Time--Information--Complexity Unified Variational Principle\\ in Computational Universes:\\ Computational Worldlines on Control--Scattering Manifold\\ and Task Information Manifold

Time–Information–Complexity Unified Variational Principle in Computational Universes: Computational Worldlines on Control–Scattering Manifold and Task Information Manifold Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In previous works on the “computational universe” series, we abstracted the universe as discrete object Ucomp = (X, T,C,I), constructing discrete complexity geometry (complexity distance, volume growth, and discrete Ricci curvature based on configuration graph) and discrete information geometry (based on task-aware relative entropy and Fisher structure) on it, and gave continuous limit of complexity geometry under unified time scale scattering mother scale: a control manifold M with Riemannian metric G. However, these geometric structures still separately characterize “time/resource cost” and “information quality/task-relevant states”, lacking a framework to unify both under a single variational principle. This paper, building on control manifold (M, G) and task information manifold (SQ, gQ), introduces joint manifold EQ=M×SQ, constructing on it a time–information–complexity joint action AQ, thereby characterizing “computational trajectories” in computational universe as minimal curves on joint manifold (computational worldlines). Specifically, we first give action at discrete level Adisc Q(γ) = X kαC(xk, xk+1) + β dinfo,Q(xk, xk+1)−γ∆IQ(xk, xk+1), proving that under appropriate scaling, this discrete action family Γ-converges as h→0 to continuous action AQ[θ(·), ϕ(·)] = ZT 01 2α2Gab(θ)˙ θa˙ θb+1 2β2gij(ϕ)˙ ϕi˙ ϕj−γ UQ(ϕ)dt, 1 where θ(t)∈ M is control trajectory, ϕ(t)∈ SQis task information state, UQis task-related information potential function (e.g., negative information quality). Then we derive Euler–Lagrange equations on joint manifold EQ, proving that minimal trajectories satisfy coupled “geodesic equations with potential”: control part evolves along geodesics of (M, G) but receives feedback from gradient of UQ with respect to ϕ; information part evolves along geodesics of (SQ, gQ) but is modulated by control trajectory θ. Furthermore, using standard variational methods and Γ-convergence theory, we prove: under unified time scale and local Lipschitz assumptions, discrete optimal computational paths converge in the limit to minimal worldlines on joint manifold, achieving rigorous correspondence between “optimal algorithms in discrete computational universe” and “continuous time–information– complexity worldlines.” This paper concludes with discussion of minimization problems with resource constraints: maximizing task information quality under fixed time budget or complexity budget. We give equivalent Lagrange multiplier form, thereby characterizing “optimal information acquisition strategy under given budget” as a class of geodesic flows with effective potential. Results of this paper provide variational foundation at intrinsic dynamics level for subsequent construction of categorical equivalence between “computational universe ↔physical universe.” Keywords: Computational universe; Variational principle; Complexity geometry; Information geometry; Joint manifold; Computational worldline; Euler–Lagrange equations; Γ-convergence; Resource constraints 1 Introduction From the “computational universe” perspective, the entire universe is abstracted as discrete dynamical system: one-step update relation Ton configuration space Xand singlestep cost Cdescribe resources needed to go from one state to another; information quality function Ievaluates at task level the “goodness” of a configuration relative to goals. Previous works showed that under axioms of finite information density and local update, (X, T,C) can be viewed as complexity graph, constructing complexity distance, complexity ball volume, complexity dimension, and discrete Ricci curvature, thereby using discrete geometry to characterize “problem difficulty” and “horizon structure”; simultaneously, through observation operator families and task-aware relative entropy, we defined information distance and information