Unified Theory of Observer--Attention--Knowledge Graph\\ in Computational Universe:\\ Cognitive Dynamics and Discrete Geometric Structure\\ Under Finite Resources

Unified Theory of Observer–Attention–Knowledge Graph in Computational Universe: Cognitive Dynamics and Discrete Geometric Structure Under Finite Resources Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In previous works on the “computational universe” series Ucomp = (X, T,C,I), we separately constructed discrete complexity geometry, discrete information geometry, control manifold (M, G) induced by unified time scale, and task information manifold (SQ, gQ), giving on joint manifold EQ=M × SQjoint variational principle for time–information–complexity. These structures characterize “geometry of computational universe itself” at ontological level, but have not yet explicitly introduced mathematical object of “internal observer”: how does observer with finite resources select attention, construct knowledge graph, and gradually accumulate information on complexity–information geometry? This paper, within framework of computational universe and its continuous geometric limit, gives unified axiomatic and geometric description of “observer– attention–knowledge graph.” We first formalize observer as class of state machine with finite memory O= (Mint,Σobs,Σact,P,U), where Mint is internal memory state space, Σobs is observation symbol space, Σact is action space, Pis attention–observation policy, Uis internal update operator. Based on this structure we define time-dependent attention operator At:X→[0,1], or equivalently visible subset Xatt t⊂X, proving: attention operator defines on complexity–information geometry of computational universe a family of timedependent “reachable sections,” thereby imposing constraints on observer’s worldline. Second, we formalize knowledge graph as Gt= (Vt, Et, wt,Φt), 1 where Vtis finite node set, Et⊂Vt×Vtare relation edges, wtare weights, Φt:Vt→ SQis embedding mapping into task information manifold. We construct knowledge graph Laplace operator ∆t, proving that in suitable limit, spectrum of ∆t approximates Laplace–Beltrami operator on (SQ, gQ), thereby viewing finite-node knowledge graph as “discrete skeleton” on information manifold. Then we introduce observer’s extended worldline on joint manifold bz(t)=(θ(t), ϕ(t), m(t),Gt, At), where (θ(t), ϕ(t)) ∈ EQis control–information state, m(t)∈Mint is internal memory, Gtand Atare knowledge graph and attention at time t. On basis of time– information–complexity joint action, we add observer internal cost and knowledge graph reconstruction cost, obtaining extended observation–computation action, deriving its Euler–Lagrange type conditions, giving variational characterization of “under finite complexity budget and finite memory, how observer selects attention and updates knowledge graph.” Finally, we prove two representative results: 1. Under local Lipschitz and finite capacity assumptions, information entropy increment observable by observer in any finite time is subject to double upper bound of complexity budget and attention bandwidth, giving class of “observer version time–information inequality”; 2. Spectral dimension of knowledge graph tends in long-time limit toward local information dimension of task information manifold, showing that “knowledge graph of rational observer almost necessarily approximates skeleton of true information geometry in infinite time limit.” This paper lays structural foundation at single-observer level for subsequent construction of “multi-observer–consensus geometry–causal network” theory, viewing observer as geometric object internal to computational universe rather than external “measurer.” Keywords: Computational universe; Observer theory; Attention mechanism; Knowledge graph; Cognitive dynamics; Finite resources; Complexity geometry; Information manifold; Spectral convergence 1 Introduction In axiomatic framework of computational universe, universe is abstracted as discrete dynamical system Ucomp = (X, T,C,I), where Xis configuration space, Tis one-step transition relation, Cis single-step cost, Iis information quality. Previous works have constructed under this framework:  Discrete complexity geometry: characterizing problem difficulty and horizons with complexity distance dcomp, complexity volume, and discrete Ricci curvature;  Discrete information geometry and task information manifold (SQ, gQ,ΦQ): embedding task-relevant visible states into information manifold through observation operator families and relative entropy structure;  Control manifold (M, G) induced by unified time scale: constructing complexity metric through scattering mother scale κ(ω) and group delay matrix Q(ω;θ); 2  Joint variational principle for time–information–complexity on joint manifold EQ= M×SQ: geometrizing “optimal algorithms” as minimal worldlines. These structures essentially describe “how universe evolves” and “how information is stored and propagates in universe,” but have not yet explicitly described “how observer internal to universe acts on these structures.” Observers have following characteristics: 1. Finite attention: at any moment, can only access small part of X, or analyze local region