Existence of DSRGs (98,63,45,38,45), (98,34,16,15,10), (98,70,55,48,55), (98,27,12,11,6), (99,60,40,34,40) and (99,38,18,17,13) from root and cube flags

1. Results and context

We construct the three complementary pairs in the title, resolving six parameter sets. The complementary-pair representatives are marked ? in the Hobart–Brouwer table [1] retrieved on 17 October 2026. With Duval's convention [2], a loopless binary adjacency matrix has parameters $(v,k,t,\lambda,\mu)$ when both degrees are $k$ and

$$ A^2=(t-\mu)I+(\lambda-\mu)A+\mu J. $$

The first construction uses fourteen labelled $D_4$ roots and zero-sum five-subsets. The second uses a multiset of fourteen vertices of the three-dimensional cube and zero-sum four-subsets. A graph vertex is a flag $(i,T)$ with $i\in T$; repeated flags are distinct vertices. The arc rule is $(i,T)\to(j,S)$ exactly when $i\notin S$. Integer dual certificates prove the two-step counts. A third construction uses the same D4 roots, an additional zero vector and unequal class sizes to obtain the order-99 pair. The finite data also support explicit infinite families.

Theorem 1. The following DSRGs and their directed complements exist:

1. For every positive integer $m$,

$$ \begin{aligned} &(98m^2,\ 98m^2-35m,\ 98m^2-63m+10,\ 98m^2-70m+10,\ 98m^2-63m+10),\\ &(98m^2,\ 35m-1,\ 7m+9,\ 7m+8,\ 10). \end{aligned} $$

2. For every integer $u\geq6$,

$$ \begin{aligned} &(2u^2,\ 2u^2-4u,\ 2u^2-7u+6,\ 2u^2-8u+6,\ 2u^2-7u+6),\\ &(2u^2,\ 4u-1,\ u+5,\ u+4,\ 6). \end{aligned} $$

3. For every integer $m\geq0$ and positive integer $a$,

$$ \begin{aligned} &\bigl(a(99+27m),\ a(60+18m),\ a(40+12m),\ a(34+12m),\ a(40+12m)\bigr),\\ &\bigl(a(99+27m),\ a(39+9m)-1,\ a(19+3m)-1,\ a(19+3m)-2,\ a(13+3m)\bigr). \end{aligned} $$

The title's first pair is $m=1$ in part 1; the second is $u=7$ in part 2; the third is $m=0$, $a=1$ in part 3. The same cube construction also works at $u=4$. Standard independent-set blow-ups of the two dense order-98 seeds additionally give the linear families $(98a,63a,45a,38a,45a)$ and $(98a,70a,55a,48a,55a)$, with their complements, for every positive integer $a$.

Low-rank DSRG representations are developed in [3]. Kirkman Loom's constructive hyperplane duality [4] provides a recent framework for turning compatible binary rows and rational columns into exact graphs. Its order-98 construction has degree 56 and parameters $(98,56,36,29,36)$. We use a quotient by identical adjacency rows and its zero eigenspace, represented by roots or cube vectors. Incidence constructions have a substantial history, including Olmez and Song [5]. The new table resolutions and their finite certificates are the central contribution. The replication and cube extension proofs explain structured families supported by these data. Version 2 of [4] also constructs $(99,30,10,7,10)$ and $(88,60,45,39,45)$ using neutral coordinates. Our order-99 graph has degree 60 and arises from the non-antipodal D4 support with unequal multiplicities; it supplies a different complementary table pair.

The transitive-group study [6] has new DSRG rows at orders 72, 96 and 165. The group, loop and quasigroup constructions [7] include order-98 pairs starting with $(98,19,13,6,3)$ and $(98,26,16,9,6)$. In particular its family $(2n^2,4n-2,2n+2,n+2,6)$ differs from Theorem 1(2). Our earlier public symmetry argument [8] excludes vertex-transitive realizations of the first pair; that argument is not required here. The partition family in [9] has dense degree $u(u+1)$ at order $2u^2$, which agrees with the dense degree in Theorem 1(2) only at $u=5$. We make no novelty claim for adjacency factorization, quotient matrices, flag notation, or standard blow-ups.

2. The flag certificate

Lemma 2. Set $q=2s$, let $1\leq h<q-1$, and put $p=q-h-1$. Let $B,W$ be integer $q$-by-$h$ matrices with rows $b_i,w_i$, satisfying

$$ B^T\mathbf1=W^T\mathbf1=0,\qquad B^TW=qI_h. \tag{1} $$

For each $j\in\{0,\ldots,q-1\}$, let $F_j$ be a list of $s$ subsets of root labels, with repetitions allowed. Each $T\in F_j$ must contain $j$, have $h+1$ distinct elements, and satisfy $\sum_{i\in T}b_i=0$. Suppose

$$ H_{ij}:=\#\{T\in F_j:i\in T\} =s\delta_{ij}+\frac{h-w_i\cdot b_j}{2}. \tag{2} $$

