# Automated Generation of Arbitrarily Many Kochen-Specker and Other Contextual
Sets in Odd Dimensional Hilbert Spaces

Mladen Pavičić<EMAIL_ADDRESS>Center of Excellence for Advanced Materials
and Sensing Devices (CEMS), Photonics and Quantum Optics Unit, Rud er Bošković
Institute and Institute of Physics, Zagreb, Croatia.  Norman D. Megill  Passed
away on December 9, 2021 Boston Information Group, Lexington, MA 02420, U.S.A.

###### Abstract

Development of quantum computation and communication, recently shown to be
supported by contextuality, arguably asks for a requisite supply of contextual
sets. While that has been achieved in even dimensional spaces, in odd
dimensional spaces only a dozen contextual critical Kochen-Specker (KS) sets
have been found so far. In this paper we give three methods for automated
generation of arbitrarily many contextual KS and non-KS sets in any dimension
for possible future application and implementation and we employ them to
obtain millions of KS and other contextual sets in dimensions 3, 5, 7, and 9
where previously only a handful of sets have been found. Also, no explicit
vectors for the original Kochen-Specker set were known so far, while we now
generate them from 24 vector components.

quantum contextuality, MMP hypergraphs, Kochen-Specker sets, NBMMPH

###### pacs:

03.67.Dd, 03.67.Ac, 42.50.Ex

Contextuality is paving the road for applications in quantum computation
Howard et al. (2014); Bartlett (2014), quantum steering Tavakoli and Uola
(2020), and quantum communication Saha et al. (2019), as proved by the methods
of processing, preparing, and measuring of qudits (quantum dits). Qudits are
the units of quantum information carried by a quantum system whose number of
states ($d$) is an integer greater than one. Qubits (quantum bits) are two-
dimensional qudits ($d=2$) and their tensor products build the corresponding
even-dimensional Hilbert spaces. The smallest contextual sets from these
spaces have been implemented in a series of experiments, while in Waegell and
Aravind (2015); Pavičić (2017); Pavičić and Megill (2018); Pavičić (2019);
Pavičić et al. (2019) billions of contextual sets in 4-, 6-, 8-,
$16{\textrm{-}},$ and 32-dim Hilbert spaces, predominantly related to qubits,
were generated by means of vector component algorithms (called method M3
below), particular symmetries, geometries, polytope correlations, parity
filtering, Pauli operators, qubit states, and dimensional upscaling.

In contrast, far fewer contextual sets based on qudits in odd-dim spaces have
been obtained so far. In particular, of Kochen-Specker contextual sets only
four in the 3-dim space Pavičić et al. (2005); Pavičić (2019), four in the
5-dim space Cabello et al. (2005),(Waegell and Aravind, 2017, Supp. Material),
five in the 7-dim space Zimba and Penrose (1993); Cabello et al.
(2005),(Waegell and Aravind, 2017, Supp. Material), two in the 9-dim space
(Waegell and Aravind, 2017, Supp. Material), and two in the 11-dim space
(Waegell and Aravind, 2017, Supp. Material). General methods for automated
generation of sets of a chosen structure did not exist.

Since the quantum communication and computation are supported by contextuality
Howard et al. (2014); Bartlett (2014), the actual potential use of a large
supply of contextual sets is twofold. First, quantum computation algorithms
which would rely on contextual sets would arguably rely on a variety of such
sets and on their automated generation. Second, structural properties of
contextual sets differ according to their coordinatization, parities,
dimensions, sizes, etc., and that can lead us to better understanding and
applications of the sets.

In this paper, we offer universal and general algorithms for automated
generation of arbitrarily many contextual sets in any dimension. In contrast
to them, the programs we wrote to implement them are computationally demanding
and therefore, here, we use them to generate sets that have not been generated
so far: billions of KS and contextual non-KS sets in $n=3,5,7,9$-dim spaces.
The programs themselves are freely available from our repository and technical
details of their previous versions are given in Waegell and Aravind (2015);
Pavičić (2017); Pavičić and Megill (2018); Pavičić (2019); Pavičić et al.
(2019).

To describe and handle contextual sets we make use of McKay-Megill-Pavičić-
hypergraph (MMPH) language Pavičić et al. (2005); Pavičić (2021). An MMPH is a
connected $n$-dim hypergraph $k$-$l$ with $k$ vertices and $l$ hyperedges in
which (i) every vertex belongs to at least one hyperedge; (ii) every hyperedge
contains at least $2$ and at most $n$ vertices; (iii) no hyperedge shares only
one vertex with another hyperedge; (iv) hyperedges may intersect each other in
at most $n-2$ vertices. Graphically, vertices are represented as dots and
hyperedges as (curved) lines passing through them.

We encode MMPHs by means of the following 90 ASCII characters: 1 2 … 9 A B … Z a b … z ! " # $ % & ’ ( ) * - / : ; < = > ? @ [ $\backslash$ ] ^ _ ‘ { | } $\sim$ Pavičić et al. (2005). When all 90 characters are exhausted, we reuse them prefixed by ‘+’ (91st character), then again by ‘++’, and so on. So encoded single ASCII characters (possibly prefixed by +’s) represent vertices; e.g., 1 or +++A. Put together they represent hyperedges; e.g., 123 or 1+1+++1Dd. To represent an MMPH, hyperedges are organized in a string in which they are separated by commas; each string ends with a period; e.g., the string 123,345,567,789,9A1. represents a noncontextual 3-dim MMPH pentagon. There is no limit on the size of an MMPH. An MMPH is a special kind of a general hypergraph in the sense that none of the aforementioned points (i-iv) holds for it.

Of course, instead of ASCII characters we could have used Unicode characters
or 16-bit integers but 20 years ago we decided to proceed with the ASCII
characters to encode MMPH strings and design our algorithms and programs which
in turn yield all properties and features of MMPHs as well as their figures
within the MMPH language. All our papers since 2000 McKay et al. (2000) make
use of the ASCII characters for the purpose.

The MMPHs above are defined without a coordinatization. The meaning of
coordinatization is that a vector is assigned to each vertex and that all
vectors assigned to vertices belonging to a common hyperedge are orthogonal to
each other. Thus, in the MMPHs above, neither their vertices nor their
hyperedges are related to either vectors or operators. We say that an MMPH is
in an $n$-dim space, called MMPH space, when either all its hyperedges contain
$n$ vertices or when we might add vertices to hyperedges so that each contains
$n$ vertices. Orthogonality between vertices in an MMPH space just means that
they are contained in common hyperedges. Our programs may handle MMPHs without
any reference to either vectors or projectors. In an MMPH with a
coordinatization, i.e. with vectors assigned to vertices, an $n$-dim MMPH
space becomes an $n$-dim Hilbert space spanned by a maximal number of
(possibly added) vectors within hyperedges. Whether we speak about an MMPH
without or with a coordinatization will be clear from the context.

A non-binary MMPH (NBMMPH) is an $n$-dim $(n\geq 3)$ $k$-$l$ MMPH, whose
$i$-th hyperedge contains $\kappa(i)$ vertices $(2\leq\kappa(i)\leq n$,
$i=1,\dots,l)$, to which it is impossible to assign 1s and 0s in such a way
that the following rules hold: (I) no two vertices in any hyperedge are both
assigned the value $1$ and (II) in any hyperedge, not all of the vertices are
assigned the value $0$. A binary MMPH (BMMPH) is an MMPH for which such an
assignment is possible. NBMMPHs are nonclassical and contextual since they do
not allow assignments of predefined 0s and 1s to their vertices. BMMPHs are
classical and noncontextual since they do allow such an assignment. A KS MMPH
is an NBMMPH with $\kappa(i)=n,\ \forall i$. The assignments of 0s and 1s do
not require a coordinatization but an implementation of (N)BMMPHs does require
it as well as their filled MMPHs, i.e. those in which to all hyperedges with
$\kappa(i)\leq n$ we add $n-\kappa(i)$ vertices so as to satisfy the mutual
orthogonalities. An example of a non-KS NBMMPH without a coordinatization is
the 33-27 in Fig. 1(d) in the Supplemental Material (SM).

When either state-dependent or state-independent tests of operators defined on
vertices of an NBMMPH with $\kappa(i)\leq n$ confirm the contextuality, e.g.
Yu and Oh (2012); Cabello et al. (2012, 2018), then the NBMMPH turns out to be
contextual in all considered cases so far.

A critical NBMMPH is an NBMMPH which after removing any of its hyperedges
becomes a BMMPH.

To generate (N)BMMPHs in the odd-dim spaces we make use of three
methods—M1-M3.

M1 consists in an automated dropping of vertices contained in single
hyperedges (multiplicity $m=1$) Pavičić (2021) of MMPHs and a possible
subsequent stripping of their hyperedges. The obtained smaller MMPHs are often
NBMMPH although never KS.

M2 consists in an automated random addition of hyperedges to MMPHs so as to
obtain larger ones which then serve us to generate smaller KS MMPHs by
stripping hyperedges randomly again;

M3 consists in combining simple vector components so as to exhaust all
possible collections of $n$ mutually orthogonal $n$-dim vectors. These form
big master MMPHs which consist of single or multiple MMPHs of different sizes.
Master MMPHs may or may not be NBMMPH, what we find out by applying filters to
them. NBMMPHs serve us to massively generate a class of smaller MMPHs via our
algorithms and programs.

We carry out methods M1-M3 and by means of our programs MMPSstrip (for
stripping and adding hyperedges), States01 (for verifying the contextuality),
MMPShuffle (for reorganizing MMPHs), and ONE (for evaluating the structural
properties of MMPHs Pavičić (2021)).

We combine all three methods to obtain a large number of NBMMPHs in odd-dim
spaces.

3-dim case. So far there have been only four known KS MMPHs, and as we show in
Pavičić (2019) none of their varieties with stripped $m=1$ vertices is
critical. Via M1, i.e. by stripping their edges, we can obtain thousands of
smaller non-KS NBMMPHs and BMMPHs down to a pentagonPavičić (2019). But this
means that they are limited in size by the size of four original MMPHs. To get
larger MMPHs we have to apply M2,3. The final distribution of critical KS
MMPHs we generated is given in Fig. 1.

Figure 1: The distribution of critical 3-dim KS MMPHs obtained via M2,3 with
the given vector components. Abscissa is $l$ (number of hyperedges); ordinate
is $k$ (number of vertices). Dots represent $(k,l)$. Consecutive dots (same
$l$) are shown as strips. The same applies to Figs. 2 and 3. See text. MMPH
strings, figures, and vectors are given in SM.

M2 consists in adding hyperedges to Bub, Conway-Kochen, and Peres’ KS MMPHs
(for citations and figures see Pavičić (2017)) using our program MMPSstrip,
filtering out KS MMPHs, and stripping them to the critical KS MMPHs by
States01. The latter critical MMPHs build their coordinatization from the two
vectors component sets of the original three MMPHs.

These vector components also serve us to obtain the same critical KS MMPHs by
employing M3 so as to exhaust all possible collections of 3 mutually
orthogonal vectors representing hyperedges interwoven in MMPHs.

When applying M3 we obtain that Bub’s is the only 49-36 NBMMPH and that there
are no smaller KS ones for the considered vector components. There are no
other critical KS MMPHs between 49-36 and 51-37; between 51-37 and Peres’
57-40 we obtained the following non-isomorphic MMPHs: one 53-38, eight 54-39,
and one 55-40. Peres’ 57-40 is generated from the components $\\{0,\pm
1,\pm\sqrt{2},3\\}$ and all the smaller ones from $\\{0,\pm 1,\pm 2,5\\}$.
When we apply M2 so as to add sufficiently many hyperedges to any of the three
original MMPHs (Bub, Conway-Kochen, or Peres’) and then strip them back down
in a search for smaller critical MMPHs, we always obtain the other two among
the generated critical KS MMPHs.

The more components we use with M3, the larger the master files and the more
critical KS MMPHs we obtain. For instance with the help of $\\{0,\pm
1,\sqrt{2},3\\}$ (Peres’) components we obtain the master 81-52 which contains
just one single critical set—Peres’ 57-40; $\\{0,\pm 1,\pm 2,5\\}$ yield the
master 97-64 which generates 20 critical KS MMPHs from 49-36 to 55-40; in
contrast, $\\{0,\pm 1,\sqrt{2},\pm 2,\pm 3,5\\}$ yield the master 301-184
which generates 81 critical KS MMPHs from 49-36 to 92-66;
$\\{0,\pm\omega,2\omega,\pm\omega^{2},2\omega^{2}\\}$, where $\omega$=$e^{2\pi
i/3}$, yield 514 criticals from 69-50 to 106-79, etc. Several smallest MMPHs
from these classes are shown in Table 1 and SM.

dimen. | Smallest | No. of |  Methods | Smallest | Vector
---|---|---|---|---|---
critical | non-isom | master | components
MMPHs | MMPHs
3D MMPHs | 5-5† | 1 | M1 | 40-23 | $\\{0,\pm 1,2\\}$
19-13 | 1 | M2 | 20-14 | none
39-27 | 1 | M2 | 39-30 | ?
49-36 | 1 | $\\!\\!\\!\\!\\!\\!\\!\begin{rcases*}\\\ \\\ \\\ \end{rcases*}$ M2-3 |  |
51-37 | 9 | 97-64 | $\\{0,\pm 1,\pm 2,5\\}$
53-38 | 1 |  |
57-40 | 1 | M2-3 | 81-52 | $\\{0,\pm 1,\sqrt{2},\pm 3\\}$
69-50 | 3 | M3 | 169-120 | $\\!\\!(0,\pm\omega,2\omega,\pm\omega^{2},2\omega^{2}\\}$
5D | 29-16 | 2 | M3 | 105-136 | $\\{0,\pm 1\\}$
7D | 13-4 | 1 | M1,2 | 24-6 | ?
28-14† | 1 | M2,3 | 805-9936 | $\\{0,\pm 1\\}$
34-14 | 1 | M3 | 805-9936 | $\\{0,\pm 1\\}$
207-97 | 1 | M3 | 805-9936 | $\\{0,\pm 1\\}$
9D | 19-5 | 2 | M2 | 20-6 | ?
47-16 | 2 | M3 | 9586-12068704 | $\\{0,\pm 1\\}$
7-6† | 1 | M1,3 | 9586-12068704 | $\\{0,\pm 1\\}$
Table 1: Some of smaller NBMMPHs obtained by methods M1-3; ‘${\dagger}$’
indicates that the MMPH is a non-KS NBMMPH—all the others are KS NBMMPHs; ‘?’
indicates that obtaining the coordinatization is too demanding and that we
were not able to carry it out on our supercomputers or that it might not exist
at all.

We did not find simple real vector components which would yield KS MMPHs
smaller than 49-36, although we are able to generate many smaller KS MMPHs
down to 19-13 or 39-27 shown in SM, although without a coordinatization based
on components shown in Fig. 1.

The path taken in Voráček and Navara (2021) is intractable for hundreds of
small KS MMPHs we checked on our supercomputer since the number of free
variables is too high. See Fig. 1(a) in SM.

Another path was taken in 2021 by Jean-Pierre Merlet who applied the interval
analysis method of solving nonlinear equations to the 19-13 MMPH. The
equations turned out not to have a real solution and complex ones were not
calculable even on a supercomputer.

As for possible coordinatizations of smaller KS MMPHs, we draw a parallel with
the original KS set 192-118 Kochen and Specker (1967). Its trigonometric
formula in Kochen and Specker (1967) looks simple, but its coordinatization is
so complicated that a random search for them is infeasible. More specifically,
the aforementioned trigonometric formula is not sufficient to provide a
coordinatization. We, therefore, constructed apparently the first known
coordinatization of the original Kochen-Specker set from 24 components in SM.

5-dim case. In contrast to the 3-dim case, the 5-dim KS MMPHs can be obtained
from just three vector components $\\{0,\pm 1\\}$, which by method M3 generate
the 105-136 master set and its 105-136 class of KS MMPHs. These include
critical ones from the smallest 29-16 to the biggest 64-41, altogether
27,829,399 non-isomorphic MMPHs. The distribution of the MMPHs within the
class is shown in Fig. 2(a) and the 29-16 in Fig. 2(b).

Figure 2: (a) Distribution of the 5-dim critical KS MMPHs obtained by M3; the
generation of the 1185-3596 was demanding and CPU time consuming, so, we
stopped it after obtaining the upper samples which enable us to estimate the
size of the class; (b) One of the two smallest 5-dim 29-16 critical KS MMPH;
the other (29-16b) was previously obtained by the dimensional upscaling method
in Waegell and Aravind (2017). Figure 3: (a) Distribution of 42,816 7-dim
critical KS MMPHs from the class 805-9936 generated by M2,3; the master is
obtained from $\\{0,\pm 1\\}$; (b) 207-97; too interwoven to be graphically
represented in detail; (c) the non-KS NBMMPH obtained from 207-97 via
MMPShuffle; (d) critical KS 13-4 obtained by M1,2; (e) critical KS 34-14 MMPH
obtained by massive targeting MMPHs with 34 vertices—see text.

We also obtained the master (1185-3596) and a number of elements from its
class from the five components $\\{0,\pm 1,\pm 2\\}$ but we stopped the
generation of KS criticals after only 307 MMPHs. Their distribution is
indicated in Fig. 2—if fully generated it would completely include the 105-136
class and would be continuously spread over the whole ranged of hyperedges and
vertices.

7-dim case. We generated the 7-dim critical KS MMPHs from the vector
components $\\{0,\pm 1\\}$ via M3 so as to first obtain the 805-9936 master
which in turns generated the the 805-9936 class shown in Fig. 3(a). Their
vectors were automatically generated from the master set by means of our
programs MMPStrip, MMPShuffle, and States01.

The generation provided MMPHs from 207-97 to 333-159 after running 200
parallel jobs on a supercomputer for two weeks. Longer runs would give us
smaller MMPHs as proved by the 34-28 MMPH (see the top of Fig. 3(a)) obtained
in Cabello et al. (2005). Particular targeted runs might give us particular
smaller KS MMPHs, e.g. the critical 34-14 (Fig. 3(a,e) and SM); it required
stripping the master down to the MMPHs with 34 vertices via MMPStrip and then
filtering them for the KS feature via States01; since such a procedure is too
CPU-time demanding, a search for further smaller MMPHs is out of the scope of
this paper. Instead, M1 can serve for a massive automated generation of
smaller non-KS NBMMPHs with the help of MMPShuffle and MMPStrip. An outcome
(28-14) is shown in Fig. 3(c), as obtained from the 207-97 Fig. 3(a,b). The
procedure is analogous to stripping original 3-dim MMPHs Pavičić (2019).