of information manifold SQ; 2. Finite memory: capacity of internal state m(t) is finite, can only store finitedimensional summary; 3. Knowledge graph: long-term accumulated cognitive structure can be viewed as finite-node graph embedding SQ, being compressive approximation of information manifold; 4. Resource constraints: number of computational steps and information acquisition amount executable are limited by complexity budget and time budget. Therefore, to characterize observer within computational universe, we need to superimpose on existing geometric structure another layer of “cognitive geometry”: how attention selects submanifolds, how knowledge graph constructs skeleton on information manifold, how these choices are constrained by complexity geometry and information geometry, and how observer optimizes its cognitive behavior under resource constraints. Goal of this paper can be summarized as: Under premise of given unified time scale and complexity–information geometry, give unified axiomatic and variational geometric description of single observer’s attention, knowledge graph, and cognitive dynamics. Subsequent multi-observer and consensus geometry can be constructed on this basis through juxtaposition and interaction of multiple observer objects. 2 Observer Objects in Computational Universe This section defines observer objects in computational universe, giving basic interface between them and computational universe. 2.1 Internal Structure of Observer Definition 2.1 (Observer Object).In computational universe Ucomp = (X, T,C,I), an observer object O= (Mint,Σobs,Σact,P,U) consists of following components: 1. Internal memory state space Mint: countable or finite set, representing observer’s internal cognitive state; 3 2. Observation symbol space Σobs: finite set, representing symbols (or symbol vectors) obtained from single observation; 3. Action space Σact: finite set, representing control or query actions observer imposes on universe; 4. Attention–observation policy P:Mint →∆(Σact), representing distribution of action selection under internal state m∈Mint; 5. Internal update operator U:Mint ×Σobs →Mint, representing how to update internal memory under current internal state and observation result. For simplification, we assume at any discrete time step k: 1. Universe is in configuration xk∈X, observer internal state is mk∈Mint; 2. Observer draws action ak∈Σact from P(mk); 3. Universe generates observation symbol ok∈Σobs according to akand xk(its distribution determined by universe–observation coupling mechanism); 4. Observer updates internal state mk+1 =U(mk, ok). Configuration evolution xk→xk+1 of universe determined by Tand possibly control mechanism affected by ak. 2.2 Attention Operator At discrete level, we formalize observer’s “attention” as time-dependent weight function on configuration space X. Definition 2.2 (Discrete Attention Operator).At time step k, observer’s attention operator is function Ak:X→[0,1], satisfying normalization condition X x∈X Ak(x) = 1, or weaker constraint (e.g., total mass not exceeding some constant). We call Xatt k={x∈X:Ak(x)>0} visible configuration subset at time k. 4 Intuitively, Ak(x) represents observer’s current attention weight on configuration x, typically concentrated around configuration trajectory or some local region. In continuous limit, we prefer to characterize attention on task information manifold. Definition 2.3 (Attention Density on Information Manifold).Under task Q, attention can be viewed as probability density ρt(ϕ) on information manifold SQ, satisfying ρt(ϕ)≥0,ZSQ ρt(ϕ) dµgQ(ϕ) = 1, where dµgQis volume element of gQ. Under embedding ΦQ:X→ SQ, discrete Akand continuous ρtcan correspond through pushforward and sampling. 3 Knowledge Graph as Discrete Skeleton of Information Manifold This section formalizes observer’s knowledge graph, embedding it into task information manifold, obtaining bridge between discrete skeleton and continuous information geometry. 3.1 Definition of Knowledge Graph Definition 3.1 (Knowledge Graph at Time t).Observer’s knowledge graph at time tis quadruple Gt= (Vt, Et, wt,Φt), where: 1. Vtis finite node set, each node representing a “concept” or “abstract state”; 2. Et⊂Vt×Vtis directed or undirected edge set, representing relationships between concepts (such as causality, implication, similarity, etc.); 3. wt:Et→(0,∞) are edge weights, representing relationship strength; 4. Embedding mapping Φt:Vt→ SQ, embeds each node into some point in task information manifold, making knowledge graph become finite sampling skeleton of SQ. 5 3.2 Consistency of Graph Laplace and Information Laplace On knowledge graph Gtdefine undirected edge set e Etand symmetric weights ewt, constructing graph Laplace operator (∆tf)(v) = X u∼vewt(v, u)f(u)−f(v), f :Vt→R. On other hand, information manifold has Laplace–Beltrami operator ∆gQf(ϕ) = 1 pdet gQ(ϕ)∂iqdet gQ(ϕ)gij Q(ϕ)∂jf(ϕ). We hope that in limit of sufficiently large tand sufficiently dense Vt, spectrum of ∆t approximates spectrum of ∆gQ. Definition 3.2 (Spectral Approximation).Knowledge graph Gtis said to spectrally approximate on information manifold (SQ, gQ) if there exist embedding Φt:Vt→ SQ and appropriate weight normalization such that: 1. Φt(Vt) becomes dense in SQas t→ ∞; 2. Kernel weights ewtconstructed based on Φtsatisfy that graph Laplace ∆tunder appropriate scaling Γ-converges to ∆gQ. This setup is consistent with graph Laplace convergence theory in manifold learning, only here interpreted as “observer’s knowledge graph’s asymptotic approximation of information manifold.” 