Multi-Observer Consensus Geometry and Causal Network\\ in Computational Universe:\\ Observer Family, Distributed Update,\\ and Discrete Ricci Contraction Under Unified Time Scale

Multi-Observer Consensus Geometry and Causal Network in Computational Universe: Observer Family, Distributed Update, and Discrete Ricci Contraction Under Unified Time Scale Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In previous works on computational universe Ucomp = (X, T,C,I) series, we have completed construction of following structural levels: 1. Axiomatization of computational universe and discrete complexity geometry; 2. Task-aware discrete information geometry and information manifold (SQ, gQ); 3. Control manifold (M, G) and continuous complexity geometry under unified time scale scattering mother scale; 4. Time–information–complexity joint variational principle on joint manifold EQ=M×SQ; 5. Categorical equivalence between physical universe category and computational universe category on reversible QCA subclass; 6. Unified theory of single observer’s attention–knowledge graph–cognitive dynamics; 7. Boundary computation and causal diamond structure on finite blocks. However, in real universe there does not exist single observer, but rather causal network formed by multiple observers: they each carry finite memory and knowledge graphs, exchanging information through finite-bandwidth channels, gradually forming or losing consensus under constraints of unified time scale and complexity budget. To rigorously characterize this phenomenon within computational universe framework, this paper proposes and develops unified theory of multi-observer consensus geometry and causal network. We first formalize observer family as O={Oi}i∈I, 1 where each observer Oiin computational universe has its own internal memory space M(i) int, attention operator Ai,t, knowledge graph Gi,t, and individual worldline zi(t) = (θi(t), ϕi(t)) on joint manifold EQ. On this basis, we introduce timedependent directed communication graph Ct= (I, Et, ωt), using it to define consensus geometry between multi-observers: “distance” between observers is jointly determined by their information manifold embedding ΦQand spectral structure of knowledge graphs, Laplace operator of communication graph induces class of “consensus Ricci curvature” acting on observer distribution. Core results of this paper include: 1. Introduce multi-observer state space EN Q= N Y i=1 E(i) Q, defining consensus energy functional on it Econs(t) = 1 2X i,j wij(t)d2 SQ(ϕi(t), ϕj(t)), proving that under symmetric communication graph and appropriate Lipschitz conditions, Econs exhibits exponential decay under unified time scale, with decay rate controlled by “consensus Ricci curvature” lower bound. 2. Glue multi-observer knowledge graphs into joint knowledge graph Gunion t, proving that in spectral sense of graph Laplace, effective spectral dimension of this joint graph tends in long-time limit toward local information dimension of task information manifold, thereby showing that “multi-observer consensus” geometrically corresponds to skeleton approximation of same information manifold. 3. Introduce multi-observer causal network in causal diamond framework: for family of time-ordered observation–communication events, define multi-observer causal diamond ♢multi, constructing on its boundary joint boundary operator Kmulti ♢:O i B− ♢,i →O i B+ ♢,i, proving its uniqueness under local unitary gauge transformations, compatible with single-observer boundary operators through graph Schur elimination. 4. On basis of time–information–complexity joint variational principle, construct multi-observer joint action b Amulti Q=X ib A(i) Q+λcons ZT 0 Econs(t) dt, deriving its Euler–Lagrange type equations, giving variational characterization of multi-observer optimal strategy for “maximizing collective task information quality under finite complexity budget”. This paper lays rigorous computational universe foundation for subsequent unified description of causal network splicing, multi-observer consensus geometry, and social–multi-agent systems. 2 Keywords: Computational universe; Multi-observer; Consensus geometry; Causal network; Communication graph; Ricci curvature; Distributed dynamics; Joint action; Spectral dimension 1 Introduction In single-observer theory, observer is viewed as internal computational process in computational universe: it has finite memory Mint, attention operator At, knowledge graph Gt, and worldline z(t) = (θ(t), ϕ(t)) on joint manifold EQ=M × SQ. Its behavior described by minimization of time–information–complexity joint action b AQ, subject to constraints of unified time scale and complexity budget. In real universe, observers not isolated, but form dynamic causal network through finite-bandwidth channels:  At physical level, these observers may be local subsystems in physical system;  At information level, they possess finite memory and self-consistent knowledge graphs, capable of gradually forming consensus on some task Qthrough message exchange;  At complexity level, they each have finite complexity budget, communication and computation both consume resources under unified time scale. Therefore, for “computational universe” to serve as rigorous framework for unified universe description, must answer following questions at multi-observer level: 1. How to define “geometric distance” and “consensus error” between multi-observers? 2. Given communication graph and unified time scale, does collective consensus error decay over time? Is decay rate controlled by some “consensus curvature”? 3. How do multi-observer knowledge graphs glue into joint skeleton, how does approximation capability for task information manifold SQevolve over time? 4. Under finite complexity budget, what local attention selection and communication strategies can “optimally” form consensus in variational sense? Main goal of this paper is to give unified, geometric, and variational answer to these questions. 2 Multi-Observer Objects and Joint State Space This section defines multi-observer family on basis of single-observer objects, constructing multi-observer joint state space. 3 2.1 Multi-Observer Object Family Recall single-observer object O= (Mint,Σobs,Σact,P,U), where Mint is internal memory state space, Pis attention–action policy, Uis internal update operator. Definition 2.1 (Multi-Observer Family).In computational universe Ucomp = (X, T,C,I), a multi-observer family consists of set Iand observer object set O={Oi}i∈I where each Oi= (M(i) int,Σ(i) obs,Σ(i) act,P(i),U(i)). We assume: 1. Iis finite or countable; 2. Each M(i) int is finite set or direct product of finite-dimensional registers; 3. For all observers Oi, their observation and action can be represented through computational universe update relation Tand task observation operator family OQ. 2.2 Joint State Space In single-observer case, we defined joint manifold EQ=M×SQ, introducing time–information–complexity action on it. In multi-observer case, we define Definition 2.2 (Multi-Observer Joint Manifold).For N=|I|<∞case, define E(i) Q=M(i)× S(i) Q as control–information manifold of i-th observer (can take M(i)=M,S(i) Q=SQin isomorphic sense), defining joint manifold EN Q= N Y i=1 E(i) Q. For each observer i, its continuous limit worldline is zi(t) = (θi(t), ϕi(t)) ∈ E(i) Q, multi-observer joint worldline is Z(t) = (z1(t), . . . , zN(t)) ∈EN Q. Internal memory and knowledge graph can be attached as external structures, at main geometric level we focus on evolution of θi, ϕi. 4 3 Communication Graph, Consensus Energy, and Consensus Geometry This section introduces time-dependent communication graph, defining multi-observer consensus energy and consensus geometric structure on information manifold. 3.1 Time-Dependent Communication Graph Definition 3.1 (Communication Graph).At time t, multi-observer communication structure represented by directed graph Ct= (I, Et, ωt) where: 1. Vertex set is observer index set I={1, . . . , N}; 2. Directed edge (j→i)∈Etrepresents observer Ojsending information to Oiat time t; 3. Weight ωt(i, j)≥0 represents weight/bandwidth of edge j→i. Important special case of symmetric communication graph is ωt(i, j) = ωt(j, i). Communication graph induces class of graph Laplace operators: define Lt:RN→RN, for vector x∈RNhave (Ltx)i=X j ωt(i, j) (xi−xj). When ωtsymmetric, Ltis symmetric positive semidefinite matrix, whose spectral structure characterizes connectivity and “consensus contractivity” of communication structure. 3.2 Consensus Energy and Consensus Geometry To characterize “degree of consensus” of multi-observers on task information manifold (SQ, gQ), we define consensus energy functional. Definition 3.2 (Consensus Energy).At time t, for observer information states ϕi(t)∈ SQ, define consensus energy Econs(t) = 1 2X i,j∈I ωt(i, j)d2 SQϕi(t), ϕj(t), where dSQis geodesic distance induced by gQ. 5 When Econs(t) = 0, all observers completely coincide on task information manifold, achieving perfect consensus; larger Econs indicates higher degree of information dispersion. Consensus energy can be viewed as “discrete Dirichlet energy” of observer distribution on information manifold, whose gradient flow corresponds to consensus dynamics of information states on communication graph. To more geometrically characterize global structure between multi-observers, can define product metric on joint manifold EN Q G(N) t= N X i=1 α2G(i)⊕β2g(i) Q, viewing consensus energy as potential function on information factor Ucons(Z(t)) = Econs(t). From this perspective, multi-observer joint worldline satisfies “geodesic flow with coupling potential”, with potential function being precisely consensus energy. 4 Consensus Ricci Curvature and Energy Decay Theorem This section introduces “consensus Ricci curvature” concept related to communication graph and information manifold geometry, proving exponential decay theorem for consensus energy under its lower bound constraint. 