balls on configuration space, geometrizing “taskrelevant distinguishability.” Under unified time scale scattering mother scale, single-step cost of computational universe can be viewed as discrete sampling of actual physical time scale: for physically realizable computational processes, there exist control manifold Mand scattering matrix family S(ω;θ), such that control derivatives of group delay matrix Q(ω;θ) induce complexity metric G, whereby discrete complexity distance approximates geodesic distance on (M, G) in refinement limit. This result unifies discrete complexity geometry with physical time scale into a Riemannian geometric framework. However, to understand “how best to compute in finite time,” neither complexity geometry nor information geometry alone suffices:  Complexity geometry concerns “how far traveled, how much time/resource spent”; 2  Information geometry concerns “how far moved in task space, how much information gained”;  The truly meaningful question is: under given time/complexity budget, how to reach best possible endpoint in information geometry. This naturally leads to a joint variational problem: in joint space, for given task, find minimal/maximal trajectory considering both time cost and information benefit. This paper, building on control manifold (M, G) and task information manifold (SQ, gQ), constructs joint manifold EQ=M × SQ, defining on it a time–information– complexity joint action AQ. Discrete computational paths become piecewise linear approximations on joint manifold, continuous computational worldlines are smooth curves on EQ. Using Γ-convergence and classical variational methods, we prove discrete optimal paths converge in the limit to continuous minimal worldlines, thereby geometrizing the problem of “optimal algorithms” as the problem of “optimal worldlines.” 2 Unified Notation: Computational Universe, Complexity Geometry, and Information Geometry This section briefly summarizes main objects and notation used in previous works for subsequent unified reasoning. 2.1 Computational Universe Object A computational universe object is quadruple Ucomp = (X, T,C,I), where: 1. Xis countable configuration set; 2. T⊂X×Xis one-step update relation; 3. C:X×X→[0,∞] is single-step cost, with C(x, y) = ∞if (x, y)/∈T,C(x, y)∈ (0,∞) if (x, y)∈T, additive along paths; 4. I:X→Ris information quality function (may be task-dependent). Complexity distance defined as dcomp(x, y) = inf γ:x→y C(γ), where path γ= (x0, . . . , xn) satisfies x0=x, xn=y, and (xk, xk+1)∈T. 2.2 Complexity Geometry and Control Manifold Under unified time scale framework, for physically realizable computational universe there exist control manifold Mand scattering matrix family S(ω;θ), whose group delay matrix Q(ω;θ) = −iS†∂ωShas control derivatives inducing complexity metric Gab(θ) = ZΩ w(ω) tr ∂aQ(ω;θ)∂bQ(ω;θ)dω. Under appropriate positive definiteness conditions, (M, G) is Riemannian manifold, discrete complexity distance converges to geodesic distance dGin refinement limit. 3 2.3 Task Information Manifold Given task Q, through observation operator family O={Oj}j∈Jdefine visible state p(Q) x∈∆(YQ) of configuration x. Under appropriate regularity assumptions, these visible states can be embedded into some information manifold SQ:  There exist mapping ΦQ:X→ SQand embedding ΨQ:SQ,→∆(YQ), such that ΨQ(ΦQ(x)) ≈p(Q) x;  Fisher information metric gQgiven by second derivative of relative entropy, constructing Riemannian structure of (SQ, gQ);  Information distance between configurations can be represented using Jensen–Shannon distance or Fisher geodesic distance, denoted dinfo,Q(x, y)≈dSQ(ΦQ(x),ΦQ(y)). We call (SQ, gQ,ΦQ) the information geometric data for task Q. 3 Joint Time–Information–Complexity Manifold With above preparation, we construct joint manifold EQand its metric. 