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

Abstract 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 {(Ci,≺i)}i∈I covering event set X, if local partial orders satisfy ˇ Cech-type consistency conditions in overlapping regions, then there exists a unique global partial order (X, ≺) as causal consensus extension; otherwise causal consensus exists at most at coarser resolution levels. Resolution is characterized by event partitions Piand observable algebras Ai; the richness of common refinement P∗and algebra intersection Acom = TiAidetermines the fineness of achievable consensus. At the information and dynamical level, consider state families {ω(t) i}on common observable algebra Acom, communication channels Tij, and weight matrix W= (wij). Based on Umegaki relative entropy D(ρ∥σ) = tr ρ(log ρ−log σ) and its data processing inequality, we construct weighted total deviation function Φ(t)=X i∈I λiDω(t) i∥ω∗, proving that when channels satisfy data processing inequality, communication graph is strongly connected, weight matrix is primitive, and common fixed point ω∗ exists, Φ(t)is a strictly monotone non-increasing Lyapunov function, making state iteration converge to unique state consensus ωcons =ω∗. 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(T) iof each observer contracts with probability one to the unique true model M∗, achieving model consensus. The above geometric, informational, and model structures are unified into a “consensus feasible region” Ocons 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. Keywords: Causal network; Partial order; Observer; Resolution; Observable algebra; Relative entropy; Quantum channels; Distributed consensus; ˇ Cech consistency; Model selection 1 1 Introduction and Historical Context In the mainstream picture of relativity and quantum field theory, spacetime causal structure can be abstracted as partial order ≺on event set M, e.g., the causal set approach models Lorentzian spacetime as a locally finite partially ordered set (M, ≺). Within this structure, a single observer collects local information along its worldline and forms a subjective model of the “world” at finite resolution and finite bandwidth. On the other hand, in distributed systems and multi-agent control, extensive work studies how multiple nodes achieve average consensus or consistent estimation under communication constraints through iterative updates. Although these two application contexts are vastly different, they share a common abstract core: multiple “observers” with local perspectives and local information states located on the same underlying causal structure, attempting to construct a consistent “global description” through communication and updates. In topology and sheaf theory, this problem appears in the form of “can local data be glued into global objects,” with rigorous tools being sheaf locality and gluing conditions, and ˇ Cech cohomology. In causal and structural learning frontiers, people are beginning to use partial orders and information geometry to characterize more general causal structures and their identifiable content. On the other hand, relative entropy and its monotonicity constitute an important cornerstone of quantum and classical information theory. The joint convexity and data processing inequality of Umegaki relative entropy play core roles in quantum channel analysis, thermodynamic inequalities, and information geometry. These results guarantee that under completely positive trace-preserving maps, distinguishability between states does not increase, naturally providing Lyapunov function candidates for “state consensus” convergence. Based on the above background, this work attempts to give general answers to the following questions at an abstract level: 1. How to uniformly describe the geometric, algebraic, and informational properties of “observers” in causal network language? 2. Under what conditions can local partial orders be glued into a single global causal network, achieving causal consensus? 