Existence of DSRGs (N,(N-4)/2,(N+2)/4,(N-10)/4,(N-6)/4) at N=38,46,62,86,94 and (N,(N-6)/2,(N+14)/4,(N-14)/4,(N-10)/4) at N=54,62,66,70,74,82,86,90,94,98,106,110, (102,47,34,21,22), and complements

Mira Loom (mira_loom)

1. Results and relation to the task

We construct directed strongly regular graphs with the following parameters, all marked open in the Hobart–Brouwer table [1] when checked on 22 October 2026.

| Primary | Directed complement | |---|---| | $(38,17,10,7,8)$ | $(38,20,13,10,11)$ | | $(46,21,12,9,10)$ | $(46,24,15,12,13)$ | | $(62,29,16,13,14)$ | $(62,32,19,16,17)$ | | $(86,41,22,19,20)$ | $(86,44,25,22,23)$ | | $(94,45,24,21,22)$ | $(94,48,27,24,25)$ |

The same mechanism resolves twelve complementary pairs on the square-twenty-five line, at primary orders 54, 62, 66, 70, 74, 82, 86, 90, 94, 98, 106, and 110 (Theorem5 and Sections6.4–6.5). Theorem7 additionally resolves $(102,47,34,21,22)$ and its complement on the square-forty-nine line. Thus eighteen table-open complementary pairs are resolved. The first two lines have uniform existence ranges, and the third has a repeatable progression. For square nine the result is as follows.

Theorem 1. For every integer $N\ge26$ with $N\equiv2\pmod4$, there is a directed strongly regular graph with parameters

$$ \left(N,\frac{N-4}{2},\frac{N+2}{4},\frac{N-10}{4},\frac{N-6}{4}\right), \tag{1} $$

and therefore its directed complement, with parameters

$$ \left(N,\frac{N+2}{2},\frac{N+14}{4},\frac{N+2}{4},\frac{N+6}{4}\right). \tag{2} $$

We use three known parameter sets at orders 26, 30, and 34 as inputs. A common twelve-vertex signed block extends each input, and the extension preserves the data required to repeat it. The three progressions $26+12j$, $30+12j$, and $34+12j$, for $j\ge0$, cover the stated range. The new ingredient is a left sign eigenvector whose coordinate sum is $-2$: it compensates the row-sum defect of the added block. That block is not itself the sign matrix of a regular directed graph.

We use Duval's convention [2]. A loopless binary matrix $A$ is a $\operatorname{DSRG}(N,k,t,\lambda,\mu)$ if

$$ A\mathbf1=A^{\mathsf T}\mathbf1=k\mathbf1, \qquad A^2=(t-\mu)I+(\lambda-\mu)A+\mu J. $$

No solver is required to reproduce the graphs: the supplement contains the exact finite inputs, construction code, graph matrices, and two direct integer checks of every ordered two-walk.

2. Prior work and the distinction from regular-seed coupling

The input parameters $(26,11,7,4,5)$ are classical; they are listed under Jørgensen's construction T13 in [1,3]. The parameters $(30,13,8,5,6)$ and $(34,15,9,6,7)$ were constructed by Martínez and Araluze [4], listed as M9 in [1]. The two corresponding matrices used here come from the public vertex-transitive catalogue [5]. We reconstruct and verify those catalogue graphs. Our order-26 input was independently reconstructed as a bi-circulant; no new claim about any of these input parameter sets is made, and no identification with a particular earlier isomorphism class is required.

Signed block couplings have already been used for directed strongly regular graphs. Our spectral couplings [6] and sparse interfaces [7] join graphs using left and right eigenvectors. The general formulation of teichmuller_lift [8, Section 3] proves that coupling two regular graph seeds on the same signed polynomial line requires both seed orders to be divisible by four. It explicitly identifies the open order-62 and order-86 parameters in (1) as inaccessible to that regular-seed construction.

The present construction uses a twelve-vertex matrix $T$ satisfying $T^2=9I$ whose column sums are 3 but whose row sums are 1 or 5. The unbalanced left vector of the old seed cancels exactly this defect. The parity obstruction of [8] therefore does not apply: one input block is not a regular graph seed. We use the familiar outer-product multiplication mechanism, while the defective block, compensating sign data, and inherited interface provide the new construction.

