Universal Conservation of Information Rate: Unification from Quantum Cellular Automata to Relativity, Mass and Gravity

Universal Conservation of Information Rate: Unification from Quantum Cellular Automata to Relativity, Mass and Gravity Anonymous Author November 26, 2025 Abstract Within quantum cellular automaton (QCA) and finite information ontology framework, we introduce unified axiom: for any local excitation, its external group velocity vext and internal state evolution velocity vint satisfy information rate conservation v2 ext +v2 int =c2. Defining proper time τby internal evolution parameter, can directly derive special relativity’s time dilation, four-velocity normalization and Minkowski line element from this axiom. In continuum limit of linear Dirac-type QCA, internal Hamiltonian Hint gives internal frequency ωint, mass obtains information-theoretic definition mc2=ℏωint, satisfying Zitterbewegung frequency relation ωZB = 2ωint. Combining QCA’s winding number and index invariants, massive excitations can be interpreted as optical path quota bound in topologically nontrivial self-referential loops. At many-body level, introducing local information processing density ρinfo(x), from local information volume conservation derives optical metric ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj, where η(x) determines local effective light speed ceff (x) = η2(x)c, and refractive index n(x) = η−2(x). In weak-field limit, this structure recovers first-order expansion of Schwarzschild metric and standard light deflection angle, and through information–gravity variational principle obtains field equation formally equivalent to Einstein equation. Further introducing information mass MI, combining Landauer principle to analyze asymptotic rest behavior and minimal dissipation power of high information mass subjects, giving unified informationtheoretic characterization of mass, gravity and complex energetic structures, proposing testable predictions based on superconducting quantum circuits and quantum simulation platforms. Keywords: Quantum cellular automaton; Information rate conservation; Optical metric; Special relativity; General relativity; Topological mass; Zitterbewegung; Information mass; Landauer principle 1 1 Introduction & Historical Context Special and general relativity characterize universe as four-dimensional manifold (M, gµν) with Lorentz signature. Metric tensor gµν determines spacetime causal structure and geodesics, field equation Rµν −1 2Rgµν = 8πGTµν connects stress–energy tensor Tµν with curvature. Experimental tests such as gravitational redshift, light deflection and gravitational wave detection highly support this geometric narrative. Quantum theory is formulated in Hilbert space H, states as vectors or density operators, observables as self-adjoint operators, time evolution generated by unitary groups or semigroups. Statistical interpretation built on Born rule, with superposition, phase and entanglement forming core structure. Two theories splice in quantum field theory through “field operators defined on background manifold”, but ontological starting points remain separate: one side is deformable geometric stage, other side is linear space of probability amplitudes. When approaching Planck scale, continuity assumption of manifold and classical metric loses empirical support, while Hilbert space structure itself does not depend on continuous spacetime. This motivates ontology based on discrete, finite information structure as natural candidate. Quantum cellular automata (QCA) define local, causal and unitary evolution on countable lattice points, proven to emerge Dirac, Weyl and Maxwell field equations in appropriate limits, with complete structural and classification theory, serving as precise operator realization of “universe as quantum computation” framework [see references [1,2]]. On other hand, gravitational lensing theory widely uses “optical metric” geometric method, viewing light rays as geodesics in optical geometry, calculating deflection angles through Gauss– Bonnet theorem, establishing equivalence