Universal Conservation of Information Rate: From Quantum Cellular Automata to Unified Framework of Relativity, Mass, and Gravity

Universal Conservation of Information Rate: From Quantum Cellular Automata to the Unication of Relativity, Mass, and Gravity Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract In the framework of Quantum Cellular Automata (QCA) and nite information ontology, we construct an eective description of single-particle long-wavelength excitations and prove a core result based on Hilbert space geometry and unitarity: for any discrete quantum walk/QCA dened by local unitary evolution and translation invariance that emerges a one-dimensional Dirac-type Hamiltonian in the continuous limit, the external group velocity ( vext ) and internal state evolution velocity ( vint ) of its long-wavelength single-particle eigenmodes must satisfy the Information Rate Conservation Theorem: v2 ext +v2 int =c2, where c is the maximum causal propagation speed of the lattice system. This theorem is not an additional axiom but a geometric result enforced by the local unitarity of QCA and the orthogonal decomposition of the anti-commuting algebra of internal degrees of freedom under the FubiniStudy projective metric. By dening proper time ( τ ) with the internal evolution parameter, special relativity's time dilation, four-velocity normalization, and Minkowski line element can be directly derived from the Information Rate Conservation Theorem. In the continuous limit of Dirac-type QCA, the internal Hamiltonian ( Hint ) gives the internal frequency ( ωint ), and mass obtains an information-theoretic denition: mc2=ℏωint, satisfying the Zitterbewegung frequency relation: ωZB = 2ωint. Combined with the winding number and index invariants of QCA, massive excitations can be interpreted as optical path quotas bound in topologically non-trivial self-referential loops. At the many-body level, we introduce local information processing density ( ρinfo(x) ) and derive the optical metric from local conservation of information volume: ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj, where η(x) determines the local eective speed of light: ceff (x) = η2(x)c, and the refractive index: n(x) = η−2(x). In the weak-eld limit, this structure recovers the rst-order expansion of the Schwarzschild metric and the standard light deection angle, and eld equations formally equivalent to Einstein's equations can be obtained through an information-gravity variational principle. Furthermore, we introduce information mass ( MI ) and, combining with Landauer's principle, analyze the asymptotic stationary behavior and minimum dissipation power of highinformation-mass subjects, providing a unied information-theoretic characterization of mass, gravity, and complex energetic structures, and propose testable predictions based on superconducting quantum circuits and quantum simulation platforms. 1 Keywords: Quantum Cellular Automata; Conservation of Information Rate; FubiniStudy Metric; Optical Metric; Special Relativity; General Relativity; Topological Mass; Zitterbewegung; Information Mass; Landauer's Principle 1 Introduction & Historical Context Special and General Relativity describe the physical world as a four-dimensional manifold (M, gµν) with Lorentzian signature. The metric tensor gµν determines the causal structure and geodesics, and the eld equations Rµν −1 2Rgµν = 8πGTµν relate the stress-energy tensor Tµν to curvature. Experimental tests such as gravitational redshift, light deection, binary pulsar timing, and gravitational wave detection highly support this geometric narrative. Relativity is axiomatically constructed on the constancy of the speed of light and the principle of relativity, introducing the Minkowski line element and Lorentz transformations, with its geometric structure typically regarded as an a priori background. Quantum theory is formulated in Hilbert space H , where states are vectors or density operators, observables are self-adjoint operators, and time evolution is generated