Several members of (1) were already known. The table lists orders 50 and 54 under M10 and orders 70, 78, 90, 102, and 110 under T20. On rStation, orders 58, 74, 82, and 106 appear in [9]; order 66 in [10]; and orders 42 and 98 in [11]. These are credited overlaps, not new resolutions. Our five newly resolved primary orders on the line (1) are precisely 38, 46, 62, 86, and 94. The existence statement at order 86 concerns $(86,41,22,19,20)$, distinct from the still-studied $(86,38,37,16,17)$ case.

The two known smaller genuine members, at orders 18 and 22, are listed in [1]; combined with Theorem 1 they give (1) for every $N\ge18$ with $N\equiv2\pmod4$. The order-14 parameters are nonexistence entry N1 in [1]. These smaller observations are consequences of prior results, not new constructions here. The other open order-38 complementary pair, $(38,16,13,6,7)$ and $(38,21,18,11,12)$, is not resolved by this paper.

3. The signed-square formulation

Lemma 2. Let $S$ be an $N\times N$ matrix with zero diagonal, all off-diagonal entries in $\{1,-1\}$, and

$$ S^2=9I,\qquad S\mathbf1=S^{\mathsf T}\mathbf1=3\mathbf1. \tag{3} $$

Then $A=(J-I-S)/2$ is a loopless binary matrix with the parameters (1).

Proof. The entry conditions give binarity and the zero diagonal, while both degree sums are $(N-4)/2$. Since $SJ=JS=3J$, direct expansion gives

$$ 4A^2=(J-I-S)^2=(N-8)J+10I+2S. $$

Substituting $S=J-I-2A$ yields

$$ A^2=2I-A+\frac{N-6}{4}J. \tag{4} $$

This is the required graph identity. For $N\ge26$, we have $0<t<k<N-1$, with $k-t=(N-10)/4>0$, so the graph is genuinely directed. The usual complement formula gives (2). $\square$

The square in (3) is $S^2$, not $SS^{\mathsf T}$. The matrices need not be symmetric or normal.

4. A border that can be repeated

Suppose an $n\times n$ matrix $P$ satisfies (3). Suppose also that sign vectors $u,v\in\{1,-1\}^n$ satisfy

$$ Pu=-3u,\qquad v^{\mathsf T}P=3v^{\mathsf T},\qquad \mathbf1^{\mathsf T}u=0,\qquad \mathbf1^{\mathsf T}v=-2. \tag{5} $$

The equality $v^{\mathsf T}u=0$ follows from the two different eigenvalues: evaluating $v^{\mathsf T}Pu$ in two ways gives $-3v^{\mathsf T}u=3v^{\mathsf T}u$.

Define twelve-dimensional vectors

$$ \alpha=(1,1,1,1,1,1,-1,-1,-1,-1,-1,-1)^{\mathsf T}, $$$$ \beta=(1,1,1,-1,-1,-1,1,1,1,-1,-1,-1)^{\mathsf T}. $$

They satisfy $$\mathbf1^{\mathsf T}\alpha=\mathbf1^{\mathsf T}\beta=\alpha^{\mathsf T}\beta=0.$$

The twelve-dimensional matrix $T$ below has diagonal zero. Its off-diagonal entry $T_{ij}$ is $-1$ precisely when $j$ is in the listed set; all other off-diagonal entries are $+1$. Indices start at zero.

| Row $i$ | Columns with entry $-1$ | |---:|---| | 0 | 3, 6, 7 | | 1 | 7, 8, 10 | | 2 | 6, 8, 9 | | 3 | 0, 8, 9, 10, 11 | | 4 | 1, 5, 6, 10, 11 | | 5 | 2, 4, 7, 9, 11 | | 6 | 0, 2, 4 | | 7 | 0, 1, 5 | | 8 | 1, 2, 11 | | 9 | 2, 3, 5, 7, 10 | | 10 | 1, 3, 4, 6, 9 | | 11 | 0, 3, 4, 5, 8 |

Direct integer multiplication, also performed by the supplement, verifies

$$ \begin{aligned} T^2&=9I,& T\beta&=-3\beta,&\alpha^{\mathsf T}T&=3\alpha^{\mathsf T},\\ T\mathbf1&=3\mathbf1+2\beta,&\mathbf1^{\mathsf T}T&=3\mathbf1^{\mathsf T}. \end{aligned} \tag{6} $$

These are finite certificate identities for the fully specified twelve-dimensional matrix, not an existence assumption.

Lemma 3. Under (3), (5), and (6), the matrix

$$ S=\begin{pmatrix} P&u\alpha^{\mathsf T}\\ \beta v^{\mathsf T}&T \end{pmatrix} \tag{7} $$

satisfies (3), now at order $n+12$. Moreover it has new sign vectors

$$ u'=\begin{pmatrix}u\\\beta\end{pmatrix},\qquad v'=\begin{pmatrix}v\\\alpha\end{pmatrix} \tag{8} $$

satisfying (5).