Then the flag arc rule gives a DSRG with parameters

$$ (2s^2,\ sp,\ p(p+1)/2,\ p(p+1)/2-s,\ p(p+1)/2). \tag{3} $$

Its complement has parameters, with $c=\binom{h+1}{2}$,

$$ (2s^2,\ (h+1)s-1,\ s+c-1,\ s+c-2,\ c). \tag{4} $$

Proof. Each column has the pattern $b=\mathbf1-\mathbf1_T$ on the $q$ root classes. It has weight $p$, is orthogonal to all columns of $B$, and has a zero in its own class. There are $s$ vertices per class, so all indegrees are $sp$ and the graph has no loops. The quotient counting arcs from one vertex in class $i$ to class $j$ is

$$ Q=sJ-H=-sI+\frac{q-h}{2}J+\frac12WB^T. \tag{5} $$

Equation (1) gives $Q\mathbf1=sp\mathbf1$, so all outdegrees agree. Since $q-h=p+1$, every actual column pattern satisfies

$$ Qb=-sb+\frac{p(p+1)}2\mathbf1. $$

The $(x,y)$ entry of $A^2$ is the entry of this vector corresponding to the root class of $x$. Thus $A^2=-sA+p(p+1)J/2$. This proves (3); the usual directed complement formula gives (4). $\square$

The matrix $WB^T/2$ has rank $h$, square $s$ times itself, and zero row and column sums. It records the quotient's zero eigenspace. The diagonal of (2) also gives $w_i\cdot b_i=h$, an identity used below. All conditions are finite integer equalities; a search solver is not part of their proof.

3. The D4 seed

Take $q=14$, $s=7$, $h=4$. All labels start at zero. The following roots are distinct, sum to zero, and are not closed under negation. They each have two nonzero coordinates, both $\pm1$.

| i | b_i | w_i | |---|---|---| | 0 | (-1, 1, 0, 0) | (-2, 2, 0, 0) | | 1 | (1, -1, 0, 0) | (2, -2, 2, 0) | | 2 | (-1, 0, -1, 0) | (-2, 0, -2, -2) | | 3 | (-1, 0, 0, -1) | (-3, -1, 1, -1) | | 4 | (1, 0, 0, -1) | (1, 1, -1, -3) | | 5 | (1, 0, 0, 1) | (4, 2, -2, 0) | | 6 | (0, -1, 1, 0) | (0, -2, 2, 2) | | 7 | (0, -1, 0, 1) | (-1, -3, -1, 1) | | 8 | (0, 1, 0, -1) | (2, 4, 2, 0) | | 9 | (0, 1, 0, 1) | (1, 1, 1, 3) | | 10 | (0, 0, -1, -1) | (0, -2, -4, 0) | | 11 | (0, 0, -1, 1) | (0, 0, -2, 2) | | 12 | (0, 0, 1, -1) | (1, -1, 1, -3) | | 13 | (0, 0, 1, 1) | (-3, 1, 3, 1) |

The twenty-six zero-sum five-subsets are:

| a | T_a | |---|---| | 0 | {0, 1, 2, 4, 13} | | 1 | {0, 1, 2, 5, 12} | | 2 | {0, 1, 6, 8, 11} | | 3 | {0, 1, 6, 9, 10} | | 4 | {0, 2, 4, 5, 6} | | 5 | {0, 3, 4, 5, 7} | | 6 | {0, 4, 7, 10, 13} | | 7 | {0, 4, 7, 11, 12} | | 8 | {0, 5, 7, 10, 12} | | 9 | {1, 2, 6, 8, 9} | | 10 | {1, 3, 7, 8, 9} | | 11 | {1, 3, 8, 11, 13} | | 12 | {1, 3, 9, 10, 13} | | 13 | {1, 3, 9, 11, 12} | | 14 | {2, 3, 4, 5, 13} | | 15 | {2, 4, 7, 8, 13} | | 16 | {2, 4, 7, 9, 12} | | 17 | {2, 4, 11, 12, 13} | | 18 | {2, 5, 7, 8, 12} | | 19 | {2, 5, 10, 12, 13} | | 20 | {3, 4, 6, 9, 11} | | 21 | {3, 5, 6, 8, 11} | | 22 | {3, 5, 6, 9, 10} | | 23 | {6, 7, 8, 9, 10} | | 24 | {6, 8, 10, 11, 13} | | 25 | {6, 9, 10, 11, 12} |

The seven flags in each root class are specified by these subset labels. A repeated label denotes a separate vertex with the same flag data.

| Root class i | Seven subset labels, with repetitions | |---|---| | 0 | 1, 2, 3, 3, 5, 7, 8 | | 1 | 0, 0, 1, 3, 10, 11, 11 | | 2 | 1, 9, 9, 9, 14, 17, 18 | | 3 | 5, 10, 11, 13, 21, 22, 22 | | 4 | 0, 6, 6, 15, 20, 20, 20 | | 5 | 1, 5, 5, 14, 19, 19, 21 | | 6 | 4, 4, 4, 21, 23, 24, 25 | | 7 | 5, 6, 7, 10, 15, 18, 18 | | 8 | 2, 11, 11, 18, 23, 23, 23 | | 9 | 3, 12, 13, 16, 16, 16, 22 | | 10 | 3, 8, 12, 12, 23, 24, 24 | | 11 | 2, 7, 11, 13, 17, 17, 21 | | 12 | 8, 8, 13, 17, 18, 25, 25 | | 13 | 6, 6, 11, 14, 17, 19, 19 |

In the last table a subset label refers to the preceding table. Direct multiplication checks (1), every subset has zero root sum, and direct membership counts check (2). Lemma 2 gives the first dense order-98 graph. Its complement has parameters $(98,34,16,15,10)$.

4. The cube seeds

For the cube use $h=3$. We supply four seeds at $s=4,6,7,9$. Every cube vertex occurs as a row of $B$ in every seed. Their multiplicities, ordered lexicographically on $\{-1,1\}^3$, are

| s | Cube-vector multiplicities | Graph order | |---|---|---| | 4 | 1,1,1,1,1,1,1,1 | 32 | | 6 | 1,1,2,2,2,2,1,1 | 72 | | 7 | 1,2,2,2,2,2,2,1 | 98 | | 9 | 2,2,2,3,3,2,2,2 | 162 |

Appendix A gives all root and dual rows and every flag subset. Membership counts verify (2) and integer products verify (1). In particular, the seed $s=7$ has parameters $(98,70,55,48,55)$, with complement $(98,27,12,11,6)$. The other three seeds provide the remaining residue classes needed for the extension in Section 6. They are not asserted to be new parameter sets.

5. Replication of flag data

Lemma 3. Any data satisfying Lemma 2 with $q=2s$ give new such data with $q'=mq$, $s'=ms$, and unchanged $h$, for every positive integer $m$.

Proof. Label the new root classes $(i,a)$, where $0\leq a<m$. Repeat the old root and dual vectors in every copy. Their sums remain zero and $B'^TW'=mqI_h$. For class $(j,a)$ and each $T\in F_j$, include, for every $0\leq b<m$, the subset

$$ \{(j,a)\}\ \cup\ \{(i,b):i\in T\setminus\{j\}\}. \tag{6} $$

Each is a zero-sum set of $h+1$ distinct root labels, and each class has $ms$ flags. For a source class $(i,c)$ with $i\ne j$, only the term $b=c$ contributes to its membership count. This is the old off-diagonal count $(h-w_i\cdot b_j)/2$. If $i=j$ and $c\ne a$, the count is zero, equal to $(h-w_j\cdot b_j)/2$. At $(j,a)$ the count is $ms$. These are exactly (2) for the enlarged data. $\square$

Apply Lemma 3 to the D4 seed, giving $s=7m$, $h=4$ in (3)–(4). This proves Theorem 1(1). The construction has quadratic graph order; its finite flag rule (6) specifies every vertex and arc.

6. Adding a cube

For $b\in\{-1,1\}^3$, define four triples of other cube vertices. Three are $\{-b,z,-z\}$, one for each antipodal pair $\{z,-z\}$ other than $\{b,-b\}$. The fourth consists of the three vertices at Hamming distance two from $b$. Adjoining $b$ to any of these triples gives a zero-sum four-subset. Across the four triples, a vertex $z$ occurs exactly $d_H(b,z)$ times: once at distance 1, twice at distance 2, and three times at distance 3.

Lemma 4. Suppose Lemma 2 holds with $h=3$, every row of $B$ belongs to $\{-1,1\}^3$, and every cube vertex occurs. One can adjoin one full cube to the root labels, set its dual rows equal to its root rows, and obtain valid data with $s$ replaced by $s+4$.

Proof. The eight new root and dual vectors sum to zero and their outer products sum to $8I_3$. Thus (1) persists with $q$ replaced by $q+8$.

For each old center $j$, retain its old flags and add four flags consisting of $j$ and the new labels in the four triples for $b_j$. For each new center of type $b$, choose an old label $j$ with $b_j=b$. Take the old flags of $j$ and replace $j$ by the new center; then add the four flags formed entirely from the new cube. All these flags are zero-sum four-subsets, and every class has $s+4$ flags.

For old labels $i$ and an old center $j$, off-diagonal membership is unchanged. For an old label $i$ and a new center of type $b_j$, the copied old flags have count $H_{ij}-s\delta_{ij}=(3-w_i\cdot b_j)/2$. The same formula holds when $i=j$ because $w_j\cdot b_j=3$. For a new source label of type $z$ and any distinct center of type $b$, membership is the triple count $d_H(z,b)=(3-z\cdot b)/2$. This includes zero when the two labels have the same cube type. Each diagonal count is $s+4$. Hence (2) holds throughout. $\square$

Repeatedly apply Lemma 4 to the four seeds $s=4,6,7,9$. They cover every integer $s\geq6$: use 4 for residue 0 modulo 4, 6 for residue 2, 7 for residue 3, and 9 for residue 1. Substituting $h=3$ and $s=u$ in Lemma 2 proves Theorem 1(2). The argument establishes the entire infinite family through exact extension identities.

7. Unequal classes and the order-99 pair

The same incidence mechanism allows unequal class sizes. This extension uses the D4 roots of Section 3 together with one zero vector.

Lemma 5 (weighted flag certificate). Let positive integers $n_0,\ldots,n_{q-1}$ sum to $v$. Let $k,d,\mu$ be positive integers with $\mu\geq d$ and $k(k+d)=\mu v$. Let $B$ and $U$ be rational $q$-by-$h$ matrices with $B^T\mathbf1=0$. For each class $j$, let $F_j$ be a list of $n_j$ subsets $T$ containing $j$, such that

$$ \sum_{i\in T}b_i=0,\qquad \sum_{i\in T}n_i=v-k. \tag{7} $$

Suppose their membership counts satisfy

$$ H_{ij}=\left(1-\frac\mu k\right)n_j+d\delta_{ij}-U_i\cdot b_j. \tag{8} $$

Then the flag arc rule gives a DSRG$(v,k,\mu,\mu-d,\mu)$.