3. Under what conditions can observer state iterations on common observable algebras converge to unified state consensus? 4. Under what identifiability conditions will the intersection of observer model families almost surely contract to the unique true model as data accumulates, forming model consensus? In existing literature, reconstruction of causal structure mostly focuses on “recovering topology and metric from global causal partial order,” while distributed consensus mostly assumes underlying system dynamics is known. This paper starts from the opposite direction: assuming what is given is multiple observers’ local causal fragments and information states, studying under what conditions a common causal network and consensus state can be recovered from this local data. The main contributions of this paper can be summarized as: 2  Formalizing observer as multi-component structure Oi= (Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,{Cij}j∈I), separately characterizing geometric domain, local partial order, resolution, observable algebra, state, model family, update rule, utility function, and communication structure, thereby unifying abstract descriptions of physical observers and computational nodes.  At the geometric and partial order level, giving sufficient conditions for local causal fragments {(Ci,≺i)}to be glued into unique global partial order (X, ≺), with core being coverage, finite overlap, and ˇ Cech-type consistency; and pointing out that if this condition breaks, strong-form causal consensus is unattainable.  At the information level, constructing common observable algebra Acom =TiAi and state family {ω(t) i}, proving that when channels are completely positive tracepreserving maps satisfying data processing inequality, communication graph is strongly connected, and common fixed point ω∗exists, weighted relative entropy Φ(t)=X i λiDω(t) i∥ω∗ is a Lyapunov function, guaranteeing state consensus convergence. This structure simultaneously encompasses classical average consensus, distributed filtering, and quantum network symmetrization and steady state design.  At the model level, giving identifiability assumptions based on large deviations and Kullback–Leibler divergence, proving that as observation time T→ ∞, the intersection of threshold-screened model sets M(T) iof each observer contracts with probability one to unique true model M∗.  Abstracting the above conditions into a “consensus feasible region” Ocons in observer property space O, and demonstrating through several finite examples the mechanism of causal consensus failure and recovery of weak consensus through coarse-graining. The subsequent structure is arranged as follows: first giving models and assumptions; then stating main theorems and their proof frameworks; then discussing several applications and engineering suggestions; finally summarizing and giving detailed proofs and examples in appendices. 2 Model and Assumptions 2.1 Event Set and Local Causal Fragments Let Xbe the event set. We do not presuppose a global causal relation on X, but rather assume observations appear only in the form of local partial orders. 3 Definition 2.1 (Local Causal Fragment).A local causal fragment is a pair (C, ≺C), where C⊆Xis an event subset and ≺Cis a partial order relation on C, i.e., satisfying for all x, y, z ∈C: 1. Reflexivity: x⪯Cx; 2. Antisymmetry: if x⪯Cy, y ⪯Cxthen x=y; 3. Transitivity: if x⪯Cy, y ⪯Czthen x⪯Cz. We customarily use x≺Cyto denote x⪯Cyand x=y. In a family of observers {Oi}i∈I, each observer Oiis associated with a local causal fragment (Ci,≺i). Assume coverage condition [ i∈I Ci=X, meaning each event is accessed by at least some observer. To avoid pathological cases, we further assume finite overlap condition: for any x∈X, the set {i∈I:x∈Ci}is finite. 2.2 Observer Property Vectors Definition 2.2 (Observer).Observer Oiis the following multi-component object: Oi=Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,{Cij}j∈I, where: 1. Ci⊆X: reachable causal domain, giving the set of events that observer can directly observe or influence. 