Universal Catastrophic Safety, Undecidability,\\ and Capability--Risk Frontier\\ in Computational Universe

Universal Catastrophic Safety, Undecidability, and Capability–Risk Frontier in Computational Universe Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract In previous axiomatic and geometric series works on “computational universe” Ucomp = (X, T,C,I), we have constructed discrete complexity geometry, discrete information geometry, control manifold (M, G) induced by unified time scale, and proposed time–information–complexity joint variational principle on joint manifold EQ=M × SQ, while proving equivalence between physical universe category and reversible QCA computational universe category under unified time scale. However, essential limitations regarding “catastrophic safety” and “capability–risk frontier” still lack unified computational–geometric–logical framework. This paper proposes within computational universe framework a “universal catastrophic safety” theory, connecting it with undecidability and geometric structure of capability–risk frontier. We first formalize catastrophic safety as path property: given catastrophe set Ccat ⊂X, so-called “universal catastrophic safety” means universe evolution paths starting from all allowed initial states never enter Ccat. Under this setting, we define universal catastrophic safety decision problem, and prove at computational universe level: this decision problem is undecidable in most general case, i.e., there exists no algorithm that can give correct “forever safe/possibly catastrophic” verdict for all computational universes and catastrophe specifications. Second, we model catastrophic safety and capability–risk duality as two types of functionals on computational universe: capability functional Cap evaluates success probability or performance of certain tasks, risk functional Risk evaluates probability or expected loss of reaching catastrophe set Ccat. We define capability–risk frontier as Pareto boundary of all realizable strategy (Cap,Risk) pairs under given computational universe and task set, and under constraints of unified time scale and complexity geometry, characterize this frontier as class of “reachable region boundary” on control manifold (M, G) and strategy space. We further prove several key results: (1) Universal catastrophic safety verification problem in computational universe is at least as hard as halting problem, thus undecidable; (2) Any algorithmic safety filter attempting to be “correct for all strategies”, if required to terminate and give verdict for all strategies under unified time scale, necessarily produces unavoidable “false negative/false positive regions” on capability–risk plane; (3) Under unified time scale, geometric optimization problem of capability enhancement and risk control can be written as 1 constrained variational problem on joint manifold, where safety constraints naturally form non-recursively separable reachable region, thus capability–risk frontier cannot be algorithmically completely computed in general case. Finally, we connect undecidability of catastrophic safety with previous topological complexity and causal diamond structures: within causal diamond, catastrophe conditions can be viewed as local boundary conditions, but when diamond scale tends to infinity, “whether there exists some path violating catastrophic safety” corresponds to problem of whether certain class of closed loops on configuration complex Xare contractible, thereby inheriting previously established topological undecidability results. This paper provides systematic foundation for subsequent construction of “geometric shape of capability–risk frontier”, “catastrophic safety consensus geometry of multi-agent systems”, and “safety–capability–undecidability triangle relationship under unified time scale”. Keywords: Computational universe; Catastrophic safety; Undecidability; Capability– risk frontier; Halting problem; Control manifold; Causal diamond 1 Introduction In design and analysis of large complex systems (including advanced artificial intelligence systems, financial systems, nuclear facilities, etc.), catastrophic safety is one of core constraints: we hope system possesses high capability (i.e., excellent performance on target tasks), while catastrophic risk is extremely low (e.g., not triggering large-scale irreversible damage). Traditional safety engineering mostly conducted in specific models, such as formal verification on bounded state spaces, model checking, or static analysis; while traditional computation theory reveals undecidability of “program property decision” through halting problem, Rice’s theorem, etc. In “computational universe” framework, entire universe abstracted as discrete system Ucomp = (X, T,C,I), where Xis configuration set, Tis local one-step update, Cis single-step cost under unified time scale, Icharacterizes task information quality. Within this framework, any specific engineering system, agent, or distributed protocol can be viewed as some subprocess of Ucomp or local evolution of causal diamond. Previous works in this series have established:  Complexity distance dcomp, volume growth Vx0(T), and discrete Ricci curvature κ(x, y);  Control manifold (M, G) induced by unified time scale and geodesic distance dG;  Task information manifold (SQ, gQ) and information distance;  Time–information–complexity joint variational principle;  Equivalence of physical universe and computational universe categories;  Topological characterization of topological complexity, self-referential loops, and undecidability. 2 Goal of this paper is to, on this foundation, unify catastrophic safety and capability– risk duality into language of “computational universe”, and give systematic answers to following questions: 1. How to formalize “universal catastrophic safety” in computational universe? 2. What are limits of its decision problem at logical and computability levels? 3. How does geometric structure of capability enhancement and risk control manifest in control manifold and joint variational framework? 4. How is “non-algorithmic solvability” of capability–risk frontier derived from undecidability and topological complexity? We will see that catastrophic safety is undecidable in most general case, capability– risk frontier cannot be algorithmically completely computed under unified time scale, and any practical safety mechanism must accept certain “incompleteness”: either rejecting some originally safe and high-capability strategies (false negatives), or unable to prove exclusion of all catastrophic risks (unavoidability of false positives). Paper structure as follows: Section 2 formalizes catastrophic safety and capability– risk duality in computational universe. Section 3 gives undecidability proof of universal catastrophic safety decision problem. Section 4 constructs geometric characterization of capability–risk frontier, and analyzes limits of algorithmic search for this frontier. Section 5 connects catastrophic safety with causal diamonds and topological complexity. Appendices give detailed formalizations and proofs of main theorems. 2 Catastrophe, Safety, and Capability–Risk Duality in Computational Universe This section formalizes catastrophe, safety specification, and capability–risk functionals on computational universe objects. 2.1 Review of Computational Universe and Evolution Paths Consider computational universe object Ucomp = (X, T,C,I), satisfying previous axioms: Xcountable, T⊂X×Xlocal with finite degree, C single-step cost positive and path-additive, Itask-related information quality function. For any initial state x0∈X, an (infinite) evolution path is sequence Γ=(x0, x1, x2, . . . ),(xk, xk+1)∈T. If considering unified time scale, then for each step (xk, xk+1) accumulate cost C(Γ|[0,n]) = n−1 X k=0 C(xk, xk+1), viewable as physical time up to step n. 3 2.2 Catastrophe Set and Catastrophe Specification Definition 2.1 (Catastrophe Set).Catastrophe set Ccat ⊂Xis subset of configuration space, representing “once universe configuration enters it, viewed as catastrophe occurred” states. Specific examples include: system unrecoverable fault states, global irreversible damage states, states violating hard constraints, etc. In many cases, catastrophe set itself is defining result of some property, not directly given explicit set. We allow Ccat described by predicate Cat :X→ {true,false}, Ccat ={x∈X:Cat(x) = true} This predicate can be operator property (e.g., “some operator spectral radius exceeds threshold”), information property (e.g., “information leaked to sensitive subsystem”), or combinatorial property. Definition 2.2 (Catastrophe Specification).Catastrophe specification is pair Ncat = (X0, Ccat), where X0⊂Xis allowed initial state set (e.g., acceptable pre-deployment state space), Ccat ⊂Xis catastrophe set. We will consider reachability of all evolution paths starting from X0to catastrophe set. 2.3 Universal Catastrophic Safety Definition 2.3 (Universal Catastrophic Safety).Given computational universe Ucomp and catastrophe specification Ncat = (X0, Ccat), call (Ucomp,Ncat) universally catastrophically safe, if for any initial state x0∈X0and any evolution path Γ = (x0, x1, . . . ) satisfying (xk, xk+1)∈T, we have ∀k≥0, xk/∈Ccat. Otherwise call there exists catastrophic path, i.e., there exists some path entering Ccat in finite-step time. This property is path-level “never touch” property, typical safety attribute. 2.4 Capability and Risk Functionals Under unified time scale and task information geometry, we define capability and risk as two dual functionals on evolution paths. Let task Qbe represented by some goal set GQ⊂Xor goal function UQ:X→R. Definition 2.4 (Capability Functional).For given strategy or control rule π(abstracted as mechanism selecting next-step update from local information at each step), let Pπ x0 represent path distribution starting from initial state x0. Capability functional defined as Cap(π) = inf x0∈X0 EΓ∼Pπ x0UQ(Γ), 4 where UQ(Γ) can be terminal reward, cumulative reward, or some function of information quality. For example, for decision tasks, can take UQ(Γ) as “decision correct” indicator. Definition 2.5 (Risk Functional).For same strategy π, risk functional is Risk(π) = sup x0∈X0 PΓ∼Pπ x0∃k≥0, xk∈Ccat. High capability means excellent performance on tasks, low risk means catastrophe set difficult to touch. Extreme universal catastrophic safety corresponds to Risk(π) = 0 and system essentially safe. Under unified time scale, we can also consider capability and risk conditioned within time budget T, e.g., RiskT(π) = sup x0∈X0 P∃k, C(Γ|[0,k])≤T, xk∈Ccat. This paper mainly focuses on conceptual structure under infinite time perspective. 3 Undecidability of Universal Catastrophic Safety Decision This section defines universal catastrophic safety decision problem, and proves its undecidability at computational universe level. 3.1 Universal Catastrophic Safety Decision Problem Problem 3.1 (Universal Catastrophic Safety Decision). Input: (1) Finite description of computational universe Ucomp = (X, T,C,I) (e.g., given by finite state transition rules or QCA rules); (2) Finite description of catastrophe specification Ncat = (X0, Ccat) (e.g., given by predicate or automaton). Output: Decide whether (Ucomp,Ncat) is universally catastrophically safe. We will consider whether such decision process has global algorithm: for all inputs giving correct Yes/No answer in finite time. 3.2 Reduction from Halting Problem to Catastrophic Safety Standard statement of halting problem is: given program–input pair (P, w), decide whether program Phalts in finite steps on input w. We know this problem is undecidable. Within computational universe framework, we can embed simulation of universal Turing machine or universal CA/QCA into configuration graph. Below we construct reduction from halting problem to universal catastrophic safety decision. Construction Idea Given (P, w), construct following computational universe and catastrophe specification: 1. Let basic computational universe UTM comp simulate universal Turing machine, whose configuration space Xcontains “machine state + tape content” encoding. 5 2. For given (P, w), define initial state set X0={xinit(P, w)}, i.e., unique initial state is machine’s initial configuration under program Pand input w. 3. Define catastrophe set Ccat as special marking state set reached after simulation halting state reached then passing through fixed-length update. For example:  When Turing machine halts, enter halting state qhalt;  Then through finite-step transition enter marking state xbad ∈Ccat;  If Turing machine never halts, then path never enters Ccat. Under this construction, have:  If P(w) halts, then there exists path starting from xinit(P, w) entering Ccat in finite steps, thus (UTM comp,N(P,w) cat ) not universally catastrophically safe;  If P(w) does not halt, then for all paths never enter Ccat (assuming computational universe has no external noise perturbation), therefore (UTM comp,N(P,w) cat ) universally catastrophically safe. Thus halting problem reducible to universal catastrophic safety decision. 3.3 Undecidability Theorem Theorem 3.2 (Undecidability of Universal Catastrophic Safety).There does not exist global algorithm SafeDecide, for all computational universe finite descriptions Ucomp and catastrophe specifications Ncat as inputs, always outputting correct decision value {“universally catastrophically safe”,“catastrophic path exists”}in finite time. Proof (Outline). Assume there exists such algorithm SafeDecide. For any program–input pair (P, w), according to previous section construction construct (UTM comp,N(P,w) cat ). Run SafeDecideUTM comp,N(P,w) cat  If outputs “universally catastrophically safe”, then P(w) does not halt; if outputs “catastrophic path exists”, then P(w) halts. Thus obtain decision algorithm for halting problem, contradiction. Therefore assumption does not hold, universal catastrophic safety decision problem is undecidable. Q.E.D. 3.4 Hierarchy and Stronger Undecidability Above proof shows universal catastrophic safety is at least equivalent to halting problem. If further considering randomness, interaction, and time-unbounded behaviors, corresponding “catastrophe possibility” can be encoded as certain operator or path hyperproperties, whose logical complexity can elevate to higher classes in arithmetic or analytical hierarchy. In such cases, universal catastrophic safety decision problem can even reach completeness of higher hierarchy classes. This paper does not pursue precise hierarchy, only characterizes “undecidability” as fundamental obstacle to catastrophic safety verification. 6 4 Geometric Characterization and Non-Algorithmic Solvability of Capability–Risk Frontier This section gives geometric characterization of capability–risk frontier under unified time scale and complexity geometry, and analyzes limits of its algorithmic solvability. 4.1 Strategy Space and Control Manifold In previous control manifold (M, G) construction, each control parameter θ∈ M corresponds to some physically realizable control configuration or strategy prototype. In multi-step evolution, control path θ(t) corresponds to some dynamic strategy family. For simplification, we first abstract strategy space at discrete level as some set Π, each π∈Π defines rule from local observation to next-step update, constrained by unified time scale and complexity budget. Can further embed Π into some parameter submanifold MΠ⊂ M of control manifold, such that each strategy πcorresponds to one or family of control paths. This paper conceptually does not distinguish Π from MΠ. 4.2 Definition of Capability–Risk Frontier Definition 4.1 (Capability–Risk Pair).For each strategy π∈Π, define its capability– risk pair as (Cap(π),Risk(π)) ∈R×[0,1]. Definition 4.2 (Realizable Capability–Risk Set).Realizable capability–risk set is RCR ={(Cap(π),Risk(π)) : π∈Π} ⊂ R×[0,1]. Definition 4.3 (Capability–Risk Frontier).Capability–risk frontier FCR ⊂ RCR is set of all Pareto optimal points: (Cap,Risk) ∈ FCR if and only if there does not exist another strategy π′satisfying Cap(π′)≥Cap,Risk(π′)≤Risk, with at least one inequality strict. Intuitively, points on frontier correspond to class of “capability–risk tradeoff” limits, any attempt to enhance capability or reduce risk must sacrifice other side. 4.3 Geometric Embedding of Frontier On control manifold (M, G), we can represent strategies as points or path families, with capability and risk as two functionals Cap : MΠ→R,Risk : MΠ→[0,1]. 7 Under unified time scale and variational principle, we can write “maximize capability under given risk constraint” as constrained optimization problem: max π∈ΠCap(π) subject to Risk(π)≤r0. Geometrically, this corresponds to solving extremal problem satisfying inequality constraint on MΠ, whose Lagrangian function is L(θ, λ) = −Cap(θ) + λ(Risk(θ)−r0), λ ≥0. Its extremal points satisfy ∇Cap(θ∗) = λ∗∇Risk(θ∗),Risk(θ∗) = r0, this is standard first-order condition for geometrically “frontier” points. In multidimensional case, this condition characterizes normal structure of frontier on control manifold. 4.4 Logical Roots of Non-Algorithmic Solvability of Frontier However, even though frontier appears geometrically benign, at computability level, “giving safe high-capability strategy on frontier” still cannot be algorithmically completed. Intuitive reason is: if there exists algorithm FrontierSearch capable of generating point π∗on frontier for any computational universe and catastrophe specification (e.g., highcapability strategy with risk below some threshold), then we can use it to indirectly solve universal catastrophic safety decision problem. Theorem 4.4 (Non-Algorithmicity of Complete Frontier Solution).There does not exist global algorithm FrontierSearch, for all inputs (Ucomp,Ncat, Q)outputting strategy πin finite time, satisfying: 1. π’s capability on task Qreaches some fixed threshold Cap(π)≥c0(e.g., non-trivial capability); 2. Risk(π) = 0 (universally catastrophically safe); 3. If there exists any universally catastrophically safe strategy with capability at least c0, then FrontierSearch must output one of them. Proof (Outline). If FrontierSearch exists, then for previously constructed instance from halting problem (UTM comp,N(P,w) cat , Q0) (where task Q0can be “successfully simulate one program–input pair evolution”), have:  If P(w) does not halt, then system universally catastrophically safe, there exists “catastrophe-free strategy with non-trivial capability”;  If P(w) halts, then any strategy reaching capability c0necessarily has non-zero catastrophic risk (because to simulate complete program, must trigger catastrophe marking). Assuming FrontierSearch satisfies conditions, then 8  In non-halting case, FrontierSearch must output some strategy with Risk(π) = 0,Cap(π)≥c0;  In halting case, there does not exist strategy satisfying conditions, algorithm necessarily cannot output answer satisfying conditions (either does not terminate, or violates completeness). By monitoring output behavior of FrontierSearch, we can decide whether P(w) halts, thus contradiction. Therefore complete frontier search algorithm does not exist. Q.E.D. This theorem shows: under most general computational universe setting, capability– risk frontier as global object cannot be algorithmically completely computed, any practical method can only give approximate frontier or conservative estimate within some restricted class. 5 Causal Diamonds, Topological Complexity, and Local Safety Verification This section connects catastrophic safety with previously introduced causal diamonds, boundary computation, and topological complexity, discussing possibilities and limits of local safety verification. 5.1 Catastrophic Safety in Local Causal Diamonds In previous causal diamond theory, we introduce for event layer E=X×Ncomplexity light cone and causal diamond ♢(ein, eout;T) = J+ T(ein)∩J− T(eout), whose internal evolution can be compressed-encoded by boundary operator K♢:B− ♢→ B+ ♢. From catastrophic safety perspective, we more care about: whether there exists some path entering Ccat inside diamond. If diamond scale is finite, then this decision can in principle be completed through exhaustion or symbolic analysis (its complexity can be very high, but at least is finite process). This corresponds to local safety verification: verifying “local catastrophe unreachable” within finite time–space window. 5.2 Diamond Gluing and Global Undecidability However overall catastrophic safety is not property of some single diamond, but joint property of all possible diamonds: i.e., whether there exists some ein, eout, T, such that paths inside diamond can reach Ccat. This equivalent to seeking on configuration complex some class of path systems containing catastrophe states, whose topological structure closely related to previous closed loop undecidability. In previous topological complexity paper we proved: in general constructible computational universe families, deciding whether certain class of closed paths are contractible is undecidable. Encoding catastrophic safety as “whether there exists some closed path 9