between “gravity bending light” and “geometric optics in non-uniform refractive index media” [see references [3,4]]. Information thermodynamics connects logical irreversibility with heat dissipation through Landauer principle: erasing one bit of information in thermal bath at temperature Tdissipates at least heat kBTln 2, showing information update is resource consumption process constrained by physical laws [see reference [5]]. This paper proposes: in unified perspective of QCA and finite information ontology, can use “information rate conservation” as sole microscopic axiom. Each local excitation in universe possesses finite “information rate budget” at each microscopic time step, this budget corresponding to maximal propagation velocity c, allocated in Pythagorean manner between “displacement” (external motion) and “internal evolution” (local Hilbert space self-referential update). This paper will prove this axiom sufficient to derive special relativistic geometry, mass–frequency relation, optical structure of weak-field gravity, and basic constraints on information mass and dissipation, thereby providing unified information-theoretic characterization of mass, gravity and complex energetic structures. 2 Model & Assumptions 2.1 QCA Universe and Finite Information Let Λ be countable connected graph, nodes representing “spatial cells”. Each node x∈Λ carries finite-dimensional Hilbert space Hx≃Cd. For any finite subset F⋐Λ define local Hilbert space HF=O x∈FHx, local operator algebra is B(HF). Global quasilocal C∗algebra is A=[ F⋐ΛB(HF). 2 Quantum cellular automaton specified by ∗-automorphism α:A → A, requiring existence of unitary operator Usuch that α(A) = U†AU, A ∈ A, with finite propagation radius R < ∞such that for any local operator Asupported on Fhave supp α(A)⊂BR(F), where BR(F) is radius Rneighborhood of Fin graph distance sense. Given initial state ω0, discrete time evolution is ωn=ω0◦αn, n ∈Z. Assume Λ can embed into three-dimensional Euclidean space with effective lattice spacing a, single-step evolution corresponding to physical time ∆t. If R= 1, maximal propagation rate is c=a ∆t. Finite local dimension and finite propagation radius mean that in any finite spacetime window, number of distinguishable physical states is finite, universe has upper bound on information capacity in any finite region. 2.2 Single-Excitation Effective Space and External/Internal Velocities Consider local “single-excitation” mode, in appropriate approximation its effective Hilbert space can be represented as Heff ≃ HCOM ⊗Hint, where HCOM describes center coordinate or wave packet envelope, Hint describes internal degrees of freedom. In continuum limit, exists approximate position operator Xand momentum operator P on HCOM, effective Hamiltonian Heff generates coarse-grained time evolution. Define external (group) velocity vext =d dt⟨X⟩=1 iℏ⟨[X, Heff]⟩. Internal state |ψint(t)⟩∈Hint can be viewed as point on projective space CPDint−1, equipped with Fubini–Study metric ds2 FS = 41−|⟨ψ|ψ+dψ⟩|2. Define internal velocity vint := dsFS dt ≥0. 2.3 Information Rate Vector and Universal Conservation Axiom In two-dimensional “information rate space” R2 info = span{eext, eint} define information rate vector u=vexteext +vinteint,|u|2=v2 ext +v2 int. Axiom 2.1 (Information rate conservation).Exists constant c > 0such that for any local excitation and any time thave v2 ext(t) + v2 int(t) = c2. 3 Define information phase angle θ(t) := arctan vint(t) vext(t)∈[0, π/2]. θ= 0 corresponds to lightlike mode, θ=π/2 corresponds to completely local internal mode, 0< θ < π/2 corresponds to massive mode. 2.4 Proper Time and Internal Evolution For any excitation worldline define proper time τsatisfying vint dt =c dτ, i.e. vint =cdτ dt . Substituting into information rate conservation v2 ext +v2 int =c2, letting v:= vext =|dx/dt|, obtain dτ dt 2= 1 −v2 c2, taking positive root dτ dt =r1−v2 c2, i.e., special relativity time dilation relation. Proper time can be understood as natural parameter of “internal information path”, its reduction relative to coordinate time is direct result of external motion occupying optical path quota. 