Yet another way of automated generation of 7-dim MMPHs is via M2 by means of
MMPStrip, an example of which is 13-6 shown in Fig. 3(d); we were not able to
find its coordinatization and we conjecture that it does not have any. We
confirmed that it is not determined by the vector components $\\{0,\pm 1\\}$
and we work on a program which would calculate coordinatization for bigger
instances of such MMPHs we generated.

9-dim case. Two entangled qutrits live in a 9-dim space and we generated the
MMPH master from $\\{0,\pm 1\\}$ components. It consists of 9,586 vertices and
12,068,705 hyperedges and that proved to be too huge for a direct generation
of critical MMPHs (via stripping and filtering) from the master MMPH although
the KS 47-16 given in SM proves that the master is a KS MMPH.

Figure 4: (a) KS MMPH 19-5; no coordinatization was found; (b) BMMPH 44-6; (c)
NBMMPH 7-6; MMPH strings (a-c) and vectors (b,c) are given in SM.

However, smaller critical KS MMPH can be obtained from simple BMMPHs via M2,
in particular via an automated procedure of adding hyperedges and then
generating critical KS MMPHs by stripping hyperedges via M1 and filtering them
for the critical KS property. The critical KS MMPH 19-5 obtained in this way
(Fig. 4(a)) has no coordinatization from $\\{0,\pm 1\\}$ and we conjecture
that it does not have any, but it represents the proof of principle of how M2
works.

Also, a great many of MMPHs stripped of $m=1$ vertices exhibit contextuality.
Smaller ones can easily be implemented. The smallest one we obtained by M3,1
is shown in Fig. 2(c) of SM and referred to in Table 1. Its filled MMPH is
shown in Fig. 2(c) of SM. Their differences are discussed in SM.

To summarize, in this paper we give methods for generating KS as well as non-
KS NBMMPHs in odd-dimensional Hilbert spaces. Our goal is not to find “record”
smallest MMPHs but to establish general methods for automated generation of
NBMMPHs in any dimension for any possible future application and
implementation, e.g., in quantum computation and communication. The methods
are especially needed in odd dimensional Hilbert spaces since, in contrast to
even dimensional ones, we cannot make use of polytopes, Pauli operators, qubit
states, parities, and other approaches specific to qubit spaces. We propose
three such methods: M1 which consists in dropping vertices contained in single
hyperedges, M2 consists in random addition of hyperedges to MMPHs, and M3
which consists in combining simple components so as to exhaust all possible
collections of mutually orthogonal vectors. Automated generation is achieved
by means of our algorithms and programs presented above.

In the 3-dim space we generated roughly a million and a half non-isomorphic KS
ones, ranging from MMPHs with 19 vertices and 13 hyperedges (19-13) without a
coordinatization (via M3), over eleven 51-37s, up to a 232-172, all with
coordinatizations, distributed as shown in Fig. 1. Special cases are given in
Table 1 and SM. In SM we also give, for the first time, an explicit
coordinatization of the original KS set.

In the 5-dim space, from the vector components $\\{0,\pm 1\\}$, we generated
roughly 28 million KS MMPHs whose distribution is shown in Fig. 2(a), ranging
from 29-16 to 242-131.

In the 7-dim space the components $\\{0,\pm 1\\}$ generate roughly 30,000
rather big and computationally demanding KS MMPHs shown in Fig. 3(a), ranging
from 34-14 to 333-159.

The 9-dim MMPH master generated by $\\{0,\pm 1\\}$ has 9,586 vertices and
12,068,705 hyperedges and it is computationally too demanding to yield
critical KS MMPHs directly.

We also explain how a combination of methods M1-M3 can be employed to generate
targeted smaller classes of non-KS NBMMPHs. This approach is important because
the operator and projector based contextual sets, which are recently being
used in the literature, are often built by means of such NBMMPHs.

###### Acknowledgements.

Supported by the Ministry of Science and Education of Croatia through the
Center of Excellence CEMS funding, and by MSE grants Nos. KK.01.1.1.01.0001
and 533-19-15-0022. Computational support provided by the Zagreb University
Computing Centre. Repository http://puh.srce.hr/s/Qegixzz2BdjYwFL

## Supplemental Material

### 3-dim

#### One of eight 51-37 MMPHs generated by $\\{0,\pm 1,\pm 2,5\\}$,
nonisomorphic to Conway-Kochen’s MMPH

51-37
213,35b,bYh,hgR,RQN,N6C,CBD,DaF,FEG,GHI,IOP,PTU,UpK,KJL,LV8,89A,Akl,lcf,fde,eZX,XWn,nm2,456,783,MNL,STR,WDA,XYK,ZaV,ZI6,cG2,cYC,WT5,cVT,ijP,oSF,jba.
1=(-1,5,2); 2=(2,0,1); 3=(1,1,-2); 4=(1,1,2);5=(1,-1,0); 6=(1,1,-1);
7=(5,-1,2); 8=(0,2,1); 9=(5,1,-2); A=(1,-1,2); B=(1,2,-1); C=(1,0,1);
D=(-1,1,1); E=(-2,5,-1);F=(2,1,1); G=(1,0,-2); H=(2,5,1); I=(2,-1,1);
J=(5,-2,-1); K=(1,2,1); L=(0,1,-2); M=(5,2,1); N=(-1,2,1); O=(-2,1,5);
P=(1,2,0); Q=(1,-2,5); R=(2,1,0); S=(1,-2,0); T=(0,0,1); U=(2,-1,0);
V=(1,0,0); W=(1,1,0); X=(1,-1,1); Y=(1,0,-1); Z=(0,1,1); a=(0,1,-1);
b=(1,1,1); c=(0,1,0); d=(-2,5,1); e=(2,1,-1); f=(1,0,2); g=(-1,2,5);
h=(1,-2,1); i=(2,-1,5); j=(-2,1,1); k=(1,5,2); l=(2,0,-1); m=(1,5,-2);
n=(-1,1,2); o=(-2,-1,5); p=(-1,-2,5).

#### Chosen MMPHs

Figure 5: (a) BMMPH obtained by Voráček and Navara Voráček and Navara (2021);
the 18-13 MMPH which one obtains by removing the vertices o,r,u is a NBMMPH;
(b) KS MMPH which does not have a real coordinatization and most probably also
not a complex one; (c) One of nine 51-37 KS MMPHs we generated; it is a 22-gon
non-isomorphic to Conway-Kochen’s 51-37; (d) KS NBMMPH 39-27 does not have a
known coordinatization—non-KS NBMMPH 33-27 obtained by removal of all gray
vertices from the 39-27 therefore also does not have a known coordinatization.

#### 39-27 critical KS MMPH—possible coordinatization is an open problem

39-27 1st 17 triads build a 17-gon:
837,7NF,FEG,GLK,K4c,cbY,Y2S,SRT,TWX,X6H,HJI,IQV,Vd5,5UA,A9B,BDC,CZ8,123,456,MND,OPQ,UJ8,aVL,E62,dW3,P74,cW9.

#### Other MMPHs generated by $\\{0,\pm 1,\pm 2,5\\}$

53-38
213,39A,AFG,GpB,BNX,XWY,YdK,KVf,fe5,546,678,8ED,DIr,rqO,OLP,Pkl,lCH,HMa,aZb,bhJ,JSj,ji2,BC5,HI2,JEC,KIG,L63,MNL,LKJ,QRS,TUV,cd8,ghA,mnP,opO,nFE,UND,RMF.
1=(5,2,1); 2=(-1,2,1); 3=(0,1,-2);4=(5,-1,2); 5=(1,1,-2); 6=(0,2,1);
7=(5,1,-2); 8=(1,-1,2); 9=(5,-2,-1); A=(1,2,1); B=(1,1,1); C=(1,-1,0);
D=(-1,1,1); E=(1,1,0); F=(1,-1,1); G=(1,0,-1); H=(1,1,-1); I=(1,0,1);
J=(0,0,1); K=(0,1,0); L=(1,0,0); M=(0,1,1); N=(0,1,-1); O=(0,1,2); P=(0,2,-1);
Q=(2,1,5); R=(2,1,-1); S=(1,-2,0); T=(-2,5,-1); U=(2,1,1); V=(1,0,-2);
W=(2,5,-1); X=(-2,1,1); Y=(1,0,2); Z=(-2,1,5); a=(2,-1,1); b=(1,2,0);
c=(1,5,2); d=(2,0,-1); e=(-1,5,2); f=(2,0,1); g=(-1,-2,5); h=(2,-1,0);
i=(1,-2,5); j=(2,1,0); k=(5,-1,-2); l=(1,1,2); m=(5,1,2); n=(-1,1,2);
o=(5,2,-1); p=(1,-2,1); q=(5,-2,1); r=(1,2,-1).

One of the eight 54-39 MMPHs:

54-39
546,6DE,EmW,WRV,VUJ,JHI,Ipq,qTs,srG,GFC,CAB,B38,879,9ZL,LMN,NOP,PbY,Yci,ihj,jdg,gef,fXa,a25,123,KLJ,QRP,STN,XYG,bI3,cE9,cT2,dC6,dbZ,XV8,dVT,klR,nSB,oU5,laZ.
1=(-1,1,2); 2=(1,1,0); 3=(1,-1,1); 4=(5,1,-2); 5=(1,-1,2); 6=(0,2,1);
7=(1,-2,1); 8=(1,0,-1); 9=(1,1,1); A=(5,-2,-1); B=(1,2,1); C=(0,1,-2);
D=(5,-1,2); E=(1,1,-2); F=(5,2,1); G=(-1,2,1); H=(-2,5,1); I=(2,1,-1);
J=(1,0,2); K=(2,5,-1); L=(-2,1,1); M=(2,-1,5); N=(1,2,0); O=(-2,1,5);
P=(2,-1,1); Q=(2,5,1); R=(1,0,-2); S=(2,-1,0); T=(0,0,1); U=(2,0,-1);
V=(0,1,0); W=(2,0,1); X=(1,0,1); Y=(1,1,-1); Z=(0,1,-1); a=(-1,1,1);
b=(0,1,1); c=(1,-1,0); d=(1,0,0); e=(5,-2,1); f=(1,2,-1); g=(0,1,2);
h=(5,-1,-2); i=(1,1,2); j=(0,2,-1); k=(-2,5,-1); l=(2,1,1); m=(-1,5,2);
n=(-1,-2,5); o=(1,5,2); p=(2,1,5); q=(1,-2,0); r=(1,-2,5); s=(2,1,0).

55-40
123,456,789,AB6,CD9,EF3,GHB,IJD,KLF,MN9,OP6,QR3,SNH,TRJ,UPL,VW5,XY8,Za2,bcB,dcD,ecF,eaN,dYP,bWR,RPN,fgS,hiU,jkT,lmH,noJ,pqL,rM5,sQ8,tO2,mXW,qaV,oZY,gdV,keX,ibZ.
1=(5,-1,2), 2=(1,1,-2), 3=(0,2,1), 4=(2,5,1), 5=(2,-1,1), 6=(1,0,-2),
7=(-1,-2,5), 8=(1,2,1), 9=(2,-1,0), A=(-2,5,-1), B=(2,1,1), C=(1,2,5),
D=(1,2,-1), E=(5,1,-2), F=(1,-1,2), G=(-2,-1,5), H=(1,-2,0), I=(5,-2,1),
J=(0,1,2), K=(1,5,2), L=(2,0,-1), M=(1,2,0), N=(0,0,1), O=(2,0,1), P=(0,1,0),
Q=(0,1,-2), R=(1,0,0), S=(2,1,0), T=(0,2,-1), U=(1,0,2), V=(1,1,-1),
W=(0,1,1), X=(1,-1,1), Y=(1,0,-1), Z=(1,1,1), a=(1,-1,0), b=(0,1,-1),
c=(-1,1,1), d=(1,0,1), e=(1,1,0), f=(1,-2,5), g=(-1,2,1), h=(2,5,-1),
i=(-2,1,1), j=(5,1,2), k=(-1,1,2), l=(2,1,5), m=(2,1,-1), n=(5,2,-1),
o=(1,-2,1), p=(-1,5,-2), q=(1,1,2), r=(-2,1,5), s=(5,-2,-1), t=(-1,5,2)

Figure 6: (a-c) Critical 3D MMPHs generated by the components $\\{0,\pm 1,\pm
2,5\\}$; (a) the only 53-38; 22-gon; (b) one of the eight 54-39s; 23-gon; (c)
the only 55-40; 22-gon; (d) the only 57-41—the smallest MMPH generated by
$\\{0,\pm 1,2,\pm 5,\pm\omega,2\omega\\}$; 21-gon.

#### The smallest MMPH generated by $\\{0,\pm 1,2,\pm 5,\pm\omega,2\omega\\}$,
shown in Fig. 6(d)

57-41
213,398,876,645,5cd,dBa,abG,GWV,VTU,UAR,RSQ,QrO,OHP,Pmn,nJD,DlF,FjE,EpI,ItY,YCf,fe2,ABC,DC6,EB3,FGH,IJK,LK5,MK2,NH8,KHA,XSG,YZG,gZL,hbM,ijL,klM,opP,qTO,stN,uvN,vaJ.
1=(1,2,5); 2=(2,-1,0); 3=(1,2,-1); 4=(-1,2,5); 5=(2,1,0); 6=(-1,2,-1);
7=(5,2,-1); 8=(0,1,2); 9=(-5,2,-1); A=(0,1,0); B=(1,0,1); C=(1,0,-1);
D=(1,1,1); E=(-1,1,1); F=(0,1,-1); G=(0,1,1); H=(1,0,0); I=(1,1,0);
J=(1,-1,0); K=(0,0,1); L=(-1,2,0); M=(1,2,0); N=(0,2,-1); O=(0,-1,2);
P=(0,2,1); Q=(-1,2$\omega$,$\omega$); R=(1,0,$\omega$);
S=(1,$\omega$,-$\omega$);T=(1,2$\omega$,$\omega$); U=(1,0,-$\omega$);
V=(1,-$\omega$,$\omega$); W=(2,$\omega$,-$\omega$); X=(2,-$\omega$,$\omega$);
Y=(1,-1,1); Z=(2,1,-1); a=(1,1,-1); b=(2,-1,1);c=(-1,2,-5); d=(-1,2,1);
e=(1,2,-5); f=(1,2,1); g=(2,1,5); h=(2,-1,-5); i=(2,1,-5); j=(2,1,1);
k=(2,-1,5);l=(2,-1,-1); m=(5,-1,2); n=(-1,-1,2); o=(-5,-1,2); p=(1,-1,2);
q=(-5,2$\omega$,$\omega$); r=(5,2$\omega$,$\omega$); s=(5,1,2); t=(-1,1,2);
u=(-5,1,2); v=(1,1,2).

#### Original KS set

The original 192-118 KS set found by Kochen and Specker Kochen and Specker
(1967) was not accompanied by an explicit coordinatization. Instead, they give
the equation $\sin(\pi/10)=xy/\sqrt{(1+x^{2}+x^{2}y^{2})(1+y^{2}+x^{2}y^{2})}$
where $x$ and $y$, provided they satisfy this equation, are presumably
arbitrary parameters that are used to define the vectors in their starting
subgraph called $\Gamma_{1}$. The final hypergraph (called $\Gamma_{2}$),
shown in Fig. 3, consists of 15 copies of $\Gamma_{1}$, with assumed—but so
far undefined—vectors rotated (using rotation matrices) so that the copies
align correctly. Notice that Kochen and Specker in their original graphical
representation of $\Gamma_{2}$ dropped all 75 gray vertices thus arriving and
117-118 hypergraph. But as we explained in (Pavičić, 2017, Sec. XII) if one
dropped all gray vertices, the remaining hypergraph would not be a KS set any
more.

The $x,y$ satisfying the equation is not sufficient to provide a
coordinatization. E.g., one solution is $x$=$y$=$\sqrt{\varphi}$, where
$\varphi$=$(1+\sqrt{5})/2$ is the golden ratio. When we compute vectors for
just the 117 black vertices on hypergraph $\Gamma_{2}$, only 115 of them are
unique, so we cannot use it even for the truncated non-critical Pavičić (2019)
117-118 set. The complete 192-118 KS MMPH requires $192-117=75$ additional
unique vectors (each computed as the cross product of the other two vectors on
the hyperedge), placing an even more severe constraint on the possible values
of $x$ and $y$.

We found that $x$=$1/\varphi$ and $y$=$\sqrt{\varphi}$ yield 192 unique
vectors, and we used these values to compute the coordinatization. To show the
final result, let us define $f(p,q,r)=\sqrt{2^{p}\sqrt{5}^{q}\varphi^{r}}$.
Then we define 16 constants as follows: $c_{1}$=$f$(-1,0,0);
$c_{2}$=$f$(0,0,-1); $c_{3}$=$f$(-1,1,-3); $c_{4}$=$f$(-1,0,-2);
$c_{5}$=$f$(0,-1,-6); $c_{6}$=$f$(-1,-1,-5);
$c_{7}$=$f$(2,-1,0);$c_{8}$=$f$(-1,-1,-3); $c_{9}$=$f$(0,-1,-2);
$c_{10}$=$f$(0,0,-5); $c_{11}$ =$f$(-1,0,-6); $c_{12}$=$f$(2,-1,-2);
$c_{13}$=$f$(-1,-1,3);
$c_{14}$=$f$(2,-1,-4);$c_{15}$=$f(\ln(9/2-9/\sqrt{5})/\ln
2,0,0)$=$3\sqrt{(5-2\sqrt{5})/10}$; $c_{16}$=$f(\ln(17/4-31/(4\sqrt{5}))/\ln
2,0,0)$=$\frac{1}{2}\sqrt{(85-31\sqrt{5})/5}$.

Figure 7: KS MMPH of the original Kochen-Specker 192-118 set presented
according to (Pavičić et al., 2005, Fig. 6); note that the figure in (Pavičić
et al., 2005, Fig. 6) and the one-to-one correspondence to (Kochen and
Specker, 1967, $\Gamma_{2}$) presented there is obtained exclusively from the
ASCII string of the MMPH without any reference to its coordinatization; note
also that one can use the string below to assign its vertices to the vertices
of the figure by hand.

