Poset Engineering for DAGI on Quantum Hardware: Causal-Cone Mobius Decomposition and Patch-Invariant Context Signatures on IBM Torino

The DAGI program uses M"obius inversion on partially ordered sets to decompose an information observable into irreducible contributions. A central methodological question is whether the Boolean powerset is merely a convenient default, or whether the poset should be engineered to match the physical accessibility structure of the experiment. We report a hardware validation of the latter claim on IBM Quantum hardware (\texttt{ibm\_torino}). In repetition-code memory circuits with mid-circuit measurement and reset, we compare two decompositions computed on exactly the same detector-window data: a truncated subset lattice (all detector subsets with $|S|\leq 3$) and a reachability-defined causal-cone poset (truncated past cones with $r_{\max}=3$). Controlled contexts increase two-qubit exposure using barrier-separated canceling two-qubit operations while preserving the ideal logical circuit. Across a single-patch pilot and two multi-patch extensions ($P=4$ and $P=8$ far-separated parallel patches), the detector windows contain measurable context signal. The strongest positive result is not raw classification accuracy but \emph{region stability}: top cone-poset regions selected by M"obius magnitude are highly patch-invariant, whereas subset-lattice top terms are nearly random-like across patches. For $P=4$ and $P=8$, normalized top-region Jaccard stability is $\Jnorm_{\cone}=0.505$ and $0.603$, while $\Jnorm_{\sub}=0.008$ and $-0.001$. Out-of-patch classifiers remain only weakly above chance, but multinomial logistic calibration reduces expected calibration error to $0.035$--$0.079$ and removes a large Naive-Bayes calibration artifact. This weak classification is expected: predicting a global two-qubit-load context from one local 12-detector window is a low-SNR domain-generalization task. The mutual information statistic in this experiment is interpreted as an operational proxy for the context-modulated correlated-error footprint, not as a direct measurement of pure topological synergy. We therefore state a conservative conclusion: causal-cone posets expose reusable, patch-invariant macro-region signatures in this repetition-code hardware setting; they are not yet shown to improve logical decoding or to dominate arbitrary subsets in all predictive tasks. The accompanying dataset bundle contains raw detector windows, QPY circuits, explicit poset definitions, fold outputs, and paper-ready tables.