Unified Theory of Causal Structure:\\ Time Scale, Partial Order, and Generalized Entropy

Unified Theory of Causal Structure: Time Scale, Partial Order, and Generalized Entropy Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract This paper provides a unified characterization of “what is causality” within a single mathematical framework. The core thesis is: causality is not an external relation added onto spacetime or quantum states, but rather a unified object jointly defined by the compatibility of three types of structures: 1. Geometric partial order: light cone structure on globally hyperbolic Lorentzian manifolds and local partial order of small causal diamonds; 2. Unitary evolution and time scale: unified time scale consisting of scattering phase gradient, Wigner–Smith group delay, and spectral shift function; 3. Generalized entropy and information monotonicity: time arrow characterized by generalized entropy extrema on small causal diamonds and QNEC/QFCtype inequalities. On the spectral and scattering side, this paper adopts the scale identity φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω), where φis total scattering semi-phase, ρrel is relative state density, Q(ω) = −iS(ω)†∂ωS(ω) is Wigner–Smith group delay operator. This equality originates from the Birman–Kre˘ın formula and spectral shift function theory, viewing “time delay” as derivative of spectral–phase geometry. On the algebraic quantum field theory and holographic gravity side, this paper characterizes “modular time” on causal diamonds via Tomita–Takesaki modular flow and Null–Modular double cover, whose generator is weighted integral of stress–energy tensor along null boundaries, satisfying Markov inclusion-exclusion and strong subadditivity saturation on overlapping causal diamond chains. On gravity and geometry side, introducing Gibbons–Hawking–York boundary term and its null and corner generalizations ensures well-defined variation, Brown–York quasilocal stress tensor becomes Hamiltonian generator of “time translation” along boundary, geometric time determined by Hamilton–Jacobi relation. This paper establishes the following unified proposition: within the semiclassical– holographic window satisfying local quantum energy conditions, Hadamard states, and small causal diamond limit, there exists a class of unified time scale equivalence classes [τ] such that: 1  Geometric causal partial order is equivalent to existence of strictly increasing time function τ:M→R;  Scattering phase gradient and group delay trace give readout of “observable causal order” under this scale;  Extrema and monotonicity of generalized entropy Sgen on small causal diamonds are equivalent to nonlinear Einstein equations and their stability under this scale. Thus, causality can be restated as: existence of a partial order–time scale structure self-consistent on geometric, scattering, and entropic facets, topologically non-anomalous. This paper provides axiomatic definition of this structure, proposes several main theorems, and gives proof frameworks related to scale identity, information-geometric variational principle, and Null–Modular double cover in appendices. Keywords: Causal structure; Time scale; Partial order; Generalized entropy; Spectral shift function; Wigner–Smith group delay; Birman–Kre˘ın formula; Tomita–Takesaki modular flow; Quantum Null Energy Condition; Small causal diamond; Gibbons–Hawking– York boundary term; Brown–York stress tensor; Null–Modular double cover; Markov property; Information-geometric variational principle 1 Introduction and Historical Context In classical general relativity, causality is often characterized by light cones and time functions. On globally hyperbolic Lorentzian manifolds, stable causality is equivalent to existence of strictly increasing time function T:M→Rsuch that if q∈J+(p), then T(q)≥T(p). This structure guarantees well-posedness of Cauchy problem and condition of “no closed causal curves.” In quantum field theory, causality is usually expressed as commutativity of local operators at spacelike separated points, i.e., microcausality. Wightman axioms and algebraic quantum field theory frameworks construct field algebras given causal structure, but rarely reverse the question: can causal partial order be “recovered” solely from operator algebra and state structure? Development of holography and information theory introduced new perspectives characterizing causality via entropy and relative entropy. Jacobson proposed “entanglement equilibrium hypothesis”: on fixed-volume small geodesic balls, generalized entropy reaches extremum if and only if locally satisfying Einstein equations, establishing correspondence “gravitational field equations = small ball entanglement extremum condition” in semiclassical window. Subsequent work further combined Ryu–Takayanagi formula and relative entropy, relating second-order variation of generalized entropy to bulk gauge energy. On the other hand, long-term development of scattering theory and spectral theory yielded complete “phase–spectral shift–time delay” structure. Birman–Kre˘ın formula shows that for pair of self-adjoint operators (H, H0) satisfying trace-class perturbation conditions, there exists spectral shift function ξ(ω) determining phase of scattering determinant; this yields equivalence relation between derivative of total scattering phase, derivative of spectral shift function, relative state density, and Wigner–Smith group delay trace. These results indicate that time scale with “phase gradient as density” can be established in frequency domain. 2 A recent important development is proof of quantum null energy condition QNEC. QNEC relates stress–energy expectation value in null direction at a point to secondorder deformation of generalized entropy of cut surfaces passing through that point in null direction, being local quantization of ANEC. Koeller–Leichenauer and subsequent work showed that for half-spaces or regions cut by null planes, modular Hamiltonian can be written as local energy flow integral in null direction, establishing “local modular time–energy flow–QNEC” connection. Casini–Huerta–Myers systematically analyzed vacuum modular Hamiltonian for spherical regions in conformal field theory, using conformal transformation to map spherical cut surface to accelerated coordinate system of Rindler wedge, geometrizing modular flow as boost generator, deriving holographic entanglement entropy for spherical regions. Casini– Teste–Torroba studied modular Hamiltonian of general regions on null plane, proving its locality on null plane and establishing Markov property and strong subadditivity saturation. These scattered developments point to stronger unified picture: 1. On scattering and spectral side, time can be understood as parameter determined by phase gradient and group delay scale; 2. On algebraic and holographic side, modular time is determined by state–algebra pair, geometrically realizable as weighted energy flow on null boundary; 3. On gravity and geometry side, extrema and monotonicity of generalized entropy on small causal diamonds are equivalent to local gravitational field equations; 4. On null planes and causal diamond chains, locality and Markov property of modular Hamiltonians give information structure on causal chains. The goal of this paper is to incorporate these structures into single axiomatic system, defining “causality” as object jointly constituted by partial order, unified time scale, and generalized entropy monotonicity, giving equivalence theorems and topological constraints among the three. 2 Model and Assumptions This section gives geometric, scattering, algebraic, and entropic structures used in this paper, listing axioms and assumptions underlying unified causal theory. 2.1 Geometric Background and Causal Diamonds Let (M, g) be four-dimensional oriented, time-oriented Lorentzian manifold with metric signature (−,+,+,+), satisfying:  Global hyperbolicity: there exists Cauchy slice Σ ⊂Msuch that every inextendible timelike or lightlike curve intersects Σ exactly once;  Stable causality: no closed causal curves exist, and there exists smooth time function T:M→Rstrictly increasing along all future-directed timelike curves. 3 For any point p∈M, taking sufficiently small proper scale r≪Lcurv(p), define small causal diamond Dp,r =J+(p−)∩J−(p+), where p±are points at proper time ±ralong some reference timelike direction. The boundary of Dp,r consists of two families of null hypersurfaces N±generated by null geodesics and their intersection lines, constituting basic unit of local causal geometry. Axiom 1 (Geometric Causal Axiom).1. (M, g) satisfies above global hyperbolicity and stable causality; 2. For any pand sufficiently small r, small causal diamond Dp,r is homeomorphic to causal diamond in Minkowski space under normal coordinates, with curvature corrections O(r2). 2.2 Scattering System and Spectral Shift Function On Hilbert space Hconsider pair of self-adjoint operators (H, H0) satisfying:  His trace-class perturbation relative to H0, or resolvent difference is trace-class;  Wave operators W±exist and are complete, so scattering operator S=W† +W−is well-defined;  On absolutely continuous spectrum, Sfiberizes into unitary matrix family S(ω). By spectral shift function theory, there exists locally integrable function ξ(ω) such that for sufficiently smooth test function f, tr(f(H)−f(H0)) = ZR ξ(ω)f′(ω) dω, and Birman–Kre˘ın formula gives det S(ω) = exp−2πiξ(ω). Define total scattering phase Φ(ω) = arg det S(ω), φ(ω) = 1 2Φ(ω), relative state density ρrel(ω) = −ξ′(ω), and Wigner–Smith group delay operator Q(ω) = −iS(ω)†∂ωS(ω), whose trace tr Q(ω) corresponds to group delay at energy ω. From above definitions obtain scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), holding under appropriate regularity and energy window restrictions. 4 Axiom 2 (Scattering Scale Axiom).1. For considered energy window I⊂R, scale identity holds almost everywhere; 2. ρrel(ω)≥0 almost everywhere, and ρrel ≡ 0; 3. Q(ω) is positive semi-definite operator on I, and tr Q(ω) is locally integrable. Based on this define scattering time scale: Definition 2.1 (Scattering Time Scale).Relative to reference point ω0∈I, define τscatt(ω)−τscatt(ω0) = Zω ω0 ρrel(˜ω) d˜ω=1 2πZω ω0 tr Q(˜ω) d˜ω. By Axiom ??,τscatt is strictly increasing on I. 2.3 Boundary Algebra, Modular Flow, and Null–Modular Double Cover Let A∂be observable algebra on appropriate boundary, ωits faithful normal state. Tomita–Takesaki theory assigns corresponding modular operator ∆ωand modular flow σω t(A)=∆it ωA∆−it ω. For vacuum state of spherical or wedge regions in Minkowski spacetime, modular flow can be geometrized as Killing flow in corresponding causal diamond or wedge region. For causal diamond D(p, q), its boundary decomposes into two null hypersurfaces N±; removing corners yields two sheets E±. On each sheet introduce affine parameter λand transverse coordinate x⊥, constructing double cover of null boundary e ED=E+⊔E−. Results by Casini–Teste–Torroba et al. show that in conformal field theory vacuum and its appropriate deformations, modular Hamiltonian of Dcan be written as local energy flow integral along e ED: KD= 2πX σ=±ZEσ gσ(λ, x⊥)Tσσ(λ, x⊥) dλdd−2x⊥, where Tσσ is stress–energy tensor component in null direction, gσweight function determined by geometry. Axiom 3 (Modular Flow Localization Axiom).1. For small causal diamond Dp,r, above local modular Hamiltonian expression exists within semiclassical–holographic window; 2. Modular time parameter tmod is monotonically co-oriented with null direction affine parameter λ. 5 2.4 GHY Boundary Term, Brown–York Stress, and Geometric Time In gravitational system with codimension-one boundary ∂M, Einstein–Hilbert action SEH =1 16πG ZM R√−gd4x produces normal derivative terms on boundary during variation. To ensure welldefined variation under fixed boundary induced metric hab, must add Gibbons–Hawking– York boundary term SGHY =1 8πG Z∂M Kp|h|d3x, plus generalizations to null and corner terms. Brown–York quasilocal stress tensor defined as Tab BY =2 p|h| δS δhab =1 8πG(Kab −Khab) + ··· , where ··· denotes null and corner corrections. For Killing vector taalong boundary time translation, corresponding Hamiltonian is H∂=ZΣ Tab BYtanbdd−1x, where Σ is spatial slice on boundary, naits normal. Geometric time τgeom can be viewed as parameter generated by H∂, related to boundary action via Hamilton–Jacobi relation. Axiom 4 (Boundary Variation Axiom).1. Action S=SEH +SGHY +··· has welldefined variation under fixed boundary geometric data; 2. Tab BY is bounded, and H∂is well-defined on selected boundary sheet family; 3. Boundary time translation group {Φτgeom }can be conformally aligned with modular flow within semiclassical window. 2.5 Generalized Entropy and Local Quantum Conditions For cut surface Σ inside causal diamond Dp,r, define generalized entropy Sgen(Σ) = A(Σ) 4Gℏ+Sout(Σ), where Sout is von Neumann entropy of quantum field outside cut surface. Jacobson’s “entanglement equilibrium” and subsequent work show that under appropriate constraints, extremum condition of Sgen on small balls or small causal diamonds is equivalent to local Einstein equations. Quantum Null Energy Condition gives local inequality in null direction: ⟨Tkk(x)⟩ψ≥ℏ 2π d2Sout dλ2(x), where kais null vector, λits affine parameter. This inequality has been rigorously proven in broad CFT classes, closely related to local modular Hamiltonian deformation. 6 Axiom 5 (Generalized Entropy–Energy Axiom).1. For each small causal diamond Dp,r, under fixed appropriate “volume” or equivalent local conservation constraint, Sgen attains first-order extremum at a reference cut surface; 2. For all null direction deformations, second-order variation of Sgen satisfies QNEC/QFCtype inequalities; 3. Relative entropy is independent of Cauchy slice foliation, equivalent to Iyer–Wald canonical energy. 2.6 Topology and Z2Sector Assumption In systems with self-referential scattering networks or nontrivial topology, square root of scattering semi-phase gives principal Z2bundle whose holonomy ν√S(γ)∈ {±1}can be viewed as topological indicator on loops. Corresponding bulk BF theory sector class [K]∈H2(Y, ∂Y ;Z2) describes possible topological anomalies. Axiom 6 (Topological Non-anomaly Axiom).In small causal diamond limit and finite regions glued from them, satisfies [K] = 0, equivalently, for all appropriate closed loops γ,ν√S(γ) = +1. 3 Main Results: Theorems and Alignments Under above axiomatic system, this paper proposes and argues following main results. 3.1 Theorem 1: Unified Time Scale Equivalence Class Theorem 3.1 (Unified Time Scale Equivalence Class).Within semiclassical–holographic window where Axioms ??,??, and ?? hold, there exists time scale equivalence class [τ] satisfying: 1. Scattering time scale τscatt belongs to [τ]; 2. Modular time τmod belongs to [τ]; 3. Geometric time τgeom belongs to [τ]. More specifically, there exist constants a > 0, b ∈Rsuch that on considered energy window and causal diamond family, τscatt =aτmod +b, τgeom =cτmod +d, where c > 0, d ∈Rare constants. Equivalence class [τ]is represented by any of above scales, unique up to affine transformation. 7 3.2 Theorem 2: Equivalent Characterizations of Causal Partial Order Theorem 3.2 (Equivalent Characterizations of Causal Partial Order).In region where Axioms ??,??,??,?? hold, unified time scale equivalence class [τ]gives equivalent characterizations of causal partial order: For any p, q ∈M, following are equivalent: 1. Geometric causality: q∈J+(p), i.e., there exists future-directed non-spacelike curve from pto q; 2. Time scale monotonicity: there exists τ∈[τ]such that for all connections γfrom pto qalong timelike curves, τ(p)≤τ(q), with strict inequality for some connecting curve; 3. Generalized entropy monotonicity: for each sufficiently small causal diamond chain {Dj}containing p, q, generalized entropy Sgen is co-monotone with τin null direction, giving non-negative “entropy distance” in p→qlimit. Thus, causal partial order can equivalently be viewed as “monotonicity on unified time scale” or “monotonic structure of generalized entropy flow on small causal diamond chains.” 3.3 Theorem 3: Generalized Entropy Variational Principle and Local Gravitational Equations Theorem 3.3 (IGVP and Einstein Equations).When Axioms ?? and ?? hold, assuming matter fields satisfy appropriate local conservation conditions and Hadamard state conditions, generalized entropy variational condition on small causal diamonds is equivalent to local Einstein equations: For any p∈Mand sufficiently small r, if for all null direction deformations, under fixed appropriate constraints Sgen attains first-order extremum at reference cut surface with non-negative second-order variation, then at p Gab + Λgab = 8πG Tab, where Gab is Einstein tensor, Tab matter stress–energy tensor, Λconstant. Conversely, if above gravitational equations hold and matter state satisfies observed local equilibrium condition, then generalized entropy on small causal diamonds satisfies first-order extremum and second-order non-negativity required by Axiom ??. 3.4 Theorem 4: Null–Modular Double Cover, Markov Property, and Causal Chains Theorem 3.4 (Markov Structure on Causal Chains).When Axioms ??,??,?? hold in conformal field theory vacuum or its small deformations, for region family on null plane or causal diamond family {Dj}as their conformal images, modular Hamiltonians satisfy inclusion-exclusion structure 8 K∪jDj=X k≥1 (−1)k−1X j1<···<jk KDj1∩···∩Djk, with corresponding relative entropy satisfying Markov property and strong subadditivity saturation. This yields: 1. Information propagation on causal diamond chains is “no extra memory” Markov process; 2. Unified time scale τon this chain agrees with modular time, co-monotone with null direction affine parameter; 3. Generalized entropy arrow on small causal diamond chains agrees with geometric causal arrow. 3.5 Theorem 5: Equivalence of Topological Non-anomaly and Gauge Energy Non-negativity Theorem 3.5 (Topological Non-anomaly).When Axiom ?? holds, in finite regions glued from small causal diamonds, following are equivalent: 1. Z2–BF bulk sector class [K]=0; 2. For all physically allowed closed loops γ, holonomy of scattering semi-phase square root satisfies ν√S(γ) = +1; 3. On matter configurations satisfying gravitational field equations and local quantum conditions, second-order variation of gauge energy on small causal diamonds is non-negative. Conversely, if there exists [K]= 0 or loop with ν√S(γ) = −1, one can construct configuration violating gauge energy non-negativity, breaking consistency of generalized entropy monotonicity and causal arrow. 4 Proofs This section gives proof ideas and key steps for main theorems, placing technical details in appendices. 4.1 Proof Idea for Theorem ??: Unified Time Scale Equivalence Class Step 1: Existence and affine uniqueness of scattering time scale By Axiom ??, scale identity holds in energy window I, with ρrel(ω)≥0 almost everywhere, not identically zero. Define τscatt(ω)−τscatt(ω0) = Zω ω0 ρrel(˜ω) d˜ω, 9 2. Geometric partial order (M, ⪯), monotonicity on unified scale, and generalized entropy monotonicity on small causal diamond chains are equivalent, giving three complementary characterizations of causality; 3. Generalized entropy variational principle on small causal diamonds is equivalent to local Einstein equations, making gravitational field equations expression of “how entropy organizes on causal boundary”; 4. Null–Modular double cover and Markov property guarantee information propagation on causal diamond chains has locality and no extra memory structure; 5. Topological non-anomaly of Z2–BF sector is equivalent to gauge energy non-negativity, topologically constraining allowed sectors of causal structure. In this picture, spacetime metric, scattering matrix, modular flow, and generalized entropy are no longer independent objects, but manifestations of same causal structure in different projections. Time is understood as strictly monotone scale coordinate on this structure, whose arrow jointly determined by generalized entropy monotonicity and topological non-anomaly. Acknowledgements Authors thank research work in relevant fields for foundations provided to this paper, including spectral shift function and Birman–Kre˘ın theory, quantum null energy condition and local modular Hamiltonian, holographic entanglement entropy and gravitational field equations, and series of studies on Null–Modular double cover and Markov property. Code Availability All results in this paper based on analytical derivations and existing mathematical physics theorems, without specialized numerical code. If future work conducts simulations based on scattering networks and numerical relativity, corresponding code implementations will be made publicly available separately. References [1] M. Sh. Birman and M. G. Kre˘ın, “On the theory of wave and scattering operators,” Soviet Math. Dokl. 3, 740–744 (1962). [2] K. B. Sinha, “Spectral shift function and trace formula,” Proc. Indian Acad. Sci. (Math. Sci.) 104, 571–588 (1994). [3] D. Borthwick, “The Birman–Kre˘ın formula and scattering phase,” arXiv:2110.06370 (2021). [4] H. Casini, M. Huerta and R. C. Myers, “Towards a derivation of holographic entanglement entropy,” JHEP 05, 036 (2011). 16 [5] T. Jacobson, “Entanglement Equilibrium and the Einstein Equation,” Phys. Rev. Lett. 116, 201101 (2016). [6] R. Bousso, Z. Fisher, S. Leichenauer and A. C. Wall, “Proof of the Quantum Null Energy Condition,” Phys. Rev. D 93, 024017 (2016). [7] S. Balakrishnan, T. Faulkner, Z. U. Khandker and H. Wang, “A general proof of the quantum null energy condition,” JHEP 09, 020 (2019). [8] J. Koeller and S. Leichenauer, “Local modular Hamiltonians from the quantum null energy condition,” Phys. Rev. D 97, 065011 (2018). [9] H. Casini, E. Teste and G. Torroba, “Modular Hamiltonians on the null plane and the Markov property of the vacuum state,” J. Phys. A: Math. Theor. 50, 364001 (2017). [10] G. S´arosi and T. Ugajin, “Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields,” JHEP 01, 012 (2018). [11] A. Shahbazi-Moghaddam, “Aspects of Generalized Entropy and Quantum Null Energy Condition,” PhD thesis, University of California, Berkeley (2020). [12] E. Oh, I.-Y. Park and S.-J. Sin, “Complete Einstein equations from the generalized first law of entanglement,” Phys. Rev. D 98, 026020 (2018). A Scale Identity and Construction of Unified Time Scale This appendix gives derivation framework of scale identity and technical details of existence– uniqueness of unified time scale equivalence class. A.1 Spectral Shift Function and Birman–Kre˘ın Formula Let (H, H0) be self-adjoint operator pair satisfying difference is trace-class or resolvent difference is trace-class. By spectral shift function theory, there exists unique (modulo constant) function ξ(ω) such that for any f∈C∞ 0(R), tr(f(H)−f(H0)) = ZR ξ(ω)f′(ω) dω. Choosing smooth approximation of f(λ) = χ(−∞,ω](λ), can interpret ξ(ω) as trace of two spectral projection difference, i.e., ξ(ω) = trEH((−∞, ω]) −EH0((−∞, ω]). Birman–Kre˘ın formula gives relation between scattering determinant and spectral shift function: det S(ω) = exp−2πiξ(ω). Taking logarithm and differentiating yields 17 ∂ωΦ(ω) = ∂ωarg det S(ω) = −2πξ′(ω), hence define ρrel(ω) = −ξ′(ω) = 1 2πΦ′(ω). A.2 Wigner–Smith Group Delay Trace Scattering matrix S(ω) is unitary operator family on absolutely continuous spectrum; its frequency derivative gives Wigner–Smith group delay operator Q(ω) = −iS(ω)†∂ωS(ω). By unitarity of S(ω), Q(ω) is self-adjoint. Decomposing S(ω) into eigenphases and eigenvectors, S(ω) = Pne2iδn(ω)|n(ω)⟩⟨n(ω)|, then Q(ω) = 2 X n ∂δn(ω) ∂ω |n(ω)⟩⟨n(ω)|+ off-diagonal terms, hence tr Q(ω)=2X n ∂δn(ω) ∂ω =∂ωΦ(ω). This yields scale identity φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). In non-local potential or dissipative scattering cases, can generalize above formula using modified determinants and self-adjoint extension theory; form remains unchanged, only requiring renormalization of spectral shift function. A.3 Windowed Clock and Affine Uniqueness To suppress resonance and high-frequency tail effects, introduce positive definite window function h∆(e.g., Poisson kernel) h∆(ω) = ∆ π(ω2+ ∆2), define convolution Θ∆(ω) = (ρrel ∗h∆)(ω) = ZR ρrel(˜ω)h∆(ω−˜ω) d˜ω. If ρrel(ω)≥0 and not identically zero in considered energy window, then Θ∆(ω)>0 almost everywhere. Define windowed time scale t∆(ω)−t∆(ω0) = Zω ω0 Θ∆(˜ω) d˜ω, then t∆is strictly increasing and continuous. 