4.1 Local Ricci Curvature Between Two Observers In single-observer information manifold, we already defined discrete Ricci curvature based on Wasserstein distance. In multi-observer case, we focus on combination of geodesic structure on information manifold and communication Laplace. For given time t, let ϕi, ϕj∈ SQbe task information states of two observers. Consider connecting ϕi, ϕjby geodesic on SQ, defining on it local sectional curvature Kij (t). In consensus dynamics driven by graph Laplace, natural discrete Ricci curvature analog is: Definition 4.1 (Local Lower Bound of Consensus Ricci Curvature).If there exists constant κcons(t)∈R, such that for any i, j d dϵϵ=0hd2 SQϕi(t+ϵ), ϕj(t+ϵ)i≤ −2κcons(t)d2 SQϕi(t), ϕj(t), then call κcons(t) consensus Ricci curvature lower bound at time t. Intuitively, κcons(t)>0 indicates under consensus dynamics, information distance between observers exhibits exponential contraction; κcons(t)<0 indicates information distance may diverge. 6 4.2 Consensus Dynamics and Energy Decay Consider simple continuous consensus dynamics model: dϕi dt=−X j ωt(i, j) gradϕi1 2d2 SQ(ϕi, ϕj), equivalent to gradient descent for consensus energy Econs on SN Q. On Riemannian information manifold, this can be written as dϕk i dt=−X j ωt(i, j)gkl Q(ϕi)∂l1 2d2 SQ(ϕi, ϕj). Under standard assumptions, time derivative of Econs is d dtEcons(t) = −X i∇ϕiEcons(t)2 gQ. Combining consensus Ricci curvature lower bound, following theorem can be proved. Theorem 4.2 (Exponential Decay of Consensus Energy).Assume: 1. Communication graph Ctsymmetric with uniform algebraic connectivity lower bound λmin 2>0; 2. Information manifold (SQ, gQ)has Ricci curvature lower bound RicgQ≥K∈R; 3. Consensus dynamics as above, evolving under unified time scale. Then there exist constants κeff >0and C > 0, such that Econs(t)≤ Econs(0) e−2κeff t, t ≥0, where κeff given by combination of λmin 2and K. Proof in Appendix C.1. Core idea is using Otto perspective of Wasserstein–information geometry to view consensus process as kind of “discrete viscous flow”, whose energy decay rate controlled by lower bound curvature, consistent with Bakry–´ Emery gradient estimate form. 5 Multi-Observer Joint Action and Optimal Consensus Strategy This section, on basis of time–information–complexity joint variational principle, constructs multi-observer joint action and derives its Euler–Lagrange equations. 7 5.1 Multi-Observer Joint Action For each observer i, its single-body observation–computation action is b A(i) Q[zi(·)] = ZT 01 2α2 iGab(θi)˙ θa i˙ θb i+1 2β2 igjk(ϕi)˙ ϕj i˙ ϕk i−γiUQ(ϕi)dt+(knowledge graph/attention cost). We focus on control–information main term, ignoring details of knowledge graph and attention cost, absorbing them into effective potential. Definition 5.1 (Multi-Observer Joint Action).Define b Amulti Q[Z(·)] = N X i=1 b A(i) Q[zi(·)] + λcons ZT 0 Econs(t) dt. where λcons >0 controls weight of consensus energy in overall optimization. Minimizing b Amulti Qcorresponds to seeking under given complexity and time budget, optimal multi-observer strategy that can both improve individual task information quality and form collective consensus on task information manifold. 5.2 Euler–Lagrange Equations and Coupled Geodesic–Consensus Dynamics Varying θa iand ϕk irespectively gives following coupled Euler–Lagrange equation system: 1. Control part ¨ θa i+ Γa bc(θi)˙ θb i˙ θc i= 0, i.e., control variable of each observer still evolves along geodesics of (M, G) (ignoring control cooperation); 2. Information part ¨ ϕk i+ Γk mn(ϕi)˙ ϕm i˙ ϕn i=−γi β2 i gkl Q(ϕi)∂lUQ(ϕi)−λcons β2 i gkl Q(ϕi)∂l∂Econs ∂ϕi, where gradient of consensus energy with respect to ϕiis ∂Econs ∂ϕi =X j ωt(i, j)∇ϕi1 2d2 SQ(ϕi, ϕj). Therefore, multi-observer information worldline is geodesic motion with potential driven jointly by “individual task potential” and “consensus potential”. Under unified time scale and small velocity approximation, above dynamics degenerates to combination of aforementioned consensus gradient flow and single-body geodesics, minimum action path corresponds to optimal consensus–task tradeoff under complexity budget. 8 6 Multi-Observer Causal Diamond and Joint Boundary Operator This section extends causal diamond theory from previous paper to multi-observer case, constructing joint boundary operator and discussing its relationship with single-observer boundary operators. 6.1 Multi-Observer Events and Causal Network On event layer E=X×Nextend index, adding observer label: define Eobs =I×X×N, e = (i, x, k), representing “i-th observer at step kin some local perspective of universe configuration x”. Communication events defined on I×I×N, forming multi-layer causal network. For given input–output multi-observer event families {e(i) in }i∈I,{e(i) out}i∈I, and complexity budget T, define multi-observer causal diamond as ♢multi =\ i∈IJ+ Te(i) in ∩J− Te(i) out⊂Eobs. Its boundary can likewise be decomposed into incoming–outgoing parts, layered by observer index. 6.2 Joint Boundary Hilbert Space and Boundary Operator Under QCA realization, each observer layer corresponds to local Hilbert space factor. Multi-observer diamond internal Hilbert space decomposes as H♢multi =O i∈IHbulk,♢,i ⊗ B− ♢,i ⊗ B+ ♢,i, defining joint incoming and outgoing boundary Hilbert spaces B− ♢,multi =O i∈I B− ♢,i,B+ ♢,multi =O i∈I B+ ♢,i. Bulk internal evolution given by U♢multi :H♢multi → H♢multi . Choosing bulk reference state and boundary projection for each observer layer, define joint incoming embedding ι− ♢,multi :B− ♢,multi → H♢multi , joint outgoing projection Π+ ♢,multi :H♢multi → B+ ♢,multi. 9