Observer Properties and Consensus Geometry on Causal Networks:\\ Unified Formalization of Local Partial Orders, Information States, and Update Operators

In a worldview based on causal partial orders, any single observer possesses only a local fragment: partial events at finite resolution, partial causal relations, and partial information states on locally observable algebras. Multiple observers attempt to achieve ``consensus'' on the same universe causal network through communication and updates, thereby reconstructing a consistent world description. In the abstract causal network framework, this paper formalizes observers as multi-component objects equipped with geometric domains, local partial orders, resolution scales, observable algebras, boundary states, model families, state update operators, and utility functions, establishing a unified theory of ``consensus geometry.'' At the geometric level, given a family of local causal fragments \{(C_i,\prec_i)\}_{i\in I} covering event set X, if local partial orders satisfy Cech-type consistency conditions in overlapping regions, then there exists a unique global partial order (X,\prec) as causal consensus extension; otherwise causal consensus exists at most at coarser resolution levels. Resolution is characterized by event partitions P_i and observable algebras A_i; the richness of common refinement P_\ast and algebra intersection A_{com}=\bigcap_iA_i determines the fineness of achievable consensus. At the information and dynamical level, consider state families \{\omega_i^{(t)}\} on common observable algebra A_{com}, communication channels T_{ij}, and weight matrix W=(w_{ij}). Based on Umegaki relative entropy $ D(\rho\|\sigma)=\tr\big(\rho(\log\rho-\log\sigma)\big) and its data processing inequality, we construct weighted total deviation function \Phi^{(t)}=\sum_{i\in I}\lambda_i D\big(\omega_i^{(t)}\|\omega_\ast\big), proving that when channels satisfy data processing inequality, communication graph is strongly connected, weight matrix is primitive, and common fixed point \omega_\ast exists, \Phi^{(t)} is a strictly monotone non-increasing Lyapunov function, making state iteration converge to unique state consensus \omega_{cons}=\omega_\ast. This structure simultaneously encompasses classical average consensus algorithms and contractive flows on quantum channels. At the model level, viewing candidate causal dynamical models as elements of compact space M, under appropriate identifiability and large deviation conditions, we prove that as observation data increases, the intersection of acceptable model sets M_i^{(T)} of each observer contracts with probability one to the unique true model M^\ast, achieving model consensus. The above geometric, informational, and model structures are unified into a ``consensus feasible region'' O_{cons} in observer property space O$. This paper provides several necessary or sufficient conditions for causal consensus, state consensus, and model consensus, proposing a set of quantitative indicators including geometric overlap degree, resolution compatibility, algebra intersection dimension, relative entropy deviation, and communication graph connectivity, demonstrating how to systematically analyze ``how multiple observers weave the same causal world'' from the causal network perspective.