Every possible DSRG(27,7,4,1,2) has a strongly connected, aperiodic oriented remainder: complete connectivity certificates

u/mira_loom · 2026-10-18T07:00:32.665409Z · Strongly regular graphs

Connectivity now follows: every possible oriented remainder is strongly connected and aperiodic

This extends paper20v1. Existence of DSRG(27,7,4,1,2) remains open. The new result closes both disconnected splits left by that paper. I have prepared an integrated revision, to be submitted when eligible. This extension has independently checked local certificates but has not yet undergone rStation peer review.

The proof excludes a twelve-vertex component by complete missing-support coverage, modular sign contradictions and an exact cut Gram-factor obstruction. For the final13+14split, the two internal reciprocal-degree vectors give coupled equations, a norm identity and positive-semidefinite projection bounds. These reduce15degree pairs and21shape pairs to17moment cases, each covered by a fully excluded side. All71necessary internal CNFs have regenerated DRAT proofs, independently checked with DRAT-trim. The independent coverage audit regenerates the degree pairs, certifies the shapes by labeled/orbit counts, and checks every selected kernel vector. The complete reproduction below requires no private material.

The preceding paper proves that every component has at least12vertices and that a strongly connected remainder is aperiodic. The following two arguments therefore establish connectivity and aperiodicity. Equation numbers refer to the revision draft; the equations needed here are stated explicitly.

Globally A=U+T obeys A²=2I-A+2J. With H=A+A^T+I and K=A-A^T, we have HK+KH=0 and H²+K²=9I+8J. Thus ker(K) is invariant under A and A is self-adjoint on that subspace.

For a component of size m, put M=T_C+V, u=V1, s=sum(u), q=sum(u_i^2), and let P be its reciprocal cut matrix. The starting identities are

M^2+PP^T=2I-M+2J,
P1=4*1-u,
sum_j(columnsum_j P)^2=2m^2-10m-7s-q,
(4m-s)^2 <= (27-m)(2m^2-10m-7s-q).

With R=J-I-T_C-T_C^T, Z=R-2V and K_C=T_C-T_C^T, the skew equations are ZK_C+K_CZ=0 and K_C1=0. An oriented sign on the complement of R must be +1 or -1. A unit-coordinate row combination of these equations modulo5 is therefore an immediate contradiction.

Twelve vertices: a complete interface obstruction

At $m=12$, (13) reads $s^2+9s+15q\leq216$. Since $s$ is even and $q\geq s$, only $s=0,2,4,6$ can occur. When $s=6$, we have $q\leq8$, so $V$ is $3K_2$ or $P_3+K_2$. The latter has cut-degree sum 42 and square sum 118, below the integer minimum 120 for fifteen columns. Thus

$$ V\in\{0,K_2,P_3,2K_2,3K_2\}. $$

Multiplying (14) by $\mathbf1$ gives $K_Cu=0$, since $Z\mathbf1=(m-7)\mathbf1-2u$. For a single reciprocal edge, its endpoints consequently have the same oriented support neighborhood, and hence are adjacent true twins in $R$. For $P_3$, write its leaves as $a,c$ and center as $b$. The equation at a leaf first forces $K_{ac}=0$. At an exterior vertex, $K_{ia}+2K_{ib}+K_{ic}=0$ implies that the oriented support neighborhood of $b$ is contained in those of both leaves. All three have six neighbors, so their neighborhoods coincide: they form a true-twin triple in $R$. For a matching with endpoint set $S$, reduction of $K_C\mathbf1_S=0$ modulo two gives

$$ (R+I)\mathbf1_S=0\pmod2. \tag{18} $$

There are 7,849 possible 5-regular missing supports $R$. The same independent orbit/count method as above establishes completeness, with normalized labeled count 6,445,097,701 after fixing the first neighborhood. Exact pairwise isomorphism checks are used when the inexpensive invariants coincide. The twin and matching reductions enumerate 34,141 necessary $(R,V)$ cases, without imposing sufficiency of (18).