2.5 Local Information Processing Density and Optical Metric In many-body situation, introduce coarse-grained local information processing density ρinfo(x), representing average path length per unit time per unit volume walked by local Hilbert space under Fubini–Study metric, or equivalently, effective expectation value of internal Hamiltonian density. High ρinfo(x) regions can be viewed as regions where “internal computation is highly active”. Allow local rescaling of time and space scales in coordinate system (t, xi), introducing scale factors ηt(x), ηx(x) such that dteff =ηt(x)dt, dℓeff =ηx(x)dℓ. Unitarity of underlying QCA means global Hilbert volume preserved under evolution. Localizing this requirement and simplifying to “physical Hilbert volume corresponding to unit coordinate volume does not change with time”, can model with constraint ηt(x)η3 x(x)=1. In isotropic approximation, taking ηt(x) = η(x), ηx(x) = η−1(x), four-dimensional line element can be written as ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj, 4 where γij(x) is three-dimensional spatial metric. In isotropic case with γij =δij, for null geodesic ds2= 0 have 0 = −η2(x)c2dt2+η−2(x)dx2, thus coordinate light speed  dx dt =ηt(x) ηx(x)c=η2(x)c=: ceff(x), equivalent refractive index n(x) := c ceff(x)=η−2(x). High ρinfo(x) means more frequent local internal evolution; by information rate conservation, external propagation is suppressed; therefore smaller η(x), smaller local effective light speed ceff(x) = η2(x)c, corresponding to larger refractive index n(x) = η−2(x). This is compatible with weak-field gravity result n(r)≃1−2ϕ(r) c2 (ϕ < 0 is Newton potential), giving n(r)>1 for ϕ(r) = −GM/r, thus ceff(r)< c. 3 Main Results (Theorems and Alignments) On basis of above model and assumptions, summarize main mathematical results and physical correspondences of this paper. 3.1 Theorem 1 (Emergence of Special Relativity) Under information rate conservation axiom v2 ext +v2 int =c2 and proper time definition vint dt =c dτ have: 1. Proper time satisfies dτ dt 2= 1 −v2 c2, v := vext = dx dt . 2. Four-velocity of worldline xµ(τ)=(ct(τ),x(τ)) uµ=dxµ dτ =γ(v) (c, v), γ(v) = 1 p1−v2/c2, satisfies normalization under Minkowski metric ηµν = diag(−1,1,1,1) uµuµ=−c2. 3. Line element ds2:= −c2dτ2=−c2dt2+dx2 consistent with Minkowski spacetime metric. Thus special relativistic dynamics appears in this framework as direct corollary of information rate conservation and proper time definition. 5 3.2 Theorem 2 (Mass as Internal Frequency) On internal Hilbert space Hint introduce Hamiltonian iℏ∂τ|ψint(τ)⟩=Hint |ψint(τ)⟩. If exists eigenstate |ψint⟩satisfying Hint |ψint⟩=E0|ψint⟩, then internal state evolves as |ψint(τ)⟩= e−iE0τ/ℏ|ψint⟩, define internal frequency ωint := E0 ℏ. Identifying rest energy E0=mc2, obtain mass–frequency relation m=ℏωint c2. 3.3 Proposition 1 (Zitterbewegung Frequency and Internal Frequency) In continuum limit of one-dimensional Dirac-type QCA, effective Hamiltonian Heff(k)≃cℏk σz+mc2σx, eigenvalues E±(k) = ±p(cℏk)2+m2c4. In Heisenberg picture, evolution of position operator X(t) contains rapid oscillation term with frequency ωZB =2E ℏ (Zitterbewegung). In rest limit k= 0, E=mc2, thus ωZB(0) = 2mc2 ℏ= 2ωint. 3.4 Theorem 3 (Topological Stability and Nonzero Information Phase Angle) Consider one-dimensional translation-invariant QCA, single-step unitary operator U(k)∈U(N) defines closed curve in momentum space. Winding number W[U] = 1 2πiZπ/a −π/a ∂klog det U(k)dk ∈Z preserved under finite-depth local unitary transformations. 1. If W[U] = 0, exists continuous deformation reducing QCA to form containing only massless propagation modes, with internal frequency ωint possibly zero, corresponding to vint = 0, θ= 0. 2. If W[U]= 0, exist local excitations carrying nonzero topological charge, any finite-depth local unitary cannot continuously deform them to topologically trivial vacuum. To maintain topological phase winding, internal Hamiltonian of such excitations must have nonzero eigenfrequency ωint >0, thus vint >0, θ > 0. Therefore nonzero information phase angle θand existence of mass are topologically stabilized by QCA structure, massive excitations can be interpreted as macroscopic manifestation of optical path quota bound by topologically nontrivial self-referential loops. 