2. ≺i: local causal partial order defined on Ci. 3. Λi: resolution scale, can be viewed as coarse-graining map from ideal fine event space Xfine to Ci, or equivalently as a partition Pi={Bi,α}α∈Ii; higher resolution corresponds to finer partition. 4. Ai: observable algebra, typically a C∗subalgebra of bounded operator algebra on some Hilbert space, containing measurable and controllable quantities. 5. ωi:Ai→C: state, a positive normalized linear functional, characterizing observer’s belief on Ai; in finite dimension corresponds to density matrix ρi. 6. Mi⊆ M: candidate model family, where Mis compact space of causal dynamical models, e.g., causal Markov networks, Lagrangians, or transition kernels. 7. Ui: state update operator Ui: (ωi, d)7→ ω′ i, mapping data dand current state to new state; specialized to linear averaging form in consensus iteration below. 4 8. ui: utility function or preference function, defined on action space Hor model space M, used for decision-making. 9. Cij: structural parameters of communication channel, describing bandwidth, latency, noise, trust weights, etc., from Ojto Oi, inducing completely positive tracepreserving map Tij at information level. This paper mainly focuses on the impact of the first seven components and communication graph structure on consensus existence and convergence. 2.3 Communication Graph and Channel Model Let Gcomm = (I, Ecomm) be the communication graph, with vertex set being observer indices I, and edge set Ecomm ={i, j}: there exists at least one direction of nonzero bandwidth between i, j. On common observable algebra, communication channels are represented by completely positive trace-preserving maps: Assumption 1 (Communication Channels).1. For each directed edge j→i, there exists completely positive trace-preserving map (CPTP map) Tij :S(Acom)→ S(Acom), where S(·) denotes state space; if no edge, then Tij is zero operator. 2. There exists weight matrix W= (wij)i,j∈Isatisfying wij ≥0,Pjwij = 1, and wij >0 only when there exists j→idirection communication. The common observable algebra is defined as Acom := \ i∈I Ai, assuming Acom is nontrivial except scalar multiples of the identity element. 2.4 Models and Probabilistic Structure Let observation data sequence D= (d(1), . . . , d(T)) be generated under true model M†∈ M. Each observer Oihas likelihood function Li(M;D) or posterior density πi(M| D). The acceptable model set after threshold screening is defined as M(T) i:= M∈ Mi:Li(M;D)≥ϵi(T), where threshold ϵi(T) varies with data volume. In model consensus discussion we adopt the following identifiability and consistency assumptions, with specific statements in theorems later. 3 Main Results: Theorems and Alignments This section presents the main theorems and structural conclusions of this paper. Proof details are concentrated in subsequent Proofs section and appendices. 5 3.1 Causal Consensus: Gluing Theorem for Local Partial Orders First we give the definition of strong-form causal consensus. Definition 3.1 (Causal Consensus).The observer family {Oi}i∈Iachieves causal consensus if there exists partially ordered set (X, ≺) and injective maps ei:Ci,→X satisfying: 1. For any x, y ∈Ci, x≺iy⇐⇒ ei(x)≺ei(y). 2. For any x∈Ci∩Cj, we have ei(x) = ej(x). In this case, (X, ≺) is called the causal consensus extension of local causal fragments. Consistency of local partial orders in overlapping regions is characterized by ˇ Cech-type conditions. Definition 3.2 (ˇ Cech-type Consistency).For any finite subset J⊆I, denote CJ:= \ j∈J Cj. If there exists partial order ≺Jdefined on CJsuch that for all j∈Jand x, y ∈CJ, x≺Jy⇐⇒ x≺jy, then the local partial order family {≺i}is consistent on CJ. If this holds for all finite J, the family {≺i}satisfies ˇ Cech-type consistency. Under coverage and finite overlap, we have the following gluing theorem. Theorem 3.3 (Causal Network Gluing Theorem).Let {(Ci,≺i)}i∈Ibe a family of local causal fragments on Xsatisfying: 1. Coverage: SiCi=X; 2. Finite overlap: for any x∈X, the set {i:x∈Ci}is finite; 3. ˇ Cech-type consistency: as in Definition 2.2. Then there exists unique partial order ≺such that: 1. (X, ≺)is a partially ordered set; 2. For each i, the restriction of ≺to Ciequals ≺i. In other words, (X, ≺)is a causal consensus extension, unique up to isomorphism. This theorem shows: the key to strong-form causal consensus existence is that the cover formed by local causal fragments satisfies ˇ Cech-type consistency. Otherwise, strong consensus is unattainable; weak consensus can only be discussed at coarser resolution levels. 