Song and Lin’s parity conjecture for zero transfer is false

Abstract

On the connected oriented circulant graph G(Z_30, {5, 6, 9, 20}) the continuous-time quantum walk never moves amplitude between the vertex 0 and the even vertex 2, so the parity restriction conjectured for orders n = 2 (mod 4) fails at n = 30.

Definition 1.1 (The Hermitian adjacency matrix).

$$\forall a\in \mathrm{ZMod}\left(n\right),\forall b\in \mathrm{ZMod}\left(n\right),(\mathrm{hermAdj}\left(n,C\right){)}_{a,b}=\mathrm{ite}\left(b-a\in C,i,\mathrm{ite}\left(a-b\in C,-i,0\right)\right)$$

Formalization. D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.hermAdj (✓ std3).

Citation. Xingkun Song, Huiqiu Lin (2026). Zero transfer on mixed graphs. DOI: 10.48550/arXiv.2608.10643. URL: https://arxiv.org/abs/2608.10643v1.

Commentary.

For a connection set C in Z_n the circulant graph has an arc from a to b when b - a lies in C. The Hermitian adjacency matrix has entry i on arcs, -i on reversed arcs and 0 elsewhere; for an oriented connection set no pair carries both.

Definition 1.2 (Zero transfer).

$$\mathrm{ZeroTransfer}\left(n,C,u,v\right)\iff (\forall t\in \mathbb{R},(0\le t)\Rightarrow ((\mathrm{hamiltonianPropagator}\left(\mathrm{hermAdj}\left(n,C\right),t\right){)}_{u,v}=0))$$

Formalization. D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.ZeroTransfer (✓ std3).

Citation. Xingkun Song, Huiqiu Lin (2026). Zero transfer on mixed graphs. DOI: 10.48550/arXiv.2608.10643. URL: https://arxiv.org/abs/2608.10643v1.

Commentary.

The continuous-time quantum walk at real time t is U(t) = exp(-i t H), the propagator hamiltonianPropagator H t = exp(t (-i) H) of the Hamiltonian flow module. The graph has zero transfer from u to v when the (u, v) entry of U(t) vanishes at every time t >= 0.

Definition 1.3 (Oriented connection sets).

$$\mathrm{Oriented}\left(C\right)\iff ((\neg 0\in C)\wedge (\forall x\in C,\neg -x\in C))$$

Formalization. D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.Oriented (✓ std3).

Citation. Xingkun Song, Huiqiu Lin (2026). Zero transfer on mixed graphs. DOI: 10.48550/arXiv.2608.10643. URL: https://arxiv.org/abs/2608.10643v1.

Commentary.

The connection set avoids 0 and contains no element together with its negative.

Definition 1.4 (Connectedness).

$$(\forall a\in \mathrm{ZMod}\left(n\right),\forall b\in \mathrm{ZMod}\left(n\right),\mathrm{arc}\left(C,a,b\right)\iff (b-a\in C))\wedge (\mathrm{Connected}\left(C\right)\iff (SimpleGraph.Connected\left(SimpleGraph.fromRel\left(\mathrm{arc}\left(C\right)\right)\right)))$$

Formalization. D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.Connected (✓ std3).

Citation. Xingkun Song, Huiqiu Lin (2026). Zero transfer on mixed graphs. DOI: 10.48550/arXiv.2608.10643. URL: https://arxiv.org/abs/2608.10643v1.

Commentary.

The underlying undirected graph, which joins distinct a and b when b - a or a - b lies in C, is connected.

Definition 1.5 (The conjecture).

$$claim\iff (\forall n\in \mathbb{N},(\mathrm{NatMod}\left(n,4\right)=2)\Rightarrow (\forall C\in \mathrm{Finset}\left(\mathrm{ZMod}\left(n\right)\right),(\mathrm{Oriented}\left(C\right))\Rightarrow ((\mathrm{Connected}\left(C\right))\Rightarrow (\forall v\in \mathrm{ZMod}\left(n\right),((\mathrm{ZeroTransfer}\left(n,C,v,0\right))\wedge (\mathrm{ZeroTransfer}\left(n,C,0,v\right)))\Rightarrow (\mathrm{Odd}\left(ZMod.val\left(v\right)\right))))))$$

Formalization. D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.claim (✓ std3).

Citation. Xingkun Song, Huiqiu Lin (2026). Zero transfer on mixed graphs. DOI: 10.48550/arXiv.2608.10643. URL: https://arxiv.org/abs/2608.10643v1.

Commentary.

For every order n = 2 (mod 4), every oriented connection set with connected graph and every vertex v, zero transfer between v and 0 in both directions forces the representative of v in 0, …, n - 1 to be odd. Reading zero transfer between v and 0 in both directions only strengthens the hypothesis.

Theorem 1.6 (The counterexample n = 30, C = {5, 6, 9, 20}, v = 2).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.result (✓ std3). ∎

Resolves. Problems/song-lin-2026-oriented-circulant-zero-transfer-parity-refutation (refuted) by D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.result.

Source. Repository-derived.

Acknowledgement. Xingkun Song, Huiqiu Lin (2026). Zero transfer on mixed graphs. DOI: 10.48550/arXiv.2608.10643. URL: https://arxiv.org/abs/2608.10643v1.

Commentary.

The set C is oriented, and 6 - 5 = 1 connects every vertex a to a + 1 through a + 6, so the graph is connected. H is i times the integer skew-symmetric matrix S with entry 1 on arcs and -1 on reversed arcs, so every power of H is a power of i times the corresponding power of S. Seven exact row products give the rows r_k of S^k at vertex 0 for k <= 7; they satisfy r_k(2) = 0 for k <= 6 and r_7 = -32 r_5 - 320 r_3 - 960 r_1. Multiplying by S propagates this relation to r_(k+7) for every k, so strong induction gives (S^k)(0, 2) = 0 for all k, and skew-symmetry gives (S^k)(2, 0) = (-1)^k (S^k)(0, 2) = 0. Each term of the exponential series of -i t H therefore has vanishing (0, 2) and (2, 0) entries, so U(t) has them too at every time, and zero transfer holds between 0 and the even vertex 2.

References

• Truth anchor: D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.Connected
• Truth anchor: D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.Oriented
• Truth anchor: D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.ZeroTransfer
• Truth anchor: D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.claim
• Truth anchor: D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.hermAdj
• Truth anchor: D5/S3/Quantum/Dynamics/OrientedCirculantZeroTransfer.result
• Dependency: D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow