}.
Except for $r_0*$, $*\dagger$, E0* and cB0* columns, all entries are \emph{EXACT}, either \emph{integers}, or
(truncated) \emph{physical constants} or \emph{calculable,} kinematic quantities.
``Closed orbit mismatches'', from the  $\dagger$ column, are
0.71927 = 41/57   - 0.0000282, 
0.52997 = 53/100  - 0.0000299,
0.48261 = 83/172  + 0.0000518,
0.52235 = 35/67   - 0.0000380.
}
\end{table}

\setlength{\tabcolsep}{1pt}
\begin{table}[htb]\footnotesize
\centering
\begin{tabular}{|c|ccccc|cc|cccccc|c|} \hline 
 bm  &     m1 & G1      &q &$\beta_1$&  K1  & E0*  &  cB0* & m2     &  G2   & q    &$-\beta_2/\beta_1$& KE2 & Qs2 & bm \\ 
  1  &    GeV &         &  &         & MeV  & MV/m & cT/1e3& GeV    &       &      &  $*\dagger$     &  MeV &     &  2 \\ \hline
     & $r_0*=$ & 50.0\,m &  & Qs1=0  &      & {\bf FUTURE} &        &   ``  & HOLY &    &    GRAIL   &  ``  &     &    \\ \hline
   p & 0.9383 & 1.7928  & 1& 0.59840 & 233  & 8.39 &-5e-4  & 0.9383 & 1.79  &  1   &  -1/1           & 233  & 0/1 &  p \\ \hline
     & $r_0*=$ & 11.0\,m &  &        &      & {\bf FUTURE} &        &       &      &    &     PTR    &      &      &    \\ \hline
   p & 0.9383 & 1.7928  & 1& 0.27831 & 38.6 & 3.90 & 6.13  & 2.8084 &-4.18  &  2   & 107/180         & 39.2 & 0/1  &  h \\ 
   h & 2.8084 & -4.1842 & 2& 0.16544 & 39.2 & 3.90 & -6.13 & 0.9383 & 1.79  &  1   & 180/107         & 38.6 & 0/1  &  p \\  \hline
   p & 0.9383 & 1.7928  & 1& 0.40238 & 86.6 & 11.6 & 8.77  & 0.0005 & .0012 &  1   & 82/33           & 30.1 & 0/1  &  e+\\  
  e+ & 0.0005 & 0.0012  & 1& 0.99986 & 30.1 & 11.6 & -8.77 & 0.9383 & 1.79  &  1   & 33/82           & 86.7 & 0/1  &  p \\  \hline
\end{tabular}
\caption{\label{tbl:p-h}\label{tbl:p-He3}Quantities expressed as rational fractions are exact and can be phase locked to 
arbitrarily high accuracy, Rational fraction entries in the $\dagger$ column of the tables are fixed by RF synchronism; the 
orbit circumference ratios are close, but not exact; 
0.59445 = 107/180 + 0.0000055;
1.68218 = 180/107 - 0.0000629;
2.48486 = 82/33   + 0.0000115;
0.40247 = 33/82   + 0.0000310.
Precise magnetic field reversal is assured by three-fold phase locking.
}
\end{table}

\clearpage

\section{Practical limits, uncertainties, issues and ackowledgements}
The content of this paper has been largely theoretical--- not theoretical particle physics, but theoretical
experimental physics.  What has been missing is practical experimental physics including much of
which is essentially electrical engineering.  Since co-magnetometry depends on \emph{simultaneous} 
counter-circulating beams, accurate electric/magnetic superposition is essential. There are grounds for both 
optimism and pessimism in this area concerning the PTR design.

Experience with the AGS-Analogue electron ring, as described in considerable detail in references
\cite{RT-Instrumentation-paper}, \cite{EDM-Challenge}, and \cite{RT-JDT-AGS-Analogue}, provides grounds
for optimism concerning purely electric storage rings.  With a circumference of 43\,m, less than the
proposed PTR circumference in the ratio 43/102, the electron kinetic energy was 10\,MeV (which 
corresponds approximately with 20\,MeV proton kinetic energy, as regards achievable electric field value), 
the AGS-Analogue ring was built quickly, inexpensively and without much bother.  The cost of the ring 
itself was undoubtedly much smaller than the cost of the building housing it.  

Based only on the AGS-Analogue ring experience, the proposed PTR ring could be expected to meet
the requirements of the $p$, $d$, and $h$ examples spelled out in Section~\ref{sec:PracticalCombinations}.  

This is likely to remain true for PTR, especially if the voltage and current power supply circuits
remain simple.  For design purposes, in a ring with eightfold super-periodicity, one could power each
octant or quadrant individually. This would make beam steering easy.  All electrodes would be powered in parallel,
all magnet coils in series.  Even simpler, if construction tolerances allow it, would be to power 
all ring elements from just two supplies.  Finer than eightfold powering granularity would probably 
lead to serious and useless complication.

Less carefully documented is the evidence concerning electrical breakdown and, especially,
sparking ``catalyzed'' by the presence of weak local magnetic fields.  For PTR, \emph{for which superimposed electric 
and magnetic fields are required for best systematic EDM error reduction}, this is undoubtedly the area 
of greatest concern for simultaneous beam frozen spin functionality in PTR.

Experience seems to have shown that occasional sparking in high electric field bending elements
is unavoidable.  On the other hand, at least with short enough electrodes (i.e. of small capacitance) 
the breakdown causes no permanent damage. Ignoring possible damage to sensitive electronics, sparking rates 
less than one per day would only be a nuisance, but greater than one per hour would make quality data collection 
impossible. 

To reduce capacitance the length per construction unit will have to be quite short, perhaps one meter.
By butting construction lengths quite closely (perhaps a few mm separation, with resistive separation
spacers for registration) the electrode field uniformity degradation at transitions should be 
acceptably small.  

FURTHER details concerning PTR and co-magnetometry are contained in 
references
\cite{COSY-stripper-explained},
\cite{Septier},
\cite{Germain},
\cite{Germain-Rohrbach},
\cite{Germain-Tingley},
\cite{RohrbachElectricPrisms},
\cite{RT-RAST}.

At some cost in EDM systematic error coming from out-of plane spin precession 
(the so-called ``geometric phase problem'') counter-circulating orbit matching could be sequential, 
rather than simultaneous. Electric and magnetic bending could alternate on a quite short distance scale 
to preserve much the same frozen spin combinations as are possible with simltaneous beams. This would, 
however, require magnetic field reversal between runs in order to reverse the circulation direction..  
Spin tune phase locking would make this possible, as has been explained above, without the need for 
impractically precise magnetic field measurement.  This would also aggravate the problem of time 
dependent magnetic field error bursts---an important source of error also in the neutron EDM determination.

\noindent
{\bf Acknowledgements'':\ }
All acknowledgements in the companion paper\cite{RT-ICFA} to this one apply to the present paper.  Special
acknowledgement to Hans Stroeher, Frank Rathmann, Alex Nass, Kolya Nikoliev and my son John can be repeated; 
along with continuous support, all have made substantial contributions, suggestions and corrections to this
paper.  More significant yet have been the contributions of Christian Carli who, in a continuing series
of communications, has found mistakes and formulated insightful questions that have invariably resulted in 
major correction, reorganization and clarification.