3.1 Definition of Joint Manifold Definition 3.1 (Joint Manifold).For given task Q, define joint manifold EQ=M×SQ. Its point z= (θ, ϕ) simultaneously represents “control state” and “task information state”. In continuous limit, state of an observer or algorithm in computational universe can be viewed as point in EQ. 3.2 Metric Structure On EQ, we introduce product-type metric G=α2G⊕β2gQ, i.e., for tangent vector v= (vM, vSQ)∈TθM ⊕ TϕSQ, define Gz(v, v) = α2Gθ(vM, vM) + β2gQ,ϕ(vSQ, vSQ). Here α, β > 0 are weight parameters used to balance “velocity” measurement in complexity direction and information direction. Under this metric, velocity squared of joint trajectory z(t)=(θ(t), ϕ(t)) is |˙z(t)|2 G=α2Gab(θ(t)) ˙ θa˙ θb+β2gij(ϕ(t)) ˙ ϕi˙ ϕj. Pure geometric length on joint manifold is 4 LG[z] = ZT 0q|˙z(t)|2 Gdt. However, length alone is insufficient to encode “information quality” gain, we also need task-related potential function. 3.3 Information Potential Function Let information quality function of task Qon information manifold be written as IQ: SQ→R, for example IQ(ϕ) = IQ(x) when ϕ= ΦQ(x). We introduce information potential function UQ(ϕ) = V(IQ(ϕ)), where V:R→Ris monotone function, generally chosen as V(u) = uor V(u) = fsat(u) (saturation type). In this paper, for simplicity we directly take UQ(ϕ) = IQ(ϕ), viewing “information quality” as negative contribution of potential energy term (corresponding to higher information quality bringing lower action). 4 Discrete Joint Action and Continuous Limit This section constructs joint action for task Qat discrete level, proving its convergence to continuous action in refinement limit. 4.1 Discrete Joint Action Consider discrete computational path γ= (x0, x1, . . . , xn), where (xk, xk+1)∈T. Corresponding complexity increment is ∆Ck=C(xk, xk+1), information distance increment (under task Q) is ∆Dk=dinfo,Q(xk, xk+1), information quality increment is ∆Ik=IQ(ϕk+1)−IQ(ϕk), ϕk= ΦQ(xk). 5 Definition 4.1 (Discrete Joint Action).For task Qand path γ, define discrete joint action Adisc Q(γ) = n−1 X k=0 α∆Ck+β∆Dk−γ∆Ik, where α, β, γ > 0 are weight parameters. Intuitive understanding: each step update simultaneously pays complexity cost α∆Ck and information adjustment cost β∆Dk, and gains information quality increment ∆Ik, contributing −γ∆Ikto action. Optimal path is one that minimizes Adisc Qunder balance of all three. 4.2 Refinement and Standard Time Step To connect discrete and continuous, we introduce discrete time step h > 0, let path length n≈T/h, and set scaling of single-step cost and information distance as ∆Ck=h c(xk, xk+1) + o(h),∆Dk=h d(xk, xk+1) + o(h), ∆Ik=h˙ IQ(tk) + o(h), where tk=kh,c, d, ˙ IQare respectively complexity velocity, information velocity, and information quality rate of change in continuous limit. Under above scaling, discrete action can be approximated as Riemann sum Adisc Q(γ)≈ n−1 X k=0 hα ck+β dk−γ˙ IQ(tk)→ZT 0αc(t) + βd(t)−γ˙ IQ(t)dt. To match geometric structure, we represent c(t), d(t) respectively using velocity norms on (M, G) and (SQ, gQ). 4.3 Continuous Joint Action Let control path be θ: [0, T]→ M, information path be ϕ: [0, T]→ SQ, with corresponding velocity norms v2 M(t) = Gab(θ(t)) ˙ θa(t)˙ θb(t), v2 SQ(t) = gij(ϕ(t)) ˙ ϕi(t)˙ ϕj(t). We choose “energy-type” continuous action: Definition 4.2 (Continuous Joint Action). AQ[θ(·), ϕ(·)] = ZT 01 2α2v2 M(t) + 1 2β2v2 SQ(t)−γ UQ(ϕ(t))dt. where UQ(ϕ) = IQ(ϕ) or some monotone transformation thereof. 6 This is standard “kinetic minus potential” form: first two terms are kinetic energy on complexity and information geometry, latter term is task-related negative potential energy, minimal worldline maintains finite velocity while trying to enter regions of lower information potential energy. 5 Euler–Lagrange Equations and Computational Worldlines This section derives Euler–Lagrange equations on joint manifold, giving dynamical form satisfied by minimal worldlines. 5.1 Lagrangian and Variation Let Lagrangian be L(θ, ˙ θ;ϕ, ˙ ϕ) = 1 2α2Gab(θ)˙ θa˙ θb+1 2β2gij(ϕ)˙ ϕi˙ ϕj−γ UQ(ϕ). Varying θaand ϕirespectively gives Euler–Lagrange equations: For θa: d dtα2Gab(θ)˙ θb−1 2α2(∂aGbc)(θ)˙ θb˙ θc= 0, For ϕi: d dtβ2gij(ϕ)˙ ϕj−1 2β2(∂igjk)(ϕ)˙ ϕj˙ ϕk+γ ∂iUQ(ϕ)=0. where ∂aGbc =∂Gbc/∂θa,∂igjk =∂gjk/∂ϕi. 5.2 Joint Geodesic–Potential Equations In standard Riemannian geometry, geodesic equation can be written as ¨ θa+ Γa bc(θ)˙ θb˙ θc= 0, where Γa bc are Christoffel symbols of Levi–Civita connection. Here we rewrite control and information parts respectively in geodesic–potential form. For control variable θa, let Γa bc(θ) = 1 2Gad∂bGdc +∂cGdb −∂dGbc, where Gad is inverse of metric matrix. Euler–Lagrange equation can be rewritten as ¨ θa+ Γa bc(θ)˙ θb˙ θc= 0. Since control part of Lagrangian contains no explicit potential energy, control trajectory is geodesic on (M, G). For information variable ϕi, similarly defining Γi jk(ϕ) as Christoffel symbols of gQ, Euler–Lagrange equation rewrites as ¨ ϕi+ Γi jk(ϕ)˙ ϕj˙ ϕk=−γ β2gij(ϕ)∂jUQ(ϕ). 