When normalized, the 192 vectors have 73 different components. By using
unnormalized vectors, i.e. rays, we reduced the number of different components
to 24, whose values are $0,1,\pm c_{1}$, $\pm c_{2}$, $c_{3}$, $c_{4}$,
$-c_{5}$, $\pm c_{6}$, $c_{7}$, $\pm c_{8}$, $\pm c_{9}$, $-c_{10}$, $\pm
c_{11}$, $-c_{12}$, $-c_{13}$, $-c_{14}$, $-c_{15}$, and $-c_{16}$. The
coordinatization is presented below.

192-118
123,345,567,789,9AB,BC1,DEF,FGH,HIJ,JKL,LMN,NOD,PQR,RST,TUV,VWX,XYZ,ZaP,bcd,def,fgh,hij,jkl,lmb,1no,opq,qrs,stu,uvw,wx1,yz!,!"#,#$%,%&’,’(),)*y,-/:,:;<,<=>,>?@,@[\,\\]-,^_‘,‘{|,|}$\sim$,$\sim$+1+2,+2+3+4,+4+5^,+6+7+8,+8+9+A,+A+B+C,+C+D+E,+E+F+G,+G+H+6,y+I+J,+J+K+L,+L+M+N,+N+O+P,+P+Q+R,+R+Sy,+T+U+V,+V+W+X,+X+Y+Z,+Z+a+b,+b+c+d,+d+e+T,+f+g+h,+h+i+j,+j+k+l,+l+m+n,+n+o+p,+p+q+f,+r+s+t,+t+u+v,+v+w+x,+x+y+z,+z+!+",+"+#+r,+$+%+&,+&+’+(,+(+)+*,+*+-+/,+/+:+;,+;+<+$,+T+=+>,+>+?+@,+@+[+\,+\\+]+^,+^+_+‘,+‘+{+T,4A+|,GM+},SY+$\sim$,ek++1,pv++2,"(++3,;[++4,{+3++5,+9+F++6,+K+Q++7,+W+c++8,+i+o++9,+u+!++A,+’+:++B,+?+_++C,D7y,PJy,bVy,shy,-%+T,^>+T,$\sim$+C+T,+N+6+T,+f+Z1,+r+l1,+$+x1,+\\+*1,1y+T.

1=(1,0,0); 2=(0,-$c_{2}$,$c_{4}$); 3=(0,$c_{4}$,$c_{2}$);
4=($c_{8}$,-$c_{10}$,$c_{11}$); 5=($c_{1}$,$c_{9}$,-$c_{8}$);
6=(-$c_{1}$,$c_{7}$,$c_{8}$); 7=($c_{8}$,0,$c_{1}$);
8=(-$c_{1}$,-$c_{12}$,$c_{8}$); 9=($c_{4}$,-$c_{9}$,-$c_{6}$);
A=($c_{4}$,$c_{9}$,$c_{6}$); B=(0,$c_{6}$,-$c_{9}$); C=(0,$c_{9}$,$c_{6}$);
D=($c_{1}$,0,-$c_{8}$);
E=($c_{8}$,-$c_{12}$,$c_{1}$);F=($c_{6}$,$c_{9}$,$c_{4}$);
G=($c_{1}$,-$c_{2}$,$c_{3}$); H=(-$c_{13}$,-$c_{9}$,$c_{1}$);
I=($c_{8}$,-$c_{12}$,-$c_{11}$); J=($c_{3}$,0,$c_{1}$);
K=(-$c_{8}$,-$c_{14}$,$c_{11}$);L=($c_{8}$,-$c_{5}$,-$c_{11}$);
M=($c_{1}$,$c_{2}$,$c_{8}$); N=($c_{8}$,-$c_{9}$,$c_{1}$);
O=($c_{8}$,$c_{7}$,$c_{1}$); P=(-$c_{13}$,0,$c_{1}$);
Q=(-$c_{11}$,-$c_{12}$,-$c_{8}$);
R=(-$c_{1}$,$c_{9}$,-$c_{13}$);S=($c_{4}$,$c_{9}$,-$c_{6}$);
T=($c_{3}$,-$c_{2}$,-$c_{1}$); U=($c_{1}$,$c_{7}$,-$c_{13}$);
V=($c_{1}$,0,$c_{3}$); W=($c_{11}$,-$c_{12}$,-$c_{8}$);
X=($c_{1}$,$c_{9}$,-$c_{13}$);
Y=($c_{4}$,-$c_{9}$,$c_{6}$);Z=($c_{3}$,$c_{2}$,$c_{1}$);
a=(-$c_{1}$,$c_{7}$,-$c_{13}$); b=($c_{1}$,0,-$c_{13}$);
c=($c_{8}$,-$c_{14}$,$c_{11}$); d=(-$c_{8}$,-$c_{5}$,-$c_{11}$);
e=($c_{1}$,-$c_{2}$,-$c_{8}$);f=(-$c_{8}$,-$c_{9}$,$c_{1}$);
g=(-$c_{8}$,$c_{7}$,$c_{1}$); h=($c_{1}$,0,$c_{8}$);
i=(-$c_{8}$,-$c_{12}$,$c_{1}$); j=(-$c_{6}$,$c_{9}$,$c_{4}$);
k=(-$c_{1}$,-$c_{2}$,$c_{3}$);
l=(-$c_{13}$,$c_{9}$,-$c_{1}$);m=(-$c_{8}$,-$c_{12}$,-$c_{11}$);
n=(0,$c_{2}$,$c_{4}$); o=(0,$c_{4}$,-$c_{2}$);
p=($c_{8}$,-$c_{10}$,-$c_{11}$); q=($c_{1}$,$c_{9}$,$c_{8}$);
r=(-$c_{1}$,$c_{7}$,-$c_{8}$);
s=(-$c_{8}$,0,$c_{1}$);t=($c_{4}$,$c_{7}$,$c_{6}$);
u=($c_{4}$,-$c_{5}$,$c_{6}$); v=($c_{3}$,$c_{2}$,-$c_{1}$);
w=(0,$c_{1}$,$c_{2}$); x=(0,-$c_{2}$,$c_{1}$); y=(0,1,0);
z=($c_{6}$,0,$c_{9}$);!=($c_{9}$,0,-$c_{6}$); "=($c_{6}$,$c_{4}$,$c_{9}$);
#=(-$c_{6}$,$c_{4}$,-$c_{9}$); $=($c_{8}$,-$c_{1}$,-$c_{12}$);
%=($c_{1}$,$c_{8}$,0); &=($c_{8}$,-$c_{1}$,$c_{7}$);
’=(-$c_{8}$,$c_{1}$,$c_{9}$);(=($c_{11}$,$c_{8}$,-$c_{10}$);
)=($c_{2}$,0,$c_{4}$); *=($c_{4}$,0,-$c_{2}$); -=(-$c_{8}$,$c_{1}$,0);
/=($c_{1}$,$c_{8}$,-$c_{12}$); :=($c_{4}$,$c_{6}$,$c_{9}$);
;=($c_{3}$,$c_{1}$,-$c_{2}$);<=($c_{1}$,-$c_{13}$,-$c_{9}$);
==(-$c_{11}$,$c_{8}$,-$c_{12}$); >=($c_{1}$,$c_{3}$,0);
?=($c_{11}$,-$c_{8}$,-$c_{14}$); @=(-$c_{11}$,$c_{8}$,-$c_{5}$);
[=($c_{8}$,$c_{1}$,$c_{2}$);\=($c_{1}$,$c_{8}$,-$c_{9}$); ]=($c_{1}$,
$c_{8}$,$c_{7}$); ^=($c_{1}$,-$c_{13}$,0); _=($c_{8}$,$c_{11}$,-$c_{14}$);
‘=(-$c_{8}$,-$c_{11}$,-$c_{5}$);
{=(0,-$c_{9}$,$c_{4}$);|=($c_{8}$,-$c_{11}$,-$c_{5}$);
}=(-$c_{8}$,$c_{11}$,-$c_{14}$); $\sim$=($c_{3}$,$c_{1}$,0);
+1=(-$c_{13}$,$c_{1}$,$c_{7}$); +2=(-$c_{1}$,$c_{3}$,-$c_{2}$);
+3=(-$c_{6}$,$c_{4}$,$c_{9}$);+4=(-$c_{13}$,-$c_{1}$,$c_{9}$);
+5=(-$c_{8}$,-$c_{11}$,-$c_{12}$); +6=($c_{8}$,$c_{1}$,0);
+7=($c_{1}$,-$c_{8}$,-$c_{12}$); +8=($c_{4}$,-$c_{6}$,$c_{9}$);
+9=($c_{3}$,-$c_{1}$,-$c_{2}$);+A=(-$c_{1}$,-$c_{13}$,$c_{9}$);
+B=(-$c_{11}$,-$c_{8}$,-$c_{12}$); +C=(-$c_{13}$,$c_{1}$,0);
+D=($c_{11}$,$c_{8}$,-$c_{14}$); +E=(-$c_{11}$,-$c_{8}$,-$c_{5}$);
+F=(-$c_{8}$,$c_{1}$,-$c_{2}$);+G=($c_{1}$,-$c_{8}$,-$c_{9}$);
+H=($c_{1}$,-$c_{8}$,$c_{7}$); +I=($c_{1}$,0,-$c_{2}$);
+J=($c_{2}$,0,$c_{1}$); +K=(-$c_{1}$,$c_{3}$,$c_{2}$);
+L=($c_{6}$,$c_{4}$,-$c_{5}$);+M=($c_{6}$,$c_{4}$,$c_{7}$);
+N=($c_{1}$,-$c_{8}$,0); +O=(-$c_{8}$,-$c_{1}$,$c_{7}$);
+P=($c_{8}$,$c_{1}$,$c_{9}$); +Q=(-$c_{11}$,$c_{8}$,-$c_{10}$);
+R=(-$c_{2}$,0,$c_{4}$);+S=($c_{4}$,0,$c_{2}$); +T=(0,0,1);
+U=(-$c_{2}$,$c_{1}$,0); +V=($c_{1}$,$c_{2}$,0);
+W=($c_{2}$,-$c_{1}$,$c_{3}$); +X=(-$c_{5}$,$c_{6}$,$c_{4}$);
+Y=($c_{7}$,$c_{6}$,$c_{4}$);+Z=(0,$c_{1}$,-$c_{8}$);
+a=($c_{7}$,-$c_{8}$,-$c_{1}$); +b=($c_{9}$,$c_{8}$,$c_{1}$);
+c=(-$c_{10}$,-$c_{11}$,$c_{8}$); +d=($c_{4}$,-$c_{2}$,0);
+e=($c_{2}$,$c_{4}$,0);+f=(0,$c_{8}$,$c_{1}$);
+g=(-$c_{12}$,$c_{1}$,-$c_{8}$); +h=($c_{9}$,$c_{4}$, -$c_{6}$);
+i=(-$c_{2}$,$c_{3}$,-$c_{1}$); +j=($c_{9}$,-$c_{1}$,-$c_{13}$);
+k=(-$c_{12}$,-$c_{11}$,-$c_{8}$);+l=(0,-$c_{13}$,$c_{1}$); +m=(-$c_{14}$,
$c_{11}$,$c_{8}$); +n=(-$c_{5}$,-$c_{11}$,-$c_{8}$);
+o=(-$c_{2}$,-$c_{8}$,$c_{1}$); +p=(-$c_{9}$,$c_{1}$,-$c_{8}$);
+q=($c_{7}$,$c_{1}$,-$c_{8}$);+r=(0,$c_{3}$,$c_{1}$);
+s=($c_{7}$,-$c_{13}$,$c_{1}$); +t=(-$c_{2}$,-$c_{1}$,$c_{3}$);
+u=($c_{9}$,-$c_{6}$,$c_{4}$); +v=($c_{9}$,-$c_{13}$,-$c_{1}$);
+w=(-$c_{12}$,-$c_{8}$,-$c_{11}$);+x=(0,$c_{1}$,-$c_{13}$);
+y=(-$c_{14}$,$c_{8}$,$c_{11}$); +z=(-$c_{5}$,-$c_{8}$,-$c_{11}$);
+!=($c_{4}$,0,-$c_{9}$); +"=(-$c_{5}$,$c_{8}$, -$c_{11}$);
+#=(-$c_{14}$,-$c_{8}$,$c_{11}$);+$=(0,$c_{1}$,$c_{3}$);
+%=(-$c_{12}$,-$c_{11}$,$c_{8}$); +&=(-$c_{9}$,$c_{1}$,-$c_{13}$);
+’=(-$c_{2}$,$c_{3}$,$c_{1}$); +(=($c_{9}$,$c_{4}$,$c_{6}$);
+)=(-$c_{12}$,$c_{1}$,$c_{8}$);+*=(0,-$c_{8}$,$c_{1}$);
+-=($c_{7}$,$c_{1}$,$c_{8}$); +/=(-$c_{9}$,$c_{1}$,$c_{8}$);
+:=($c_{2}$,$c_{8}$,$c_{1}$); +;=(-$c_{5}$,-$c_{11}$,$c_{8}$);
+<=(-$c_{14}$,$c_{11}$,-$c_{8}$);+==($c_{9}$,$c_{6}$,0);
+>=($c_{6}$,-$c_{9}$,0); +?=($c_{9}$,$c_{6}$,$c_{4}$);
+@=(-$c_{9}$,-$c_{6}$,$c_{4}$); +[=(-$c_{12}$,$c_{8}$,-$c_{1}$);
+\=(0,$c_{1}$,$c_{8}$);+]=($c_{7}$,$c_{8}$,-$c_{1}$);
+^=($c_{9}$,-$c_{8}$,$c_{1}$); +_=(-$c_{10}$,$c_{11}$,$c_{8}$);
+‘=($c_{4}$,$c_{2}$,0); +{=(-$c_{2}$,$c_{4}$,0);
+|=($c_{4}$,-$c_{5}$,-$c_{16}$);+}=(-$c_{9}$,$c_{6}$,$c_{2}$);
+$\sim$=(0,$c_{6}$,$c_{9}$); ++1=($c_{9}$,$c_{6}$,$c_{2}$);
++2=(-$c_{1}$,-$c_{5}$,-$c_{15}$); ++3=(-$c_{16}$,$c_{4}$,-$c_{5}$);
++4=($c_{2}$,-$c_{9}$,$c_{6}$);++5=(-$c_{13}$,-$c_{11}$,-$c_{5}$);
++6=($c_{2}$,$c_{9}$,$c_{6}$); ++7=(-$c_{15}$,-$c_{1}$,-$c_{5}$);
++8=(-$c_{5}$,-$c_{15}$, -$c_{1}$);
++9=($c_{6}$,$c_{2}$,$c_{9}$);++A=(-$c_{5}$,-$c_{13}$,-$c_{11}$);
++B=($c_{6}$,$c_{2}$,-$c_{9}$); ++C=(-$c_{5}$,-$c_{16}$,$c_{4}$).

The components may be used to generate a master MMPH whose class contains the
MMPH 192-118. The master has 2416 vertices and 1432 hyperedges.

### 5-dim

29-16
HOINJ,JGSTF,FT4Q8,85679,92LOH,PQRST,KLMNO,CDEIO,ABEGT,34DMO,12BRT,237CO,146AT,279OP,468KT,5EJOT.
1=(1,-1,1,0,-1); 2=(1,0,-1,0,0); 3=(1,-1,1,1,0); 4=(0,1,1,0,0); 5=(0,0,1,0,0);
6=(1,0,0,0,1); 7=(0,1,0,1,0); 8=(1,0,0,0,-1); 9=(0,1,0,-1,0); A=(1,1,-1,0,-1);
B=(1,1,1,0,1); C=(1,1,1,-1,0); D=(1,1,-1,1,0);E=(1,-1,0,0,0); F=(1,-1,1,0,1);
G=(0,0,1,0,-1); H=(1,-1,1,-1,0); I=(0,0,1,1,0); J=(1,1,0,0,0); K=(0,1,-1,0,0);
L=(1,1,1,1,0); M=(1,0,0,-1,0); N=(1,-1,-1,1,0); O=(0,0,0,0,1); P=(1,0,1,0,0);
Q=(1,1,-1,0,1); R=(0,1,0,0,-1);S=(-1,1,1,0,1); T=(0,0,0,1,0).

31-17
12345,16789,16ABC,13DEB,14D8F,1GHF9,1IJ8K,1IHAE,LMGI5,L27NO,L2PQC,M27RS,2TJUO,2THPR,23VUN,24VPS,2PQRS.
1=(0,0,0,0,1); 2=(0,0,0,1,0); 3=(1,1,0,0,0); 4=(1,-1,0,0,0); 5=(0,0,1,0,0);
6=(1,0,0,-1,0); 7=(0,1,1,0,0); 8=(1,1,-1,1,0); 9=(1,-1,1,1,0); A=(1,1,1,1,0);
B=(1,-1,-1,1,0); C=(0,1,-1,0,0); D=(0,0,1,1,0);
E=(1,-1,1,-1,0);F=(1,1,1,-1,0); G=(0,1,0,1,0); H=(1,0,-1,0,0); I=(0,1,0,-1,0);
J=(1,0,1,0,0); K=(-1,1,1,1,0); L=(1,0,0,0,1);M=(1,0,0,0,-1); N=(1,-1,1,0,-1);
O=(1,1,-1,0,-1); P=(1,1,1,0,-1); Q=(-1,1,1,0,1); R=(1,-1,1,0,1);
S=(1,1,-1,0,1); T=(0,1,0,0,1); U=(1,-1,-1,0,1); V=(0,0,1,0,1).

### 7-dim

13-4 1234567,456789A,123ABCD,4567BCD.

