Hardware Validation of DAGI via a Modular "Ridge" Signature and High-Order Synergistic Information

We report a hardware validation of the DAGI (Directed Acyclic Graph Information) framework on IBM Quantum hardware using a small, controlled experiment whose ideal output distribution is constrained to a low-dimensional modular manifold (a “ridge”). For two $n$-bit registers $(u,v)$ with $n=4$ (modulus 16), each key instance $k$ induces an ideal relation $v\equiv k\cdot u(\mathrm{m}\mathrm{o}\mathrm{d}16)$, producing a visually distinct ridge in the joint $(u,v)$ distribution. Executed on ibm_torino in a single Sampler V2 job (8 keys, 1024 shots/key, $N=8192$ total shots), the ridge persists under hardware noise with ridge-hit probability $p_{hit}=0.1830$ (uniform baseline $1\mathrm{/}16$), corresponding to a ridge contrast of $2.93\times$ (95% bootstrap CI [2.80, 3.06]). Key recovery exceeds chance: per-shot accuracy 0.1689 (chance 0.125, 95% Wilson CI [0.1610, 0.1772]), and per-group dictionary recovery 0.375 (chance 0.125). To test the central DAGI hypothesis – that recoverable key information is predominantly high-order/synergistic rather than visible in low-order marginals – we compute a Möbius-based information decomposition of $I(K;D_S)$ over detector-bit subsets $S$ via a Möbius inversion pipeline and evaluate targeted positive synergy $CP{S}_K$ at order $k_{max}=3$. We observe $CP{S}_K(k=3)=0.08788$ with significance under label-shuffle permutation tests (accuracy $p=0.001996$, $CP{S}_K$ $p=0.004975$). Uniformity diagnostics show near-uniform single-bit marginals while correlation concentrates in specific low-order pairs, and a bootstrap reliability sweep confirms order-3 targeted synergy remains statistically reliable at the full 1024-shot target budget. These results support the claim that DAGI detects and quantifies nontrivial, hardware-resilient, higher-order information structure associated with a known global algebraic constraint.