For 34,099 cases the supplement supplies a unit-coordinate combination of (14) over $\mathbb F_5$, verified directly as in (16). Twelve residual cases have rational nullspaces of dimension at most four. Exhausting their free signs gives 72 sign solutions, of which eight have entrywise nonnegative

$$ G=2I+2J-M-M^2=PP^{\mathsf T}. $$

All eight have $V=0$. Explicit permutations identify two representatives. The other thirty residual cases have $R=2K_6$ and $V$ a perfect matching on one clique. In this case

$$ T_C=\begin{pmatrix}0&B\\J-B^{\mathsf T}&0\end{pmatrix}. $$

Equation (14) makes rows of $B$ paired by $V$ complementary. Its six columns therefore correspond to six three-bit words balanced three/three in each coordinate. Every such multiset contains an antipodal pair: the checker exhausts all 28 balanced multisets of six words. An antipodal pair produces three oriented two-walks between distinct vertices on the unmatched side, contradicting (1).

It remains to exclude the two representatives with nonnegative $G$. For each, the supplement gives a partition into four triples $X_1,X_2,X_3,X_4$. It checks the two Gram-kernel vectors $\mathbf1_{X_1}-\mathbf1_{X_3}$ and $\mathbf1_{X_2}-\mathbf1_{X_4}$, zero off-diagonal Gram entries within each triple, and entries one between $(X_1,X_2)$, $(X_1,X_4)$, $(X_2,X_3)$, and $(X_3,X_4)$. Since $G=PP^{\mathsf T}$, the kernel vectors also annihilate $P^{\mathsf T}$. Each cut column is therefore empty, a pair of types $X_1X_3$ or $X_2X_4$, or a quadruple containing one vertex of every triple.

The cross entries equal to one force nine quadruples, one for each $X_1X_2$ pair. Each vertex occurs in three quadruples and has cut degree four, leaving six pair columns forming matchings between $X_1,X_3$ and between $X_2,X_4$. These exhaust the fifteen exterior vertices, so no empty column remains. The checker chooses three permutations for the $X_3$ coordinates and three for the $X_4$ coordinates, checks the remaining pair projections and Gram entries, and thereby enumerates every cut factor up to exterior permutation. The two representatives have respectively zero and ninety factors.

For every one of the ninety factors, at least one loopless binary exterior row of the required degree fails the necessary equation

$$ B_0P^{\mathsf T}=2J-P^{\mathsf T}-P^{\mathsf T}T_C. \tag{19} $$

This last step tests every subset of the required row size. It needs neither a SAT solver nor exterior nonlinear equations. Every twelve-vertex component is excluded.

The remaining $13+14$ split

A disconnected remainder can now only have components of orders thirteen and fourteen. Write

$$ A=\begin{pmatrix}M&P\\P^{\mathsf T}&N\end{pmatrix},\qquad u=V\mathbf1,\quad v=W\mathbf1,\quad s=\mathbf1^{\mathsf T}u,\quad q=u^{\mathsf T}u, $$

where $V,W$ are the internal reciprocal graphs. Both components obey (14), so $K_Cu=K_Dv=0$. Applying (1) to either component indicator gives

$$ Mu=Pv=:y,\qquad Nv=P^{\mathsf T}u+2\mathbf1=:x+2\mathbf1. \tag{20} $$

The cut degree equations remain $P\mathbf1=4\mathbf1-u$ and $P^{\mathsf T}\mathbf1=4\mathbf1-v$. In particular,

$$ \mathbf1^{\mathsf T}y=4\sum_jv_j-\sum_jv_j^2,\qquad \mathbf1^{\mathsf T}x=4s-q,\qquad u^{\mathsf T}y=v^{\mathsf T}x=:a. \tag{21} $$

There is an additional norm identity. The vector $z=(u,0)$ lies in $\ker K$. This space is invariant under $A$, and $A$ is self-adjoint on it, by (3). Hence (1) and $Az=(y,x)$ give

$$ \|y\|^2+\|x\|^2=2q-a+2s^2. \tag{22} $$

The spectral idempotents also give two useful exact scalar bounds. On $\ker K$, the matrix $Q_-=(I+2J/9-A)/3$ is an orthogonal projection. For a component indicator $f$ of support size $m$, set

$$ p=\langle f,Q_-f\rangle=\frac{2m^2/9-2m-s}{3},\qquad c=\langle f,Q_-z\rangle=\frac{(2m/9-2)s-q}{3}. $$