34-14 1234567, 189A5BC, 189DE7F, 189GHIJ, 189KHBL, 2MNDOIP, 2MNEOCL, 2MNGK6F,
QRNSAJP, QT4U567, RTV9567, WXMS567, WYV8AJP, XY3U567. 1=(0,0,0,1,0,0,0);
2=(0,0,1,0,0,0,0); 3=(1,-1,0,0,0,0,0); 4=(1,1,0,0,0,0,0); 5=(0,0,0,0,0,0,1);
6=(0,0,0,0,1,1,0); 7=(0,0,0,0,1,-1,0); 8=(0,1,-1,0,0,0,0); 9=(0,1,1,0,0,0,0);
A=(0,0,0,0,1,0,0); B=(1,0,0,0,0,-1,0); C=(1,0,0,0,0,1,0); D=(1,0,0,0,1,1,-1);
E=(-1,0,0,0,1,1,1); F=(1,0,0,0,0,0,1); G=(1,0,0,0,1,-1,-1); H=(1,0,0,0,1,1,1);
I=(1,0,0,0,-1,0,0); J=(0,0,0,0,0,1,-1); K=(1,0,0,0,-1,1,-1);
L=(0,0,0,0,1,0,-1); M=(0,1,0,1,0,0,0); N=(0,1,0,-1,0,0,0); O=(1,0,0,0,1,-1,1);
P=(0,0,0,0,0,1,1); Q=(-1,1,1,1,0,0,0); R=(1,1,-1,1,0,0,0); S=(1,0,1,0,0,0,0);
T=(1,-1,1,1,0,0,0); U=(0,0,1,-1,0,0,0); V=(1,0,0,-1,0,0,0);
W=(1,-1,-1,1,0,0,0); X=(1,1,-1,-1,0,0,0); Y=(1,1,1,1,0,0,0).

28(58)-14 3451678, 4LF8BJK, PHG7QRS, 349A16(T), 38BCD(UV), F8QJO(WX),
4HIJK(YZ), AMNCD(ab), 3452(cde), EFG2(fgh), 9MNO(ijk), PLRS(lmn), EI7(opqr),
12(stuvw). 1=(0,1,0,0,0,0,0); 2=(1,0,0,0,0,0,0); 3=(0,0,1,1,1,-1,0);
4=(0,0,1,1,-1,1,0); 5=(0,0,1,-1,0,0,0); 6=(1,0,0,0,0,0,1); 7=(1,0,0,0,1,1,-1);
8=(-1,0,0,0,1,1,1); 9=(0,0,1,-1,1,1,0); A=(0,0,-1,1,1,1,0);
B=(1,0,-1,1,0,0,1); C=(1,-1,0,-1,1,0,0); D=(1,1,1,0,0,1,0);
E=(0,1,0,-1,-1,1,0); F=(0,1,1,0,0,-1,1); G=(0,-1,1,0,0,1,1);
H=(0,1,0,1,1,0,1); I=(0,1,1,0,0,-1,-1); J=(1,-1,1,0,1,0,0);
K=(1,1,0,-1,0,1,0); L=(0,1,0,1,1,0,-1); M=(0,1,0,0,1,-1,-1);
N=(0,1,-1,-1,0,0,1); O=(1,1,0,1,0,1,0); Q=(1,0,-1,-1,0,0,1);
P=(0,0,1,-1,1,-1,0); R=(1,1,1,0,-1,0,0); S=(1,-1,0,1,0,-1,0);
T=(1,0,0,0,0,0,-1); U=(0,1,0,-1,0,-1,1); V=(0,1,-1,0,1,0,-1);
W=(0,0,1,-1,-1,1,0); X=(0,1,0,-1,1,0,-1); Y=(1,0,-1,1,0,0,-1);
Z=(1,0,0,0,-1,-1,1); a=(1,0,0,1,0,-1,1); b=(-1,0,1,0,1,0,1);
c=(0,1,0,0,0,0,1); d=(0,1,0,0,1,1,-1); e=(0,-1,0,0,1,1,1);
f=(0,0,1,1,-1,0,-1); g=(0,0,1,-1,1,0,-1); h=(0,1,0,1,1,1,0);
i=(1,-1,-1,0,1,0,0); j=(-1,0,0,1,1,0,1); k=(1,0,1,0,0,-1,1);
l=(0,1,-1,0,0,-1,1); m=(1,0,0,0,1,1,1); n=(-1,0,1,1,0,0,1);
o=(0,0,1,-1,1,0,1); p=(1,-1,1,0,-1,0,0); q=(-1,0,1,1,0,1,0);
r=(1,1,0,0,1,0,-1); s=(0,0,0,1,1,1,-1); t=(0,0,0,1,1,-1,1);
u=(0,0,0,1,-1,1,1); v=(0,0,0,-1,1,1,1); w=(0,0,1,0,0,0,0).

207-97 1234567, 89ABCDE, 8FGHIJK, 8LMNOPE, QRS6TUV, QRS6WXY, QRSZab7, QRScde7,
QRfg6TW, QRfgZc7, Qf6hijk, QglLmn7, Qaopqrs, QetuvOw, Qxyzq!", Q#$V%&’,
Q(yuU)*, QLnX-/r, :;6hN<=, :>?@[\\], :^_‘{]$|$, }$\sim$^+1M+27,
}$\sim$+3+4mG7, }+5+6@+7+8+9, }e+6v<+A+B, }(+C+DY{+E, Rf6+F+GC[, RaoN+H+I+J,
Rbo+K+7\\+L, R+M+NX-+O+P, R+Q+R+S+7+T!, R+U_Y-+8+V, R+U+WGN+X+Y, R+ZMV%+8+V,
R9+aU%+A+I, R+b+RX-+A+I, +c+d6v+e+X+H, +c+ftuv+X+P, +cx+g+h%+i+P, +c+Znv+H!P,
+c(+Ru+j)+E, +k+f+l+m+n+O+o, +k+p+N+S+q+r+s, +td+l+K<s+B, +tl+u@+v+w+x,
+t+y+z+a+F+q+x, +!+5+"+#k’+$, +!b+%z+&+’+V, +!+(+z+)+#+X+x, +*+f+"+#+w’+-,
+*+Z+/+#+:+r+s, +*+Z+;+D‘+<+V, +=+1_U+>+?+9, $\sim$+f+@+[+\\+]+P,
$\sim$b+^h+\&+A+_, $\sim$+Q+/+mC+‘+s, f+5H+{+GC+$|$, fxyI+}+I+J, f#AI+}+O+E,
f(+$\sim$‘++1+A++2, f(y+K+7++3P, f++4_+j++5/++6, f+(M+h+>++7+$|$,
++8+dde++9++A7, ++8e++9+Sq+>+V, ++8e++A+[k+A’, ++8+MA+K+vq++B, ++8+U++C+[j"+s,
++8+U++CN+v+w+Y, ++8+U+2U++7r+L, ++8+Zm++DO"+s, ++8+ZnX+}r+L, ++8+(+zn+[j++B,
++E+f+^++Fi++G*, ++E+pnV++H\\++I, ++E(+C++J++K++3+‘, ++E+y++CX++L+$$|$,
;+5+^h=&+_, ;x+u++J+vq++M, ;#+z+C+#++N++O, +dx+gB+v[+Y, +d(+gU++7+O*,
+d(+Rp+7++3+T, +dF+zGv+X+Y, +dLMX++1/$|$, +dLMV++7+8+$|$, g+5+{u++P+:*,
g+5+{++D+H&’, g+M+Np+vq++Q, g+M+a+h++7’++R, g+Q+gB+v++K++Q, g+Q+R‘+}’++R,
glm++D+XD+s, glnI+}\\+L, g9+N‘+}+i*, gF+z+2++D+X++Q, gLnN+H++3P.

1=(0,0,0,1,1,1,-1); 2=(0,0,0,1,1,-1,1); 3=(0,0,0,1,-1,1,1);
4=(0,0,0,-1,1,1,1); 5=(0,0,1,0,0,0,0); 8=(0,0,1,1,1,1,0); Q=(0,0,1,1,1,-1,0);
:=(0,0,1,1,-1,0,1); }=(0,0,1,1,-1,0,-1); R=(0,0,1,1,-1,1,0);
+c=(0,0,1,1,-1,-1,0); S=(0,0,1,-1,0,0,0); +k=(0,0,1,-1,0,1,1);
+t=(0,0,1,-1,0,1,-1); +!=(0,0,1,-1,0,-1,1); +*=(0,0,-1,1,0,1,1);
+==(0,0,1,-1,1,0,1); $\sim$=(0,0,1,-1,1,0,-1); f=(0,0,1,-1,1,1,0);
++8=(0,0,1,-1,1,-1,0); ++E=(0,0,1,-1,-1,0,1); ;=(0,0,-1,1,1,0,1);
+d=(0,0,1,-1,-1,1,0); g=(0,0,-1,1,1,1,0); 6=(0,1,0,0,0,0,0);
Z=(0,1,0,0,0,0,1); c=(0,1,0,0,0,0,-1); d=(0,1,0,0,1,1,1); a=(0,1,0,0,1,1,-1);
+f=(0,1,0,0,1,-1,1); +5=(0,1,0,0,1,-1,-1); >=(0,1,0,0,-1,1,-1);
e=(0,1,0,0,-1,-1,1); b=(0,-1,0,0,1,1,1); x=(0,1,0,1,0,1,-1);
+M=(0,1,0,1,0,-1,1); +Q=(0,1,0,1,0,-1,-1); +U=(0,1,0,1,1,0,1);
+Z=(0,1,0,1,1,0,-1); ^=(0,1,0,1,1,1,0); +3=(0,1,0,1,1,-1,0);
l=(0,1,0,1,-1,0,-1); +p=(0,1,0,1,-1,1,0); 9=(0,1,0,-1,0,1,1);
+b=(0,1,0,-1,0,1,-1); #=(0,1,0,-1,0,-1,1); (=(0,-1,0,1,0,1,1);
F=(0,1,0,-1,1,0,1); L=(0,1,0,-1,1,0,-1); +y=(0,1,0,-1,1,-1,0);
++4=(0,1,0,-1,-1,0,1); +(=(0,-1,0,1,1,0,1); +1=(0,1,0,-1,-1,1,0);
+4=(0,-1,0,1,1,1,0); m=(0,1,1,0,0,1,1); ++C=(0,1,1,0,0,1,-1);
M=(0,1,1,0,0,-1,1); _=(0,1,1,0,0,-1,-1); +g=(0,1,1,0,1,0,1);
+N=(0,1,1,0,1,0,-1); +u=(0,1,1,0,1,-1,0); +$\sim$=(0,1,1,0,-1,0,1);
A=(0,1,1,0,-1,0,-1); +/=(0,1,1,0,-1,1,0); ++9=(0,1,1,1,0,0,-1);
+@=(0,1,1,1,0,1,0); +%=(0,1,1,1,1,0,0); +l=(0,1,1,1,-1,0,0);
+z=(-1,1,1,1,0,0,0); +6=(0,1,1,-1,0,1,0); +W=(1,-1,-1,1,0,0,0);
G=(0,1,-1,0,0,1,-1); n=(0,1,-1,0,0,-1,1); +2=(0,-1,1,0,0,1,1);
y=(0,1,-1,0,1,0,1); $=(0,1,-1,0,1,0,-1); +)=(0,1,-1,0,1,-1,0);
+R=(0,1,-1,0,-1,0,1); +a=(0,-1,1,0,1,0,1); +C=(0,1,-1,0,-1,1,0);
+;=(0,-1,1,0,1,1,0); o=(0,1,-1,1,0,0,1); t=(0,1,-1,1,0,0,-1);
?=(0,1,-1,1,0,-1,0); H=(1,-1,-1,0,1,0,0); +{=(0,1,-1,-1,0,0,1);
++A=(0,-1,1,1,0,0,1); +^=(0,1,-1,-1,0,1,0); u=(-1,1,1,0,0,1,0);
+"=(0,-1,1,1,1,0,0); +D=(-1,1,1,0,0,0,1); ++F=(1,-1,-1,0,0,0,1);
7=(1,0,0,0,0,0,0); T=(1,0,0,0,0,0,1); W=(1,0,0,0,0,0,-1); X=(1,0,0,0,1,1,1);
U=(1,0,0,0,1,1,-1); ‘=(1,0,0,0,1,-1,1); +h=(1,0,0,0,1,-1,-1);
I=(1,0,0,0,-1,1,1); +j=(1,0,0,0,-1,1,-1); Y=(1,0,0,0,-1,-1,1);
V=(-1,0,0,0,1,1,1); v=(1,0,0,1,0,1,1); h=(1,0,0,1,0,1,-1);
++D=(1,0,0,1,0,-1,1); +[=(1,0,0,1,0,-1,-1); +S=(1,0,0,1,1,0,1);
+F=(1,0,0,1,1,0,-1); +m=(1,0,0,1,1,1,0); z=(1,0,0,1,-1,0,1);
+#=(1,0,0,1,-1,-1,0); N=(1,0,0,-1,0,1,1); ++P=(1,0,0,-1,0,1,-1);
+e=(1,0,0,-1,0,-1,1); i=(-1,0,0,1,0,1,1); p=(1,0,0,-1,1,0,1);
B=(1,0,0,-1,1,0,-1); ++J=(1,0,0,-1,1,1,0); +K=(1,0,0,-1,-1,0,1);
+G=(-1,0,0,1,1,0,1); @=(-1,0,0,1,1,1,0); ++N=(1,0,1,0,0,1,1);
C=(1,0,1,0,0,-1,1); +:=(1,0,1,0,1,0,1); +X=(1,0,1,0,1,0,-1);
++G=(1,0,1,0,1,1,0); <=(1,0,1,0,1,-1,0); j=(1,0,1,0,-1,0,1);
O=(1,0,1,0,-1,0,-1); +q=(1,0,1,0,-1,-1,0); ++5=(1,0,1,1,0,0,1);
++1=(1,0,1,1,0,0,-1); ++L=(1,0,1,1,0,-1,0); +<=(1,0,1,1,-1,0,0);
)=(1,0,1,-1,0,0,1); J=(1,0,1,-1,0,0,-1); {=(1,0,1,-1,0,1,0);
+v=(1,-1,1,1,0,0,0); [=(1,0,-1,0,0,1,1); +7=(1,0,-1,0,0,1,-1);
++K=(1,0,-1,0,0,-1,1); q=(-1,0,1,0,0,1,1); k=(1,0,-1,0,1,0,1);
+w=(1,0,-1,0,1,0,-1); +\=(1,0,-1,0,1,1,0); +&=(1,0,-1,0,1,-1,0);
+n=(1,0,-1,0,-1,0,1); +H=(-1,0,1,0,1,0,1); ==(-1,0,1,0,1,1,0);
%=(1,0,-1,1,0,0,1); -=(1,0,-1,1,0,0,-1); +>=(1,-1,1,0,-1,0,0);
++7=(1,0,-1,-1,0,0,1); +}=(-1,0,1,1,0,0,1); ++H=(1,0,-1,-1,0,1,0);
+?=(-1,0,1,1,0,1,0); +8=(1,-1,1,0,1,0,0); +O=(1,-1,1,0,0,-1,0);
+i=(1,-1,1,0,0,1,0); +’=(1,-1,1,0,0,0,-1); +x=(1,1,0,0,0,1,1);
++O=(1,1,0,0,1,0,-1); ++B=(1,1,0,0,1,1,0); ++M=(1,1,0,0,-1,0,1);
++Q=(1,1,0,0,-1,1,0); +Y=(1,1,0,0,-1,-1,0); +9=(1,1,0,1,0,0,1);
+$|$=(1,1,0,1,0,1,0); K=(1,1,0,1,0,-1,0); +E=(1,1,0,1,1,0,0);
+]=(1,-1,1,0,0,0,1); *=(1,1,0,1,-1,0,0); +A=(-1,1,0,1,1,0,0);
+o=(1,1,0,-1,0,0,-1); +V=(1,1,0,-1,0,1,0); w=(1,1,0,-1,1,0,0);
&=(1,-1,0,-1,1,0,0); +P=(1,1,0,-1,-1,0,0); /=(-1,1,0,1,0,1,0);
+_=(1,1,1,0,0,0,1); ’=(1,1,1,0,0,1,0); +I=(1,1,1,0,0,-1,0); \=(1,1,1,0,1,0,0);
+$=(-1,1,0,1,0,0,1); +-=(1,-1,0,-1,0,0,1); r=(1,1,1,0,-1,0,0);
++3=(1,1,1,1,0,0,0); ++R=(1,-1,0,1,-1,0,0); +J=(1,-1,0,1,1,0,0);
+L=(1,-1,0,1,0,-1,0); s=(1,-1,0,1,0,1,0); ]=(1,-1,0,1,0,0,-1);
++I=(1,-1,0,1,0,0,1); +T=(-1,1,0,0,1,1,0); !=(1,1,1,-1,0,0,0);
D=(1,-1,0,0,-1,1,0); +‘=(-1,1,0,0,1,0,1); P=(1,-1,0,0,1,-1,0);
"=(1,-1,0,0,1,1,0); +r=(1,-1,0,0,0,1,-1); +B=(1,1,-1,0,0,0,-1);
++2=(1,1,-1,0,0,1,0); ++6=(1,1,-1,0,1,0,0); $|$=(1,1,-1,0,-1,0,0);
E=(1,1,-1,1,0,0,0); +s=(1,1,-1,-1,0,0,0).

