Causality Sum Rules in Conventional Scattering Matrices

Pearl's structural causal model (SCM) framework, built on directed acyclic graphs (DAGs) and the do-calculus, is the dominant formal language for causal reasoning. Yet it carries two structural restrictions: every relationship must be pre-specified as a directed causal edge, and feedback cycles are forbidden. This paper examines two classes of phenomena that strain these restrictions. First, symmetric physical and economic constraints, the ideal gas law being the canonical case, carry no intrinsic causal direction. Direction emerges only under intervention, and which variable is solved for must be specified as part of the intervention. We formalize such constraints as causal zeros within an Extended Causal Model by adding an activation operator, subject to local solvability and graph-admissibility conditions. Second, for the class of finite-propagation state-space systems considered here, we treat apparent instantaneous cycles as artifacts of suppressed time and ground both causal zeros and feedback in Causal Differential Equations (CDEs). In these, the transient regime is a time-unrolled acyclic causal process, and causal zeros arise as the defining functions of attracting equilibrium manifolds; periodic and chaotic attractors define further regimes of the same dynamics, treated through attractor-relative intervention. We give the extended do-calculus, identifiability conditions, counterfactual semantics, and open problems.

Query-separated computation forces a representation to play an operational role: data are encoded before a query is known, and a later decoder can answer only through the intermediate interface. In this regime the representation functions as a message rather than merely as a feature map. We formalize this observation by embedding information causality (IC) into representation learning, obtaining a framework called neural information causality (Neural-IC). The revised formulation separates two logically distinct statements. First, every query-separated architecture induces a random-access communication experiment and obeys the embedding inequality $I_{\mathrm{N\text{-}RAC}}\le I(\vec a:H,B)$. Second, any independently certified physical capacity bound on the interface, such as a hard $m$-bit alphabet, a finite-precision register, or a power-constrained noisy channel, implies $I_{\mathrm{N\text{-}RAC}}\le C_H$. This separation avoids treating capacity as a post hoc definition and makes Neural-IC an operational diagnostic for query leakage, precision leakage, and episode-specific memory. We also provide an exact one-bit classical RAC benchmark, showing explicitly that the relevant quantum enhancement is not total information beyond the bottleneck, but fair query-conditioned access. For CHSH-type correlation layers, nested Neural-RAC protocols multiply correlation biases across depth; requiring stability of a one-bit bottleneck for arbitrary depth selects the Tsirelson threshold. We extend the analysis to asymmetric seed biases, to multi-capacity finite-depth phase diagrams, and to correlated data via a conditional information score. Controlled simulations, including straight-through binary bottlenecks and deliberately leaky ablations, verify that apparent violations are accounted for by broken query separation or undercounted capacity.

We demonstrate that AI-driven program synthesis can autonomously discover fundamental strategies for decomposing unitary matrices in photonic networks. By extending DreamCoder to complex-valued linear algebra, the system generates decomposition programs achieving the minimal $N(N-1)/2$ Mach-Zehnder interferometers, distinct from both Reck and Clements architectures. Learned programs encode dimension-agnostic invariants: strategies discovered for $5 \times 5$ matrices generalize to higher dimensions such as $64 \times 64$. The discovered programs encode interpretable, dimension-agnostic construction rules. These rules generalize across matrix sizes without retraining, demonstrating that autonomous program synthesis can serve as a scalable paradigm for algorithm discovery and the automated design of universal unitary operators. Beyond universal decompositions, the system automatically exploits matrix structure to reduce the interferometer count below the universal theoretical bound. For instance, for Householder matrices, it discovers a dimension-independent rule that requires only $2N-3$ MZIs. This achieves linear, rather than quadratic, scaling and generalizes to arbitrary $N$ without retraining. For matrices obtained from the singular value decomposition of sparse matrices, reductions generally increase with sparsity, reaching up to 38% fewer MZIs than the universal theoretical bound $N(N-1)/2$ at 95% sparsity. These MZI reductions translate directly into practical hardware benefits for scalable photonic implementations. Taken together, the system functions as a single unified engine that discovers both universal decomposition rules and matrix-specific optimizations, without being provided with the structural or analytical properties of the input matrices.