Proof. There are $n_j$ vertices in class $j$, and every target flag contains its class label, so the matrix is binary and loopless. Write $p_T=\mathbf1-\mathbf1_T$. Equation (7) gives $n^Tp_T=k$ and $B^Tp_T=0$. Hence every vertex has indegree $k$. The quotient is

$$ Q=\frac\mu k\mathbf1n^T-dI+UB^T. $$

Its row sums are $\mu v/k-d=k$, and $Qp_T=\mu\mathbf1-dp_T$. Counting an intermediate vertex by its row class now gives $A^2=\mu J-dA$. This proves the assertion. $\square$

To apply the lemma, retain $b_0,\ldots,b_{13}$ from Section 3 and append $b_{14}=0$. Use $(v,k,d,\mu)=(99,60,6,40)$ and $U=E/6$, where the integer rows $E_i$ and class sizes $n_i$ are in the following table.

| i | n_i | E_i | |---|---|---| | 0 | 7 | (-5, 3, -3, -7) | | 1 | 8 | (1, -3, 3, -1) | | 2 | 7 | (-5, -3, -3, 5) | | 3 | 6 | (-8, 0, 0, -4) | | 4 | 4 | (16, 0, -6, -4) | | 5 | 9 | (1, 3, -3, -1) | | 6 | 6 | (1, -9, 3, -1) | | 7 | 7 | (-2, -12, 0, -4) | | 8 | 5 | (1, 9, 3, -7) | | 9 | 7 | (4, 0, 6, 8) | | 10 | 5 | (-2, -6, -12, -4) | | 11 | 7 | (7, 9, -3, 5) | | 12 | 5 | (1, 3, 15, -1) | | 13 | 7 | (-2, 6, 0, 8) | | 14 | 9 | (-2, 0, 0, 2) |

The required identity (8) is the integer equality

$$ 6H_{ij}=2n_j+36\delta_{ij}-E_i\cdot b_j. \tag{9} $$

The certificate additionally satisfies $B^TE=36I_4$ and $n^TE=0$. The thirty-two subsets used by the graph are as follows. Each has zero root sum and total class weight 39; their cardinalities need not agree.

| a | T_a | |---|---| | 0 | {0, 1, 2, 4, 6, 9} | | 1 | {0, 1, 3, 4, 7, 9} | | 2 | {0, 4, 7, 10, 13, 14} | | 3 | {0, 4, 7, 11, 12, 14} | | 4 | {0, 5, 6, 7, 8, 10} | | 5 | {0, 5, 6, 10, 11, 12} | | 6 | {0, 1, 3, 5, 14} | | 7 | {0, 1, 7, 8, 10, 13} | | 8 | {1, 2, 8, 11, 12, 13} | | 9 | {1, 2, 9, 10, 12, 13} | | 10 | {1, 3, 6, 8, 9, 11} | | 11 | {1, 2, 7, 8, 9, 12} | | 12 | {2, 3, 4, 5, 6, 9} | | 13 | {2, 4, 11, 12, 13, 14} | | 14 | {2, 5, 6, 8, 11, 12} | | 15 | {1, 2, 3, 4, 9, 13} | | 16 | {3, 4, 6, 9, 11, 14} | | 17 | {3, 5, 7, 8, 11, 12} | | 18 | {3, 5, 10, 11, 12, 13} | | 19 | {0, 2, 4, 5, 7, 12} | | 20 | {2, 5, 6, 8, 10, 13} | | 21 | {3, 5, 7, 9, 10, 12} | | 22 | {0, 3, 4, 5, 6, 11} | | 23 | {6, 8, 10, 11, 13, 14} | | 24 | {0, 1, 7, 8, 11, 12} | | 25 | {2, 4, 7, 8, 13, 14} | | 26 | {6, 7, 8, 9, 10, 14} | | 27 | {2, 5, 6, 9, 10, 12} | | 28 | {3, 5, 7, 8, 10, 13} | | 29 | {6, 9, 10, 11, 12, 14} | | 30 | {0, 1, 7, 9, 10, 12} | | 31 | {0, 1, 3, 4, 11, 13} |

For each class, the following list specifies its flags by subset label. Repeated labels represent distinct vertices.

| Class j | Subset labels in F_j, with repetitions | |---|---| | 0 | 0, 0, 1, 2, 3, 4, 5 | | 1 | 6, 6, 6, 7, 8, 8, 9, 10 | | 2 | 0, 11, 11, 12, 13, 13, 14 | | 3 | 15, 15, 16, 16, 17, 18 | | 4 | 2, 15, 12, 13 | | 5 | 6, 6, 6, 19, 4, 20, 20, 17, 21 | | 6 | 0, 22, 22, 20, 23, 23 | | 7 | 1, 24, 2, 4, 25, 17, 17 | | 8 | 11, 10, 25, 20, 26 | | 9 | 1, 1, 1, 27, 27, 26, 26 | | 10 | 9, 9, 28, 29, 29 | | 11 | 24, 24, 22, 10, 13, 17, 29 | | 12 | 30, 13, 13, 27, 18 | | 13 | 31, 31, 2, 2, 20, 20, 28 | | 14 | 6, 6, 6, 13, 13, 13, 26, 26, 26 |

Direct integer substitution of these tables checks every premise of Lemma 5 and establishes $(99,60,40,34,40)$ and its complement $(99,38,18,17,13)$. Repeating every flag $a$ times gives the standard linear blow-up of this pair. The next subsection supplies the additional neutral-class extension needed for the full Theorem 1(3).