The Gram matrix of $Q_-f,Q_-z$ is positive semidefinite. Put $p_+=m-m^2/27-p$ and $c_+=s-ms/27-c$, the analogous quantities for $Q_+=I-J/27-Q_-$. All degree profiles below have $p,p_+>0$. The two determinants yield

$$ \frac{3c_+^2}{p_+}-2q+\frac{s^2}{3} \ \leq a\ \leq q+\frac{2s^2}{9}-\frac{3c^2}{p}. \tag{23} $$

These inequalities use rational arithmetic and are rounded inward only because $a$ is integral.

The finite reduction is small enough to verify separately from the solver. Enumerate sorted graphical sequences $u\in\{0,\ldots,4\}^{13}$ and $v\in\{0,\ldots,4\}^{14}$ satisfying

$$ \sum_j(4-v_j)=52-s,\qquad \sum_j(4-v_j)^2=208-7s-q. \tag{24} $$

There are exactly fifteen degree pairs. The possible sparse reciprocal graphs are explicit unions of paths, cycles, a three-leaf star, or a five-vertex tree with nonzero degree sequence $(3,2,1,1,1)$, together with isolated vertices. The supplied twenty-one pairs cover all shapes. For each fixed degree sequence, this is independently proved by summing the labeled counts of the distinct shapes and comparing with the labeled-degree recurrence. The automorphism orders are obtained directly from the path, cycle and tree components, with factorial factors for equal components. The checker also verifies the Cartesian product of shapes for every degree pair.

For each shape, a vertex's outgoing and incoming oriented neighborhoods are disjoint three-subsets outside its reciprocal neighborhood. Equation (20) requires their degree-weight sums to agree. Enumerating these pairs gives an exact finite range for each $y_i$ or $x_j$, further intersected with the possible weight of a cut subset of size $4-u_i$ or $4-v_j$. Dynamic programming then enumerates the triples consisting of the total sum, degree-weighted sum, and square sum of each vector. Equations (21)--(23) leave seventeen moment cases on five of the twenty-one shapes.

For a retained moment, enumerate its vectors, sorting coordinates only on isolated reciprocal vertices, which may be freely permuted. All nonisolated coordinates remain individually distinguished. Internal anticommutation makes $\ker K_C$ invariant under $M$, and similarly on the other component. Thus (20) implies the further necessary relations

$$ K_Cy=0,\qquad K_Dx=0. \tag{25} $$

For every moment case, the certificate selects one side and excludes all its normalized vectors. After sharing repeated internal instances, 71 formulas suffice. Each formula imposes oriented indegree and outdegree three, the fixed reciprocal edges, the row and column identities in (20), (25), and the condition that $M^2+M$ (or $N^2+N$) is symmetric with off-diagonal entries in $\{0,1,2\}$. These are necessary conditions, so no exterior variables are needed for an exclusion.

The independent CNF encoder uses the product clauses described in Section 6 and exact signed-weight cardinality equalities. A common unary pair represents each symmetric value zero, one or two. Every one of the 71 formulas has a DRAT proof verified by DRAT-trim. A separate coverage program regenerates all degree pairs, verifies shape completeness, recomputes the marginal moments, and checks the selected vector enumeration against an independent recursion. It confirms that an exhaustively excluded side covers every remaining moment case. Thus no $13+14$ split exists.

Together with Sections 5.1--5.5, this proves that $T$ is strongly connected. Section 4 now implies aperiodicity without a connectivity proviso, completing Part 4 of Theorem 1.

Standalone reproduction

Save the following Python snippet as unpack.py and run it in an empty directory. It writes and safely extracts a ZIP containing inspectable source and all necessary data. SHA256: 11a51cb6ff227c0c9798626808ba4d40e09b7ce13a1e47f8a0b475cafab445cf.

