Symmetry

Within the framework of a discrete-time chronon model, we consider a dual description of physical time. In this description, macroscopic time is a continuous parameter, while a microscopic integer chronon index labels elementary updates of the system. On this basis, a hierarchy of temporal layers ${\mathrm{Ch}}_N$ (Chronon) is introduced. The simple layers $\mathrm{Ch}2$, $\mathrm{Ch}3$ and $\mathrm{Ch}4$ are analysed, and it is shown that they naturally support $U\left(1\right)$ (Unitary group), (Special Unitary group) and a pair-locked (Special Unitary group) symmetry, respectively. Special attention is paid to the twelve-slot layer $\mathrm{Ch}12$. This layer is the minimal one which simultaneously separates partitions into four triads and three quartets. For $\mathrm{Ch}12$, we demonstrate that the intersection of the corresponding commutants in ${\mathbb{C}}^3\otimes {\mathbb{C}}^4$ reproduces the Standard Model gauge algebra and the pattern of hypercharges and anomaly cancellation. The appearance of three fermion generations is interpreted in terms of three inequivalent embeddings of a triad into the dodecad which preserve the quartet structure. Possible connections of the chronon dynamics with the hierarchy of masses (via Floquet-type quasi-energies), with dark sectors associated with misaligned layers, and with a temporal interpretation of entanglement are briefly discussed on a qualitative level. These questions are formulated as open problems for further study.

Using a global rotation by $\theta$ about the z-axis in the spin sector of the Jordan–Wigner transformation rotates Pauli matrices $\widehat{X}$ and $\widehat{Y}$ in the $x-y$-plane, while it adds a global complex phase to fermionic quantum states that have a fixed number of particles. With the right choice of angles, this relates expectation values of Pauli strings containing products of $\widehat{X}$ and $\widehat{Y}$ to different products, which can be employed to reduce the number of measurements needed when simulating fermionic systems on a quantum computer. Here, we derive this symmetry and show how it can be applied to systems in Physics and Chemistry that involve Hamiltonians with only single-particle (hopping) and two-particle (interaction) terms. We also discuss the consequences of this for finding efficient measurement circuits in variational ground state preparation.

Industrial PLC programming faces persistent difficulties: lengthy development cycles, low fault tolerance, and cross-platform incompatibility among vendors. While LLMs show promise for automated code generation, their direct application is hindered by the gap between ambiguous natural language and the strict determinism required by control logic. This paper proposes MPC-Coder, a dual-knowledge enhanced multi-agent system that addresses this gap. The system combines a structured knowledge graph that imposes hard constraints on process parameters and equipment specifications with a vector database that offers implementation references such as code templates and function blocks. These two knowledge sources form a symmetric complementary architecture. A closed-loop “generation–verification–repair” mechanism leverages formal verification tools to iteratively refine the generated code. Experiments demonstrate that MPC-Coder achieves 100% syntactic correctness and 78% functional consistency, significantly outperforming general-purpose LLMs. The results indicate that the complementary fusion of domain knowledge and closed-loop verification effectively enhances the reliability of code generation, offering a viable technical pathway for the reliable application of LLMs in industrial control systems.