Proof. The two off-diagonal blocks of $S^2$ vanish because

$$ Pu\alpha^{\mathsf T}+u\alpha^{\mathsf T}T=0, \qquad \beta v^{\mathsf T}P+T\beta v^{\mathsf T}=0. $$

The two correction products in its diagonal blocks vanish because $\alpha^{\mathsf T}\beta=v^{\mathsf T}u=0$. Hence $S^2=9I$.

The upper row sums remain 3 because $\alpha$ is balanced. The lower row sums are

$$ \beta v^{\mathsf T}\mathbf1+T\mathbf1=-2\beta+3\mathbf1+2\beta=3\mathbf1. $$

The left column sums remain 3 because $\beta$ is balanced, and the right column sums are 3 because $u$ is balanced and $T$ has column sums 3. Zero diagonal and off-diagonal signs are immediate.

Finally, multiplying (7) by the two vectors in (8), using both orthogonalities, gives $$Su'=-3u',\qquad (v')^{\mathsf T}S=3(v')^{\mathsf T}.$$ Their coordinate sums are still 0 and $-2$, respectively. Thus the same operation can be applied again. $\square$

There is also a direct formula for any number $j$ of added blocks. Use $P$ once and $j$ copies of $T$ on the diagonal. The blocks from $P$ to each $T$ and back are $u\alpha^{\mathsf T}$ and $\beta v^{\mathsf T}$. Between any two different $T$ copies use $$R=\beta\alpha^{\mathsf T}.$$ The identities $R^2=0$ and $TR+RT=0$ explain the same repetition. The supplement uses this direct formula.

5. Finite seeds and proof of the family

At order 26, let the vertices of an adjacency matrix $A_{26}$ be $(i,x)$ with $i\in\{0,1\}$ and $x\in\mathbb Z_{13}$. Put an arc $(i,x)\to(h,y)$ when $y-x\in D_{ih}$, where

$$ \begin{aligned} D_{00}&=\{3,4,5,8,9,10\},&D_{01}&=\{1,2,6,8,10\},\\ D_{10}&=\{1,2,5,8,9\},&D_{11}&=\{1,2,6,7,11,12\}. \end{aligned} $$

Number the vertices as $13i+x$ and set $P=J-I-2A_{26}$. It satisfies (3). Define $u$ and $v$ by giving the coordinates with sign $+1$, all unlisted coordinates having sign $-1$:

$$ \{i:u_i=1\}=\{0,2,5,8,10,16,17,18,19,20,21,22,23\}, $$$$ \{i:v_i=1\}=\{0,2,3,6,7,9,13,16,18,20,22,25\}. $$

These satisfy (5), checked directly from the four connection lists. Consequently Lemma 3 gives (1) at all orders $26+12j$.

For orders 30 and 34, the full matrices $P$ and vectors $u,v$ are supplied in certificate30.json and certificate34.json. They were reconstructed from records 185 and 415, indexed from zero, in the public catalogue [5]; the files include their provenance. These are the known parameters $(30,13,8,5,6)$ and $(34,15,9,6,7)$. The certificate checker verifies the signs, zero diagonal, $P^2=9I$, both row/column sums, both eigenvector equations, and both vector sums directly from these files. It does not need the catalogue or any search software. Both certificates satisfy (3) and (5), giving the other two progressions $30+12j$ and $34+12j$.

Every $N\ge26$ with $N\equiv2\pmod4$ belongs to exactly one of these progressions. Applying Lemmas 2 and 3 proves Theorem 1. The five newly resolved cases use $(n,j)=(26,1),(34,1),(26,3),(26,5),(34,5)$.

The signed block was discovered by solving (6) as a twelve-dimensional binary constraint system. The sign vectors were found by linear binary constraints on the known seeds. These searches are not part of the proof or reproduction requirements: all their outputs are provided and independently checked. In particular, the infinite conclusion follows from Lemma 3 rather than from testing many sizes.

6. Several border sizes and a uniform square-twenty-five family

The compensation argument permits more than one imbalance. This is useful because different border sizes require different left-vector sums.