### 9-dim

47-16 123PQSUkl, 12BCJLVWb, 13DEMOYab, 1BQRVXdei, 1CPRZafgi, 1DMNWXghj,
1EKLYZcdj, 1IJKTUfhl, 1INOSTcek, 23GHJLVWb, 45EFJKTUb, 46GIJKTUb, 56ABPQSUb,
78ACPQSUb, 79HINOSTb, 89DFQRVXb. 1=(0,0,0,0,0,0,0,0,1); 2=(1,0,0,0,0,0,0,0,0);
3=(0,0,1,0,0,0,0,0,0); 4=(1,-1,1,0,0,0,0,0,1); 5=(-1,1,1,0,0,0,0,0,1);
6=(1,1,1,0,0,0,0,0,-1); 7=(1,-1,1,0,0,0,0,0,-1); 8=(1,1,-1,0,0,0,0,0,-1);
9=(1,1,1,0,0,0,0,0,1); A=(1,0,0,0,0,0,0,0,1); B=(0,1,-1,0,0,0,0,0,0);
C=(0,1,1,0,0,0,0,0,0); D=(1,-1,0,0,0,0,0,0,0); E=(1,1,0,0,0,0,0,0,0);
F=(0,0,1,0,0,0,0,0,-1); G=(0,1,0,0,0,0,0,0,1); H=(0,1,0,0,0,0,0,0,-1);
I=(1,0,-1,0,0,0,0,0,0); J=(0,0,0,1,1,0,0,0,0); K=(0,0,0,1,-1,0,0,0,0);
L=(0,0,0,0,0,1,1,0,0); M=(0,0,0,1,0,1,0,0,0); N=(0,0,0,0,1,0,1,0,0);
O=(0,0,0,0,1,0,-1,0,0); P=(0,0,0,0,1,-1,0,0,0); Q=(0,0,0,0,1,1,0,0,0);
R=(0,0,0,1,0,0,1,0,0); S=(0,0,0,1,0,0,0,0,0); T=(0,0,0,0,0,1,0,0,0);
U=(0,0,0,0,0,0,1,0,0); V=(0,0,0,1,-1,1,-1,0,0); W=(0,0,0,1,-1,-1,1,0,0);
X=(0,0,0,1,1,-1,-1,0,0); Y=(0,0,0,1,1,-1,1,0,0); Z=(0,0,0,1,1,1,-1,0,0);
a=(0,0,0,-1,1,1,1,0,0); b=(0,0,0,0,0,0,0,1,0); c=(1,-1,1,0,0,0,0,-1,0);
d=(1,-1,-1,0,0,0,0,1,0); e=(1,1,1,0,0,0,0,1,0); f=(1,-1,1,0,0,0,0,1,0);
g=(1,1,-1,0,0,0,0,1,0); h=(1,1,1,0,0,0,0,-1,0); i=(1,0,0,0,0,0,0,-1,0);
j=(0,0,1,0,0,0,0,1,0); k=(0,1,0,0,0,0,0,-1,0); l=(0,1,0,0,0,0,0,1,0).

19-5 678125493, 349CDFGHE, EFGHIJ786, ABCD56789, IJAB12789.

7(44)-6 SU(CDEFOQ6), 1G42U(8H95), 1S(AIMSVXbi), 472(acefhK), G72(LWYZdg),
4U(BJ3NPRT). 1=(0,0,0,0,0,0,0,1,0); 2=(0,0,0,0,0,0,1,0,0);
3=(0,0,0,0,1,1,0,0,0); 4=(0,0,0,1,0,0,0,0,1); 5=(0,0,0,1,0,0,0,0,-1);
6=(0,0,0,1,0,0,-1,1,0); 7=(0,0,1,0,0,0,0,1,0); 8=(0,0,1,0,0,1,0,0,0);
9=(0,0,1,0,0,-1,0,0,0); A=(0,0,1,0,-1,0,1,0,0); B=(0,0,1,0,-1,1,1,0,0);
C=(0,0,-1,-1,1,1,0,1,1); D=(0,0,1,-1,-1,-1,0,1,1); H=(0,1,0,0,1,0,0,0,0);
E=(0,1,0,0,1,-1,1,1,-1); F=(0,1,0,0,1,-1,-1,-1,1); G=(0,1,0,0,-1,0,0,0,0);
I=(0,1,0,1,1,1,1,0,1); J=(0,1,0,-1,0,0,0,-1,1); K=(1,-1,1,1,-1,1,0,-1,-1);
L=(0,1,0,-1,1,1,0,0,0); M=(0,1,0,-1,1,-1,1,0,-1); N=(0,1,-1,0,0,0,1,1,0);
O=(0,1,1,1,0,1,1,0,1); P=(0,1,1,1,1,-1,1,-1,-1); Q=(0,1,1,-1,0,1,-1,0,-1);
R=(0,1,-1,0,0,0,1,1,0); S=(0,-1,1,0,1,0,0,0,0); U=(1,0,0,0,0,0,0,0,0);
V=(1,0,0,0,0,-1,0,0,1); W=(1,0,0,1,0,1,0,0,1); X=(1,0,0,-1,0,1,0,0,0);
Y=(1,0,0,-1,0,-1,0,0,1); Z=(1,0,1,0,0,0,0,-1,-1); a=(1,1,0,0,-1,-1,0,0,0);
b=(1,1,1,1,0,0,-1,0,-1); c=(1,1,1,-1,1,1,0,-1,1); d=(-1,1,1,1,1,-1,0,-1,1);
e=(1,-1,-1,-1,-1,1,0,1,1); f=(1,1,-1,1,1,1,0,1,-1); g=(1,1,-1,1,1,-1,0,1,-1);
h=(1,-1,0,0,1,-1,0,0,0); i=(1,-1,-1,1,0,0,1,0,-1).

MMPHs in Fig. 4(b,c) (44-6 and 7-6 above) of the main body of the paper
illustrate why all vertices are needed for a consistent implementation of
MMPHs. Let us assign ‘1’ to S and 2 in Fig. 4(c), i.e. to 7-6. The hyperedge
4U then provides us with the contextual contradiction since ‘0’ is assigned to
both vertices in it. But the hyperedge 4U would not be a hyperedge without all
the other vertices (B,J,3,N,P,R,T) necessary for preparing and measuring it,
since without them it would be just the orthogonality between 4 and U and
would belong to the hyperedge 1G42U, i.e., would not be a separate hyperedge.
It is also indicative that Fig. 4(c), i.e. 7-6, is critical with any or all of
a,c,e,f,h,K considered in hyperedge 472 while it ceases to be contextual as
soon as we consider any of B,J,3,N,P,R,T in 4U since we can then assign a 3rd
‘1’ to one of them.

## References

  * Howard et al. (2014) M. Howard, J. Wallman, V. Veitech, and J. Emerson, Nature 510, 351 (2014).
  * Bartlett (2014) S. D. Bartlett, Nature 510, 345 (2014).
  * Tavakoli and Uola (2020) A. Tavakoli and R. Uola, Phys. Rev. Research 2, 013011 (2020).
  * Saha et al. (2019) D. Saha, P. Horodecki, and M. Pawłowski, New J. Phys. 21, 093057 (2019).
  * Waegell and Aravind (2015) M. Waegell and P. K. Aravind, J. Phys. A 48, 225301 (2015).
  * Pavičić (2017) M. Pavičić, Phys. Rev. A 95, 062121 (2017).
  * Pavičić and Megill (2018) M. Pavičić and N. D. Megill, Entropy 20 (2018).
  * Pavičić (2019) M. Pavičić, Entropy 21(11), 1107 (2019).
  * Pavičić et al. (2019) M. Pavičić, M. Waegel, N. D. Megill, and P. Aravind, Scientific Reports 9, 6765 (2019).
  * Pavičić et al. (2005) M. Pavičić, J.-P. Merlet, B. D. McKay, and N. D. Megill, J. Phys. A 38, 1577 (2005).
  * Cabello et al. (2005) A. Cabello, J. M. Estebaranz, and G. García-Alcaine, Phys. Lett. A 339, 425 (2005).
  * Waegell and Aravind (2017) M. Waegell and P. K. Aravind, Phys. Rev. A 95, 050101 (2017), Supplemental Material https://journals.aps.org/ pra/supplemental/10.1103/PhysRevA.95.050101.
  * Zimba and Penrose (1993) J. Zimba and R. Penrose, Stud. Hist. Phil. Sci. 24, 697 (1993).
  * Pavičić (2021) M. Pavičić, _Quantum Contextuality; arXiv:2108.00967_ (Unpublished, 2021).
  * McKay et al. (2000) B. D. McKay, N. D. Megill, and M. Pavičić, Int. J. Theor. Phys. 39, 2381 (2000).
  * Yu and Oh (2012) S. Yu and C. H. Oh, Phys. Rev. Lett. 108, 030402 (2012).
  * Cabello et al. (2012) A. Cabello, E. Amselem, K. Blanchfield, M. Bourennane, and I. Bengtsson, Phys. Rev. A 85, 032108 (2012).
  * Cabello et al. (2018) A. Cabello, J. R. Portillo, A. Solís, and K. Svozil, Phys. Rev. A 98, 012106 (2018).
  * Voráček and Navara (2021) V. Voráček and M. Navara, Found. Phys. 51, 67 (2021).
  * Kochen and Specker (1967) S. Kochen and E. P. Specker, J. Math. Mech. 17, 59 (1967).

## Appendix 1: Physical Review Letter Editor’s 1st decision, 1st referee
report, and my response to it

[Appendix 3 below contains Physical Review Letter Editor’s 2nd decision, 2nd
referee report, and my response to it; Appendices 4 and 5 at the end contains
Physical Review Letter Editor’s 3rd decision and my comment on it.]

#### Mladen Pavičić

### Editor’s 1st decision

Journal: Physical Review Letters; Paper: LP17301; Received: 17Feb2022

16Jun22 16Jun22 Editorial decision

19May22 16Jun22 Review editor concludes response unlikely

08Mar22 19May22 Review request to referee; report received

21Apr22 22Apr22 Review request to referee; message received (not a report)

29Mar22 30Mar22 Review request to referee; message received (not a report)

08Mar22 09Mar22 Review request to referee; message received (not a report)

Date: Thu, 16 Jun 2022 14:00:15 +0000

Re: LP17301 Automated generation of arbitrarily many Kochen-Specker and other
contextual sets in odd dimensional Hilbert spaces by Mladen Pavičić and Norman
D. Megill

Dear Prof. Pavicic,

This manuscript has been reviewed by our referee. A critique from the report
appears below. Based on this we judge that the work probably warrants
publication in some form, but does not meet the Physical Review Letters
criteria of impact, innovation, and interest.

The paper, with revision as appropriate, might be suitable for publication in
one of the topical Physical Review journals (e.g., Physical Review A) or
Physical Review Research. The editors of that journal will make the decision
on publication, and may seek further review. However, our complete file is
available.

Yours sincerely,

Robert Garisto (he/him/his) Editor Physical Review Letters

P.S. Another referee we consulted was not able to review your manuscript.

————————— Report of the 1st Referee – LP17301/Pavicic —————————

*OVERALL EVALUATION*

This article aims to present new methods to generate large numbers of examples
of Kochen-Specker sets in odd-dimensional Hilbert spaces. I found the
manuscript lacking in a number of aspects, which I now highlight. (In the
“Detailed Comments” section below, I go into more detail on some of these
issues, illustrating them with concrete passages, among other minor comments.)

1\. The manuscript does not provide enough motivation nor does it try to make
a case for the relevance of this work and its appeal to a broad audience such
as PRL’s. It claims that ”development of quantum computation and
communication, recently shown to be supported by contextuality, arguably asks
for an abundant supply of contextual sets”, but the authors do not bother to
explain why such sets might be necessary. I’m not claiming that there isn’t a
case to be made. There likely is, but the point is that the manuscript does
this very weakly.

2\. The presentation falls short of the standard expected in a research
article, especially in a journal as good as PRL, in terms of clarity and
rigor. Various passages are written in a confusing fashion. For example, it is
often unclear from the text that a passage is to be taken as the definition of
a new concept (e.g. when the notion of ”coordinatization” is introduced). On
the other hand, e.g. in the definition of MMPH hypergraph, some extra
commentary is written as part of the definition when it does not belong there,
namely item (v). Overall, the presentation is often imprecise (see detailed
comments below for more examples).

3\. The choice of material included in the text is odd, and it is inadequate
for a journal article. At points, the authors choose to give too much
information on irrelevant implementation details (e.g., the internal ASCII
encoding of the graphs being manipulated, or the name of functions in their
computer code). In contrast, and more worryingly, the text is quite vague in
the description of the actual methods/algorithms being implemented (especially
M3 which seems to be the less obvious of them). This is problematic for two
reasons. First, it detracts from the interest that the manuscript may have to
a journal’s audience: most readers would read the paper to learn the main
ideas underlying the methods so they could replicate and adapt them (anyone
keen on learning implementation details would refer to the code’s
documentation and/or to the code itself). Moreover, this vagueness makes it
hard for referees to judge the authors’ contributions in terms of correctness,
novelty, etc.

For the above reasons, I am unfortunately not able to recommend this article
for publication in PRL. I believe that the results may be of interest to a
more specialized audience. I would therefore encourage the authors to submit a
carefully *revised* version, with a more detailed presentation of the main
novel technical contributions, to a journal such as PRA.

*DETAILED COMMENTS*

(some more important than others, in order of appearance in the manuscript)

– Abstract, line 1-2:

”Development of quantum computation and communication, recently shown to be
supported by contextuality, *arguably asks for an abundant supply of
contextual sets*”

Contextuality has been shown to underlie quantum advantage in various
computation and communication tasks and protocols. However, the second part of
the sentence does not follow directly from this. That is, why does the
development of quantum computation and communication ask for an *abundant
supply of contextual sets*? This may well be the case, but (1) it is not an
obvious consequence of the results mentioned from the literature, and (2) the
authors fail to argue for this. Given that this is presented as the main
motivation for this work and for its broad appeal, I would expect to see a
better case made for this.

– Page 1, col. 1, l. 7:

”Qubits (quantum bits) are two-dimensional qudits (d = 2) and their tensor
products build the corresponding even-dimensional Hilbert spaces.”

While the Hilbert space corresponding to an n-qubit system (i.e., an n-fold
tensor of $C^{2}$) is even dimensional, not all even-dimensional Hilbert
spaces are of this form. The Hilbert space of multiple qubits has dimension a
power of two. One does not get, for example, a 6-dimensional Hilbert space in
this fashion. Some of the discussion around even vs odd dimension in the
article seems to misleadingly conflate even dimensionality with being
realizable with qubits.

– Page 1, col. 1, l. 21:

”and *several* in 5- to 11-dim spaces”

This is a rather vague statement: what is meant by ”several”?

– Page 1, col. 1, l. 22:

”[…]of Kochen-Specker contextual sets only four […] were explicitly provided
for particular sets”

It is unclear what the authors mean by ”were explicitly provided for
particular sets”?

– Page 1, col. 1, l. 38:

”An MMPH is a connected hypergraph k-l with k vertices and l hyperedges in
which […] every hyperedge contains at least 2 and *at most n* vertices”

The number ”n” here appears out of nowhere. Only ”k” and ”l” are bound by the
definition of ”hypergraph k-l”, but ”n” means nothing in this context and
therefore the requirement that every hyperedge contain at most n vertices
seems to be an empty one.

– Page 1, col.2, l. 1:

”(v) graphically, vertices are represented as dots and hyperedges as (curved)
lines passing through them.”

This is given as one of the conditions in the definition of MMPH graphs, but
in reality it is not a condition on hypergraphs. It is just an explanation of
the convention used to draw hypergraphs to ease the readability of Figure 2,
for example. It should be separated from the definition of MMPH hypergraph.

– Page 1, col. 2, l. 4-10:

”We encode MMPHs by means of the following 90 ASCII characters: [list of ASCII
characters] […]”

This is the kind of implementation detail that is completely irrelevant to the
potential readers of this paper. Its inclusion is a waste of space that could
be used to convey much more interesting information about the methods
implemented. This does not concern any substantial/significant algorithmic
choice, but simply a matter of convention chosen for the internal
representation. It’s the kind of detail that is typically hidden from the user
even in the documentation of a well-written library, and it’s even less
relevant in the context of a journal article, which ought to focus on
explaining the methods instead.

– Page 1, col. 2, l. 14-16:

”An MMPH is a special kind of a general hypergraph in the sens that none of
the aforementioned points (i-v) holds for it.”

The meaning of this sentence is unclear. What is the referent of the pronoun
”it” at the end of the sentence? (By the way, there is also a minor typo:
”sens” should be spelt ”sense”.)

– Page 1, col. 2, l. 23-25:

”Orthogonality between vertices in an MMPH space just means that the vertices
are contained in their hyperedges”.

This is yet another sentence whose meaning isn’t clear. What is meant by ”the
vertices are contained in *their* hyperedges”? In fact, this whole paragraph
about ”coordinatization” is written in a confusing fashion. It seems that the
intention is for it to be a definition of MMPH with coordinatization, but the
definition proper appears as seemingly an afterthought, in the clause starting
with ”i.e.”.

– Page 1, col. 2, l. 34 (and after):

The acronym ”MMPH” is long and awkward enough as it is. The strings ”NBMMPH”
and ”(N)BMMPH”, and later even ”KS NBMMPH”, are beyond what is reasonable to
expect a reader to parse. It would be much better to keep the full phrase
”non-binary”. I would even go further to suggest finding an alternative to
MMPH. Also because two of the initials in this acronym refer to the authors of
the present paper…

– Page 2, col. 1, l. 2-6:

”NBMMPHs are nonclassical since they do not allow assignments of predefined 0s
and 1s to their vertices; therefore, we consider them to be contextual. BMMPHs
are classical since they do allow such an assignment; therefore, we consider
them to be noncontextual.”

This sentence reads awkwardly. What is meant by ”*we consider* them to be
(non)contextual”. Is this supposed to be a definition?

– Page 2, col. 1, l. 11–12:

”When conditional or unconditional contextual operators […]”

The terms conditionally contextual, unconditionally contextual, and
conditionally noncontextual are never defined or even explained informally.

– Page 2, col. 1, l. 23-43:

”To generate (N)BMMPHs in the odd-dim spaces we make use of three methods […]”

These short paragraphs are what is included as a description of the methods
used. The descriptions of each of M1 to M3 provided are extremely brief and
lacking in detail. A more detailed exposition of the methods and some examples
illustrating them, especially for M3, would be useful and make the paper more
interesting to potential readers. Also, it would be good to see more details
on how the code deals with / finds / manipulates coordinatizations (i.e.,
realizations with quantum states and measurements) for the graphs in question.

–Page 2, col. 1, l. 57 - col. 2, l. 1:

”using our program *MMPSstrip*” […] ”stripping them to the critical KS MMPHs
by *States01*”

The name of the program or routine that performs a given task (such as
”MMPSstrip” or ”States01”) is totally irrelevant here. On the other hand, it
would be more interesting to convey what these programs are actually doing:
how are the edges added? randomly?, how are they stripped to critical sets?,
etc.

– Page 2, col. 2, Fig. 1:

This figure is hard to interpret. The caption does not adequately explain,
e.g., what do the axes represent, or what are the shaded black stripes. The
same applies to Fig 3 and Fig 4 later on.

– Page 2, col. 2, l. 13-14:

”When applying M3 we obtain that Bub’s is the only 49-36 NBMMPH and that there
are no smaller KS ones for the considered vector components.”

Again, this statement is unnecessarily vague. What is meant by ”for the
considered vector components”? Which vector components are being considered
which allow the authors to make this statement? This goes back, I believe, to
the overly vague explanation given of M3.