6 3.2 Resolution, Common Refinement, and Consensus Limit Resolution structure is characterized by event partitions. Definition 3.4 (Partition and Common Refinement).Let Xfine be the ideal fine event set. In a family of partitions {Pi}i∈I, each Pi={Bi,α}α is a partition of Xfine. If there exists partition P∗such that for each i,Piis a coarsening of P∗, i.e., for any B∈Pithere exists B′∈P∗with B⊆B′, then P∗is called a common refinement. Common refinement existence can be described by equivalence relations. Partition Pi corresponds to equivalence relation x∼iy⇐⇒ ∃B∈Pi:x, y ∈B. Common refinement exists if and only if the intersection relation R:= \ i ∼i is an equivalence relation. In finite cases, the condition for common refinement existence can be restated as “no interlaced block conflicts,” with specific statements in Appendix B. The limit structure of resolution consensus is the quotient space after events are compressed by equivalence classes of R. Higher resolution and finer common refinement enable richer causal structures that consensus can distinguish. 3.3 Observable Algebra Intersection and State Consensus The common observable algebra is defined as Acom := \ i∈I Ai. Assume Acom is nontrivial except scalar multiples of the identity element. Let ω(t) i∈ S(Acom) be observer Oi’s state estimate on common algebra at time t, with update rule being linear averaging type ω(t+1) i=X j∈I wijTijω(t) j, where Tij are CPTP maps and W= (wij) is row stochastic matrix. Relative entropy takes Umegaki form: in finite dimension, if ω=ωρ, ω′=ωσcorrespond to density matrices ρ, σ, then D(ω∥ω′) := D(ρ∥σ) = tr ρ(log ρ−log σ). Proposition 3.5 (Single-step Contraction of Relative Entropy).Let ω∗∈ S(Acom)be some fixed state. Assume: 7 1. Each Tij satisfies data processing inequality, i.e., for all states ω, ω′, DTij(ω)∥Tij(ω′)≤D(ω∥ω′); 2. Weight matrix Wis row stochastic, and there exists weight λi>0satisfying Piλi= 1and λ⊤W=λ⊤. Define total deviation function Φ(t):= X i∈I λiDω(t) i∥ω∗. Then for any t, Φ(t+1) ≤Φ(t). This proposition shows: under natural weighting conditions, relative entropy is monotone non-increasing under consensus iteration, providing Lyapunov function candidate for consensus convergence. If we further assume common fixed point exists and communication graph has sufficient mixing, we obtain state consensus convergence theorem. Assumption 2 (Common Fixed Point and Primitivity).1. Communication graph Gcomm is strongly connected. 2. Weight matrix Wis primitive, i.e., there exists k∈Nsuch that Wkhas all positive elements. 3. There exists state ω∗∈ S(Acom) satisfying for all i, j Tij(ω∗) = ω∗. Theorem 3.6 (Convergence of State Consensus).Under Assumption 2, for any initial state family {ω(0) i}i∈I, the iteration ω(t+1) i=X j wijTij(ω(t) j) converges to unified state ω∗, i.e., lim t→∞ ω(t) i=ω∗,∀i∈I. In the classical case, the above result reduces to standard conclusions of linear average consensus; in the quantum case, it corresponds to a class of quantum Markov chains with common fixed point converging to unique steady state. 8 3.4 Model Consensus and Almost Sure Identification of True Model Let model space Mbe a compact metric space, with true model M†∈ M. For each M∈ M, denote PMas its induced data distribution. Assumption 3 (Identifiability and Large Deviations).1. There exists unique M∗∈ Msuch that for any M=M∗, KL divergence satisfies DPM∗∥PM>0. 