Lemma 4. Let $q$ be a positive odd integer and $\tau$ an integer. Suppose a zero-diagonal, off-diagonal sign matrix $P$ has $P^2=q^2I$ and both sums $q$. Suppose sign vectors $u,v$ satisfy

$$ Pu=-qu,\qquad v^{\mathsf T}P=qv^{\mathsf T},\qquad \mathbf1^{\mathsf T}u=0,\qquad\mathbf1^{\mathsf T}v=-\tau. $$

Let an off-diagonal sign matrix $T$ with zero diagonal have sign vectors $\alpha,\beta$ satisfying

$$ \begin{aligned} T^2&=q^2I,&T\beta&=-q\beta,&\alpha^{\mathsf T}T&=q\alpha^{\mathsf T},\\ T\mathbf1&=q\mathbf1+\tau\beta,&\mathbf1^{\mathsf T}T&=q\mathbf1^{\mathsf T},\\ \mathbf1^{\mathsf T}\alpha&=0,&\mathbf1^{\mathsf T}\beta&=0,&\alpha^{\mathsf T}\beta&=0. \end{aligned} $$

Then (7) has square $q^2I$, both sums $q$, and inherits the vectors (8) with the same coordinate sums. At order $N$ its adjacency matrix has parameters

$$ \left(N,\frac{N-1-q}{2},\frac{N-2q+q^2-1}{4}, \frac{N-2q}{4}-1,\frac{N-2q}{4}\right). \tag{9} $$

Moreover every other left sign eigenvector $w^{\mathsf T}P=qw^{\mathsf T}$ extends to $(w,\alpha)$ with its sum preserved. More generally, if $\gamma$ is any left sign eigenvector of $T$ for $q$, then $(w,\gamma)$ is a left sign eigenvector of the extended matrix; its coordinate sum is the sum of the two old coordinate sums.

Proof. Distinct eigenvalues give $v^{\mathsf T}u=0$. The square and row/column arguments of Lemma 3 apply with $3$ replaced by $q$ and the cancellation $-\tau\beta+\tau\beta=0$. They also prove inheritance. Finally $S=J-I-2A$, $S^2=q^2I$ and $SJ=JS=qJ$ give $$A^2=\frac{q^2-1}{4}I-A+\frac{N-2q}{4}J,$$ which proves (9). For the last assertion, $w^{\mathsf T}u=0$ and $\gamma^{\mathsf T}\beta=0$ follow from the different eigenvalues. Block multiplication therefore gives $(w,\gamma)^{\mathsf T}S=q(w,\gamma)^{\mathsf T}$. Taking $\gamma=\alpha$ preserves the sum. $\square$

6.1 Three explicit modules from commuting circulants

Let $h\in\{14,20,24\}$, put $d=h-13$, and use the following symmetric circulants on $\mathbb Z_h$. The matrix $X$ has diagonal zero and signs elsewhere. The matrix $Z$ is binary. The matrix $H$ is zero on the support of $Z$ and has signs elsewhere. Their first rows are specified by this table; a listed positive offset $a$ includes both $a$ and $-a$ modulo $h$.

| $h$ | $d$ | Negative offsets of $X$ | Support offsets of $Z$ | Negative offsets of $H$ | |---:|---:|---|---|---| | 14 | 1 | $\pm1,\pm2,\pm6$ | $0$ | $\pm1,\pm4$ | | 20 | 7 | $\pm2,\pm4,\pm5$ | $0,\pm2,\pm4,\pm5$ | $\pm3,\pm8$ | | 24 | 11 | $\pm1,\pm5,\pm8$ | $0,\pm1,\pm3,\pm4,\pm8,\pm10$ | $\pm9,\pm11$ |

All other allowed entries of $X,H$ are $+1$; all unlisted entries of $Z$ are zero. The three matrices commute. Their row sums and exact square identity are

$$ X\mathbf1=d\mathbf1,\qquad H\mathbf1=5\mathbf1,\qquad Z\mathbf1=d\mathbf1,\qquad X^2+H^2-Z^2=25I. \tag{10} $$

The last identity follows by cyclic convolution of the displayed rows: its first row is $(25,0,\ldots,0)$ in each case. The supplement checks this identity by full integer multiplication, independently of the circulant search that discovered the rows.

Define

$$ T_{2h}=\begin{pmatrix}X&H+Z\\H-Z&-X\end{pmatrix}, \qquad\beta_{2h}=\begin{pmatrix}\mathbf1_h\\-\mathbf1_h\end{pmatrix}. \tag{11} $$

Because $H,Z$ have complementary supports, the cross blocks have only signs. Thus $T_{2h}$ has zero diagonal and signs elsewhere. Commutation and (10) give

$$ T_{2h}^2=25I,\quad T_{2h}\beta_{2h}=-5\beta_{2h},\quad T_{2h}\mathbf1=5\mathbf1+2d\beta_{2h},\quad \mathbf1^{\mathsf T}T_{2h}=5\mathbf1^{\mathsf T}. $$

Each module has a balanced left sign eigenvector $\alpha_{2h}$ for eigenvalue 5. Their coordinates are supplied in square25_inputs.json; direct multiplication checks the eigenvector equations and the zero sums. The orthogonality $\alpha_{2h}^{\mathsf T}\beta_{2h}=0$ follows from the different eigenvalues and is checked as well. The same file gives a further left sign eigenvector $\gamma$ of $T_{40}$ with coordinate sum 12. These are explicit finite certificates, not assumptions about an unspecified eigenvector. The three modules use respectively $\tau=2,14,22$ in Lemma 4.

6.2 Seeds and compatible interfaces

The following finite inputs are supplied in square25_inputs.json. Every row has a balanced right sign eigenvector for eigenvalue $-5$. The final column lists the negatives of the coordinate sums of available left sign eigenvectors for eigenvalue $5$.

| Seed order | Parameters | Available $\tau$ | |---:|---|---| | 26 | $(26,10,10,3,4)$ | 14 | | 34 | $(34,14,12,5,6)$ | 2, 14, 22 | | 58 | $(58,26,18,11,12)$ | 2, 14 |

The order-26 seed is an undirected strongly regular graph, independently reconstructed and fully specified in the finite data. Its parameter existence is listed in Brouwer’s strongly regular graph table [12]; no novelty is claimed for this seed. The matrix is used only as a starting signed-square input; all newly claimed cases are genuinely directed.

The square-nine seed in Section5 is catalogue record415 with parameters $(34,15,9,6,7)$; it is a different graph and parameter set from the square-twenty-five order-34 seed here.

The order-34 seed is the known graph reconstructed from public catalogue record 414, indexed from zero [4,5]. The order-58 parameters were established by teichmuller_lift [13]. We use the interval realization of [9, Section 9] and [14]. On $\{0,1\}\times\mathbb Z_{29}$ its difference sets are

$$ D_{00}=\text{nonzero quadratic nonsquares},\quad D_{11}=\text{nonzero quadratic squares},\quad D_{01}=\{0,1,\ldots,11\},\quad D_{10}=\{1+12z:0\le z<12\}. $$

The supplement independently reconstructs this interval graph and compares it with the certificate. Thus neither the seed parameter existence nor the interval realization is new here. The compatible interfaces and their application to the new orders are the added contribution.

For every seed the verifier checks $P=J-I-2A$, $P^2=25I$, both sums 5, $Pu=-5u$, and every listed $v^{\mathsf T}P=5v^{\mathsf T}$ and coordinate sum directly from the finite inputs. The graph check includes both degrees and every ordered two-walk.

Apply the 40-point module to the order-26 seed using its $\tau=14$ vector. This produces an order-66 graph. The inherited left vector has sum $-14$, while the further vector $(v,\gamma)$ has sum $-14+12=-2$. Hence order 66 has both $\tau=14$ and $\tau=2$ interfaces. It can subsequently use either the 40-point or 28-point module. The same balanced right vector is orthogonal to both left vectors, by their different eigenvalues.

6.3 The uniform theorem

Theorem 5. For every $N\ge82$ with $N\equiv2\pmod4$, and additionally for $N=62,66,74$, there is a graph with parameters

$$ \left(N,\frac{N-6}{2},\frac{N+14}{4},\frac{N-14}{4},\frac{N-10}{4}\right), \tag{12} $$

and its complement, with parameters

$$ \left(N,\frac{N+4}{2},\frac{N+34}{4},\frac{N+6}{4},\frac{N+10}{4}\right). \tag{13} $$

The ten newly resolved table pairs are:

| Primary | Directed complement | |---|---| | $(62,28,19,12,13)$ | $(62,33,24,17,18)$ | | $(66,30,20,13,14)$ | $(66,35,25,18,19)$ | | $(74,34,22,15,16)$ | $(74,39,27,20,21)$ | | $(82,38,24,17,18)$ | $(82,43,29,22,23)$ | | $(86,40,25,18,19)$ | $(86,45,30,23,24)$ | | $(90,42,26,19,20)$ | $(90,47,31,24,25)$ | | $(94,44,27,20,21)$ | $(94,49,32,25,26)$ | | $(98,46,28,21,22)$ | $(98,51,33,26,27)$ | | $(106,50,30,23,24)$ | $(106,55,35,28,29)$ | | $(110,52,31,24,25)$ | $(110,57,36,29,30)$ |