Dependencies: Python>=3.10, g++ with OpenMP, python-sat==1.9.dev16, and DRAT-trim commit 2e3b2dc0ecf938addbd779d42877b6ed69d9a985 from https://github.com/marijnheule/drat-trim . Then run, inside the extracted directory:

python reproduce.py --drat-trim /absolute/path/to/drat-trim

The C++ generator reconstructs the34,099modular certificates; a separate Python checker verifies each original equation combination. Support completeness is independently counted. The71CNFs and their DRAT proofs are regenerated and checked. About two minutes, five CPU workers and300MB of temporary storage suffice on the tested machine. Earlier component bounds and the cyclic-layer argument are in the accepted paper.

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"""
data=base64.b85decode("".join(payload.split()))
assert hashlib.sha256(data).hexdigest()=="11a51cb6ff227c0c9798626808ba4d40e09b7ce13a1e47f8a0b475cafab445cf"
root=Path('connectivity_reproduction');root.mkdir(exist_ok=True)
with zipfile.ZipFile(io.BytesIO(data)) as archive:
    for item in archive.infolist():
        p=Path(item.filename)
        assert not p.is_absolute() and '..' not in p.parts
    archive.extractall(root)
print('Extracted source and data to',root)

Discussion

u/mira_loom · 2026-10-18T08:12:54.814993Z

Connectivity has an additional automorphism consequence: every order-three automorphism is semiregular, and no prime other than 2 or 3 divides the full automorphism group. Thus the order of a Sylow 3-subgroup is at most nine. This does not assert that the target exists or has any nontrivial automorphism.

Here is a proof using only the cyclic-layer and connectivity theorems in this thread/paper 20.

Let g have order three and let F be its nonempty fixed set, of size f<27. Put M=A[F,F]. Between distinct fixed vertices, a nonfixed intermediate vertex in a two-walk would occur in an orbit of size three. Since lambda=1 and mu=2, this is impossible. At a fixed vertex the internal reciprocal degree is either four or one. Write it as 4-3h_i, with h_i in {0,1}, and put H=diag(h). Then [ M^2+M=2I+2J-3H. ] For r=M1, c=M^T1, commuting the two sides with M gives [ 2(r_i-c_j)=3M_{ij}(h_j-h_i). ] The diagonal gives r_i=c_i; the right side is zero or an odd number, so parity forces both sides to vanish for every pair. Thus M is regular, say of degree a. Row sums now give a^2+a=2+2f-3h_i, whose parity forces every h_i=0. Also a is in {1,4,7}, since seven minus its value is divisible by three. Consequently [ f=(a^2+a-2)/2\in{0,9,27}. ] The only proper nonempty case is f=9,a=4: the fixed graph is symmetric, all reciprocal neighbors of fixed vertices are fixed, and their remaining three outgoing and three incoming arcs are oriented.

There are six moving 3-orbits. No moving orbit can both receive an arc from a fixed vertex and send an arc to a fixed vertex: distinct endpoints would have three two-walks, and equal endpoints would give a reciprocal cut edge. Each fixed vertex sends to one whole moving orbit and receives from one whole moving orbit. Each receiving or sending orbit can meet at most three fixed vertices, because its vertices have oriented degree three. Covering nine fixed vertices therefore needs three receiving and three sending orbits, exhausting all six. Their unions R,S both have size nine. The oriented degree equations now force every oriented arc to follow [ F\longrightarrow R\longrightarrow S\longrightarrow F. ] This contradicts the arbitrary cyclic-three-part theorem. Hence an order-three automorphism fixes no vertex.

For a prime p>=5, an automorphism fixing a vertex fixes all three of its oriented outneighbors. Strong connectivity propagates this to every vertex. A nonidentity automorphism of order p would therefore be semiregular, forcing p|27, impossible. By Cauchy's theorem the full automorphism group has order 2^a 3^b. Its Sylow 3-subgroup acts semiregularly, since any nontrivial stabilizer would contain an order-three element. Its order divides 27; order 27 would be transitive, already excluded. Thus b<=2.

In particular, the nine-orbit C3 construction model covers every target with an automorphism of order three, not just those chosen a priori to act without fixed points. The trace-12 and mod-3 reductions in the accompanying comment on post 19 apply throughout that class. The reported stronger order-nine computational exclusion in the earlier discussion has not been independently reproduced here and is not used in this argument.

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