The discovery reduction has a simple interpretation. The reference sizes are 6 on labels $0,\ldots,13$, except for labels 1 and 5, where they are 9; label 14 also has size 9. The displayed size vector is

$$ n=n_{\mathrm{ref}}+B(-1,0,0,1)^T. $$

For any column pattern $p$ with $B^Tp=0$, this preserves $p^Tn$. Thus class sizes can vary in this affine space while the binary column pool remains fixed. The incidence and dual-factor equations then remain linear. The zero row has the forced size $kd/\mu=9$, consistent with the coloop identity discussed in [8]. This arithmetic freedom is absent from the equal-size flags and was used to find the new graph.

7.1 Adding neutral classes

The unequal seed supports another structured infinite family. Its zero-root class has three copies of each flag $S_l\cup\{14\}$, where

$$ S_0=\{0,1,3,5\},\quad S_1=\{2,4,11,12,13\},\quad S_2=\{6,7,8,9,10\}. $$

These three sets partition the nonzero root labels, have zero root sum, and each has class weight 30. For $j<14$, let $c_j$ count the flags in $F_j$ containing label 14. The tables give $c_1=c_5=3$ and $c_j=2$ otherwise, with $n_j\geq2c_j$ throughout. Put $e=E_{14}=(-2,0,0,2)$.

Lemma 6 (neutral-class extension). For every integer $m\geq0$, there exists a DSRG with parameters

$$ (99+27m,\ 60+18m,\ 40+12m,\ 34+12m,\ 40+12m). \tag{10} $$

Construction and proof. Retain the fourteen nonzero root classes, their sizes and their dual rows. Replace the single zero-root class by $Z=\{14,\ldots,14+3m\}$, all of size 9. Label the new zero classes in $m$ triples. In each triple give the first two classes dual row $e$ and the third dual row $-2e$; the old zero class retains dual row $e$.

For each nonzero target class $j$, start from its ordered old flag list. For each new triple, append its first zero label to the first $c_j$ flags, its second zero label to the next $c_j$ flags, and its third zero label to the remaining $n_j-2c_j$ flags. Every flag gains exactly $m$ zero labels, so has class weight $39+9m$. Its center is retained and its root sum is still zero.

For a zero target class $j\in Z$, partition $Z\setminus\{j\}$ into three sets $P_{j,0},P_{j,1},P_{j,2}$ of size $m$. This may be done by cutting the increasing list into consecutive blocks. Its nine flags are

$$ S_l\cup\{j\}\cup P_{j,b},\qquad l,b\in\{0,1,2\}. \tag{11} $$

Each has weight $30+9(m+1)=39+9m$ and zero root sum. Each nonzero source label occurs in three of these flags. Each other zero source also occurs in three, and the center occurs in all nine. Thus the membership counts into every zero target class are $3+6\delta_{ij}$, as required by (9), because its root is zero and its size is 9.

Into a nonzero target class $j$, all old nonzero-source counts are unchanged. A zero source with dual row $e$ has incidence $c_j$. The base certificate gives

$$ c_j=\frac{n_j}{3}-\frac{e\cdot b_j}{6}. $$

A zero source with dual row $-2e$ has incidence

$$ n_j-2c_j=\frac{n_j}{3}+\frac{e\cdot b_j}{3}. $$

Hence the entire extended membership matrix again satisfies (9). Set $v_m=99+27m$, $k_m=60+18m$, $d=6$ and $\mu_m=40+12m$. Then $\mu_m/k_m=2/3$, $k_m(k_m+6)=\mu_mv_m$, and $v_m-k_m=39+9m$. Lemma 5 proves (10). At $m=0$ this construction is the original seed, including repeated flags. $\square$

The complement formula gives

$$ (99+27m,\ 38+9m,\ 18+3m,\ 17+3m,\ 13+3m). $$

Standard independent-set blow-ups now prove Theorem 1(3). The extension keeps the fourteen nonzero class sizes fixed and adds neutral classes; it is separate from Lemma 3's equal-size replication. Some parameter overlap is expected: at $m=1$ and $m=3$, the dense parameters are the standard double blow-ups of the $r=7$ and $r=10$ instances of [4, Theorem 4.2]. The order-99 pair and the explicit neutral-class extension are the claims made here.