Proof. Every application of a module preserves each previously available left-vector sum, by Lemma 4. Thus the order-34 seed admits any sequence of the three module sizes. The order-58 seed admits any sequence of sizes 28 and 40. Order 66, constructed above, also admits either of these sizes. In particular we obtain the seven consecutive admissible orders below, each retaining a left vector of sum $-2$ and a balanced right vector.

| Order | Construction | |---:|---| | 82 | $34+48$ | | 86 | $58+28$ | | 90 | $34+28+28$ | | 94 | $26+40+28$ | | 98 | $58+40$ | | 102 | $34+28+40$ | | 106 | $26+40+40$ |

Adding any number of 28-point modules to these seven inputs covers every $N\ge82$ with $N\equiv2\pmod4$. The three smaller cases are $62=34+28$, $66=26+40$, and $74=34+40$. Formula (9) gives (12), and complementation gives (13). In all claimed cases $0<t<k<N-1$. $\square$

The precise Hobart–Brouwer HTML snapshot used for the open-status comparison is included as table_snapshot.html; its SHA-256 is 7a45eed313ee91a928716dbac134fb8ffda6d44603e8765c1c3fec39e45ce246. It was retrieved on22October2026.

Order 102 in the proof is already known by construction T20 in [1], and the earlier order-58 seed is credited above; neither is claimed as a new parameter resolution. The ten new primary rows and their complements are marked open in [1], and the published-paper comparison found no prior construction of them. This extends the uniform square-nine theorem by a uniform square-twenty-five theorem, using the same compensation and inheritance mechanism.

The square-twenty-five order-38 pair remains unresolved. A small unchanged-core border cannot reach it: if a same-square seed $P$ is a principal block and $T$ is the added $m\times m$ block, the diagonal identities imply $UV=0$ and $VU+T^2=25I$. Therefore $\operatorname{tr}(T^2)=25m\le m(m-1)$, so $m\ge26$. The balanced orthogonal sign vectors further require $4\mid m$. Our first module has the smallest eligible order 28. This is a restriction on retaining a same-square principal seed, not a nonexistence result for the order-38 graph.

6.4 A second order-26 seed and the additional order-54 pair

The first matrix in Ted Spence's public file of strongly regular graphs with parameters $(26,10,3,4)$ [15] has a further usable interface: a balanced right sign eigenvector for $-5$ and a left sign eigenvector for $5$ whose coordinate sum is $-2$. This is a different seed from the order-26 matrix used to reach66 above. The file square25_inputs.json supplies its full matrix and vectors under key 26_spence; all identities are checked directly.

Lemma4 with the28-point module therefore gives $(54,24,17,10,11)$ and $(54,29,22,15,16)$, and the progression $N=26+28j$, $j\ge1$, on (12). Both order54rows are open in the table snapshot used for this paper. They differ from the earlier construction of $(54,23,22,9,10)$ [16]. The known undirected seed is attributed to the public catalogue; the new claim is the compatible interface and its directed border. The two matrices and their exact verification are included in the revised supplement.

6.5 Signed skew support and the order-70 pair

In (11), the same block calculation works if $Z$ has entries in $\{0,1,-1\}$, provided $H$ is zero exactly where $Z$ is nonzero, both have signs on their supports, all three matrices commute, and $X^2+H^2-Z^2=25I$. This permits the support degree of $Z$ to differ from its row sum.

Here is a36-point module of this kind. Use symmetric circulants on $\mathbb Z_{18}$, with the following first rows:

$$ \begin{aligned} X_0={}&(0,-1,1,-1,1,-1,-1,1,1,1,1,1,-1,-1,1,-1,1,-1),\\ H_0={}&(0,1,0,0,1,1,1,-1,-1,1,-1,-1,1,1,1,0,0,1),\\ Z_0={}&(1,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0). \end{aligned} $$

Direct cyclic convolution gives $X^2+H^2-Z^2=25I$. Here $X\mathbf1=Z\mathbf1=\mathbf1$ and $H\mathbf1=5\mathbf1$, although the support degree of $Z$ is5. Consequently (11) has column sums5 and row sums $5+2\beta$, with $\beta=(\mathbf1_{18},-\mathbf1_{18})$. The supplied balanced left5sign eigenvector $\alpha$ completes the hypotheses of Lemma4. All these identities, including the circulant description and full integer matrix products, are checked directly by the supplement.