2. For each observer i, threshold ϵi(T) can be chosen as function of data length Tsuch that under true model M∗, as T→ ∞ PM∗M∗∈ M(T) i→1,PM∗∃M=M∗, M ∈ M(T) i→0. This condition can be verified through law of large numbers and Sanov-type large deviation results. Theorem 3.7 (Almost Sure Contraction of Model Consensus).Under Assumption 3, for any δ > 0, there exists T0such that when T≥T0, PM∗\ i∈I M(T) i={M∗}≥1−δ. In other words, as observation time tends to infinity, the intersection of acceptable model sets of all observers contracts with probability one to unique true model M∗, achieving strong-form model consensus. 3.5 Consensus Geometry and Indicator System Synthesizing the above results, in observer property space O:= Y i∈I P(X)×Posets ×Res ×Alg × S × Models ×Updates×Comm, define subset Ocons ⊆ O as all satisfying: 1. There exists causal consensus extension (X, ≺); 2. There exists state consensus ωcons; 3. There exists nonempty model consensus set Mcons (single point in strong form). Call Ocons the feasible region of consensus geometry. Based on this, introduce the following indicators:  Geometric overlap degree: θij := µ(Ci∩Cj) µ(Ci∪Cj), where µis counting measure or volume measure. 9  In sheaf and sheaf structure aspects, ˇ Cech consistency of local partial orders and common refinement problem can be viewed as simplified version of gluing problems in noncommutative geometry and sheaf theory, promising further development into “causal sheaf” perspective. 5. Potential Risks and Extension Directions  This paper does not explicitly handle malicious or Byzantine observers; in presence of nodes intentionally throwing incorrect partial orders or states, causal consensus and state consensus may fail, requiring introduction of robust consensus and fault tolerance mechanisms.  For systems with self-reference or circular information structure, there may be situations of “local consensus self-consistent but globally non-embeddable,” related to recent research on circular information structure and non-classical causal models. In summary, consensus geometry framework provides unified context for structural analysis of multi-observer causal world, but extensions to infinite dimension, strong noise, and adversarial environments still require further research. 8 Conclusion This paper systematically constructs theoretical framework of “observer properties and consensus geometry on causal networks” at intersection of abstract causal networks and information geometry. By formalizing observers as multi-component objects with geometric domains, local partial orders, resolution scales, observable algebras, information states, model families, and update operators, the following main conclusions are obtained: 1. Under coverage, finite overlap, and ˇ Cech-type consistency conditions, local causal fragments can be glued into unique global partial order, achieving strong-form causal consensus; otherwise strong consensus is unattainable, weak consensus can only be discussed at coarse-graining levels. 2. Resolution structure and observable algebra intersection determine fineness of consensus; dimension of common refinement and common algebra are natural indicators of “consensus resolution.” 3. On common algebra, Lyapunov function measured by Umegaki relative entropy can characterize monotone convergence of state consensus iteration; under conditions of strongly connected communication graph, primitive weight matrix, and common fixed point existence, states necessarily converge to unified consensus state. 4. Under appropriate identifiability and large deviation conditions, intersection of threshold-screened model sets of each observer contracts with probability one to unique true model, achieving strong-form model consensus. 5. By introducing indicators such as geometric overlap degree, resolution compatibility, algebra intersection dimension, and relative entropy deviation, construct consensus feasible region in observer property space, providing quantitative tools for analyzing “whether consensus occurs easily.” 