Geometrization of Observer Consensus: \\ Conflict Metrics, Information Geometry, and Boundary Time Structure

Geometrization of Observer Consensus: Conflict Metrics, Information Geometry, and Boundary Time Structure Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In the timeless block universe perspective, the universe is modeled as a topological structure consisting of a causal partial order, while any concrete observer can only access a finite region and carries a predictive model about the global causal network. Descriptions by different observers of the same causal region generate conflicts at multiple levels: directed cycles appear when locally gluing partial orders, time scale functions are inconsistent, generalized entropy arrows and modular flow directions are inconsistent, and there is Z2sector mismatch in the Null–Modular double cover. This paper constructs a unified “consensus geometry space” embedding observers’ statistical models, causal sets, time scales, and boundary states into a product manifold with a Riemannian metric, and defines a total potential energy function encoding the above conflict metrics as geometric potential. Under the synergy of linear and nonlinear information geometry, causal set geometry, and quantum state space geometry, we prove: under well-posedness assumptions such as completeness and strong convexity, the gradient flow of this potential gives a natural “consensus dynamics” making all conflict metrics monotonically decrease and converge to a “consensus manifold.” On the consensus manifold, local partial orders can be consistently glued into a global causal partial order, unified mother scale functions differ only by affine rescaling, generalized entropy arrows and modular flow directions agree in overlapping regions, and all observers inhabit the same Z2 topological sector. Finally, we couple this geometrization framework with boundary time geometry, unified time scales, and Null–Modular double covers, proposing applications and engineering implementation pathways in multi-observer quantum field theory, holographic information, and multi-agent systems. Keywords: Causal network; Information geometry; Riemannian consensus; Boundary time geometry; Modular flow; Bures metric; Null–Modular double cover; Multi-observer systems 1 1 Introduction and Historical Context 1.1 Timeless Picture of Time and Causality In classical relativity, spacetime is characterized as a Lorentzian manifold with causal cone structure, where time appears as a manifold parameter; while in the causal set approach, space and time are replaced by a discrete set of events and their partial ordering, with geometry determined by “order + counting.” This perspective strips “time” from fundamental structure, retaining only the causal partial order (E, ⪯) between events, with time arrows and scales derived from partial order and metric structure. At the intersection of quantum field theory and gravity, boundary methods and holographic thinking show that much dynamical information can be compressed to boundaries: the energy derivative of the scattering matrix yields Wigner–Smith group delay and phase time, defining scattering time scales; the Gibbons–Hawking–York boundary term and Brown–York quasilocal quantities indicate that well-defined variation of gravitational action and energy definition are “boundary phenomena” at crucial levels. Moreover, Tomita–Takesaki modular theory and the Connes–Rovelli thermal time hypothesis characterize time as an intrinsic modular flow parameter of state–algebra pairs, providing an algebraic foundation for “boundary time geometry.” 