– Page 5, col.1, l. 26-30

”The methods are especially needed in odd dimensional Hilbert spaces since, in
contrast to even dimensional ones, we cannot make use of polytopes, Pauli
operators, qubit states, parities, and other approaches specific to qubit
spaces”.

Are analogous methods not available for qudits? Is there a reason why this is
the case?

——————— Author’s response to the Report of the 1st Referee – LP17301/Pavicic
———————

The referee is not at home with the theory of hypergraphs on which our article
is based; none of her/his comments holds water. So, an acceptance of her/his
report as the only ground for a rejection of the present article would likely
be detrimental for the reputation of the Physical Review Letters in this
field.

In the revised version of the article ( available at
https://arxiv.org/abs/2202.08197 ) I made some changes that clarify the points
the referee misinterpreted, misunderstood, or twisted. In my response below, I
indicate such changes by “*****************”. Referee’s comments I indicate by
“$>$”.

$>$ *OVERALL EVALUATION*

$>$ This article aims to present new methods to generate large numbers of
examples of Kochen-Specker sets in

$>$ odd-dimensional Hilbert spaces.

Definitely not. In the summary we clearly stated our aim: “Our goal is …to
establish general methods for automated generation of NBMMPHs [of which the
Kochen-Specker sets are the subsets] in any dimension for any possible future
application and implementation, e.g., in quantum computation and
communication.” In the present article we focus on the odd-dimensional spaces
since we obtained analogous results for the even-dimensional spaces in the
previous publications of ours. Also, in the even-dimensional spaces there are
numerous other methods by which one can achieve automated generation of
NBMMPHs, while in the odd-dimensional spaces apparently only the ones
presented in this article are available.

$>$ 1\. The manuscript does not provide enough motivation nor does it try to
make a case for the relevance of this

$>$ work and its appeal to a broad audience such as PRL’s. It claims that
“development of quantum computation

$>$ and communication, recently shown to be supported by contextuality,
arguably asks for an abundant supply of

$>$ contextual sets,” but the authors do not bother to explain why such sets
might be necessary. I’m not claiming

$>$ that there isn’t a case to be made. There likely is, but the point is that
the manuscript does this very weakly.

On the contrary, the motivation is clearly expressed in the second paragraph
of the paper: “none of the methods [previously employed in the even-
dimensional spaces] is available in the odd-dimensional spaces. In this paper,
we offer several methods for automated generation of arbitrarily many KS sets
as well as contextual non-KS ones in $n=3,5,7,9$ dimensional spaces.”

The sentence: “quantum computation and communication …arguably asks for an
abundant supply of contextual sets” is taken from the abstract which is
limited in size. In the very body of the article, it is substantiated by the
motivation sentences cited above.

**** In the present version of the paper I expanded the Abstract so as to make
the first and the last sentences read: ****

************** Development of quantum computation and communication, recently
shown to be supported by contextuality, arguably asks for a requisite supply
of contextual sets. **************

************** In this paper, we offer three methods for automated generation
of arbitrarily many contextual KS and non-KS sets in any dimension for
possible future application and implementation, e.g., in quantum computation
and communication, which can be applied in both even and odd dimensional
spaces and we employ them to obtain millions of KS and other contextual sets
in dimensions 3, 5, 7, and 9. **************

$>$ 2\. The presentation falls short of the standard expected in a research
article, especially in a journal as good as

$>$ PRL, in terms of clarity and rigor. Various passages are written in a
confusing fashion.

After so many articles we published in Phys. Rev. Lett., Scientific Reports,
Phys. Rev. A $\&$ D, Optics Express, J. Math. Phys., J. Phys. A, J. Opt. Soc.
Am. B, Entropy, Phys. Lett. A, Found. Phys., Ann. H. Poincare … over almost
half a century, enters a referee to announce that we do not know how to write
an article. The article is written up to the standards of the PRL and nothing
is written in a confusing fashion, only the referee does not grasp the meaning
of the presented material since she/he is not at home with the theory of
hypergraphs.

$>$ For example, it is often unclear from the text that a passage is to be
taken as the definition of a new concept

$>$ (e.g. when the notion of “coordinatization” is introduced).

Wiktionary has a single item for: coordinatization (plural coordinatizations):
(mathematics) “The application of coordinates to a system,” and in the 3rd
paragraph of the right column on p. 1 we additionally clarify: “The MMPHs
above are defined without a coordinatization, i,e., neither their vertices nor
their hyperedges are related to either vectors or operators.” So, no “new
concept” is introduced. The “coordinatization” is a word from a dictionary the
referee should have consulted.

$>$ On the other hand, e.g. in the definition of MMPH hypergraph, some extra
commentary is written as a part of

$>$ the definition when it does not belong there, namely item (v).

The referee is obviously not at home with the theory of hypergraphs. E.g.,
Berge, Hypergraphs: Combinatorics of Finite Sets, North-Holland (1989); Ch. 1,
Sec. 1, p. 1, 4th paragraph: “A hypergraph may be drawn as a set of points
representing the vertices [and a hyper]edge is represented …by a single closed
curve enclosing the vertices”—as a part of his definition. There is no
particular kind of a “definition” of ours which would differ from Berge’s kind
except that we introduce, in point (v), a curve passing through vertices
instead of Berge’s curve which encloses them.

$>$ Overall, the presentation is often imprecise (see detailed comments below
for more examples).

The referee fails to substantiate her/his claim in any of her/his comments.

$>$ 3\. The choice of material included in the text is odd, and it is
inadequate for a journal article. At points, the

$>$ authors choose to give too much information on irrelevant implementation
details (e.g., the internal ASCII

$>$ encoding of the graphs being manipulated, or the name of functions in
their computer code).

Again, the referee clearly shows that she/he is not at home with the theory of
hypergraphs. Berge, Ch. 1, Sec. 1, p.1, 5th paragraph: “One may also define a
hypergraph by its incidence matrix, with columns representing the [hyper]edges
and rows representing the vertices (cf. Figure 1, p. 2).” We, instead of an
incidence matrix, make use of MMPH ASCII strings. Some kind of codification is
unavoidable and our MMPH string codification is an integral part of the MMPH
language and these strings are generated by our algorithms and programs and
they then serve us as further direct inputs to our algorithms and programs
whose outputs are all MMPH properties and features including graphical
representations of the obtained MMPHs. The difference between incidence
matrices and MMPHs is obvious; for instance, the MMPH string of Fig. 2(c) can
be written in a single line, while the incidence matrix takes 37 columns and
51 lines. Also an MMPH string can be put in any of our programs while no such
programs for arbitrary incidence matrices are known to us. And then enters the
referee to proclaim the MMPH codification “irrelevant.”

**** In the present version of the paper I add the following sentences to the
right column, p. 1: ****

************** So encoded single ASCII characters (possibly prefixed by +’s)
represent vertices; e.g., 1, or q, or +++A. Put one after another (without
spaces) they represent hyperedges; e.g., 123, or CZk1@=a, or 1+1+++1Dd. In a
representation of an MMPH, hyperedges are organized in a string in which they
separated by commas; each string ends with a period; e.g., the string
123,345,567,789,9A1. represents a noncontextual 3-dim MMPH pentagon.
**************

************** We stress here that MMPH strings are generated by our
algorithms and programs and then processed to yield all properties and
features of MMPHs, including their graphical representations. **************

$>$ In contrast, and more worryingly, the text is quite vague in the
description of the actual methods/algorithms

$>$ being implemented (especially M3 which seems to be the less obvious of
them).

I’m flabbergasted. What can be simpler than, p. 2: “M3 consists in combining
simple vector components so as to exhaust all possible collections of $n$
mutually orthogonal $n$-dim vectors.” For instance 3-dim components $(0,1)$
yield: 123. 1=(0,0,1), 2=(0,1,0), 3=(1,0,0); 4-dim $(0,\pm 1)$ yield Peres’
set 24-24; etc.

$>$ This is problematic for two reasons. First, it detracts from the interest
that the manuscript may have to a

$>$ journal’s audience: most readers would read the paper to learn the main
ideas underlying the methods so they

$>$ could replicate and adapt them (anyone keen on learning implementation
details would refer to the code’s

$>$ documentation and/or to the code itself). Moreover, this vagueness makes
it hard for referees to judge the

$>$ authors’ contributions in terms of correctness, novelty, etc.

What is problematic here are the motives for such twists on the part of the
referee. The first sentence seems to contradict itself: “most readers would
read the paper to learn the main ideas underlying the methods so they could
replicate and adapt them,” but they would not like “to learn the code itself.”
Note that without the MMPH language the results obtained in the article are
inaccessible as they were in the last half of century.

In the second sentence, “this vagueness”—read MMPH language—makes it hard for
[malevolent] referees to judge the authors’ contributions in terms of
correctness, novelty, etc. As if it is not transparent that the correctness
and novelty of all results can be verified directly at the results presented
in the article and its appendices or at our repository
http://puh.srce.hr/s/Qegixzz2BdjYwFL where all algorithms and programs are
freely available.

$>$ For the above reasons, I am unfortunately not able to recommend this
article for publication in PRL.

$>$ I believe that the results may be of interest to a more specialized
audience.

The researchers were striving to obtain these results for at least quarter of
century and it is regretful that the editor bought such a recommendation.

$>$ I would therefore encourage the authors to submit a carefully *revised*
version, with a more detailed

$>$ presentation of the main novel technical contributions, to a journal such
as PRA.

Our algorithms do require sophisticated programs but that does not make the
contributions “technical,” because there is no other way to obtain them.

$>$ *DETAILED COMMENTS*

$>$ – Abstract, line 1-2:

$>$ ”Development of quantum computation and communication, recently shown to
be supported by contextuality,

$>$ *arguably asks for an abundant supply of contextual sets*” Contextuality
has been shown to underlie quantum

$>$ advantage in various computation and communication tasks and protocols.
However, the second part of the

$>$ sentence does not follow directly from this. That is, why does the
development of quantum computation and

$>$ communication ask for an *abundant supply of contextual sets*? This may
well be the case, but (1) it is not an

$>$ obvious consequence of the results mentioned from the literature, and (2)
the authors fail to argue for this. Given

$>$ that this is presented as the main motivation for this work and for its
broad appeal, I would expect to see a

$>$ better case made for this.

Common scientific sense tells us that when “what gives quantum computers that
extra oomph over their classical digital counterparts …is an intrinsic,
measurable aspect of quantum mechanics called contextuality” [1,2] we should
have more than a few sets per dimension. Those recent result is so well-known
that we considered it far outside the scope of the present paper to elaborate
on it or to even add references.

**** Still in the present version of the paper I added references [1,2,3,4]
and I substituted “requisite” for “abundant” ****

$>$ – Page 1, col. 1, l. 7:

$>$ ”Qubits (quantum bits) are two-dimensional qudits ($d=2$) and their tensor
products build the corresponding

$>$ even-dimensional Hilbert spaces.” While the Hilbert space corresponding to
an n-qubit system (i.e., an $n$-fold

$>$ tensor of $C^{2}$) is even dimensional, not all even-dimensional Hilbert
spaces are of this form. The Hilbert space

$>$ of multiple qubits has dimension a power of two. One does not get, for
example, a 6-dimensional Hilbert space

$>$ in this fashion. Some of the discussion around even vs odd dimension in
the article seems to misleadingly conflate

$>$ even dimensionality with being realizable with qubits.

This is ridiculous. We first write: “[Qubit] tensor product build the
corresponding even-dimensional Hilbert spaces.” Every undergraduate student
understands that that means: $2^{n}$, $n=2,3,4,\dots$. Then we write: “in
[5–9] billions of contextual sets in 4-, 6-, 8-, 16-, and 32-dim Hilbert
spaces, predominantly related to qubits, were generated.” And again every
undergraduate student would immediately recognize that “predominantly” refers
to 4, 8, 16, and 32, but not to 6. It seems to be a failed weaponized pedantry
?

$>$ – Page 1, col. 1, l. 21:

$>$ “and *several* in 5- to 11-dim spaces” This is a rather vague statement:
what is meant by “several”?

Again, ridiculous. By Webster Dictionary “several” means “more than two but
fewer than many.”

$>$ – Page 1, col. 1, l. 22:

$>$ “[…] of Kochen-Specker contextual sets only four […] were explicitly
provided for particular sets”

$>$ It is unclear what the authors mean by “were explicitly provided for
particular sets?”

Is the referee serious? The full sentence reads: “of Kochen-Specker contextual
sets only four in the 3-dim space [8,10] and several in 5- to 11-dim spaces
[11–13] were explicitly provided for particular sets,” meaning that in those
references sets with explicitly given vectors were provided. I gave the
sentence to a group of students and no one had any problem with understanding
the meaning of “explicitly,” so I presume the reader would not have it either.

$>$ – Page 1, col. 1, l. 38:

$>$ “An MMPH is a connected hypergraph k-l with k vertices and l hyperedges in
which […] every hyperedge

$>$ contains at least 2 and *at most n* vertices.” The number ”n” here appears
out of nowhere. Only ”k” and ”l”

$>$ are bound by the definition of ”hypergraph k-l”, but ”n” means nothing in
this context and therefore the

$>$ requirement that every hyperedge contain at most n vertices seems to be an
empty one.

The referee again shows that she/he is not at home with the theory of
hypergraphs. The $n$ does not “appear out of nowhere,” but out of the one but
preceding sentence: “In this paper, we offer several methods for automated
generation of arbitrarily many KS sets as well as contextual non-KS ones in
$n=3,5,7,9$” dimensional spaces.”

**** Still in the present version of the paper I added “$n$” to the cited
sentence so as to now read: ****

************** An MMPH is a connected $n$-dim hypergraph $k$-$l$ with $k$
vertices and $l$ hyperedges … **************

$>$ – Page 1, col.2, l. 1:

$>$ ”(v) graphically, vertices are represented as dots and hyperedges as
(curved) lines passing through them.”

$>$ This is given as one of the conditions in the definition of MMPH graphs,
but in reality it is not a condition on

$>$ hypergraphs. It is just an explanation of the convention used to draw
hypergraphs to ease the readability of

$>$ Figure 2, for example. It should be separated from the definition of MMPH
hypergraph.

Nowhere do we speak about a “condition.” It is just the point within a
definition in the same way in which similar points are used in many hypergraph
textbooks. We already clarified that point (v) above when we compare it with a
similar point used by Berge, Ch. 1, Sec. 1, p. 1, 4th paragraph.

$>$ – Page 1, col. 2, l. 4-10:

$>$ ”We encode MMPHs by means of the following 90 ASCII characters: [list of
ASCII characters] […]”

$>$ This is the kind of implementation detail that is completely irrelevant to
the potential readers of this paper.

$>$ Its inclusion is a waste of space that could be used to convey much more
interesting information about the

$>$ methods implemented. This does not concern any substantial/significant
algorithmic choice, but simply a

$>$ matter of convention chosen for the internal representation. It’s the kind
of detail that is typically hidden

$>$ from the user even in the documentation of a well-written library, and
it’s even less relevant in the context

$>$ of a journal article, which ought to focus on explaining the methods
instead.

Once again, the referee clearly shows that she/he is not at home with the
theory of hypergraphs at all. “ASCII “completely irrelevant,” “waste of
space,” “matter of convention,” “a journal article ought to focus on
explaining the methods instead”…” There are no methods which could generate
the contextual sets in odd dimensional spaces outside the MMPH language based
on ASCII codification either in our minds or anywhere in the literature. MMPH
language IS the method. A “method instead” does not exist. As the title of the
paper stresses, the method consists in automated computer-program-enabled
generation of such contextual sets based on the MMPH language.

$>$ – Page 1, col. 2, l. 14-16:

$>$ ”An MMPH is a special kind of a general hypergraph in the sense that none
of the aforementioned points (i-v)

$>$ holds for it.” The meaning of this sentence is unclear. What is the
referent of the pronoun ”it” at the end of the

$>$ sentence?

“it” = a general hypergraph. It is clear because points (i-v) obviously hold
for MMPH.

$>$ – Page 1, col. 2, l. 23-25:

$>$ ”Orthogonality between vertices in an MMPH space just means that the
vertices are contained in their

$>$ hyperedges”. This is yet another sentence whose meaning isn’t clear. What
is meant by ”the vertices are

$>$ contained in *their* hyperedges”?

For anyone who reads the paper so as to find out how MMPHs and the
orthogonality within them are structured it is clear what is meant by *their*.
Explanation is crystal clear: Let ABC be a hyperedge; we can say, for vertices
A,B,C, the hyperedge is *their* hyperedge; so, the vertices A,B,C are mutually
orthogonal just by being contained in *their* hyperedge. I’m convinced the
reader will have much greater imagination and insight than the present
referee.

$>$ In fact, this whole paragraph about ”coordinatization” is written in a
confusing fashion. It seems that the

$>$ intention is for it to be a definition of MMPH with coordinatization, but
the definition proper appears as

$>$ seemingly an afterthought, in the clause starting with ”i.e.”.

Confusion is seemingly with the referee. Our paragraph clearly states that
MMPHs are defined without any specification what their vertices might be. When
vectors, states, projectors, or operators are assigned to vertices, then we
speak about MMPHs with coordinatization. The former MMPHs with bare
unspecified vertices are consequently MMPHs without coordinatization.

$>$ – Page 1, col. 2, l. 34 (and after):

$>$ The acronym ”MMPH” is long and awkward enough as it is. The strings
”NBMMPH” and ”(N)BMMPH”, and

$>$ later even ”KS NBMMPH”, are beyond what is reasonable to expect a reader
to parse. It would be much better

$>$ to keep the full phrase ”non-binary”.

The term “non-binary MMPH” introduced and abbreviated to NBMMPH on p. 2, is
specific for this short article and the reader will want to go back to its
definition a couple of times while reading it. So, “non-binary MMPH” vs. NBMMP
would not make a difference in this respect; the full phrase would only make
the article longer.

$>$ I would even go further to suggest finding an alternative to MMPH. Also
because two of the initials in this

$>$ acronym refer to the authors of the present paper…

Is the referee serious? The acronym appears in more than 20 papers since 2005
and no one of the former referees has ever brought it into question. Besides,
one of the authors (N.M.) is unfortunately not alive any more.