7 Right-hand-side term is covariant lift of potential energy gradient on information manifold, representing “driving force” of “information potential” on information trajectory. Therefore, joint worldline satisfies following coupled system: 1. Control part: evolves along geodesics of (M, G); 2. Information part: evolves along geodesics of (SQ, gQ), but driven away from geodesics by gradient of UQ. This can be viewed as special case of “geodesic with potential on complexity–information product manifold.” 6Γ-Convergence for Discrete–Continuous Consistency To prove discrete optimal paths converge in the limit to continuous minimal worldlines, we use Γ-convergence theory. Only structural theorem and proof idea given here, technical details placed in appendix. 6.1 Action Functional Family Consider family of discrete time steps h=T/n, embed discrete path γ(h)= (x0, . . . , xn) into piecewise constant or piecewise linear curve z(h): [0, T]→ EQ, such that z(h)(t) = (θ(h)(t), ϕ(h)(t)), t ∈[kh, (k+ 1)h), and z(h)(kh)=(θk, ϕk) corresponds to xk. Define discrete action A(h) Q[z(h)] = n−1 X k=0 1 2α2∆s2 M,k h+1 2β2∆s2 SQ,k h−γ UQ(ϕk)h, where ∆s2 M,k =dM(θk, θk+1)2, ∆s2 SQ,k =dSQ(ϕk, ϕk+1)2. Under local consistency assumptions, ∆sM,k ≈hqGab(θ)˙ θa˙ θb, ∆sSQ,k ≈hqgij(ϕ)˙ ϕi˙ ϕj. 6.2 Γ-Convergence Theorem Theorem 6.1 (Γ-Convergence, Schematic).Under unified time scale and local regularity assumptions, discrete action functional family {A(h) Q}h>0Γ-converges under appropriate topology (e.g., weak H1topology of z(h)⇀ z) to continuous action functional AQ[z] = ZT 01 2α2Gab(θ)˙ θa˙ θb+1 2β2gij(ϕ)˙ ϕi˙ ϕj−γ UQ(ϕ)dt. In particular, any limit point of discrete minimal sequences is a minimal curve of continuous action. Proof idea in Appendix B.2, based on standard “energy-type functional discretization” Γ-convergence framework: lower semicontinuity given by convex structure and weak topology lower semicontinuity, recovery sequence constructed through time discretization of continuous trajectory. 8 7 Optimal Computational Worldlines Under Resource Constraints In practical problems, we often care about following optimization:  Under given time budget Tor complexity budget Cmax, maximize terminal information quality IQ(ϕ(T));  Or under given terminal information quality requirement Itarget, minimize required time or complexity. Using Lagrange multiplier method, resource constraints can be absorbed into joint action. For example, maximizing IQ(ϕ(T)) under given T, equivalent to minimizing under free endpoint condition e AQ[z] = ZT 01 2α2v2 M+1 2β2v2 SQdt−γ IQ(ϕ(T)), which differs from previous action only in potential energy term. Corresponding Euler–Lagrange equations in bulk region same as before, but at endpoint add natural boundary condition β2gij(ϕ(T)) ˙ ϕj(T) = γ ∂iIQ(ϕ(T)). This boundary condition can be viewed as “endpoint reflection condition”: at endpoint, ratio of information velocity to information quality gradient controlled by parameter γ/β2, reflecting preference strength for endpoint information quality. Similarly, minimizing time under given information quality target can be obtained through constraint IQ(ϕ(T)) = Itarget and introducing multiplier λto get equivalent free problem, whereby obtaining set of geodesic–potential equations with global constraints. These variational problems provide geometric perspective for “optimal algorithm design”: seeking minimal curves on joint manifold EQsatisfying resource constraints and endpoint information constraints is precisely seeking optimal computational worldlines in computational universe. A Derivation of Euler–Lagrange Under Metric and Potential A.1 Details of Variational Derivation Let L(θ, ˙ θ;ϕ, ˙ ϕ) = 1 2α2Gab(θ)˙ θa˙ θb+1 2β2gij(ϕ)˙ ϕi˙ ϕj−γ UQ(ϕ). For variation θa7→ θa+εηa(with ηa(T0) = ηa(T1) = 0) we have δL =1 2α2(∂cGab)ηc˙ θa˙ θb+α2Gab ˙ θa˙ηb. After integration 9