1.2 Information Geometry, Quantum State Geometry, and Consensus Algorithms In statistics and information theory, the Fisher information metric and the information geometry developed by Amari–Nagaoka view parametrized statistical models as Riemannian manifolds with the Fisher–Rao metric, providing a natural geometric background for divergences and gradient flows. In quantum state space, the Bures metric and quantum Fisher information provide natural Riemannian structure for density matrix spaces, closely related to quantum estimation theory and geometric phases. On the other hand, average consensus algorithms in multi-agent systems have been generalized from Euclidean space to Riemannian manifolds, forming Riemannian consensus theory. Tron et al. constructed Riemannian consensus algorithms for Fr´echet means on manifolds with bounded curvature and gave convergence conditions; subsequent work analyzed pathologies and limitations of gradient flow consensus in more general settings. These studies revealed early mathematical structures for “achieving consensus on curved geometry.” 1.3 Multi-Observer, Consistency, and Boundary Time Geometry In the block universe or causal set picture, there exists a conceptual tension: on one hand, the universe’s causal network itself is viewed as a global structure independent of observers; on the other hand, any actual observer can only access a finite causal region, obtain finite-precision boundary scattering data or modular flow information, thus can only construct incomplete predictions of the global structure. Judgments by different observers about the same causal region—causal order, time scales, generalized entropy arrows, topological sectors—may be inconsistent. 2 Traditionally, for multi-observer consistency, much discussion focuses on: under given dynamical laws, how to recover a common macroscopic geometry from local observations; in multi-agent learning, how to enable all agents to converge to the same model through communication and update algorithms. However, elevating these problems to the “boundary time geometry” context requires simultaneously handling: 1. Information geometry on statistical model spaces; 2. Combinatorial geometry on causal set moduli spaces; 3. Hilbert geometry on mother scale function spaces; 4. Bures geometry and modular flow structure on boundary quantum state spaces; 5. Z2topological sectors on Null–Modular double covers. The goal of this paper is: within the unified time scale and boundary time geometry framework, to provide a rigorous geometric characterization of multi-observer conflicts and consensus, construct a “consensus geometry space,” and define a natural potential and gradient flow on it, such that “resolving conflicts” corresponds to a geometric contraction process on this space. 1.4 Main Contributions Against the above background, this paper’s contributions can be summarized as: 1. Propose a family of quantitative metrics for multi-observer conflicts, separately characterizing statistical model divergence, causal partial order gluing conflicts, mother scale function inconsistency, mismatch between generalized entropy arrows and modular flow directions, and Z2sector mismatch in Null–Modular double covers. 2. Construct a unified consensus geometry space M, gluing statistical manifolds, causal set moduli spaces, mother scale Hilbert spaces, and boundary state spaces through weighted direct sum metrics to form a Riemannian manifold suitable for describing multi-observer states. 3. Define on (M, G) a total potential energy function Fencoding the above conflict metrics; prove that along the gradient flow ˙ X=−gradGF, potential energy decreases monotonically, and under strong convexity and completeness conditions converges exponentially to a “consensus manifold” Mcons. 4. On the consensus manifold, give a geometric–physical characterization of “complete consensus”: there exists a global causal partial order, unified mother scale, unified entropy arrow and modular flow, unified Z2sector, and all observer states can be embedded in the same boundary time geometry and Null–Modular structure. 5. Through simple models, demonstrate the application potential of this framework in finite causal sets, multi-agent learning, and boundary scattering networks, and propose preliminary engineering implementation schemes. 3 2 Model and Assumptions 2.1 Universe Causal Network and Causal Sets Let Ebe the set of all events in the universe. The causal relation is given by a partial order ⪯⊂ E×Esatisfying: 1. Reflexivity: for any e∈E, we have e⪯e; 2. Antisymmetry: if e⪯fand f⪯e, then e=f; 3. Transitivity: if e⪯fand f⪯g, then e⪯g. The pair (E, ⪯) is called the universe causal network or causal set. Similar to causal set theory, under appropriate assumptions one can relate (E, ⪯) to continuous spacetime geometry, but this paper does not presuppose a specific continuum limit, using only partial order structure. For any event e∈E, define its causal future and past as J+(e) := {f∈E|e⪯f}, J−(e) := {f∈E|f⪯e}. The topology generated by the set family {J+(e)∩J−(f)|e⪯f}is called the Alexandrov topology, viewing the causal set as a topological space. 2.2 Observers, Local Horizons, and Predictive Models Define the observer set O={O1, . . . , ON}. For each observer Oi: 1. There exists a visible event subset Ei⊂E, called its causal horizon; 2. On Eia local partial order ⪯i⊂Ei×Eiis given, satisfying partial order axioms; 3. A predictive model about the global causal network is provided. Predictive models have two equivalent representations:  Probabilistic representation: on the space of candidate causal networks Cau, a probability measure is given µi:Cau →[0,1],X C∈Cau µi(C)=1, where C= (E, ⪯(C)) is a candidate causal structure;  Parametric representation: there exists a statistical model pθand parameter manifold Θ, with observer model specified by parameter point θi∈Θ. One can convert between the two representations via the map θi7→ µi. The statistical model (Θ, pθ) is equipped with the Fisher–Rao metric, making Θ a statistical manifold. 4 Definition 2.1 (Observer State).The state of observer Oiat a “consensus step parameter” τis defined as the quadruple Xi(τ) := (Ei,⪯i, θi(τ), ωi(τ)), where θi(τ)∈Θ is its statistical model parameter, and ωi(τ) is its state on the boundary algebra (to be specified below). The multi-observer state family {Xi(τ)}constitutes a multiple local description of the universe causal network. 2.3 Unified Time Scale and Mother Scale Function In the unified time scale and boundary time geometry framework, the total scattering semi-phase φ(ω), relative state density ρrel(ω), and the trace of the Wigner–Smith delay operator Q(ω) = −iS†(ω)∂ωS(ω) are unified as the mother scale density function κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where S(ω) is the scattering matrix. This scale simultaneously encodes scattering delay, state density, and group delay, is the core scale object of boundary time geometry, and is compatible with recent studies of Wigner time delay and group delay. Different observers, through their accessible scattering experiments and boundary states, can reconstruct local mother scale functions κi(ω) on some energy window I⊂R. Ideally, there exist constants ai>0, bi∈Rsuch that κi(ω) = aiκ(ω) + bi. To eliminate affine freedom, rescaling of each κion Iis needed. 2.4 Boundary Algebra, Modular Flow, and Null–Modular Double Cover Let the boundary observable algebra be a C∗or von Neumann algebra A∂, with states ωibeing positive normalized linear functionals. Tomita–Takesaki theory guarantees a one-to-one correspondence with modular groups {σ(ωi) t}, whose generator is the modular Hamiltonian K(i), i.e., d dtσ(ωi) t(A)t=0 = i[K(i), A]. The Connes–Rovelli thermal time hypothesis views the modular parameter tas a time scale, making time an intrinsic object of the state–algebra pair. The modular Hamiltonian allows affine transformation K(i)7→ aK(i)+b1without changing physical interpretation. On small causal diamonds and their Null boundaries, changes in generalized entropy Sgen together with modular flow direction characterize the “time arrow.” The Null– Modular double cover lifts the geometry and modular flow of small causal diamonds to a Z2double cover space, whose sector is determined by the cohomology class [K]∈ H2(Y, ∂Y ;Z2). The sectors seen by different observers in their respective covering regions Uiare denoted [Ki]. 5 3 Main Results: Theorems and Alignments Under the above models and assumptions, this section presents the main results of this paper. We first construct conflict metrics, then define the consensus geometry space and potential energy function, and finally state the theorem on gradient flow convergence to the consensus manifold. 3.1 Construction of Conflict Metrics Definition 3.1 (Model Divergence and Consensus Speed).For two observers Oi, Oj, let µi, µjbe their probability measures on Cau. Define the Kullback–Leibler divergence DKL(µi∥µj) := X C∈Cau µi(C) log µi(C) µj(C), and the Jensen–Shannon divergence DJS(µi, µj) := 1 2DKL(µi∥¯µ) + 1 2DKL(µj∥¯µ),¯µ=1 2(µi+µj). If observer states evolve with parameter τ, i.e., µi=µi(τ), define consensus speed as vij(τ) := −d dτDJSµi(τ), µj(τ). Definition 3.2 (Cycle-Breaking Cost).Let R:= SN i=1 ⪯ibe the merged relation of local partial orders, and ⪯glue its transitive closure. Assign to each directed edge e→f∈Ra weight w(e→f)≥0. The cycle-breaking cost is defined as Vcycle := min   X (e→f)∈S w(e→f)S⊂R, R \Shas transitive closure that is a partial order  . Clearly Vcycle ≥0, and Vcycle = 0 if and only if ⪯glue itself is a partial order. Definition 3.3 (Rescaled Mother Scale and Scale Divergence).On energy window I⊂R, choose weight function w(ω)>0 integrable. For observer Oi, choose ai>0, bisuch that ZIκi(ω)−aiκref(ω)−bi2w(ω) dω is minimized, where κref is a reference scale. Denote the rescaled scale κren i(ω) := aiκi(ω) + bi, and the average scale ¯κ(ω) := 1 N N X i=1 κren i(ω). Define the scale divergence ∆2 κ:= ZI 1 N N X i=1 κren i(ω)−¯κ(ω)2!w(ω) dω. 6 Definition 3.4 (Entropy Arrow Mismatch Rate and Modular Flow Difference).On overlapping Null generators, let λbe an affine parameter, and S(i) gen(λ) the generalized entropy computed by observer Oi. Define the entropy arrow mismatch rate Ξij := Roverlap 1sign ∂λS(i) gen = sign ∂λS(j) gendµ Roverlap dµ, where dµis a natural measure and 1(·) is the indicator function. The difference of modular Hamiltonians K(i), K(j)in overlapping regions is defined as ∆(ij) mod := inf a>0,b∈R K(i)−aK(j)−b1 , where the norm can be operator norm or Hilbert–Schmidt norm. Definition 3.5 (Topological Sector Conflict Metric).Let [Ki]∈H2(Y, ∂Y ;Z2) be the Null–Modular double cover sector of observer Oi. Define the topological sector count ∆topo := #{[Ki]|i= 1, . . . , N}, and the pairwise indicator function δ(ij) topo := (0,[Ki] = [Kj], 1,[Ki]= [Kj]. When ∆topo = 1, all observers are in the same Z2sector. 3.2 Consensus Geometry Space and Potential Energy Function The statistical model family {pθ|θ∈Θ}under the Fisher–Rao metric gFR ab (θ) := Eθ∂alog pθ(X)∂blog pθ(X) forms a Riemannian statistical manifold (Θ, gFR). Assume there exists a causal set moduli space (C, dC) with a Riemannian structure gCcompatible with the metric; the mother scale function space is the Hilbert space Hκ=L2(I, w(ω) dω), equipped with standard inner product and metric gκ; the boundary state space in finite dimension can be taken as the set of density matrices S, equipped with Bures metric dBures and corresponding Riemannian structure gBures. Definition 3.6 (Consensus Geometry Space and Total Metric).For Nobservers, define M:= ΘN× C × HN κ× SN, with general element denoted X= (θ1, . . . , θN;C;κ1, . . . , κN;ρ1, . . . , ρN). On Mdefine the total metric 7 G:= α N M i=1 gFR (i)+β gC+γ N M i=1 gκ (i)+δ N M i=1 gBures (i), where α, β, γ, δ > 0 are weight parameters. Definition 3.7 (Consensus Potential Energy Function).Given weights wmodel ij , wΞ ij, wmod ij ≥ 0 and λposet, λκ, λtopo >0, define Fmodel := X 1≤i<j≤N wmodel ij DJS(θi, θj), Fposet := λposetVcycle(C;{⪯i}), Fκ:= λκ∆2 κ({κi}), Fmod := X 1≤i<j≤NwΞ ij Ξij +wmod ij ∆(ij) mod, Ftopo := λtopo∆topo −12. The total potential is defined as F:= Fmodel +Fposet +Fκ+Fmod +Ftopo. Definition 3.8 (Consensus Manifold).The consensus manifold is defined as Mcons := {X∈M|F(X) = 0}. By construction, F≥0, and F= 0 if and only if Fmodel =Fposet =Fκ=Fmod = Ftopo = 0. This extremal condition corresponds to a “complete consensus” state, whose geometric and physical meaning will be analyzed later and in appendices. 3.3 Consensus Gradient Flow and Convergence Theorem On (M, G) consider the gradient flow of potential F. Definition 3.9 (Consensus Gradient Flow).Given initial state X(0) ∈ M, the consensus gradient flow is defined as d dτX(τ) = −gradGFX(τ), X(0) given, where gradGis the gradient with respect to metric G. Proposition 3.10 (Potential Monotonicity).Along the consensus gradient flow, d dτFX(τ)=−gradGFX(τ)2 G≤0. The proof relies on the general formula for gradient flows on Riemannian manifolds, given in Appendix A. Its direct implication is: the consensus evolution process always lowers total conflict potential energy along the “steepest descent direction.” To discuss convergence, we introduce the following assumptions. Assumption 1 (Completeness and Strong Convexity).1. (M, G) is a complete Riemannian manifold; 8 2. Potential F:M → [0,∞) is a C2function and bounded below; 3. There exists constant m > 0 such that on some geodesically convex subset containing the gradient flow trajectory, for any tangent vector v∈TXM, HessGF(X)[v, v]≥m GX(v, v), i.e., Fis m-strongly convex on that subset; 4. The consensus manifold Mcons is nonempty and is a closed geodesically convex subset. Under this assumption, we obtain the following main result. Theorem 3.11 (Exponential Convergence of Consensus Gradient Flow).When Assumption 1 holds, for any initial value X(0) ∈ M, the consensus gradient flow has a unique global solution X(τ). Along this solution: 1. Potential FX(τ)decreases monotonically and converges to a minimum F∗≥0; 2. If F∗= 0, then there exists a unique point X∗∈ Mcons such that distGX(τ), X∗≤Ce−mτ , where constant C > 0depends only on initial conditions and local geometry of F. This theorem shows that under appropriately constructed geometry and potential, the process of “multi-observer conflict resolution” can be understood as a gradient flow line on the consensus geometry space, along which all conflict metrics decrease monotonically and converge exponentially to a point on the consensus manifold. Proposition 3.12 (Equivalence of Cycle-Breaking Cost and Global Partial Order).Under local consistency conditions (judgments on overlapping regions Ei∩Ejby each ⪯iare consistent), the following are equivalent: 1. Vcycle = 0; 2. ⪯glue is a partial order; 3. There exists a global partial order ⪯such that for all i,⪯ |Ei=⪯i. Thus, vanishing of cycle-breaking cost is equivalent to the existence of a global causal structure compatible with all local perspectives. The proof of Proposition ?? is in Appendix B. Proposition 3.13 (Geometric–Physical Meaning of Simultaneous Vanishing of Conflict Metrics).If at some state X∈ M we have Fmodel =Fposet =Fκ=Fmod =Ftopo = 0, then there exist: 9 [7] J. Markdahl, S. E. Tuna, J. M. Hendrickx, “Pathologies of consensus seeking gradient descent flows on manifolds,” Automatica 132, 2021. [8] L. Mi, J. Gon¸calves, M. Dahl, “Riemannian polarization of multi-agent gradient flows,” 2023. [9] S. Chen, X. Liu, “Consensus on complete Riemannian manifolds in finite time,” J. Math. Anal. Appl., 2013. [10] R. D. Sorkin, “Spacetime and Causal Sets,” in Relativity and Gravitation: Classical and Quantum, World Scientific, 1991. [11] B. F. Dribus, “On the Axioms of Causal Set Theory,” arXiv:1311.2148, 2013. [12] S. Baron, “Causal Set Theory is (Strongly) Causal,” Found. Phys., 2025. [13] R. Bourgain, D. Faccio, “Direct measurement of the Wigner time delay for light,” Opt. Lett. 38, 1963 (2013). [14] B. Feti´c et al., “Wigner time delay revisited,” Ann. Phys. 460, 2024. [15] G. W. Gibbons, S. W. Hawking, “Action integrals and partition functions in quantum gravity,” Phys. Rev. D 15, 2752 (1977). [16] J. W. York, “Role of conformal three-geometry in the dynamics of gravitation,” Phys. Rev. Lett. 28, 1082 (1972); “Quasilocal energy in general relativity,” 1992. [17] K. Bhattacharya, “Boundary terms and Brown–York quasi-local parameters in GR and ST gravity,” 2023. [18] A. Parvizi et al., “Freelance holography, part I: Setting boundary conditions in holography,” SciPost Phys. 19, 043 (2025). [19] A. Teimouri, S. Talaganis, J. Edholm, A. Mazumdar, “Generalised boundary terms for higher derivative theories of gravity,” JHEP 08, 144 (2016). [20] K. Sun, G. Lebanon, S. Sra, “An Information Geometry of Statistical Manifold Learning,” AISTATS 2014. A Technical Proof of Gradient Flow Convergence Let (N, h) be a complete Riemannian manifold, and f:N → RaC2function. The gradient flow is defined as d dτx(τ) = −gradhfx(τ). 16 A.1 Uniqueness of Minimum Point Assume there exists m > 0 such that for all x∈ N and v∈TxN, Hesshf(x)[v, v]≥m hx(v, v), i.e., fis m-strongly convex. If x∗, y∗are two minimum points, then gradhf(x∗) = gradhf(y∗) = 0, and f(x∗) = f(y∗). Take geodesic γ: [0,1] → N connecting x∗, y∗, and consider g(s) := f(γ(s)). Then g′′(s) = Hesshf˙γ(s),˙γ(s)≥m h( ˙γ(s),˙γ(s)) ≥0. Thus gis strongly convex, and unless x∗=y∗, it cannot attain minimum at both interval endpoints, so the minimum point is unique. A.2 Existence and Uniqueness of Gradient Flow Solution If gradhfis Lipschitz on bounded sets, then through ODE theory on Riemannian manifolds, one can construct local solutions near any initial value. Completeness and gradient boundedness guarantee the solution cannot escape to infinity in finite time, so the solution extends to all τ≥0. A.3 Exponential Convergence Estimate Let x(τ) be a gradient flow solution, and x∗the unique minimum point. Define Φ(τ) := f(x(τ)) −f(x∗)≥0. Strong convexity and Taylor expansion give f(y)≥f(x) + ⟨gradhf(x),exp−1 xy⟩h+m 2|exp−1 xy|2 h. Taking x=x(τ), y =x∗, noting gradhf(x∗) = 0, we get Φ(τ)≤ −⟨gradhf(x(τ)),exp−1 x(τ)x∗⟩h−m 2|exp−1 x(τ)x∗|2 h. On the other hand, d dτΦ(τ) = ⟨gradhf(x(τ)),˙x(τ)⟩h=−| gradhf(x(τ))|2 h. Combining the above estimates, we obtain d dτΦ(τ)≤ −2mΦ(τ), hence Φ(τ)≤e−2mτ Φ(0). Further using strong convexity, one can relate Φ(τ) with Riemannian distance disth(x(τ), x∗), obtaining disth(x(τ), x∗)≤Ce−mτ , 17 where Cdepends only on initial conditions and local geometry of f. This completes the proof framework for Theorem ??. B Cycle-Breaking Cost and Global Partial Order Existence Given event set Eand local partial order family {⪯i}N i=1, the merged relation R= N [ i=1 ⪯i has transitive closure denoted ⪯glue. Assume for all i, j, judgments on overlapping region Ei∩Ejare consistent, i.e., x⪯iy⇐⇒ x⪯jy, ∀x, y ∈Ei∩Ej. B.1 Equivalence of Zero Cycle-Breaking Cost and Transitive Closure Being Partial Order By definition of cycle-breaking cost: Vcycle = min   X (e→f)∈S w(e→f)|S⊂R, R \Shas transitive closure that is partial order  . Clearly, if there exists a directed cycle, at least one edge must be deleted, so Vcycle >0; conversely, if ⪯glue is already a partial order, no edge deletion is needed, taking S=∅ suffices, so Vcycle = 0. Thus Vcycle = 0 ⇐⇒ ⪯glue