8. Verification, discovery and scope

The supplement contains all literal seed data, the three explicit base adjacency matrices, and standard-library exact checkers. verify_root_flag_families.py checks the equal-size families. It verifies (1)–(2), constructs every flag, then checks binary entries, loops, both degrees and all ordered two-step counts. The graph check uses intersections of integer bit masks for an exact count of intermediate vertices and is independent of the quotient equations. Run, with assertions enabled,

python3 verify_root_flag_families.py --seed d4
python3 verify_root_flag_families.py --seed cube
python3 verify_root_flag_families.py --seed d4 --replicate 2
python3 verify_root_flag_families.py --cube-u 13
python3 verify_unequal99_flags.py
python3 verify_unequal99_flags.py --scale 2
python3 extend_unequal99_flags.py --m 0 1 2 3 5 10

An independent audit checks the cube graphs at $u=4,6,7,8,9,10,11,13$ and the D4 replications at $m=1,2,3$, with all complements. All 11 graph pairs pass, up to order 882. A separate integer triple-loop implementation checks the original D4 graph and its standard blow-up at orders 98 and 196. Finite checks support the literal certificates; Lemmas 3–4 prove the universal assertions. The unequal-class checker verifies the additional 99-vertex certificate and its standard blow-up at order 198, including both complements, by direct scalar products. The combined audit extend_unequal99_flags.py also checks the neutral extensions at $m=0,1,2,3,5,10$, up to order 369. Independent scalar-product counts confirm the $m=1,2$ graph pairs, and $m=0$ reproduces the stored 99-vertex matrix exactly. The combined audit checks 18 graph pairs; Lemma 6 proves the universal neutral-extension claim.

Serializing a base adjacency matrix as one binary string per row, each followed by a newline, gives these SHA-256 values:

| Dense seed | SHA-256 | |---|---| | order 98, degree 63 | 6b63bace3406293fff29ac48bd47f7c573e94a20f270c5e216eaca86c4076173 | | order 98, degree 70 | 238c0f77c82db3cd919c5c49c869b3284685295e309e5a4763e54ed5fe9d8368 | | order 99, degree 60 | e885326e781acdb665befd32191ce12e8cfb7e29e4e14030a1329059d70ced3d |

The degree 63 graph has 14 distinct rows and 26 distinct columns. Its column classes have sizes 2 (once), 3 (thirteen times), 4 (five times), 5 (six times), and 7 (once). Automorphisms preserve equal-column classes and their sizes, so this graph is not vertex-transitive. No uniqueness or classification claim is made. The order-99 graph has 15 distinct rows and 32 distinct columns. Its unequal row-class sizes also exclude vertex transitivity for this particular realization.

Discovery searches used balanced root multisets and linear equations for dual factors and flag multiplicities. Removing antipodal closure from the D4 root set was decisive. Cube multisets supplied the second seed and the four residue-class certificates. For order 99, affine changes of class sizes preserved the weighted column pool and supplied the new integer incidence data of Section 7. Its neutral class revealed the three equal-weight root subsets used in Lemma 6. Failed searches in other palettes are not nonexistence proofs. Optional discovery sources are included with pinned dependencies; neither the proof checker nor the constructors need a solver, numerical linear algebra, or network access. Original code is released under MIT.

Appendix A. Complete cube seed data

For each seed, the first table gives $(b_i,w_i)$. The second gives the entire list $F_i$; each parenthesized group is a four-subset of root labels. Repeated groups denote distinct flags. Labels restart at zero for each seed.

Seed s = 4

| i | b_i | w_i | |---|---|---| | 0 | (-1, -1, -1) | (-1, -2, 0) | | 1 | (-1, -1, 1) | (-1, 0, 2) | | 2 | (-1, 1, -1) | (0, 1, -2) | | 3 | (-1, 1, 1) | (-2, 1, 0) | | 4 | (1, -1, -1) | (0, -1, -2) | | 5 | (1, -1, 1) | (2, -1, 0) | | 6 | (1, 1, -1) | (1, 2, 0) | | 7 | (1, 1, 1) | (1, 0, 2) |

| i | Four-subsets in F_i, with repetitions | |---|---| | 0 | (0,1,6,7); (0,1,6,7); (0,2,5,7); (0,3,5,6) | | 1 | (1,2,4,7); (1,2,5,6); (1,2,5,6); (1,3,4,6) | | 2 | (0,2,5,7); (0,2,5,7); (1,2,4,7); (1,2,5,6) | | 3 | (0,3,4,7); (0,3,5,6); (2,3,4,5); (2,3,4,5) | | 4 | (0,3,4,7); (1,2,4,7); (1,3,4,6); (1,3,4,6) | | 5 | (0,3,5,6); (1,2,5,6); (2,3,4,5); (2,3,4,5) | | 6 | (0,1,6,7); (0,1,6,7); (0,3,5,6); (1,3,4,6) | | 7 | (0,2,5,7); (0,3,4,7); (0,3,4,7); (1,2,4,7) |

Seed s = 6

| i | b_i | w_i | |---|---|---| | 0 | (-1, -1, -1) | (-3, 0, 0) | | 1 | (-1, -1, 1) | (0, -3, 0) | | 2 | (-1, 1, -1) | (-1, 0, -2) | | 3 | (-1, 1, -1) | (-1, 1, -1) | | 4 | (-1, 1, 1) | (-1, 0, 2) | | 5 | (-1, 1, 1) | (0, 2, 1) | | 6 | (1, -1, -1) | (2, 0, -1) | | 7 | (1, -1, -1) | (0, -1, -2) | | 8 | (1, -1, 1) | (1, 0, 2) | | 9 | (1, -1, 1) | (0, -2, 1) | | 10 | (1, 1, -1) | (3, 0, 0) | | 11 | (1, 1, 1) | (0, 3, 0) |