6 3.5 Theorem 4 (Optical Metric and Weak-Field Gravity) In isotropic assumption, taking ηt(x) = η(x), ηx(x) = η−1(x), construct optical metric ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj. In weak-field limit η(x) = 1 + ϵ(x),|ϵ(x)| ≪ 1, first-order expansion gives g00 ≃ −(1 + 2ϵ)c2, gij ≃(1 −2ϵ)γij. In static spherically symmetric case with isotropic coordinates, identifying ϵ(r) = ϕ(r) c2, where ϕ(r) is Newton potential, then g00 ≃ −1+2ϕ/c2c2, gij ≃1−2ϕ/c2δij, consistent with first-order expansion of Schwarzschild metric in isotropic coordinates. Refractive index n(r) = η−2(r)≃1−2ϕ(r) c2, if ϕ(r) = −GM/r, then n(r)≃1 + 2GM c2r>1, ceff(r) = c n(r)< c. Solving for null geodesics under this metric gives light deflection angle ∆θ=4GM c2b, where bis impact parameter, consistent with standard general relativity result. 3.6 Theorem 5 (Information–Gravity Variational Principle) Consider action Stot[g, ρinfo] = 1 16πG ZM √−g R[g]d4x+ZM √−gLinfo[ρinfo, g]d4x. Varying with respect to gµν (ignoring boundary terms) gives Rµν −1 2Rgµν = 8πG T(info) µν , where T(info) µν := −2 √−g δ(√−gLinfo) δgµν . If choose Linfo such that T(info) µν consistent with standard matter stress–energy tensor in lowenergy limit, then this equation formally equivalent to Einstein field equation. 7 3.7 Theorem 6 (Information Mass and Asymptotic Rest) For systems with internal models and self-referential mechanisms, introduce information mass MI(σ) = fK(σ), D(σ), Sent(σ), where Kis Kolmogorov complexity, Dis logical depth, Sent is internal entanglement entropy, f is monotonically increasing function. Assume average internal information rate vint(MI) needed to maintain given MIis monotonically increasing and bounded by c. From information rate conservation get v2 ext(MI) = c2−v2 int(MI). If lim MI→∞ vint(MI) = c, then lim MI→∞ vext(MI)=0, i.e., high information mass subjects tend to asymptotic rest in external geometry. 3.8 Proposition 2 (Landauer Cost of Maintaining Information Mass) Let system update internal model at rate Rupd, each update erasing average ∆Ibits of old information, then information erasure per unit time ˙ Ierase =Rupd∆I. In thermal bath at temperature T, according to Landauer principle, erasing one bit dissipates at least heat kBTln 2, minimal power consumption is Pmin =kBTln 2 ˙ Ierase =kBTln 2 Rupd∆I. 4 Proofs This section gives derivation structure of above main results, with detailed calculations in appendices. 4.1 Proof of Theorem 1 From proper time definition vint dt =c dτ ⇒vint =cdτ dt , substituting into v2+v2 int =c2, v := vext, obtain v2+c2dτ dt 2=c2, i.e. dτ dt 2= 1 −v2 c2. Taking positive root gives time dilation relation dτ dt =r1−v2 c2. 8 Define Lorentz factor γ(v) = dt dτ =1 p1−v2/c2, four-velocity uµ=dxµ dτ =γ(v) (c, v). Under Minkowski metric ηµν = diag(−1,1,1,1), uµuµ=−γ2c2+γ2v2=−γ2c21−v2 c2=−c2. Define line element ds2=−c2dτ2, substituting dτ2=dt2−dx2 c2 obtain ds2=−c2dt2+dx2, i.e., Minkowski metric. 4.2 Proof Outline of Theorem 2 and Proposition 1 Internal evolution equation iℏ∂τ|ψint(τ)⟩=Hint |ψint(τ)⟩ has eigenstate solution satisfying |ψint(τ)⟩= e−iE0τ/ℏ|ψint⟩, Hint |ψint⟩=E0|ψint⟩. Define ωint =E0 ℏ, if identifying rest energy E0=mc2, then obtain m=ℏωint c2, i.e., Theorem 2. In continuum limit of Dirac-type QCA, effective Hamiltonian Heff(k)≃cℏkσz+mc2σx, eigenvalues E±(k) = ±p(cℏk)2+m2c4 completely consistent with Dirac Hamiltonian. In Heisenberg picture, position operator X(t) satisfies dX dt =i ℏ[H, X] = c α, dα dt =i ℏ[H, α], where αgenerally denoted as Dirac matrix. Integrating gives X(t) = X(0) + c2H−1Pt +iℏc 2H−1e−2iHt/ℏ−1α(0) −cH−1P, second term is uniform motion, third term is rapid oscillation with frequency 2E/ℏ, i.e., Zitterbewegung, where E= +p(cP)2+m2c4. In rest limit P= 0, E=mc2, oscillation frequency ωZB(0) = 2mc2 ℏ= 2ωint, obtaining Proposition 1. 9