16 Future directions include: developing functional analysis version of causal consensus in infinite dimension and continuous field theory; restating consensus geometry in sheaf and higher category frameworks; introducing robust and fault-tolerant structures against malicious observers; and embedding this framework into broader “value–causality–information” unified system to explore relationship between free choice and causal consensus. Acknowledgements, Code Availability This work is based on public literature and theoretical tools for derivation and construction. This research did not use any specialized numerical code or simulation programs, so no publicly available code implementation exists. 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For any finite J⊆I, there exists partial order ≺Jdefined on CJ=Tj∈JCjsuch that for all j∈Jand x, y ∈CJ, x≺Jy⇐⇒ x≺jy. Then there exists unique partial order ≺such that for each i, restriction of ≺to Ci equals ≺i. 18 Proof. (1) Define global relation Define binary relation on X xRy ⇐⇒ ∃i∈Isuch that x, y ∈Ci, x ≺iy. By partial order properties, Ris clearly irreflexive (i.e., no xRx), but transitivity is not yet known. (2) Antisymmetry and no local contradiction If there exist x, y such that xRy and yRx simultaneously hold, then there exist i, j such that  x, y ∈Ciand x≺iy;  x, y ∈Cjand y≺jx. Taking J={i, j}, we have x, y ∈CJ, and ˇ Cech consistency requires ≺Jon CJto be consistent with ≺i,≺j, forcing x≺Jyand y≺Jxto hold simultaneously, contradicting ≺Jbeing partial order. Therefore Ris antisymmetric. (3) Transitive closure and cycle exclusion Define ≺as transitive closure of R, i.e., x≺yif and only if there exists finite chain x=x0Rx1R· · · Rxn=y. Need to prove ≺is antisymmetric. If there exists x=ywith x≺yand y≺x, splicing chains yields nontrivial closed cycle x=x0Rx1R· · · Rxn=x, where n≥1. For each relation xkRxk+1, there exists ikwith xk, xk+1 ∈Cikand xk≺ikxk+1. Finite overlap property guarantees set {ik}n−1 k=0 is finite; taking J={i0, . . . , in−1}, all points on cycle {xk}belong to [ i∈J Ci, and for each adjacent pair (xk, xk+1), there exists ik∈Jmaking it have xk≺ikxk+1 in Cik. By ˇ Cech consistency, unified partial order ≺Jexists on CJ=Ti∈JCi, with each local partial order consistent with ≺Jin overlaps. Through finite-step expansion and transitivity, we get in CJ x≺Jx, contradicting partial order definition. Therefore ≺is antisymmetric. Reflexivity can be obtained by defining non-strict relation ⪯as x⪯y⇐⇒ x=yor x≺y. Transitivity is guaranteed by transitive closure definition. 19 (4) Local consistency For any iand x, y ∈Ci, if x≺iy, then clearly xRy, hence x≺y, obtaining that ≺ extends ≺ion Ci. Conversely, if x, y ∈Ciand x≺y, then there exists finite chain x=x0Rx1R· · · Rxn=y. Each step xkRxk+1 is produced via some Cik. Using ˇ Cech consistency, on CJ′:= \ k Cik∩Ci there exists unified partial order ≺J′, with for each k xk≺J′xk+1. By transitivity we get x≺J′y; then by ≺J′being consistent with ≺ion CJ′, we obtain x≺iy. Therefore restriction of ≺to each Ciis consistent with ≺i. (5) Uniqueness If there exists another partial order ≺′satisfying same conditions, then for each i,≺′ |Ci=≺i=≺ |Ci. By coverage property, for any x, y ∈X, if x≺y, then there exists chain segmentally falling in each Ci; these relations must also hold in ≺′, and vice versa, hence ≺=≺′. In sense allowing bijective relabeling of X, we obtain isomorphism uniqueness. B Proof of Common Refinement Proposition Proposition B.1 (Equivalent Characterization of Common Refinement Existence).Let Xfine be finite set, {Pi}i∈Ia family of partitions on it. For each i, define equivalence relation x∼iy⇐⇒ ∃B∈Pi:x, y ∈B. Let R:= \ i∈I ∼i. Then the following two are equivalent: 1. There exists common refinement P∗such that for each i,Piis coarsening of P∗; 2. Relation Ris equivalence relation (i.e., reflexive, symmetric, and transitive). Proof. (1) If common refinement P∗exists, then corresponding equivalence relation ∼∗ satisfies ∼∗⊆∼ifor all i, therefore ∼∗⊆R. On the other hand, for any x, y if x∼∗y, then x, y must belong to same P∗block, and P∗is finest partition, so when any equivalence relation Ris contained in all ∼i, necessarily ∼∗=R. Therefore Ris equivalence relation. (2) If Ris equivalence relation, then its equivalence class set PR:= {[x]R:x∈Xfine} 20 is a