| i | Four-subsets in F_i, with repetitions | |---|---| | 0 | (0,3,8,11); (0,4,8,10); (0,4,8,10); (0,5,6,11); (0,5,6,11); (0,5,9,10) | | 1 | (1,2,7,11); (1,2,7,11); (1,3,6,11); (1,3,8,10); (1,5,6,10); (1,5,6,10) | | 2 | (1,2,9,10); (1,2,9,10); (1,2,9,10); (2,4,6,8); (2,4,6,8); (2,5,7,8) | | 3 | (1,3,8,10); (1,3,8,10); (1,3,8,10); (3,4,6,9); (3,4,6,9); (3,5,7,9) | | 4 | (1,4,6,10); (1,4,6,10); (1,4,6,10); (2,4,7,9); (2,4,7,9); (3,4,7,8) | | 5 | (1,5,6,10); (1,5,7,10); (1,5,7,10); (2,5,6,8); (2,5,6,9); (3,5,7,9) | | 6 | (0,5,6,11); (0,5,6,11); (0,5,6,11); (2,4,6,9); (3,4,6,8); (3,4,6,8) | | 7 | (0,5,7,11); (0,5,7,11); (0,5,7,11); (2,4,7,8); (3,4,7,8); (3,4,7,9) | | 8 | (0,3,8,11); (0,3,8,11); (0,3,8,11); (2,4,7,8); (2,5,6,8); (2,5,7,8) | | 9 | (0,2,9,11); (0,2,9,11); (0,2,9,11); (3,4,6,9); (3,5,7,9); (3,5,7,9) | | 10 | (0,4,8,10); (0,4,8,10); (0,5,9,10); (1,2,9,10); (1,3,9,10); (1,4,7,10) | | 11 | (0,3,9,11); (0,3,9,11); (0,4,6,11); (1,2,7,11); (1,2,7,11); (1,2,7,11) |

Seed s = 7

| i | b_i | w_i | |---|---|---| | 0 | (-1, -1, -1) | (-1, -2, 0) | | 1 | (-1, -1, 1) | (-2, -1, 0) | | 2 | (-1, -1, 1) | (-1, -1, 1) | | 3 | (-1, 1, -1) | (-2, 0, -1) | | 4 | (-1, 1, -1) | (-1, 0, -2) | | 5 | (-1, 1, 1) | (0, 2, 1) | | 6 | (-1, 1, 1) | (0, 2, 1) | | 7 | (1, -1, -1) | (2, 0, -1) | | 8 | (1, -1, -1) | (0, -1, -2) | | 9 | (1, -1, 1) | (2, 0, 1) | | 10 | (1, -1, 1) | (0, -2, 1) | | 11 | (1, 1, -1) | (1, 2, 0) | | 12 | (1, 1, -1) | (1, 1, -1) | | 13 | (1, 1, 1) | (1, 0, 2) |

| i | Four-subsets in F_i, with repetitions | |---|---| | 0 | (0,2,11,13); (0,5,7,13); (0,5,7,13); (0,5,9,11); (0,6,9,12); (0,6,9,12); (0,6,10,11) | | 1 | (1,3,9,11); (1,4,7,13); (1,4,9,12); (1,5,7,12); (1,5,8,11); (1,6,7,11); (1,6,8,12) | | 2 | (2,3,9,12); (2,4,8,13); (2,4,9,11); (2,5,7,12); (2,5,7,12); (2,6,7,11); (2,6,8,11) | | 3 | (0,3,9,13); (0,3,9,13); (1,3,10,11); (2,3,7,13); (2,3,10,12); (3,5,8,9); (3,6,7,10) | | 4 | (0,4,10,13); (0,4,10,13); (1,4,9,12); (2,4,7,13); (2,4,9,11); (4,5,7,10); (4,6,8,9) | | 5 | (0,5,7,13); (0,5,9,12); (1,5,7,12); (2,5,8,11); (3,5,7,10); (4,5,8,9); (4,5,8,10) | | 6 | (0,6,7,13); (0,6,9,11); (1,6,8,12); (2,6,8,12); (3,6,7,10); (4,6,7,9); (4,6,8,10) | | 7 | (0,6,7,13); (1,3,7,13); (1,5,7,11); (2,5,7,12); (2,6,7,11); (3,5,7,9); (4,6,7,10) | | 8 | (0,6,8,13); (1,5,8,11); (1,5,8,12); (2,4,8,13); (2,6,8,11); (3,5,8,9); (3,6,8,10) | | 9 | (0,6,9,12); (1,4,9,11); (1,4,9,11); (2,4,9,12); (3,5,7,9); (3,5,8,9); (3,6,8,9) | | 10 | (0,6,10,12); (1,3,10,11); (1,3,10,11); (2,3,10,12); (4,5,8,10); (4,5,8,10); (4,6,7,10) | | 11 | (0,2,11,13); (0,2,11,13); (0,6,10,11); (1,3,10,11); (1,3,10,11); (1,5,8,11); (2,4,9,11) | | 12 | (0,1,12,13); (0,1,12,13); (0,6,10,12); (1,5,8,12); (2,3,10,12); (2,3,10,12); (2,4,9,12) | | 13 | (0,2,12,13); (0,3,10,13); (0,3,10,13); (1,3,7,13); (1,4,8,13); (1,4,8,13); (2,4,8,13) |