18 For any function ˜ t∆satisfying same window condition with derivative almost everywhere proportional to Θ∆, there exist a > 0, b ∈Rsuch that ˜ t∆=at∆+b. Thus on energy window I, scale of “scattering clock” is unique up to affine transformation. B Local Derivation of IGVP and Local Einstein Equations B.1 Small Causal Diamond Geometry and Area Variation In Riemann normal coordinates at point p, metric expands as gµν(x) = ηµν −1 3Rµανβ(p)xαxβ+O(|x|3). Considering small ball or small causal diamond centered at pwith radius r, expansion of boundary area and volume contains Rµν contributions. For example d-dimensional small ball volume has V(Bp,r) = Ωd−1 drdh1−R(p) 6(d+ 2)r2+O(r4)i. Along null vector kagenerated geodesic family, expansion θsatisfies Raychaudhuri equation dθ dλ=−1 2θ2−σabσab −Rabkakb. For sufficiently small rand appropriate initial conditions, area second-order variation can be expressed as term containing Rkk plus non-negative contributions like θ2, σ2; in r→0 limit, Rkk dominates area variation. B.2 Generalized Entropy Variation and Local First Law First-order generalized entropy variation is dSgen dλ=1 4Gℏ dA dλ+dSout dλ. First-order relative entropy S(ρ||σ) variation with respect to source state ρgives local first law δSout =δ⟨Kmod⟩, where Kmod is modular Hamiltonian of reference state σ. For spherical region or small causal diamond, can localize Kmod as stress–energy tensor integral, yielding dSout dλ∝ZTkk dλdd−2x⊥. Requiring dSgen/dλ= 0 under fixed volume or equivalent constraint, taking r→0, using proportional relation between Rkk term in area variation and above Tkk term, can obtain 19 Rkk = 8πG Tkk. Holding for all null directions, combined with Bianchi identity and energy–momentum conservation, recovers complete Einstein equations. B.3 Second-order Variation and Gauge Energy Non-negativity Second-order generalized entropy variation relative to reference state can be written as gauge energy E=δ2Srel, whose non-negativity is equivalent to Hollands–Wald gauge energy non-negativity, quantified in QNEC/QFC results. Gauge energy non-negativity ensures IGVP extremum on small causal diamonds is stable extremum, guaranteeing stability of local Einstein equations and consistency of causal arrow. C Null–Modular Double Cover and Markov Structure C.1 Modular Hamiltonian on Null Plane On null plane Pof Minkowski space, consider half-space described by lightlike coordinates (u, v, x⊥), e.g., v≥f(x⊥). Casini–Teste–Torroba give local modular Hamiltonian expression in vacuum state: KA= 2πZA (λ−f(x⊥)) Tvv(λ, x⊥) dλdd−2x⊥, where Ais region on null plane. This expression shows modular flow translates points in null direction, modular time linearly related to affine parameter λ. C.2 Inclusion-exclusion Property and Markov Property For region family {Aj}on null plane, using linear superposition of stress–energy tensor and modular Hamiltonian definition, can prove K∪jAj=X k≥1 (−1)k−1X j1<···<jk KAj1∩···∩Ajk. Relative entropy S(ρA||σA) satisfies strong subadditivity under tensor product structure; combined with above modular Hamiltonian inclusion-exclusion relation, vacuum state of null plane region family satisfies Markov property, i.e., conditional mutual information zero. This means information propagation in causal chain along null direction carries no extra memory, depending only on adjacent segment information. 20 C.3 Conformal Mapping to Causal Diamond Families Through conformal transformation, can map null plane region families to causal diamond families {Dj}in Minkowski space or more general backgrounds. Conformal invariance ensures local structure and inclusion-exclusion properties of modular Hamiltonians preserved, establishing Markov structure on causal diamond chains. Unified time scale τ on this chain co-monotone with null direction affine parameter, unifying causal arrow, modular time arrow, and generalized entropy arrow. 21