$>$ – Page 2, col. 1, l. 2-6:

$>$ ”NBMMPHs are nonclassical since they do not allow assignments of
predefined 0s and 1s to their vertices;

$>$ therefore, we consider them to be contextual. BMMPHs are classical since
they do allow such an assignment;

$>$ therefore, we consider them to be noncontextual.” This sentence reads
awkwardly. What is meant by

$>$ ”*we consider* them to be (non)contextual”. Is this supposed to be a
definition?

What is that about? We simply state that we *consider* them in the same way as
they are *considered* to be of such a kind in the literature over and over.
For instance, “In noncontextual hidden-variable theories the predetermined
results of an observable are independent of any other observables that are
measured jointly with it”[C. Brukner and M. Zukowski, Bell’s Inequalities:
Foundations and Quantum Communication, in Handbook of Natural Computing, Eds:
G. Rozenberg, T.H.W. Baeck, and J.N. Kok, Springer (2010), pp. 1413-1450].
Contextual sets as ones that do not admit predetermined values and
noncontextual as ones that do go back to the original Kochen and Specker paper
[20].

$>$ – Page 2, col. 1, l. 11–12:

$>$ ”When conditional or unconditional contextual operators […]” The terms
conditionally contextual,

$>$ unconditionally contextual, and conditionally noncontextual are never
defined or even explained informally.

Yes, the terms might have required some comments.

**** Instead, in the present version of the paper I substituted the following
sentence for the sentences containing the disputed terms (3rd paragraph of the
left column, on p. 2): ****

************** When either state-dependent or state-independent tests of
operators defined on vertices of an NBMMPH with $\kappa(i)\leq n$ confirm the
contextuality, e.g. Yu and Oh (2012); Cabello et al. (2012, 2018), then the
NBMMPH turns out to be contextual in all considered cases so far.
**************

State-independent contextuality (SIC) is a well-known term which does not
require any further explanation.

$>$ – Page 2, col. 1, l. 23-43:

$>$ ”To generate (N)BMMPHs in the odd-dim spaces we make use of three methods
[…]” These short paragraphs

$>$ are what is included as a description of the methods used. The
descriptions of each of M1 to M3 provided are

$>$ extremely brief and lacking in detail.

They are brief but not lacking in detail. M1: dropping of $m=1$ vertices means
automated dropping of $m=1$; there is nothing more to it. M2: automated random
addition of hyperedges means automated random addition of hyperedges; there is
nothing more to it; M3: combining simple vector components so as to exhaust
all possible collections of $n$ mutually orthogonal $n$-dim vectors is just
that; e.g., (0,1) components in a 3-dim space yield vectors: A=(0,0,1),
B=(0,1,0), C=(1,0,0) which form the hyperedge ABC; there is nothing more to
it. The referee apparently cannot believe that everything turns out to be so
simple.

$>$ A more detailed exposition of the methods and some examples illustrating
them, especially for M3, would be

$>$ useful and make the paper more interesting to potential readers. Also, it
would be good to see more details on

$>$ how the code deals with / finds / manipulates coordinatizations (i.e.,
realizations with quantum states and

$>$ measurements) for the graphs in question.

On the contrary. As I just stressed, what is useful and interesting to the
reader is to see how simple and straightforward the algorithms are. Detailed
examples of the three methods in all considered dimensions are given in the
body of the article.

**** As for the realizations/implementations, in the present version of the
paper I add the following sentences at the end of the 2nd paragraph of the
left column, on p. 2): ****

************** The most direct implementation of an MMPH can be carried out by
measuring its vectors/states coming out of the gates determined by its
hyperedges. Operator based implementation, e.g. the one in which projectors
determined by states/vertices are projected on many other chosen states, is
more demanding. **************

$>$ – Page 2, col. 1, l. 57 - col. 2, l. 1:

$>$ ”using our program *MMPSstrip*” […] ”stripping them to the critical KS
MMPHs by *States01*.” The name

$>$ of the program or routine that performs a given task (such as ”MMPSstrip”
or ”States01”) is totally irrelevant

$>$ here. On the other hand, it would be more interesting to convey what these
programs are actually doing: how

$>$ are the edges added? randomly?, how are they stripped to critical sets?,
etc.

Again and again. The MMPH language and its algorithms and programs is a sine
qua non for generating the MMPHs. How? Randomly? On p. 2 the referee can read:
“M2 consists in an automated random addition of hyperedges…” How are they
stripped to critical sets? Also on p. 2 (left column, bottom): “A critical
NBMMPH is an NBMMPH which after removing any of its hyperedges becomes a
BMMPH.” E.g. 1234,4567,789A,ABCD,DEFG,GHI1,29BI,35CE,68FH,F2JA. is a KS
NBMMPH, but not a critical one. By removing the hyperedge F2JA it becomes a
critical KS NBMMPH (Cabello’s 18-9). How do we know that? Because by removing
any of the remaining hyperedges it stops being an NBMMPH (contextual) and
turns into a BMMPH (noncontextual).

$>$ – Page 2, col. 2, Fig. 1:

$>$ This figure is hard to interpret. The caption does not adequately explain,
e.g., what do the axes represent, or

$>$ what are the shaded black stripes. The same applies to Fig 3 and Fig 4
later on.

The referee is bold again. Abscissa = $l$ (number of hyperedges); ordinate =
$k$ (number of vertices). Dots represent ($k,l$). Consecutive dots (same $l$)
are shown as strips. Analogous figures have been presented in more than 20
papers of ours and all referees or those who cited or commented on our papers
or to whom the papers were presented to conferences have obviously immediately
grasped what the axes represent since no one has ever posed such a question.

$>$ – Page 2, col. 2, l. 13-14:

$>$ ”When applying M3 we obtain that Bub’s is the only 49-36 NBMMPH and that
there are no smaller KS ones

$>$ for the considered vector components.” Again, this statement is
unnecessarily vague. What is meant by ”for the

$>$ considered vector components”? Which vector components are being
considered which allow the authors to make

$>$ this statement?

Another failed weaponized pedantry. Vector components are shown in Fig. 1
($0,\pm 1,\dots,2\omega^{2}$).

$>$ This goes back, I believe, to the overly vague explanation given of M3.

Such comments, I’m sure, do not go anywhere.

$>$ – Page 5, col.1, l. 26-30

$>$ ”The methods are especially needed in odd dimensional Hilbert spaces
since, in contrast to even dimensional

$>$ ones, we cannot make use of polytopes, Pauli operators, qubit states,
parities, and other approaches specific

$>$ to qubit spaces”. Are analogous methods not available for qudits? Is there
a reason why this is the case?

It is so good that this is the last referee’s comment. We cannot make use of
them because they are “specific to qubit spaces” for whatever reason. We do
not want to speculate about reasons. It is outside the scope of the article.
It suffices that there no such methods in the literature and this is
transparent from the fact that so far there were only a dozen of MMPHs found
in the odd-dimensional spaces.

## Appendix 2: Physical Review Letter Editor’s 2nd decision, 2nd referee
report, and my response to it

#### Mladen Pavičić

### Editor’s 2nd decision

Journal: Physical Review Letters; Paper: LP17301; Received: 17Feb2022

16Sep22 16Jun22 Editorial decision

29Jul22 01Sep22 Review request to referee; report received

Date: 14 Sep 2022

Re: LP17301 Automated generation of arbitrarily many Kochen-Specker and other
contextual sets in odd dimensional Hilbert spaces by Mladen Pavičić and Norman
D. Megill

Dear Prof. Pavicic,

This manuscript has been reviewed by our [2nd] referee. A critique from the
report appears below. Based on this we judge that the work probably warrants
publication in some form, but does not meet the Physical Review Letters
criteria of impact, innovation, and interest. In accordance with our standard
practice, this concludes our review of your manuscript.

The paper, with revision as appropriate, might be suitable for publication in
one of the topical Physical Review journals or Physical Review Research. The
editors of that journal will make the decision on publication, and may seek
further review.

Yours sincerely,

Robert Garisto (he/him/his) Editor Physical Review Letters

————————— Report of the 2nd Referee – LP17301/Pavicic —————————

The work “Automated generation of arbitrarily many Kochen-Specker and other
contextual sets in odd-dimensional Hilbert spaces” by Pavicic and Megill
introduces numerical methods to obtain Kochen-Specker (KS) sets (or in general
contextual sets) in odd dimensions. Their methods yield a huge number of KS
sets in dimensions 3, 5, 7, and 9 that are not known before. KS set is a set
of projectors that cannot be assigned pre-determined values independent of the
context of the observables in which the projector belongs. KS sets are genuine
signatures of nonclassicality and have found applications in quantum
computation and communication. There are, in general, two ways that contextual
sets are associated with quantum information. Contextuality is shown to be
necessary for quantum advantages over classical systems in certain aspects,
that is, an advantage by quantum systems in certain aspects of information
processing over classical systems implies some particular proofs of
contextuality (for example, Ref. [1] in the manuscript). On the other hand, it
has also been pointed out that contextual sets are sufficient for quantum
advantage in some particular information processing tasks. In other words,
uncovering any KS set immediately implies the existence of some information
processing tasks with quantum advantages (for example, Ref. [4] in the
manuscript). In the latter sense, findings of new KS sets have implications
for quantum information. It is also fundamentally interesting to know about
novel KS sets as they give rise to compelling nonclassical correlations. And
only a few KS sets are indeed known so far in odd dimensions. So I think the
question addressed here in this work is important.

Before I present my opinion regarding the suitability of this work for
publishing in PRL, I would like to pose a few queries and suggestions.

1\. KS MMPH is defined as the NBMMPH with k(i) =n for all i. If an NBMMPH does
not have coordinatization then the graph has no quantum realization. However,
one may get confused with the terminology of ‘KS MMPH’ since the “KS set” is
usually referred to a set that already has a quantum realization. I would
recommend adding one line in the text to clarify this. Are there examples of
non-KS NBMMPH without known coordinatization?

2\. Do non-KS NBMMPHs always give rise to proof of contextuality? In other
words, given any non-KS NBMMPH, is it always possible to propose a
noncontextuality inequality that quantum theory violates?

3\. I would request the authors to mention some details about the software
or/and tools used to execute the numerics so that it would be convenient to
study this technique in the future.

4\. A huge number of critical KS sets are found, and it would be nice if some
sort of comparison is made among those critical sets. For instance, one can
consider any measure of robustness for contextuality and study which set is
more robust, or, one can see which set has the highest ratio between the
chromatic number of the graph and n (the dim of the Hilbert space). This is
just a suggestion.

Overall I find this work interesting. But, there is a substantial limitation
of this work. The proposed numerical methods are computationally demanding for
higher dimensions, and the KS sets are found only in a few small dimensions.
Thus the proposed methods for automated generation of contextual sets are
applicable to a certain extent. This is the reason I think the paper is
suitable for specialized audiences, and I am unable to give a reason why the
manuscript deserves publication in PRL. Given the queries mentioned above are
appropriately addressed, this work can be published in a specialized journal.

——————— Author’s response to the Report of the 2nd Referee – LP17301/Pavicic
———————

In the revised version of the article ( available at
https://arxiv.org/abs/2202.08197 ) I made some changes suggested by the
referee. In my response below, I indicate such changes by “*****************”.
Referee’s comments I indicate by “$>$”.

$>$ The work “Automated generation of arbitrarily many Kochen-Specker and
other contextual sets in odd-

$>$ dimensional Hilbert spaces” by Pavicic and Megill introduces numerical
methods to obtain Kochen-Specker (KS)

$>$ sets (or in general contextual sets) in odd dimensions. Their methods
yield a huge number of KS sets in

$>$ dimensions 3, 5, 7, and 9 that are not known before. KS set is a set of
projectors that cannot be assigned

$>$ pre-determined values independent of the context of the observables in
which the projector belongs. KS sets

$>$ are genuine signatures of nonclassicality and have found applications in
quantum computation and

$>$ communication. There are, in general, two ways that contextual sets are
associated with quantum information.

$>$ Contextuality is shown to be necessary for quantum advantages over
classical systems in certain aspects, that is,

$>$ an advantage by quantum systems in certain aspects of information
processing over classical systems implies

$>$ some particular proofs of contextuality (for example, Ref. [1] in the
manuscript). On the other hand, it has also

$>$ been pointed out that contextual sets are sufficient for quantum advantage
in some particular information

$>$ processing tasks. In other words, uncovering any KS set immediately
implies the existence of some information

$>$ processing tasks with quantum advantages (for example, Ref. [4] in the
manuscript). In the latter sense, findings

$>$ of new KS sets have implications for quantum information. It is also
fundamentally interesting to know about

$>$ novel KS sets as they give rise to compelling nonclassical correlations.
And only a few KS sets are indeed known

$>$ so far in odd dimensions. So I think the question addressed here in this
work is important.

In contrast to the 1st referee, the present referee is at home with both the
hypergraph theory and contextual sets. Her/his understanding and evaluation of
the results achieved in the paper are accurate—up to the last point
below—which is not.

$>$ Before I present my opinion regarding the suitability of this work for
publishing in PRL, I would like to pose a

$>$ few queries and suggestions.

$>$ 1\. KS MMPH is defined as the NBMMPH with $k(i)=n$ for all $i$. If an
NBMMPH does not have coordinatization

$>$ then the graph has no quantum realization. However, one may get confused
with the terminology of ‘KS MMPH’

$>$ since the “KS set” is usually referred to a set that already has a quantum
realization. I would recommend adding

$>$ one line in the text to clarify this. Are there examples of non-KS NBMMPH
without known coordinatization?

**** In the 2nd paragraph of the left column on p. 2, the following sentences
are added: ****

************** The assignments of 0s and 1s do not require a coordinatization
… **************

************** An example of a non-KS NBMMPH without known coordinatization is
the 33-27 in Fig. 1(d) in the Supplemental Material (SM). **************

$>$ 2\. Do non-KS NBMMPHs always give rise to proof of contextuality? In other
words, given any non-KS

$>$ NBMMPH, is it always possible to propose a noncontextuality inequality
that quantum theory violates?

**** Yes. See Table IX in [14]. ****

$>$ 3\. I would request the authors to mention some details about the software
or/and tools used to execute the

$>$ numerics so that it would be convenient to study this technique in the
future.

**** What is of the primary interest here are the algorithms. They are
universal and they hold in any dimension. I added the following paragraph to
the left column on p. 1: ****

************** In this paper, we offer universal and general algorithms for
automated generation of arbitrarily many contextual sets in any dimension. In
contrast to them, the programs we wrote to implement them are computationally
demanding and therefore, here, we use them to generate sets that have not been
generated so far: billions of KS and contextual non-KS sets in $n=3,5,7,9$-dim
spaces. The programs themselves are freely available from our repository and
technical details of their previous versions are given in [5-9].
**************

$>$ 4\. A huge number of critical KS sets are found, and it would be nice if
some sort of comparison is made among

$>$ those critical sets. For instance, one can consider any measure of
robustness for contextuality and study which

$>$ set is more robust, or, one can see which set has the highest ratio
between the chromatic number of the graph

$>$ and $n$ (the dim of the Hilbert space). This is just a suggestion.

**** I have done this in [14]. ****

$>$ Overall I find this work interesting. But, there is a substantial
limitation of this work. The proposed numerical

$>$ methods are computationally demanding for higher dimensions, and the KS
sets are found only in a few small

$>$ dimensions. Thus the proposed methods for automated generation of
contextual sets are applicable to a certain

$>$ extent. This is the reason I think the paper is suitable for specialized
audiences, and I am unable to give a reason

$>$ why the manuscript deserves publication in PRL. Given the queries
mentioned above are appropriately addressed,

$>$ this work can be published in a specialized journal.

**** The referee misunderstood the main accomplishment of the paper. Our
algorithm and method are applicable to any dimension and are therefore
universal and general—they are not “computationally demanding”—our programs
based on them are. The present computational barrier which prevents us from
providing outputs in higher dimensions is not their “limitation” in any way
that might diminish their importance for the PRL readership. Following our
algorithms one can obtain outputs in higher dimensions by making use of more
powerful supercomputers or new linear programs one might invent in the future.
Also, a quantum computer would allow for outputs in any dimension. Is the
discovery of the DNA structure unimportant just because we still cannot
calculate how to manipulate all of its sequences? ****

**** The algorithm we propose is the only universal method for generating
arbitrary contextual sets known in the literature and that, in my opinion,
warrants its publication in PRL. I hope the referee will concur.****

## Appendix 3: Physical Review Letter Editor’s 3rd decision

#### Mladen Pavičić

### Acknowledgement LP17301 Pavicic

Date: Mon, 19 Sep 2022 11:49:44 +0000

From<EMAIL_ADDRESS>
Subject: Acknowledgement LP17301 Pavicic

Re: LP17301 Automated generation of arbitrarily many Kochen-Specker and other
contextual sets in odd dimensional Hilbert spaces by Mladen Pavičić and Norman
D. Megill

Dear Prof. Pavicic,

We have successfully generated output of the above manuscript that you sent
for resubmission. The manuscript has been forwarded for further processing.

Yours sincerely,

Editorial Systems Physical Review Letters

### Editor’s 3rd decision

Date: Fri, 30 Sep 2022 14:53:08 +0000

Subject: Your manuscript LP17301 Pavicic

Re: LP17301

Dear Prof. Pavicic,

This is in response to your resubmission.

Reopening review at this stage would be an extraordinary step, and we see
insufficient cause for that.

Thus, we turn the matter back to you. We suggest that you submit to a more
specialized journal, perhaps Physical Review A.

Yours sincerely,

Robert Garisto (he/him/his) Editor Physical Review Letters

## Appendix 4: Comment on Physical Review Letter Editor’s 3rd decision

#### Mladen Pavičić

Date: Sat, 1 Oct 2022 23:25:42 +0200

Subject: To the exclusive attention of Dr. Garisto - LP17301 Pavicic

Dear Dr. Garisto,

On Fri, Sep 30, 2022 you wrote:

$>$ Reopening review at this stage would be an extraordinary step, and we see
insufficient cause for that.

What is extraordinary here is that I have received no comments on my responses
and changes from either of the referees.

With the best regards,

Mladen Pavicic.