Seed s = 9

| i | b_i | w_i | |---|---|---| | 0 | (-1, -1, -1) | (-1, -2, 0) | | 1 | (-1, -1, -1) | (0, -1, -2) | | 2 | (-1, -1, 1) | (-2, 0, 1) | | 3 | (-1, -1, 1) | (-1, -1, 1) | | 4 | (-1, 1, -1) | (0, 2, -1) | | 5 | (-1, 1, -1) | (-1, 0, -2) | | 6 | (-1, 1, 1) | (-1, 1, 1) | | 7 | (-1, 1, 1) | (-2, 0, 1) | | 8 | (-1, 1, 1) | (-1, 1, 1) | | 9 | (1, -1, -1) | (1, -1, -1) | | 10 | (1, -1, -1) | (1, -1, -1) | | 11 | (1, -1, -1) | (0, -2, -1) | | 12 | (1, -1, 1) | (1, -1, 1) | | 13 | (1, -1, 1) | (1, 0, 2) | | 14 | (1, 1, -1) | (0, 2, -1) | | 15 | (1, 1, -1) | (0, 3, 0) | | 16 | (1, 1, 1) | (3, 0, 0) | | 17 | (1, 1, 1) | (2, 0, 1) |

| i | Four-subsets in F_i, with repetitions | |---|---| | 0 | (0,2,15,16); (0,3,15,17); (0,4,13,17); (0,4,13,17); (0,6,10,16); (0,6,13,15); (0,7,9,16); (0,8,12,14); (0,8,12,14) | | 1 | (1,2,14,16); (1,3,15,17); (1,4,13,17); (1,4,13,17); (1,6,12,14); (1,6,13,15); (1,7,10,16); (1,8,9,16); (1,8,12,15) | | 2 | (1,2,14,17); (1,2,15,16); (2,4,10,16); (2,4,10,16); (2,4,13,15); (2,5,9,17); (2,5,12,14); (2,6,9,14); (2,8,11,15) | | 3 | (1,3,15,16); (1,3,15,16); (3,4,9,17); (3,4,9,17); (3,4,12,14); (3,5,11,16); (3,5,13,14); (3,6,10,15); (3,8,10,14) | | 4 | (0,4,12,17); (0,4,13,16); (1,4,13,17); (2,4,10,17); (3,4,9,16); (3,4,9,16); (4,6,11,13); (4,7,10,12); (4,8,11,12) | | 5 | (0,5,12,16); (0,5,12,16); (1,5,13,17); (2,5,10,17); (3,5,9,16); (3,5,11,17); (5,6,11,13); (5,7,9,12); (5,8,10,13) | | 6 | (0,6,11,16); (0,6,11,16); (1,6,9,16); (1,6,10,17); (1,6,10,17); (3,6,9,14); (4,6,9,12); (5,6,10,12); (5,6,11,13) | | 7 | (0,7,11,16); (0,7,11,16); (1,7,9,16); (1,7,9,17); (1,7,9,17); (3,7,11,14); (4,7,10,12); (5,7,10,12); (5,7,10,13) | | 8 | (0,8,11,16); (0,8,11,17); (1,8,10,16); (1,8,10,16); (1,8,11,17); (3,8,9,14); (4,8,10,13); (5,8,9,12); (5,8,9,12) | | 9 | (0,6,9,17); (2,6,9,15); (2,7,9,14); (2,8,9,15); (3,6,9,14); (3,8,9,15); (4,7,9,12); (4,7,9,13); (5,8,9,13) | | 10 | (0,8,10,17); (2,6,10,15); (2,6,10,15); (2,7,10,14); (3,6,10,14); (3,7,10,15); (4,7,10,12); (4,8,10,13); (5,8,10,13) | | 11 | (0,6,11,17); (2,8,11,15); (2,8,11,15); (2,8,11,15); (3,6,11,14); (3,6,11,14); (4,7,11,13); (4,7,11,13); (5,7,11,12) | | 12 | (0,6,12,15); (1,6,12,15); (1,7,12,14); (2,5,12,14); (2,5,12,14); (3,4,12,15); (4,7,11,12); (4,8,9,12); (5,8,10,12) | | 13 | (0,7,13,15); (1,6,13,15); (1,6,13,15); (2,4,13,14); (2,4,13,14); (3,5,13,14); (4,8,11,13); (5,7,9,13); (5,8,10,13) | | 14 | (0,2,14,17); (0,6,13,14); (0,6,13,14); (1,8,12,14); (2,5,12,14); (2,7,11,14); (3,7,9,14); (3,7,10,14); (3,8,11,14) | | 15 | (0,2,15,17); (0,7,13,15); (0,7,13,15); (1,8,12,15); (2,6,9,15); (2,7,11,15); (3,5,12,15); (3,6,11,15); (3,8,10,15) | | 16 | (0,3,14,16); (0,7,9,16); (0,8,11,16); (1,5,12,16); (1,6,10,16); (1,7,11,16); (2,4,11,16); (2,5,9,16); (3,5,10,16) | | 17 | (0,3,14,17); (0,6,11,17); (0,8,11,17); (1,4,12,17); (1,7,10,17); (1,7,11,17); (2,5,9,17); (2,5,9,17); (3,5,10,17) |

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