by the unitary group. Statistical interpretation is built on the Born rule, with superposition, intrinsic phase, and entanglement constituting core structures. The two theories are stitched together in Quantum Field Theory by dening "eld operators on a background manifold," but their ontological starting points remain separated: one side is a continuous, curvable spacetime manifold, and the other is an abstract linear Hilbert space. When approaching the Planck scale, the assumptions of continuous manifolds and classical metrics lose empirical support, while the Hilbert space structure itself does not depend on continuous spacetime. Quantum Cellular Automata (QCA) provide an alternative formulation with discrete structure as ontology: dening nite-dimensional local Hilbert spaces and local unitary evolution on countable lattices, requiring strict causality and nite propagation radius. Existing research has shown that in appropriate continuous limits, Dirac, Weyl, and Maxwell equations can emerge from local unitary evolution of QCA, and QCA possesses systematic topological classication and index theory. On the other hand, Hilbert space itself has a natural projective geometric structure. The projective Hilbert space CPn is equipped with the FubiniStudy metric, whose arc length gives the natural distance between quantum states. For unitary evolution driven by a time-independent Hamiltonian H , the "velocity" of the state vector under the FubiniStudy metric is determined by the energy uncertainty ∆H , and the "path length" of quantum evolution can be viewed as a measure of information update. This paper attempts to unify the above three threads under an information-theoretic perspective: 1. Assume the universe is microscopically described by a local unitary, translation-invariant QCA with a maximum propagation speed c ; 2. Treat single-particle long-wavelength excitations as a class of eective modes in QCA, whose external motion is described by group velocity ( vext ) and internal state self-referential evolution is described by geometric velocity ( vint ) in projective Hilbert space; 3. Prove that in the continuous limit of Dirac-type QCA, the orthogonal decomposition induced by the anti-commuting structure of the Hamiltonian and the Fubini Study metric necessarily yields the Information Rate Conservation Theorem: v2 ext +v2 int =c2, thereby elevating "conservation of optical path length" from an assumption to a theorem. On this basis, special relativity can no longer be viewed as an independent axiom but as an emergent result of QCA unitarity and Hilbert geometry; mass can be interpreted as the coecient 2 of internal frequency ωint ; gravitational geometry can be interpreted as a manifestation of local information processing density and optical metric structure; and the "information mass" of complex energetic systems can be linked to Landauer's principle, providing a unied picture of mass, gravity, and complexity. 2 Model Assumptions 2.1 QCA Universe and Local Unitarity Let Λ be a countable connected graph, whose nodes represent "spatial cells." Each cell x∈Λ carries a nite-dimensional Hilbert space Hx≃Cd . For any nite subset F⋐Λ , dene the local Hilbert space HF=O x∈FHx, and the local operator algebra B(HF) . The global quasi-local C∗ -algebra is A=[ F⋐ΛB(HF). A Quantum Cellular Automaton is specied by a ∗ -automorphism α:A→A , requiring the existence of a unitary operator U such that α(A) = U†AU, A ∈ A, and the existence of a nite propagation radius R < ∞ , such that for any local operator A supported on F , supp α(A)⊂BR(F), where BR(F) is the R -neighborhood of F in the sense of graph distance. Given an initial state ω0 , the discrete time evolution is ωn=ω0◦αn, n ∈Z. Assume Λ can be embedded in three-dimensional Euclidean space with eective lattice spacing a , and a single step of evolution corresponds to physical time ∆t . If R= 1 , the maximum propagation speed is c=a ∆t. Finite local dimension and nite propagation radius imply that the number of distinguishable physical states in any nite spacetime window is nite, and the information capacity of the universe in any nite region has an upper bound. 2.2 Eective Space of Single Excitations and External Velocity Consider a local "single-excitation" mode, whose eective Hilbert space under appropriate approximation can be represented as Heff ≃ HCOM ⊗Hint, where HCOM describes the center-of-mass coordinate or wave packet envelope, and Hint describes internal degrees of freedom. In the continuous limit, approximate position operator X and momentum operator P exist on HCOM , and the eective Hamiltonian Heff generates coarse-grained time evolution. Dene the external (group) velocity vext =d dt⟨X⟩=1 iℏ⟨[X, Heff]⟩. 3 In symmetric cases, long-wavelength single-particle eigenmodes can be labeled by momentum, |ψp⟩ satisfying Heff|ψp⟩=E(p)|ψp⟩, and the group velocity of this mode is vext(p) = dE dp . 2.3 Internal Hilbert Space and FubiniStudy Metric The internal state |ψint(t)⟩ ∈ Hint can be viewed as a point on the projective space CPDint−1 . The FubiniStudy metric ds2 FS = 41−|⟨ψ|ψ+dψ⟩|2 gives the natural distance between two states in projective Hilbert space. For unitary evolution driven by a time-independent Hamiltonian H : iℏ∂t|ψ(t)⟩=H|ψ(t)⟩, the FubiniStudy velocity can be dened as vFS := dsFS dt . For a general state, vFS is related to energy uncertainty ∆H , while for energy eigenstates, vFS = 0 . In the framework of this paper, the focus is not on vFS on the global H , but on decomposing H into two mutually orthogonal generators corresponding to external translation and internal self-reference, thereby dening "internal evolution velocity" on the internal projective space: vint := ds(int) FS dt ≥0. This velocity characterizes the geometric motion rate of the internal state in CPDint−1 , and its denition depends on the orthogonal decomposition of the Hamiltonian. 2.4 Dirac-Type QCA and Orthogonal Decomposition of Hamiltonian Take the one-dimensional Dirac-type QCA as a concrete model. In the long-wavelength limit, its eective Hamiltonian can be written as Heff(p) = cˆp σz+mc2σx, where σx, σz are Pauli matrices, ˆp=−iℏ∂x , and m is the eective mass parameter. Decompose it into HT=cˆp σz, HM=mc2σx, H =HT+HM. HT generates external translation, and HM generates internal self-referential rotation. Pauli matrices satisfy the anti-commutation relation {σz, σx}=σzσx+σxσz= 0, and σ2 x=σ2 z=I . Therefore H2=H2 T+H2 M= (c2ˆp2+m2c4)I. 4 This gives the operator origin of the relativistic energy-momentum relation E2=p2c2+m2c4. In the Bloch sphere description, internal states correspond to unit vectors on S2 , and the Hamiltonian Heff(p) corresponds to the angular velocity vector on the Bloch sphere: Ω(p) = 2 ℏmc2,0, cp, whose magnitude |Ω(p)|=2E(p) ℏ gives the total geometric velocity in the internal projective space. Due to the orthogonality of the commutator and anti-commutator structures of σx and σz in the Lie algebra, the "velocity components" corresponding to HT and HM can be understood as two mutually orthogonal directions, the sum of whose squares gives the square of the total speed. This structure is the algebraic and geometric basis for the Information Rate Conservation Theorem derived later. 3 Main Results (Theorems and Alignments) Under the above model framework, this paper presents the following main results. 3.1 Theorem 1 (Information Rate Conservation Theorem) In any discrete quantum walk/QCA system satisfying local unitarity and translation invariance that emerges a one-dimensional Dirac-type eective Hamiltonian in the long-wavelength limit, for any positive-energy single-particle eigenmode, denote the external group velocity as vext(p) = dE dp , and dene the internal evolution velocity in the internal projective Hilbert space as vint(p) := cmc2 E(p), then it must hold that v2 ext(p) + v2 int(p) = c2, where c is the maximum causal propagation speed of the QCA. This theorem is guaranteed jointly by the anti-commuting decomposition of the Hamiltonian and the orthogonality of generators under the FubiniStudy metric, and is an inevitable result of local unitarity and Dirac structure, not an additional assumption. 3.2 Corollary 1 (Emergence of Special Relativity) Dene proper time τ using the internal evolution parameter such that vint dt =c dτ. From Theorem 1, we obtain dτ dt 2= 1 −v2 c2, v := vext. 5 Dening four-velocity uµ=dxµ dτ =γ(v) (c, v), γ(v) = 1 p1−v2/c2, then under the Minkowski metric ηµν = diag(−1,1,1,1) , the normalization condition holds: uµuµ=−c2, and the corresponding line element is ds2=−c2dτ2=−c2dt2+dx2. Time dilation and velocity normalization of special relativity emerge directly from conservation of information rate. 3.3 Theorem 2 (Mass as Internal Frequency) Introduce a Hamiltonian on the internal Hilbert space Hint : iℏ∂τ|ψint(τ)⟩=Hint|ψint(τ)⟩. If there exists a stationary state |ψint⟩ satisfying Hint|ψint⟩=E0|ψint⟩, the internal state evolves as |ψint(τ)⟩= e−iE0τ/ℏ|ψint⟩. Dene internal frequency ωint =E0 ℏ. Identifying E0 as the rest energy mc2 , we obtain m=ℏωint c2. Mass is given by the internal frequency, expressing the extent to which the internal self-referential structure occupies the optical path quota. 3.4 Proposition 1 (Zitterbewegung Frequency and Internal Frequency) In the continuous limit of one-dimensional Dirac-type QCA, the eective Hamiltonian is Heff(k) = cℏkσz+mc2σx, eigenvalues are E±(k) = ±p(cℏk)2+m2c4. In the Heisenberg picture, the evolution of the position operator X(t) contains a rapidly oscillating term with frequency ωZB(k) = 2E+(k) ℏ. In the rest limit ( k= 0 ), E+(0) = mc2 , so ωZB(0) = 2mc2 ℏ= 2ωint. The Zitterbewegung frequency is twice the internal frequency. 6 3.5 Theorem 3 (Topological Stability and Non-Zero Information Phase Angle) Consider a one-dimensional translation-invariant QCA whose single-step unitary operator U(k)∈ U(N) denes a closed curve in momentum space. The winding number W[U] = 1 2πiZπ/a −π/a ∂klog det U(k)dk ∈Z remains invariant under nite-depth local unitary transformations. If W[U]= 0 , there exist local excitations carrying non-zero topological charge, which cannot be continuously deformed to the topologically trivial vacuum by any nite-depth local unitary transformation. To maintain topological phase winding, the internal Hamiltonian of such excitations must have a non-zero eigenfrequency ωint >0 , thus vint >0 , and the information phase angle θ= arctan(vint/vext) is non-zero. The existence of mass is therefore stabilized by the topological structure of the QCA. 3.6 Theorem 4 (Optical Metric and Weak-Field Gravity) In the many-body case, introduce coarse-grained local information processing density ρinfo(x) , representing the average path length traversed by the internal Hilbert space under the Fubini Study metric per unit time per unit volume. Allow local rescaling of time and spatial scales in the coordinate system (t, xi) , introducing a scale factor η(x) such that the line element can be written as ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj, where γij(x) is the three-dimensional spatial metric. Dene coordinate speed of light ceff(x) :=  dx dt =η2(x)c, and refractive index n(x) := c ceff(x)=η−2(x). In the static, spherically symmetric, weak-eld limit η(x) = 1 + ϵ(x),|ϵ(x)| ≪ 1, taking ϵ(r) = ϕ(r)/c2 , where ϕ(r) = −GM/r is the Newtonian potential, we obtain g00 ≃ −(1 + 2ϕ/c2)c2, gij ≃(1 −2ϕ/c2)δij, consistent with the rst-order expansion of the Schwarzschild metric in isotropic coordinates. The refractive index n(r)≃1−2ϕ(r) c2≃1 + 2GM c2r>1, gives the light deection angle in gravitational lensing theory based on the GaussBonnet theorem: ∆θ=4GM c2b, where b is the impact parameter, consistent with the standard result of General Relativity. 7 3.7 Theorem 5 (Information-Gravity Variational Principle) Consider the action Stot[g, ρinfo] = 1 16πG ZM √−g R[g]d4x+ZM √−gLinfo[ρinfo, g]d4x. Varying with respect to gµν (ignoring boundary terms) yields Rµν −1 2Rgµν = 8πG T (info) µν , where T(info) µν := −2 √−g δ(√−gLinfo) δgµν . If in the low-energy limit the choice of Linfo makes T(info) µν consistent with the stress-energy tensor of standard matter, this equation is formally equivalent to Einstein's eld equations. 3.8 Theorem 6 (Information Mass and Asymptotic Stationarity) For systems with internal models and self-referential mechanisms, introduce information mass MI(σ) = fK(σ), D(σ), Sent(σ), where K is Kolmogorov complexity, D is logical depth, Sent is internal entanglement entropy, and f is a monotonically increasing function. Assume the average internal information rate vint(MI) required to maintain a given MI is monotonically increasing and bounded by c . From Theorem 1, we have v2 ext(MI) = c2−v2 int(MI). If lim MI→∞ vint(MI) = c, then lim MI→∞ vext(MI)=0, meaning high-information-mass subjects tend to be asymptotically stationary in external geometry. 3.9 Proposition 2 (Landauer Cost of Maintaining Information Mass) Assume a system updates its internal model at a rate Rupd , erasing ∆I bits of old information on average per update. The rate of information erasure per unit time is ˙ Ierase =Rupd∆I. In a heat bath at temperature T , according to Landauer's principle, erasing one bit of information dissipates at least kBTln 2 heat. The minimum power consumption is Pmin =kBTln 2 ˙ Ierase =kBTln 2 Rupd∆I. The continued existence of high-information-mass systems is necessarily accompanied by nonzero minimum power consumption and entropy ux output. 4 Proofs This section provides proofs or proof ideas for the main theorems. Detailed calculations and model details are placed in the appendices. 8 4.1 Proof of Theorem 1 (Information Rate Conservation Theorem) Consider the long-wavelength limit of a one-dimensional Dirac-type QCA with eective Hamiltonian H(p) = HT(p) + HM, HT(p) = c p σz, HM=mc2σx, where p is momentum. Pauli matrices satisfy σ2 x=σ2 z=I,{σz, σx}= 0. Therefore H2(p) = H2 T(p) + H2 M= (c2p2+m2c4)I. For a positive-energy eigenmode |ψp⟩ , we have H(p)|ψp⟩=E(p)|ψp⟩, E2(p) = c2p2+m2c4. The external group velocity is dened as vext(p) = dE dp . From E2=c2p2+m2c4 , we get 2EdE dp = 2c2p, so vext(p) = dE dp =c2p E(p). The internal part is given by HM=mc2σx . Its corresponding "energy share" relative to total energy E(p) is EM E=mc2 E(p). Dene internal velocity as vint(p) := cEM E=cmc2 E(p). This denition can be understood as: in the total information rate budget c , the component occupied by internal evolution is weighted by the "mass energy share." Thus we have v2 ext(p) c2=c4p2 c2E2(p)=c2p2 E2(p),v2 int(p) c2=m2c4 E2(p). Adding them gives v2 ext(p) c2+v2 int(p) c2=c2p2+m2c4 E2(p)= 1, i.e., v2 ext(p) + v2 int(p) = c2. Algebraically, this is a direct result of the Hamiltonian decomposing into two anti-commuting operators, leading to the Pythagorean form of energy squared. Geometrically, on the internal two-dimensional Hilbert space, the Hamiltonian can be written as H(p) = n(p)·σ,n(p) = (mc2,0, cp), corresponding to the angular velocity vector on the Bloch sphere Ω(p) = 2 ℏn(p) = 2 ℏmc2,0, cp. 9 References [1] T. Farrelly, "A Review of Quantum Cellular Automata", Quantum 4, 368 (2020). [2] A. Bisio, G. M. D'Ariano, A. Tosini, "Dirac Quantum Cellular Automaton in One Dimension: Zitterbewegung and Scattering from Potential", Phys. Rev. A 88, 032301 (2013). [3] G. W. Gibbons, M. C. Werner, "Applications of the GaussBonnet Theorem to Gravitational Lensing", Class. Quantum Grav. 25, 235009 (2008). [4] M. Halla, "Application of the GaussBonnet Theorem to Lensing in Static Spherically Symmetric Spacetimes", Gen. Relativ. Gravit. 52, 95 (2020). [5] R. Landauer, "Irreversibility and Heat Generation in the Computing Process", IBM J. Res. Dev. 5, 183191 (1961). [6] For other literature reviews on QCA topological classication, quantum simulation platforms, and optical metric methods, see [14] and references therein. A Appendix A: Continuous Limit and Information Rate Conservation of One-Dimensional Dirac-QCA A.1 A.1 Model Denition Consider a 1D lattice Λ = aZ , where each site x carries a two-component spin ψx=ψx,L ψx,R. Dene the conditional shift operator S as (Sψ)x,L=ψx+a,L,(Sψ)x,R=ψx−a,R, Internal rotation W(θ) = cos θ−i sin θ −i sin θcos θ. Single-step QCA evolution is ψ(t+ ∆t) = U(θ)ψ(t), U(θ) := W(θ)S. In momentum representation, dening ψk=X x e−ikxψx, we have S(k) = eikaσz, U(θ;k) = W(θ)S(k). A.2 A.2 Eective Hamiltonian and Dirac Equation Taking a, ∆t, θ to be small simultaneously, dene eective Hamiltonian Heff(k) = iℏ ∆tlog U(θ;k). Expanding for small parameters gives log U(θ;k)≃ikaσz−iθσx, 16 thus Heff(k)≃ℏ ∆t(kaσz+θσx). Letting c=a ∆t, mc2=ℏθ ∆t, we obtain Heff(k)≃cℏkσz+mc2σx, whose position space form is the one-dimensional Dirac equation iℏ∂tψ(x, t) = −iℏcσz∂x+mc2σxψ(x, t). Dispersion relation is E±(k) = ±p(cℏk)2+m2c4. Group velocity vext(k) = 1 ℏ ∂E+ ∂k =c2k pk2+ (mc/ℏ)2< c. A.3 A.3 Explicit Realization of Internal Velocity and Information Rate Conservation For xed k , the internal state is an eigenstate |u+(k)⟩ in the 2D spin space, whose internal phase evolves with frequency E+(k)/ℏ . In Bloch sphere representation, the qubit state corresponds to a unit vector r∈S2 , and Hamiltonian Heff(k) = n(k)·σ generates uniform rotation around n(k) with angular speed magnitude |Ω(k)|=2|n(k)| ℏ=2E+(k) ℏ. The FubiniStudy metric on CP1 is equivalent to the standard metric on the Bloch sphere, so internal geometric velocity is proportional to |Ω(k)| . In Dirac-type QCA n(k)=(mc2,0, cℏk). Can be written as |n(k)|2= (mc2)2+ (cℏk)2. Decomposing the total "angular velocity vector" into Ω(k) = ΩM(k) + ΩT(k),ΩM(k) = 2 ℏ(mc2,0,0),ΩT(k) = 2 ℏ(0,0, cℏk). The two components are orthogonal, |Ω(k)|2=|ΩM(k)|2+|ΩT(k)|2 . Normalizing the internal velocity vint(k) as vint(k) c=|ΩM(k)| |Ω(k)|=mc2 E+(k), and external velocity vext(k) given by group velocity vext(k) c=cℏk E+(k). Then v2 ext(k) c2+v2 int(k) c2=c2ℏ2k2+m2c4 E2 +(k)= 1, i.e., v2 ext(k) + v2 int(k) = c2. This explicitly realizes the Information Rate Conservation described in Theorem 1 in the DiracQCA model. 17 B Appendix B: Optical Metric, Local Volume Conservation, and Light Deection B.1 B.1 Local Volume Element and Scale Factor Constraints Under the metric ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj the four-volume element is dV4=√−g d4x. Under isotropy assumption, the 3D spatial metric can be written as γijdxidxj= Ψ4(x)dx2, then √−g∝η(x)η−3(x)Ψ6(x) = η−2(x)Ψ6(x). Local Hilbert volume conservation can be simplied as "physical Hilbert volume corresponding to unit coordinate volume is invariant," expressed by constraint ηt(x)η3 x(x)=1. Under isotropy, taking ηt(x) = η(x), ηx(x) = η−1(x), yields the optical metric form ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj. B.2 B.2 Weak-Field Expansion and Schwarzschild Metric In static, spherically symmetric case, Schwarzschild metric in isotropic coordinates (t, r, θ, φ) is ds2=−h1−GM 2c2r 1 + GM 2c2ri2c2dt2+1 + GM 2c2r4(dr2+r2dΩ2). Expanding under weak-eld approximation GM c2r≪1 gives g00 ≃ −(1 −2GM c2r)c2=−(1 + 2ϕ/c2)c2, gij ≃(1 + 2GM c2r)δij = (1 −2ϕ/c2)δij, where ϕ(r) = −GM/r is the Newtonian potential. Starting from the optical metric, taking η(r) = 1 + ϕ(r) c2, γij =δij, expanding gives g00 =−η2c2≃ −(1 + 2ϕ/c2)c2, gij =η−2δij ≃(1 −2ϕ/c2)δij, consistent with the weak-eld expansion of the Schwarzschild metric. 18 B.3 B.3 Refractive Index and Light Deection Angle For null geodesics ds2= 0 under the optical metric, η2(r)c2dt2=η−2(r)dx2, Coordinate speed of light ceff(r) =  dx dt =η2(r)c. Refractive index n(r) := c ceff(r)=η−2(r). In the weak-eld limit η(r) = 1 + ϕ(r) c2, ϕ(r) c2≪1, rst-order expansion gives n(r) = η−2(r)≃1−2ϕ(r) c2. Taking ϕ(r) = −GM/r gives n(r)≃1 + 2GM c2r>1, ceff(r) = c n(r)< c, consistent with the physical picture of light slowing down in weak gravity. In the GibbonsWerner method, the optical metric can be viewed as a 2D Riemann surface, and light trajectories as geodesics on this surface. Calculating the deection angle via the Gauss Bonnet theorem in the weak deection limit yields ∆θ≃4GM c2b, consistent with Einstein's prediction. If only g00 is modied while keeping spatial metric at, the corresponding refractive index is only n(r)≃1−ϕ(r) c2, and the deection angle would be half of the above value, showing that simultaneously deforming time and spatial scales (i.e., introducing optical metric) is crucial for recovering the correct deection factor. C Appendix C: Zitterbewegung and Internal Frequency in Dirac Theory C.1 C.1 Dirac Equation and Plane Wave Solution Consider 1D Dirac equation iℏ∂tψ= (cαp +βmc2)ψ, taking representation α=σz , β=σx , p=−iℏ∂x . Plane wave solutions are ψk,±(x, t) = u±(k) ei(kx−ω±t), ω±=±r(ck)2+m2c4 ℏ2. General wave packet can be written as ψ(x, t) = Za+(k)ψk,+(x, t) + a−(k)ψk,−(x, t)dk. 19 C.2 C.2 Position Operator in Heisenberg Picture In Heisenberg picture, position operator evolves as X(t)=eiHt/ℏX(0)e−iHt/ℏ, H =cαp +βmc2. Heisenberg equations give dX dt =i ℏ[H, X] = cα, dα dt =i ℏ[H, α]. Solving for α(t) and substituting back into X(t) gives X(t) = X(0) + c2H−1Pt +iℏc 2H−1e−2iHt/ℏ−1α(0) −cH−1P, where the second term is uniform motion, and the third term is a rapidly oscillating term with frequency 2E/ℏ , i.e., Zitterbewegung, where E=p(cP)2+m2c4 is the positive energy branch. In the rest limit ( P= 0 ), E=mc2 , oscillation frequency ωZB(0) = 2mc2 ℏ. In the framework of this paper, internal frequency is dened as ωint =mc2 ℏ, thus ωZB(0) = 2ωint. D Appendix D: Information Mass, Landauer's Principle, and Minimum Power Consumption D.1 D.1 Information Erasure and Minimum Dissipation Assume a system's internal state is σ , and its information mass MI(σ) is related to the scale and complexity of its internal model. To maintain model validity, the system must regularly update internal representations with new observations, a process necessarily involving erasure of some old information. Assume the system updates the model at rate Rupd , erasing ∆I bits of old information on average per update. The amount of information erased per unit time is ˙ Ierase =Rupd∆I. In a heat bath at temperature T , according to Landauer's principle, erasing one bit of information must dissipate at least kBTln 2 heat into the environment; minimum dissipated power is Pmin =kBTln 2 ˙ Ierase =kBTln 2 Rupd∆I. This result is independent of the specic physical implementation of the system, depending only on the number of bits erased and environmental temperature, serving as a universal lower bound for power consumption required to implement any high-information-mass system. 20 D.2 D.2 High Information Mass and High Dissipation If information mass MI(σ) increases with the scale, structural complexity, and update frequency of the internal model, then larger MI typically requires larger Rupd and ∆I to continuously discard obsolete information and introduce new information. Therefore, in general Pmin(MI) = kBTln 2 Rupd(MI) ∆I(MI) will increase with increasing MI . Combining with the Information Rate Conservation Theorem of this paper, we arrive at a unied picture: 1. To maintain high MI , the system must allocate a large amount of optical path quota to internal evolution (large vint ), thereby limiting external motion speed vext , manifesting as asymptotic stationarity; 2. Meanwhile, frequent internal updates and erasures lead to continuous entropy ow to the outside, making the system a signicant heat source, manifesting as strong dissipative characteristics; 3. Macroscopically, these two eects often appear in regions with deep gravitational potentials: stellar interiors maintain high information density and complex structure via nuclear reactions while emitting massive radiation; biological and neural systems maintain low entropy structures via metabolism while outputting heat to the environment. Therefore, from the perspective of Information Rate Conservation, the connection between mass, gravity, and complex energetic structures can be uniedly understood as: to maintain a highly ordered, topologically stable structure internally, one must continuously consume optical path quota and output entropy and energy, and this process manifests geometrically as curvature and dynamically as gravity. 21