## Appendix 5: Quantum Journal Editor’s decision and my response to it

### Publication Decision from Quantum

#### Automated Generation of Arbitrarily Many Kochen-Specker and Other
Contextual Sets in Odd Dimensional Hilbert Spaces

Decision made on October 20th, 2022 Editorial board’s determination: Reject

#### Comments from the Admin

Dear Mladen Pavicic,

Thank you for submitting to Quantum. Unfortunately, we have to inform you that
the editorial board of Quantum have decided not to publish your paper
“Automated Generation of Arbitrarily Many Kochen-Specker and Other Contextual
Sets in Odd Dimensional Hilbert Spaces”. This decision is due to the following
reason:

We have noticed that you have made public, in arXiv:2202.08197, the referee
reports which you received for this manuscript. The reports were anonymous and
formed a confidential communication between author and editor. By making them
public, including your denigrating comments on the scientific expertise of one
of the referees, you have breached this confidentiality. Moreover, by this
conduct you have put an undue pressure on the refereeing process at Quantum,
since potential referees can expect a similar action in response to their own
reports, if these happen to be critical as well. (This case of “pressure” by
the authors on editors or referees is explicitly mentioned in our editorial
policies, as a ground for desk rejection.)

For this reason we are not able to consider you paper for publication in
Quantum.

We hope that this quick response will allow you to submit to another journal,
and we look forward to receiving your future work. Thank you for supporting
our efforts towards open, community-led and fair science publishing.

Best regards,

Lukas Schalleck (Admin) on behalf of the Editorial Board of Quantum

This rejection letter is final and this discussion is now closed. Under
certain conditions authors can appeal against editorial decisions.

### My response to the Decision from Quantum

Mladen Pavicic Oct 25, 2022 - 12:12 pm CEST

Dear Mr. Schalleck,

Lukas Schalleck (Admin) replied to a discussion on the manuscript:

Automated Generation of Arbitrarily Many Kochen-Specker and Other Contextual
Sets in Odd Dimensional Hilbert Spaces. (Quantum)

$>$ Further to this decision and as previously indicated, we would like to
inform you that desk-rejections are not

$>$ eligible for appeals.

OK. But you write: “This case of “pressure” by the authors on editors or
referees is explicitly mentioned in our editorial policies, as a ground for
desk rejection.”

Where exactly? I was unable to spot it in your documents. Please, help me with
that and direct me, as an author, to the relevant link and lines in it.

With the best regards,

M. Pavicic.

### Admin’s response

Lukas Schalleck (admin) Oct 25, 2022 - 12:24 pm CEST

Dear Mladen Pavicic,

Thanks for reaching out again. Happy to help, of course: this may be found
under “Acceptance criteria” $\to$ “Editorial pre-selection” on the following
page: https://quantum-journal.org/editorial-policies/

The pertinent lines state: “External pressure by the authors on editors or
referees outside the due editorial process may also result in editorial
rejection.”

Best regards, Lukas Schalleck

### My response to Admin’s response

Mladen Pavicic, Oct 25, 2022 - 1:26 pm CEST

Dear Mr. Schalleck,

I fail to grasp how my publication of anonymous previous reports can be
possibly rendered as: “External pressure by the authors on editors or
referees” when there is a standard procedure to send previous reports to next
referees of a paper in most journals and to send previous reports to other
journals of the same publisher.

By these standards, insight into previous reports can only be helpful and by
no means a “pressure.”

With the best regards,

M. Pavicic.

## Appendix 6: Physical Review A Editor’s letter, 3 referee reports, my
responses to them, and Editor’s decision

Physical Review A has taken the manuscript and the reports over from the
Physical Review Letters

### Editor’s letter

Date: Tue, 13 Dec 2022

Re: LP17301AR

Automated generation of arbitrarily many Kochen-Specker and other contextual
sets in odd-dimensional Hilbert spaces by Mladen Pavičić and Norman D. Megill

Dear Prof. Pavicic,

The above manuscript has been reviewed by three of our referees. Comments from
the reports appear below for your consideration.

When you resubmit your manuscript, please include a summary of the changes
made and a brief response to all recommendations and criticisms.

Yours sincerely,

Dr. Erin Knutson Associate Editor Physical Review A

P.S. We are taking referee A’s comments seriously. It will be easy for the
review process to become (more) argumentative at this point, and we want to
avoid that outcome. Please be sure to thoroughly and politely address their
comments, which are designed to improve the impact and readability of your
manuscript.

### Three reports

———————————————————————

Second Report of Referee A – LP17301AR/Pavicic

———————————————————————

Overview

This is a resubmission of a manuscript originally submitted to PRL. In my
original referee report I raised some concerns about the paper and made a
number of suggestions for improvement, encouraging the author to submit a
carefully revised version to e.g. PRA. When I received a revised submission I
was convinced this would be an easy case of recommending acceptance to PRA,
with at most only some minor remarks.

However, I am sad to say that I maintain most of my reservations about the
manuscript. Unfortunately the author seems to have followed only the last part
of the recommendation and resubmitted an almost identical version to PRA. I
was disappointed to see that they made very little effort to address or engage
with almost all of the points raised in the report.

Even regarding fairly minimal comments (such as a request to clarify the
caption of a figure, see point 11 below), the author simply did not bother to
make the very small, harmless change suggested to improve readability (instead
suggesting in their reply that the referee should have “immediately grasped
what the axes represent”). This was of course a very minor and unimportant
point in the scheme of the paper, but I think this reply is indicative of the
author’s failure to take the peer-review process seriously enough and to make
an effort to engage with criticism and suggestions.

More worryingly, their reply to some of the more substantial points seems to
betray some basic misunderstandings. The most shocking is the insistence on
presenting unimportant implementation details, such as keeping the list of
ASCII characters used to encode hypergraphs in their programs as “an integral
part of the MMPH language,” and even doubling down on this. This seems to show
a failure to grasp the distinction between (high-level) algorithms (which are
relevant to the reader) and (low-level) implementation choices (which are
incidental and completely irrelevant). I had commented that choosing to
present such implementation details obfuscates the actual contributions of the
paper—however, this reply made me think that the distinction is not even clear
to them.

As another example, similar remarks apply to the suggestion regarding the
definition of MMPH hypergraphs. The author’s reply mentions a standard
textbook to counter the suggestion raised in the report, claiming that in that
book the definition appears in a similar form as in their manuscript. Looking
at the book, one actually finds out that this is not the case: in the book the
presentation follows the suggestion I had given in my report. Once again, no
change was made to the manuscript regarding this point.

In the remainder of this review, I go into more detail into the issues
mentioned above and consider the author’s reply point by point. But to
summarize: in addition to the reservations I had expressed about the adequacy
of the way their methods are presented (which were not addressed), the replies
quoted above and the underlying misconceptions made me have bigger doubts
about the quality of this paper.

For the reasons just outlined, I believe that the manuscript should not be
accepted in its current form. As I still believe that there is worthwhile work
behind this paper, I recommend giving the author another opportunity to revise
the manuscript and resubmitting it as a PRA Letter, in the hope that they’ll
engage with criticism and suggestions in a more constructive fashion to
improve the presentation.

Detailed comments

Quotes in pink are from the previous referee report and in blue from the
author’s response.

  1. 1.

$<$ ‘[…] It claims that “development of quantum computation and communication,
recently shown to be supported by contextuality, arguably asks for an abundant
supply of contextual sets,” but the authors do not bother to explain why such
sets might be necessary […]’

$>$ ‘On the contrary, the motivation is expressed in the second paragraph of
the paper: “none of the methods [previously employed in the even-dimensional
spaces] is available in the odd-dimensional spaces. In this paper, we offer
several methods for automated generation of arbitrarily many KS sets as well
as contextual non-KS ones in $n=3,5,7,9$ dimensional spaces.” The sentence:
“quantum computation and communication … arguably asks for an abundant supply
of contextual sets” is taken from the abstract which is limited in size. In
the very body of the article, it is substantiated by the motivation sentences
cited above.’

These sentences motivate why one would look at odd-dimensional subspaces given
that prior work has focused on even-dimensional subspaces. These sentences do
not address the following point I was referring to: why does the ‘development
of quantum computation and communication’ ask for a requisite/abundant supply
of contextual sets? I understand that quantum communication and computation
are ‘supported by contextuality’, but what is the actual use of having a large
supply of contextual sets? Again, I am not claiming that finding such
contextual sets is not necessary. I am simply stating that the article does
not argue for this need, beyond a this flimsy reference to contextuality
supporting communication and communication. More motivation on this point
would make the manuscript stronger and more broadly appealing. Having said
that, I think this is a less important point than previously for PRL.

  2. 2.

$<$ ‘The presentation falls short of the standard expected in a research
article, especially in a journal as good as PRL, in terms of clarity and
rigor. Various passages are written in a confusing fashion.’

$>$ ‘We published in Phys. Rev. Lett., Scientific Reports, Phys. Rev. A & D,
Optics Express, J. Math. Phys., J. Phys. A, J. Opt. Soc. Am. B, Entropy, Phys.
Lett. A, Found. Phys., Ann. H. Poincare … over almost half a century, and we
do not think we wrote this paper in a “confusing fashion.”’

The authors have produced great research and published excellent papers over
their careers. But as far as I am concerned their prior publishing record is
irrelevant in this context where we’re only being asked to judge the one
manuscript at hand. I fail to understand the relevance of bringing this up. As
for the substance of the criticism, the overview paragraph (as well as the
detailed comments in my first report) highlight various passages where the
presentation was, in my opinion, confusing. The author would do better to
respond to these concerns instead of bragging about their publication list.

  3. 3.

$<$ ‘For example, it is often unclear from the text that a passage is to be
taken as the definition of a new concept (e.g. when the notion of
coordinatization is introduced).’

$>$ ‘Wiktionary has a single item for: coordinatization (plural
coordinatizations): (mathematics) “The application of coordinates to a
system,” and in the 3rd paragraph of the right column on p. 1 we additionally
clarify: “The MMPHs above are defined without a coordinatization, i,e.,
neither their vertices nor their hyperedges are related to either vectors or
operators.” So, no “new concept” is introduced. The “coordinatization” is a
word from a dictionary the referee should have consulted.’

The dictionary definition is characteristically vague, as suits a dictionary
of a natural language. It does not explain what the word means in specific
contexts where is is used with a precise, formal, mathematical meaning. There
are many ways one could instantiate this broad natural-language definition
(‘applying coordinates to a system’) in the concrete setting of hypergraphs.
‘Coordinatization of a MMPH’ is a concept that is first (i.e. newly)
introduced in this text in the last paragraph of page 1. There, its meaning is
not precisely set out. One infers from the way the sentence is written (and
from one’s knowledge of the English language) that a coordinatization of an
MMPH is a map from vertices or from hyperedges (which?) to vectors or
operators (which?!). This is sloppy and imprecise. Which is it? Do you mean
that a coordinatization associates to each vertex a vector or an operator? Or
are such vectors (or operators) associated to hyperedges instead of vertices?
Are such associations supposed to satisfy some constraints, such as the
vectors needing to be orthogonal when they belong to the same hyperedge (as
one seems to infer in between the lines from reading a few sentences ahead)?

Concepts should be defined precisely and fully in scientific (especially
mathematical) text. The reader is not supposed to infer a definition by trying
to reconstruct what was in the authors’ head from pieces of sentences here and
there.

  4. 4.

$<$ ‘On the other hand, e.g. in the definition of MMPH hypergraph, some extra
commentary is written as a part of the definition when it does not belong
there, namely item (v).’

$>$ ‘In the theory of hypergraphs, e.g., Berge, Hypergraphs, North Holland
(1989), Ch. 1, Sec. 1, p. 1, 4th paragraph it is assumed that “A hypergraph
may be drawn as a set of points representing the vertices [and a hyper]edge is
represented … by a single closed curve enclosing the vertices is a part of the
definition.” There is no particular kind of a “definition” of ours which would
differ from Berge’s or any other standardly accepted definition except that we
introduce, in point (v), a curve passing through vertices instead of Berge’s
curve which encloses them.’

$<$ ‘ “(v) graphically, vertices are represented as dots and hyperedges as
(curved) lines passing through them.” This is given as one of the conditions
in the definition of MMPH graphs, but in reality it is not a condition on
hypergraphs. It is just an explanation of the convention used to draw
hypergraphs to ease the readability of Figure 2, for example. It should be
separated from the definition of MMPH hypergraph.’

$>$ ‘Nowhere do we speak about a condition. It is just the point within a
definition in the same way in which similar points are used in many hypergraph
textbooks. We already clarified that point (v) above when we compared it with
a similar point used by Berge, Ch. 1, Sec. 1, p. 1, 4th paragraph.’

I located a copy of the book mentioned. It is surprising that the author chose
to point to it. The beginning of Chapter 1, which introduces the notion of
hypergraph, is indeed clearly written. But it does not do what I criticised in
this manuscript; instead, the defining conditions and additional commentary on
notational or pictorial conventions are clearly delineated from each other. I
suggest that the author actually take this passage of the book as a model.

Berge gives the definition of a hypergraph (‘A hypergraph on $X$ is a family …
such that (1) … (2) … A simple hypergraph … is a hypergraph … such that (3)
…’). Then in the following paragraph Berge goes on to explaining that elements
of $X$ are called vertices etc. as well as setting out some other conventions.
Then the paragraph after that contains the sentence quoted, explaining the
book’s convention for depicting/drawing hypergraphs. This is most definitely
not part of the definition, which Berge clearly demarcates.

Quite differently, the manuscript reads: ‘An MMPH is a connected $n$-dim
hypergraph $k$-$l$ with $k$ vertices and $l$ hyperedges in which (i) …; (ii)
…; (iii) …; (iv) …; (v) graphically, vertices are represented …’. Here, these
are presented as five conditions (on hypergraphs) defining what it means to be
an MMPH hypergraph. Item (v) is not (as in Berge’s book) presented as what it
is: a convention on how to depict MMPH hypergraphs, rather than an item that
is part of the definition, i.e. of what it means to be an MMPH hypergraph. It
should, as in Berge’s text, be clearly separated.

The author claims that this is never spoken about as a ‘condition’. But in
fact the paragraph defining MMPH hypergraphs is phrased in a way that places
(v) on par with the other conditions defining what it means to be a
hypergraph. And moreover, later in the text one reads: ‘An MMPH is a special
kind of a general hypergraph in the sense that none of the aforementioned
points (i-v) holds for it’, which implicitly says that (i-v) are to be taken
as conditions on hypergraphs that define what it means to be an MMPH
hypergraph.

  5. 5.

$<$ ‘The choice of material included in the text is odd, and it is inadequate
for a journal article. At points, the authors choose to give too much
information on irrelevant implementation details (e.g., the internal ASCII
encoding of the graphs being manipulated, or the name of functions in their
computer code).’

$>$ ‘ Again, Berge, Ch. 1, Sec. 1, p. 1, 5th paragraph: “One may also define a
hypergraph by its incidence matrix, with columns representing the [hyper]edges
and rows representing the vertices (cf. Figure 1, p. 2).++ We, instead of an
incidence matrix, make use of MMPH ASCII strings. Some kind of codification is
unavoidable and our MMPH string codification is an integral part of the MMPH
language and these strings are generated by our algorithms and programs and
they then serve us as further direct inputs to our algorithms and programs
whose outputs are all MMPH properties and features including graphical
representations of the obtained MMPHs. The difference between incidence
matrices and MMPHs is obvious; for instance, the MMPH string of Fig. 2(c) can
be written in a single line, while the incidence matrix takes 37 columns and
51 lines. Also an MMPH string can be put in any of our programs while no such
programs for arbitrary incidence matrices are known to us.’

$<$ ‘This is the kind of implementation detail that is completely irrelevant
to the potential readers of this paper. Its inclusion is a waste of space that
could be used to convey much more interesting information about the methods
implemented. This does not concern any substantial/significant algorithmic
choice, but simply a matter of convention chosen for the internal
representation. Its the kind of detail that is typically hidden from the user
even in the documentation of a well-written library, and its even less
relevant in the context of a journal article, which ought to focus on
explaining the methods instead.’

$>$ ‘There are no methods which could generate the contextual sets in odd
dimensional spaces outside the MMPH language based on ASCII codification
either in our minds or anywhere in the literature. The MMPH language is the
method. A “method instead” does not exist.’

$<$ ‘This is problematic for two reasons. First, it detracts from the interest
that the manuscript may have to a journals audience: most readers would read
the paper to learn the main ideas underlying the methods so they could
replicate and adapt them (anyone keen on learning implementation details would
refer to the codes documentation and/or to the code itself )…’

$>$ ‘The 2nd sentence seems to contradict itself: “most readers would read the
paper to learn the main ideas underlying the methods so they could replicate
and adapt them,” but they would not like “to learn the code itself.” Note that
without the MMPH language the results obtained in the article are inaccessible
as they were in the last half of century.’

These replies contributed to seriously aggravate my misgivings, in particular
the claims that the string codification chosen is ‘an integral part of the
MMPH language’, or that ‘there are no methods … outside the MMPH language
based on ASCII codification’ and that this ‘MMPH language is the method,’ or
the outrageous suggestion that ‘without the MMPH language the results obtained
in the article are inaccessible’. These seems to reveal some very basic
misunderstanding on the author’s part of the distinction between the level of
ideas, methods, algorithms and that of concrete implementation details. The
higher level abstracts from such irrelevant details and it should be conveyed
in a way that is oblivious to it.

For illustration, let me first remark the obvious point that the use of ASCII
characters is totally immaterial. If instead the author had used Unicode
characters or 16-bit integers to encode hypergraphs, nothing of relevance
would change whatsoever (except that it would take longer for the available
‘characters’ to be exhausted).

Similarly, the point about a string being written ‘in a single line while the
incidence matrix takes 37 columns and 51 lines’ is misguided. For example,
many graph algorithms are typically presented in terms of incidence matrices
(or even more abstractly), but of course ultimately everything is represented
in the computer as a sequence of bits, which can in theory be displayed in a
single line. One doesn’t usually do this for a reason: such encodings are
cumbersome and distract from the essence of the algorithms. The great power of
abstraction is precisely that it allows one to focus on the relevant structure
at each level, without concerning oneself with irrelevant, low-level details.

Finally, regarding comment that ‘an MMPH string can be put in any of our
programs while no such programs for arbitrary incidence matrices are known to
us’ let me just reiterate that a research article is not a manual documenting
the usage of a computer program. In such a context, it would be natural to
describe the format in which the program expects to receive its input, so that
a potential user could interface with the program. But that’s completely
beside the point for a journal article.

  6. 6.

$<$ ‘I would therefore encourage the authors to submit a carefully *revised*
version, with a more detailed presentation of the main novel technical