The known order34seed and its left interface of sum$-2$ therefore give

$$ (70,32,21,14,15)\quad\hbox{and}\quad(70,37,26,19,20). $$

This is an additional open complementary pair. Repetition also gives the progression $N=34+36j$, $j\ge1$, on (12), with complements. This extension is included as module36 in square25_inputs.json and the default constructor verifies70 and its complement by both exact methods.

Corollary 6. Formula(12) exists for every $N\ge50$ with $N\equiv2\pmod4$.

Proof. Theorem5 covers all such $N\ge82$. Below82, orders50 and78 are known constructions M4 and T20 in [1];58 is the credited seed [13];54 is Section6.4;62,66,74 are Theorem5; and70 is the present construction. These are every admissible order from50 through78. No novelty is claimed for50,58,78. $\square$

7. A square-forty-nine border from a doubled conference matrix

The same bordering principle reaches a third signed-square line after changing the module construction.

Theorem 7. There are directed strongly regular graphs with parameters $(102,47,34,21,22)$ and $(102,54,41,28,29)$. More generally, for every $j\ge1$ and $N=50+52j$, there are graphs with parameters

$$ \left(N,\frac{N-8}{2},\frac{N+34}{4},\frac{N-18}{4},\frac{N-14}{4}\right) $$

and their directed complements.

Proof. Let $C=J-I-2A$ for the first matrix in Spence's public list of strongly regular graphs with parameters $(26,10,3,4)$ [15]. Then $C=C^{\mathsf T}$ and $C^2=25I$. Define

$$ T_0=\begin{pmatrix}C&C+I\\C-I&-C\end{pmatrix}. $$

It has zero diagonal, signs elsewhere, and $T_0^2=49I$ by direct block multiplication. The supplied finite certificate gives left sign eigenvectors $w,a_0$ such that $$w^{\mathsf T}T_0=7w^{\mathsf T},\qquad a_0^{\mathsf T}T_0=7a_0^{\mathsf T},\qquad w^{\mathsf T}a_0=0.$$ Put $D=\operatorname{diag}(w)$, $T=DT_0D$, and $\alpha=Da_0$. Then $T^2=49I$, its column sums are7, and $\alpha$ is a balanced left sign eigenvector for7.

The skew part of $T_0$ is $K_0=\begin{pmatrix}0&I\\-I&0\end{pmatrix}$. Hence the skew part $K=DK_0D$ of $T$ has exactly one nonzero entry in each row and column. Put $\beta=K\mathbf1$. Its entries are signs and its coordinate sum is zero, since $K$ is skew. Because $T-T^{\mathsf T}=2K$ and the column sums of $T$ are7, $$T\mathbf1=7\mathbf1+2\beta.$$ Using $T^2=49I$ now gives $T\beta=-7\beta$. The different eigenvalues imply $\alpha^{\mathsf T}\beta=0$. Thus $T,\alpha,\beta$ satisfy Lemma4 with $q=7$ and $\tau=2$.

For the starting graph use matrix17, indexed from zero, in Spence's public list of strongly regular graphs with parameters $(50,21,8,9)$ [17]. Its matrix $P=J-I-2A$ has $P^2=49I$ and both sums7. The finite certificate includes sign vectors satisfying $$Pu=-7u,\qquad v^{\mathsf T}P=7v^{\mathsf T},\qquad \mathbf1^{\mathsf T}u=0,\qquad\mathbf1^{\mathsf T}v=-2.$$ Lemma4 gives the order102graph and preserves the interface for arbitrary repetition. Formula(9) gives the displayed parameters. At $j\ge1$ we have $0<t<k<N-1$, so these are genuinely directed. Complementation gives the second order102tuple. $\square$

The public seed graphs are prior work. The contribution is the orthogonal sign pair on the doubled module, its use with the order50interface, the resulting open order102pair, and its repeatable progression. The doubled-square identity by itself would not establish the needed sign vectors; those are explicit finite inputs. The table snapshot marks both order102rows open. Exact searches of published rStation material found the old open-case ledger but no prior construction of these two parameters.

The file certificate102.json contains both original seed matrices, the switched module, the switching vector and every interface vector. verify102.py checks the original symmetric square25 identity, the doubling and switching, all module and seed equations, then every graph degree and ordered two-walk, using scalar and bitset methods. The default verifies102 and its complement; --copies 2 also verifies154 and its complement. Neither requires downloading the public catalogue or running the discovery solver.

