Impossibility of superluminal signalling rules out causal loops in conical spacetimes
Source: arXiv:2606.20476 · Published 2026-06-18 · By Maarten Grothus, V. Vilasini
TL;DR
This paper addresses a foundational question at the interface of information-theoretic and spacetime (relativistic) notions of causality: can operationally detectable causal loops exist without violating the no-superluminal-signalling (NSS) principle in spacetimes with spatial dimensions greater than one? Prior work showed such loops are possible in (1+1)-dimensional Minkowski spacetime, but it was an open problem whether this extends to higher spatial dimensions. The authors resolve this by proving a general no-go theorem: in a broad class of "conical" spacetimes—which includes (d+1)-Minkowski spacetime for any d>1—NSS rules out all operationally detectable causal loops in classical, quantum, and post-quantum theories. This sharply contrasts with the known (1+1) dimensional case, showing that the relationship between NSS and the absence of causal loops depends critically on the geometric property of conicality in spacetime. Technically, the paper shows that higher-order signalling relations can be reduced to simpler ones, then uses the conicality property to prove incompatibility of nondegenerate embeddings of signalling loops. The result clarifies how relativistic causality constraints interface with cyclic (information-theoretic) causal structures and highlights a new form of fine-tuning required for exotic causal loops to coexist with NSS in non-conical spacetimes.
Key findings
- Operationally detectable causal loops (affects causal loops, ACLs) can exist in (1+1)-Minkowski spacetime without violating no superluminal signalling, as shown previously.
- For any d > 1 spatial dimensions—in particular in conical spacetimes such as (d+1)-Minkowski—no superluminal signalling rules out all operationally detectable causal loops.
- Higher-order affects relations reduce to 0th-order affects relations with equivalent causal inference and compatibility conditions in conical spacetimes (Lemma A.7).
- In conical spacetime embeddings, recursive compatibility and conicality imply strict nesting of future light cones that leads to contradictions if causal loops were to embed nondegenerately.
- Nondegenerate embeddings of ACLs in conical spacetimes are impossible; all compatible embeddings of ACLs are degenerate (Theorem 1).
- Causal loops consistent with NSS require fine-tuning both in the causal model and in the spacetime embedding, corresponding to non-conical embeddings and information-theoretic clustering of affects relations.
- The geometric property of conicality uniquely holds in dimensions d > 1, fails for d=1, explaining the dimensionality dependence.
- These results hold generally across classical, quantum, and post-quantum theories without assuming details of states or channels.
Threat model
The adversary consists of any operational agent attempting to produce or detect signalling causal loops embedded in spacetime, potentially exploiting fine-tuned causal mechanisms and spacetime embeddings that mask superluminal signalling. They cannot violate the no-superluminal-signalling principle as enforced by the relativistic causal order (light cones and partial orders). The proof excludes all such loops that are operationally detectable (via affects relations) and compatibly embedded in conical spacetimes without degeneracies.
Methodology — deep read
Threat Model & Assumptions: The adversary is conceptualized as any operational procedure attempting to demonstrate an observable causal loop (ACL) without violating the no-superluminal-signalling (NSS) principle embedded in a spacetime poset (partially ordered set) representing causal order in spacetime. It includes classical, quantum, and post-quantum theories. The adversary does not break relativistic causality but may exploit subtle causal structures and fine-tuning to realize causal loops without signalling faster than light. The no-go theorem aims to exclude operationally detectable causal loops compatible with NSS in conical spacetimes.
Data & Formal Structures: The framework uses the "affects framework"—an axiomatic, theory-independent formalism to describe causal models as directed graphs with sets of observed random variables (RVs) and unobserved systems. Instead of relying on specific physical states or channels, it focuses on signalling/causal relations defined via affects relations between sets of RVs (X ⊨ Y | do(Z)) and their embedding into spacetime represented by a poset T. Key properties like d-separation and irreducibility (Irred1 and Irred3) of affects relations guide causal inference.
Architecture/Algorithmic Steps: The core technical approach involves:
- Formalizing operational detectability of causal loops via affects causal loops (ACLs), combinations of signalling relations implying cyclic causal influence.
- Embedding RVs as ordered RVs (ORVs) into a spacetime poset respecting the partial order (light cones).
- Characterizing the geometric property of conicality of the spacetime poset, which uniquely holds for d>1 Minkowski space.
- Reducing higher-order affects relations to equivalent 0th-order ones preserving causal inference and NSS compatibility in conical spacetimes (Lemma A.7).
- Proving that compatible embeddings of these ACLs induce strict inclusion relations between future cones, which through conicality imply contradictions, ruling out nondegenerate embeddings (Theorem A.9 and Theorem 1).
Training Regime: Not applicable, as this is a theoretical, mathematical proof paper rather than empirical machine learning.
Evaluation Protocol: The evaluation is a mathematical proof framework that encompasses all compatible embeddings of ACLs in conical spacetimes. The authors exemplify the argument via concrete causal loops (such as Fig. 1) embedding into (1+1) versus (d+1) Minkowski spacetime to highlight the role of conicality. Lemmas and theorems are rigorously stated and proven, ensuring logical completeness. The approach applies across all classical, quantum, and post-quantum models satisfying d-separation.
Reproducibility: The theoretical results are fully rigorous and based on well-defined mathematical frameworks. The paper cites prior open-source formal frameworks (affects framework) and standard causal inference concepts (d-separation). No code or datasets are required. The proofs are provided in detail in the appendices for verification.
Technical innovations
- Proving that conicality of spacetime (a precise order-theoretic property) uniquely ensures no operationally detectable causal loops exist without superluminal signalling, resolving the gap between 1+1 and higher dimensions.
- Introducing a reduction from higher-order affects relations to 0th-order relations that preserve causal inference and NSS compatibility conditions in conical spacetimes (Lemma A.7), greatly simplifying causal loop analysis.
- Generalizing the concept of conical embeddings within possibly non-conical spacetimes, revealing that causal loops require fine-tuned, non-conical embeddings.
- Demonstrating that NSS compatibility conditions translate to strict nesting relations of joint future cones that lead to contradiction when nondegenerate operational causal loops are assumed.
Baselines vs proposed
- (1+1)-Minkowski spacetime: Operationally detectable causal loops compatible with NSS are known (from prior work).
- (d+1)-Minkowski spacetime with d > 1: Theorem 1 proves all compatible embeddings of such loops are degenerate (i.e., no operationally detectable ACLs exist).
Limitations
- The results rely on the affects framework and d-separation; cyclic causal models violating d-separation may not be covered and present an open question.
- The no-go theorem assumes nondegenerate embeddings; it does not exclude fine-tuned degenerate embeddings that may be physically less realistic.
- The framework fixes a background spacetime and does not consider dynamical or quantum gravity scenarios where spacetime itself emerges or fluctuates.
- The causal inference is theory-independent but may be limited if system cardinalities or additional structure is imposed.
- The subtle notion of information-theoretic fine-tuning in causal models and embeddings merits further study to understand physical realizability.
- Extensions to frameworks using other separation criteria (σ-separation, p-separation) for cyclic causal models are not addressed.
Open questions / follow-ons
- Does the no-go theorem extend to cyclic causal models that violate d-separation, e.g., those that satisfy σ- or p-separation?
- Can stronger causal inference rules be derived when system size or cardinalities are fixed, potentially uncovering more causal loops?
- How can the notions of fine-tuning in both causal models and spacetime embeddings be quantified or physically characterized?
- Can these results be extended to dynamical or emergent spacetime scenarios, such as in quantum gravity or holography?
Why it matters for bot defense
For practitioners focused on bot-defense and CAPTCHA, this paper is more foundational than directly applied, but it clarifies fundamental limits on embedding cyclic causal structures without superluminal signalling in standard spacetime geometries. Understanding that operational causal loops compatible with no-superluminal signalling cannot exist in higher dimensional Minkowski-type spacetimes reinforces assumptions about the causal order implicit in secure protocol designs that rely on relativistic causality. The paper highlights subtle fine-tuning conditions that an attacker would need to exploit to realize causally cyclic signalling patterns without violating relativistic causality—an insight that can guide threat model design against exotic or post-quantum adversaries. Moreover, the affects framework's fine-grained causal inference approach may inspire new methods to detect or rule out hidden signalling channels in distributed interactive security protocols under relativistic constraints.
Cite
@article{arxiv2606_20476,
title={ Impossibility of superluminal signalling rules out causal loops in conical spacetimes },
author={ Maarten Grothus and V. Vilasini },
journal={arXiv preprint arXiv:2606.20476},
year={ 2026 },
url={https://arxiv.org/abs/2606.20476}
}