is a partial order. B.2 Equivalence of Transitive Closure Being Partial Order and Global Partial Order Existence Define global relation ⪯:=⪯glue. Clearly restriction ⪯ |Eicontains ⪯i. Since local partial orders are consistent on overlapping regions, any relation derived within Eifrom other ⪯jmust be compatible with ⪯i, hence ⪯ |Ei=⪯i. Therefore, if ⪯glue is a partial order, it is a global partial order realizing all local partial orders. Conversely, if there exists global partial order b ⪯satisfying b ⪯|Ei=⪯i, then the graph of b ⪯contains R, its transitive closure is itself, and since b ⪯is a partial order, it is acyclic, hence Vcycle = 0. In summary, Proposition ?? is proved. 18 C Geometric and Physical Meaning of Simultaneous Vanishing of Conflict Metrics Assume at some state X∈ M, all conflict metrics simultaneously vanish: Fmodel =Fposet =Fκ=Fmod =Ftopo = 0. C.1 Consistency of Statistical Models Fmodel = 0 means for all i, j, we have DJS(θi, θj) = 0. Jensen–Shannon divergence is zero if and only if the two distributions are almost everywhere identical, hence all pθiare consistent. In parameter space Θ, this means there exists unique parameter point θ∗such that θi=θ∗for all i. C.2 Existence of Global Causal Partial Order Fposet = 0 i.e., Vcycle = 0; under local consistency assumptions, Appendix B shows there exists global partial order ⪯embedding all local partial orders ⪯i. Thus there exists global causal network (E, ⪯) whose restriction to each observer horizon Eiis the partial order structure. C.3 Unified Mother Scale and Time Scale Consensus Fκ= 0 means scale divergence ∆κ= 0, i.e., for all ω∈I, κren i(ω) = ¯κ(ω). Thus there exists unified mother scale function κ∗(ω) := ¯κ(ω) such that each observer’s local scale coincides with it after appropriate affine rescaling. In the unified time scale framework, this corresponds to all observers adopting the same scale identity among scattering phase derivative, relative state density, and group delay. C.4 Consistency of Entropy Arrow and Modular Flow Direction Fmod = 0 implies for all i, j, Ξij = 0 and ∆(ij) mod = 0. The former means on overlapping Null generators, ∂λS(i) gen and ∂λS(j) gen have consistent signs, so entropy arrow directions agree; the latter means there exist aij >0, bij ∈Rsuch that K(i)=aijK(j)+bij1. Through transitivity, one can choose global coefficients ai>0, bi∈Rsuch that all K(i)are affinely equivalent to some unified modular Hamiltonian K∗. Thus in consensus state, all observers achieve consistency in modular flow and generalized entropy arrow, making time arrow definition a global boundary time geometric structure. C.5 Consistency of Null–Modular Double Cover Sector Ftopo = 0 gives ∆topo = 1, i.e., all [Ki] are identical. By Z2principal bundle classification in cohomology, this shows there exists a global double cover sector [K] such that [Ki] = [K]|Ui. This means Null–Modular double cover topological structure is compatible over all observer horizons, with no sector mismatch on consensus manifold. 19 C.6 Summary Combining C.1–C.5, we give the following picture:  There exists unified statistical model point θ∗describing all observers’ probabilistic predictions of causal network;  There exists unified global causal partial order ⪯accommodating all local partial orders;  There exists unified mother scale function κ∗(ω) and unified modular Hamiltonian K∗, after appropriate prefactors and zero-point choices, all observers’ time scales and modular flows align with them;  There exists unified Null–Modular double cover sector [K]; Thus multi-observer system is in a state completely consistent at all levels of causality, time, entropy, and topology. This state geometrically corresponds to a point or orbit on consensus manifold Mcons, the limiting form of “observer consensus geometrization” in this framework. 20