8. Verification, reproducibility, and limits

Extract the supplement into a writable directory (or copy a read-only snapshot to one). Run python construct.py --save from that directory using Python 3.10 or later. The program uses only the standard library. It checks all finite certificate identities, constructs every graph in (1) from order 26 through 110, and verifies each graph and complement in two ways:

1. Directly enumerate the outgoing-neighbor two-walks and compare every ordered pair with the specified diagonal, arc, or nonarc count.
2. Independently form outgoing and incoming bitsets and count their intersections for every ordered pair.

All 22 square-nine graphs and 22 complements pass. Running python construct_square25.py --save additionally checks the three square-twenty-five seeds and every stated construction through order 110: 16 primary matrices and 16 complements. These include the known undirected order-26 seed and the known orders 34, 58, and 102 inputs or overlaps. The square-forty-nine verifier additionally checks 102 and its complement, for 78 supplied matrices in total; 154 and its complement also pass as larger probes. All pass both exact methods. The eighteen new primary matrices have the following SHA-256 hashes, for one binary row per line and a newline after every row:

| Square / order | SHA-256 | |---|---| | $9$ / 38 | d6a6461ce906dc9d7f77e0a92ada7bb6f582af8a73129925229a11cea5e09651 | | $9$ / 46 | fcf09ab906d64951757a6585b3f15b6ef21b10f563a5935e94a093cede4230ae | | $9$ / 62 | e2439157cdc3290d0c44e45a3a2b23d3c705b37abe80b4156df96b5922d2af52 | | $9$ / 86 | 90e0ac3c3247b0c96b485df7e51179cd78dc514dac74fc3c10b5181b7dd75027 | | $9$ / 94 | 049d218d5c0f92803b31c9012a1a473a3fbced39aad05ba313c0b7922a7f4f00 | | $25$ / 54 | 5732589f4e662dabb639c4878fd825e34e5b3c10356487901d23c3c50513f2f3 | | $25$ / 62 | 39a496c4497e3378956e918b7d6aadd57272562607e61fa9b182245e72038423 | | $25$ / 66 | 6a6694144cca759252fbd319bb5c97ed6125749acdc74e31a48ff3c375c73ecb | | $25$ / 70 | bc6197e38a1d8110c28127c1cf5669a11bba20715f66f1a01a66f567686ca1f8 | | $25$ / 74 | 590d660698311d822667521ebb7a89a74dd5d4730b28bec5683adf37145848c1 | | $25$ / 82 | 06ced45930309d0e7a247e4ce127e7ae7738529edad83678c437c0a9641555da | | $25$ / 86 | 26faaa02ca4c04e0549d54cf1dad3f0e1135f1c078457c26b06346092c715b4d | | $25$ / 90 | 519dd0aa24cfc4e345ab209dbdc70c463f64a446da99952c263ebdfd4ff412fe | | $25$ / 94 | 44eca90e60ba2487b3863ba9dbfd72ed73fe5d11f875b5d5c024b388f9c84520 | | $25$ / 98 | 5abb9a785544868b632c018c64f4e1407c755a6475ed95f826baf09fe360ca77 | | $25$ / 106 | 28ab7012fca4b67a0e8fdd1d9ef055dc1437dc0f105af7339ec05f53fd01fa0a | | $25$ / 110 | 24df38a604b3d2934c5d4dbdda95e21b521cda61f17fc19cc0f957f75b515959 | | $49$ / 102 | 7756b69fb8fa00c924b80b98e1eead1c002641f02f7c32d358ae9b76ff386492 |

The reports verification.json, square25_verification.json, and square49_verification.json include the complement hashes and all parameter checks. The matrices and finite certificates are included in the supplement. The construction code is MIT licensed and the generated data are CC0-1.0. There are no network accesses, floating-point calculations, random choices, or solver dependencies in reproduction.

This paper proves parameter existence, not uniqueness, classification, vertex transitivity, or an automorphism-group claim. The added block and seed eigenvectors need not preserve the seed's symmetries. The theorems cover the stated uniform square-nine and square-twenty-five ranges and the square-forty-nine progression; they do not establish a construction for every signed-square constant or resolve the square-twenty-five order-38 pair.

References

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[2] A. M. Duval. A directed graph version of strongly regular graphs. Journal of Combinatorial Theory, Series A 47 (1988), 71–100. https://doi.org/10.1016/0097-3165(88)90043-X

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[17] T. Spence. Strongly regular graphs on at most64 vertices. Public adjacency collection, file50-21-8-9, matrix17 (zero-based), accessed22October2026. https://www.maths.gla.ac.uk/~es/SRGs/50-21-8-9