4 Observer Extended Worldline and Cognitive Dynamics This section combines observer with control–information geometry, obtaining extended joint state space and worldline. 4.1 Extended State Space Define extended state space of observer–universe joint b EQ=M×SQ×Mint ×G×A, where: 1. Mis control manifold, SQis task information manifold; 2. Mint is internal memory state space; 3. Gis collection of all finite knowledge graphs; 4. Ais collection of all attention configurations (e.g., probability density ρtor discrete weights Ak). Under time parametrization, joint trajectory of observer–universe is bz(t) = θ(t), ϕ(t), m(t),Gt, At. 6 4.2 Observation–Computation Action We add observer internal cost and knowledge graph update cost on basis of previous time–information–complexity action AQ. Let v2 M(t) = Gab(θ(t)) ˙ θa˙ θb, v2 SQ(t) = gij(ϕ(t)) ˙ ϕi˙ ϕj. Define following terms: 1. Complexity kinetic energy term Kcomp(t) = 1 2α2v2 M(t); 2. Information kinetic energy term Kinfo(t) = 1 2β2v2 SQ(t); 3. Knowledge potential energy term UQ(ϕ(t)) = IQ(ϕ(t)), where IQis task information quality function; 4. Knowledge graph update cost term RKG(t) = λKG DGt+dt,Gt, where Dis distance between graphs (e.g., spectral distance or Gromov–Wasserstein distance); 5. Attention configuration cost term Ratt(t) = λatt Catt(At), e.g., in form of entropy regularization or bandwidth constraint. Definition 4.1 (Observer–Computation Joint Action). b AQ[bz(·)] = ZT 0Kcomp(t) + Kinfo(t)−γ UQ(ϕ(t)) + RKG(t) + Ratt(t)dt. Minimizing b AQgives “optimal” observation–computation–learning strategy under finite resources. 5 Information Accumulation and Attention–Complexity Inequality This section gives representative “observer version time–information inequality”: under complexity budget and attention bandwidth constraints, information amount observer can accumulate in finite time has upper bound. 7 5.1 Information Accumulation Rate Let HQ(t) represent observer’s knowledge amount under task Q, can be taken as sum of information entropy or relative entropy on internal knowledge graph nodes, e.g., HQ(t) = X v∈Vt πt(v)IQ(Φt(v)), where πtis weight distribution on knowledge graph nodes. Information accumulation rate is ˙ HQ(t) = d dtHQ(t). We connect this with complexity velocity and attention bandwidth. 5.2 Attention Bandwidth and Fisher Rate Assume at each moment t, observer samples information manifold through attention density ρt(ϕ), its single-step Fisher information acquisition rate J(t) associated with attention bandwidth, e.g., J(t) = ZSQ ρt(ϕ)∇IQ(ϕ)2 gQdµgQ(ϕ). Under complexity–information joint variational framework, Lipschitz relationship exists between v2 SQ(t) and J(t). 5.3 Information Accumulation Inequality Under local Lipschitz conditions and finite attention bandwidth constraints, following inequality can be proved. Theorem 5.1 (Observer Information Accumulation Upper Bound).Assume: 1. Task information quality function IQis Lipschitz on SQwith bounded gradient: there exist LI, CI>0such that ∇IQ(ϕ)gQ≤CI,∀ϕ∈ SQ; 2. Second moment of observer attention density ρtis bounded, i.e., there exists Batt >0 such that ZSQ ρt(ϕ)d2 SQ(ϕ, ¯ ϕ) dµgQ(ϕ)≤Batt, for some fixed point ¯ ϕand all t∈[0, T]; 3. Observer’s complexity budget is Cmax =ZT 0qGab(θ(t)) ˙ θa˙ θbdt. 8 Then there exists constant K > 0, depending only on CI, Batt and joint geometric structure, such that HQ(T)−HQ(0) ≤K Cmax. Proof in Appendix D.1. This inequality states: under unified time scale and geometric constraints, information amount observer can accumulate has linear upper bound with respect to available complexity resources, attention only changes proportionality constant without changing linear form. 6 Knowledge Graph Dimension Convergence and Information Manifold Skeleton This section proves that under suitable conditions, spectral dimension of observer’s knowledge graph converges in long-time limit to local information dimension of task information manifold. 6.1 Spectral Dimension of Knowledge Graph For knowledge graph Gt, let λ(t) 1≤λ(t) 2≤ · · · be eigenvalue sequence of graph Laplace operator −∆t. Define spectral dimension dspec(t) = −2 lim ε↓0 log Tr exp(ε∆t) log ε, if this limit exists. Intuitively, dspec(t) describes effective dimension of graph at small scales. 6.2 Local Information Dimension of Information Manifold On information manifold (SQ, gQ), local information dimension can be defined as dinfo,Q(ϕ0) = lim R→0 log µgQBR(ϕ0) log R, where BR(ϕ0) is geodesic ball of radius Rnear ϕ0. 6.3 Convergence Theorem Theorem 6.1 (Convergence of Knowledge Graph Spectral Dimension).Assume: 1. Observer’s knowledge graph Gt= (Vt, Et, wt,Φt)spectrally approximates on (SQ, gQ) as t→ ∞; 2. Observer’s long-term attention covers compact region K⊂ SQ, and Φt(Vt)⊂Kfor sufficiently large t; 3. For any ϕ0in K, local information dimension dinfo,Q(ϕ0)exists and is constant dinfo,Q. 9