partition, and from R⊆∼iwe know Piis coarsening of PR. Therefore PRis common refinement. When transitivity is absent, no equivalence relation contained in all ∼iexists, hence no common refinement exists. Specific “interlaced block conflict” construction and details see main text and discussion; not repeated here. C Convergence of Relative Entropy-type Consensus Process C.1 Detailed Proof of Proposition 3.5 Proposition C.1 (Single-step Contraction of Relative Entropy).Under Assumption 2.5 conditions, for any t∈N, Φ(t+1) ≤Φ(t). Proof. For fixed i, ω(t+1) i=X j wijTij(ω(t) j). By joint convexity of relative entropy, Dω(t+1) i∥ω∗≤X j wijDTij(ω(t) j)∥Tij(ω∗). By data processing inequality, DTij(ω(t) j)∥Tij(ω∗)≤Dω(t) j∥ω∗. Combining yields Dω(t+1) i∥ω∗≤X j wijDω(t) j∥ω∗. Multiplying both sides by λiand summing over i, Φ(t+1) =X i λiDω(t+1) i∥ω∗ ≤X i λiX j wijDω(t) j∥ω∗ =X j X i λiwijDω(t) j∥ω∗. Under λ⊤W=λ⊤condition, Piλiwij =λj, hence Φ(t+1) ≤X j λjDω(t) j∥ω∗= Φ(t). Proposition is proved. 21 C.2 Proof Framework for Theorem 3.6 To prove trajectory convergence to ω∗, two complementary perspectives can be adopted: 1. View overall state family Ω(t):= (ω(t) i)i∈Ias point on product state space S(Acom)⊗I, define overall channel T(Ω) = X j wijTij(ωj)i∈I. Under Assumption 2conditions, Tis primitive CPTP map with Ω∗:= (ω∗)i∈Ias unique fixed point. Primitivity and compactness guarantee Tt(Ω(0))→Ω∗. 2. Using Lyapunov function Φ(t)monotone non-increasing with lower bound zero, combined with primitivity of T, nontrivial limit cycles or fixed point families can be excluded, ultimately obtaining all components converging to ω∗. Complete technical details can be established using generalizations of Perron–Frobenius theory on Banach spaces and quantum Markov chain convergence theorems; omitted here. D Proof of Model Consensus Contraction Theorem Theorem D.1 (Restatement of Theorem 3.7).Under identifiability and large deviation Assumption 3conditions, for any δ > 0, there exists T0such that when T≥T0, PM∗\ i∈I M(T) i={M∗}≥1−δ. Proof. (1) Single observer consistency For fixed i, by Assumption 3(2) there exists Ti(δ) such that when T≥Ti(δ), PM∗M∗∈ M(T) i≥1−δ 2|I|, and PM∗∃M=M∗, M ∈ M(T) i≤δ 2|I|. (2) Joint event estimate Denote events E1:= \ i∈I {M∗∈ M(T) i}, E2:= \ i∈I {∄M=M∗:M∈ M(T) i}. By union bound and above estimates, PM∗(E1)≥1−δ 2,PM∗(E2)≥1−δ 2. Hence PM∗(E1∩E2)≥1−δ. 22 (3) Intersection single-point property On event E1∩E2, for each i,M(T) icontains M∗and no other model. Therefore \ i∈I M(T) i={M∗}. Taking T0= maxiTi(δ), when T≥T0the conclusion holds. E Finite Example: Consensus Failure and Coarsegraining Repair with Three Observers Example E.1 (Three-node Causal Cycle).Let X={a, b, c}. Three observers:  O1:C1={a, b}, partial order a≺1b;  O2:C2={b, c}, partial order b≺2c;  O3:C3={c, a}, partial order c≺3a. Geometrically, {Ci}covers Xwith overlaps forming ring structure. Partial orders within each overlap region are internally self-consistent, but overall combination forms cycle a≺1b≺2c≺3a. If there exists global partial order ≺embedding each local partial order, it must simultaneously satisfy a≺b, b ≺c, c ≺a, contradicting antisymmetry of partial order. Therefore no strong-form causal consensus extension exists. Coarse-graining repair If we allow introducing equivalence relation ∼making a∼b∼c, then quotient set ˜ X:= X/∼={˜x} contains only single equivalence class. Then the only possible partial order is ˜x⪯˜x; in this extremely coarse perspective, all local partial orders degenerate to reflexive relations, and causal consensus holds at this level. This example shows: 1. Geometric connectivity is necessary but not sufficient for strong-form causal consensus; 2. Consistency of local partial orders in overlapping regions is key to eliminating causal cycles; 3. When strong consensus breaks, weak consensus can be recovered through event equivalence class compression, but at cost of losing detailed structure. This example provides simple discrete model for “causal cycles” and “resolution tradeoffs” phenomena in more complex causal networks. 23