Anthony Kobanda

C^{1} trajectories γ:[0,T]→𝒳\gamma:[0,T]\to\mathcal{X}, with γ​(0)=x\gamma(0)=x and γ​(T)=y\gamma(T)=y, ∀T>0\forall T>0. Let L:T​𝒳→ℝ¯+L:\mathrm{T}\mathcal{X}\to\bar{\mathbb{R}}^{+} be a local effort density, veryfing L​(x,v)≥c⋅∥v∥L(x,v)\geq c\cdot\lVert v\rVert for some norm ∥⋅∥\lVert\cdot\rVert and a constant c>0c>0. Defining the action of a trajectory as : 𝖠𝖼𝗍​(γ)=∫, Inria Scool, Université de Lille, France Abstract Joint-Embedding Predictive Architectures (JEPAs) aim to learn representations by predicting target embeddings from context embeddings, inducing a scalarcompatibility energy in a latent space. In contrast, Quasimetric Reinforcement Learning (QRL) studies goal-conditioned control through directed distance values(cost-to-go) that support reaching goals under asymmetric dynamics. In this short article, we connect these viewpoints by restricting attention to a principled class of JEPA energy functions : intrinsic (least-action) energies, defined as infima ofaccumulated local effort over admissible trajectories between two states. Undermild closure and additivity assumptions, any intrinsic energy is a quasimetric. In goal-reaching control, optimal cost-to-go functions admit exactly this intrinsic form ; inversely, JEPAs trained to model intrinsic energies lie in the quasimetric value class targeted by QRL. Moreover, we observe why symmetric finite energies are structurally mismatched with one-way reachability, motivating asymmetric (quasimetric) energies when directionality matters. 1 Introduction Joint-Embedding Predictive Architectures (JEPAs) (LeCun, 2022; Assran et al., 2023) recently emerged as a compelling self-supervised paradigm for representation learning in which modelspredict target embeddings from context embeddings rather than reconstructing raw observations. From a control viewpoint, the resulting training score can be interpreted as a scalar compatibilityenergy between inputs : low energy meaning compatible, while high energy means incompatible. Independently, Quasimetric Reinforcement Learning (QRL) (Wang et al., 2023) represents goal-conditioned control through the geometry of a learned cost-to-go. A central fact is that reaching costs induce a directed notion of distance : going from ss to gg can be easy, while returning from gg to ss may be difficult or impossible, and multi-step composition should be consistent. Hence, QRL formalizes this by targeting function classes with quasimetric structure (Wang and Isola, 2022), and proposes a learning framework and objectives designed for that regime. This short article addresses a narrow structural question : Under what conditions does a JEPA-induced energy behave like a cost-to-go ? A key insight is that sequential compositionality is not a modelling preference, but usually required when reasoning (Johnson-Laird, 2010; Plaat et al., 2025). If a state uu (e.g., a position, a proposition, a concept) is accessible (e.g., reachable, demonstrable, explainable) on the way from xx to yy, then an energy meant to support multi-step reasoning should satisfy a consistency inequality of the form : E​(x,g)≤E​(x,u)+E​(u,g),E(x,g)\;\leq\;E(x,u)+E(u,g), which is precisely a triangle inequality. In physics and optimal control, the canonical way to obtain such a global, compositional energy is to define it intrinsically via a least-action principle: the energy between two states is the infimum of accumulated local effort over admissible trajectories connecting them (De Sapio et al., 2008; Ho, 2018; Terekhovich, 2018). This construction yields triangle inequality by definition (via concatenation), and it naturally produces asymmetry whenever admissibility or local effort is directed. Our main message is simple : If a JEPA is defined according to an intrinsic (least-action) energy, then it induces a quasimetric space. In goal-conditioned tasks, costs-to-go can be interpreted as intrinsic energies, hence such Intrinsic-Energy JEPAs fall into the same quasimetric value-functions class that QRL is designed to model. 2 Minimal Background JEPA (Latent Representations and Induced Energies). Joint-Embedding Predictive Architectures learn representations by predicting target embeddings from context embeddings (LeCun, 2022). Given a context encoder fϕf_{\phi} and a target encoder fϕ¯f_{\bar{\phi}}, JEPA forms zx=fϕ​(x),zy=fϕ¯​(y),z_{x}=f_{\phi}(x),z_{y}=f_{\bar{\phi}}(y), and a predictor pθp_{\theta} produces a predicted target embedding z^y=pθ​(zx;c)\hat{z}_{y}=p_{\theta}(z_{x};\,c), where cc denotes a conditioning. Training uses a comparator function D​(⋅,⋅)D(\cdot,\cdot) in the embedding space : ℒJEPA​(ϕ,ϕ¯,θ)=𝔼(x,y)∼𝒟​[D​(z^y,sg​(zy))],\mathcal{L}_{\text{JEPA}}(\phi,\bar{\phi},\theta)~=~\mathbb{E}_{(x,y)\sim\mathcal{D}}\!\left[\,D\!\left(\hat{z}_{y},\texttt{sg}(z_{y})\right)\right], Even when the comparator form is fixed, the model learns an induced energy landscape over inputs through the learned representation and predictor: pairs (x,y)(x,y) that are compatible are precisely those for which the latent prediction error is small. This motivates interpreting JEPA training as learning a scalar energy (compatibility) between xx and yy, in line with the energy-based perspective on JEPA. QRL (Goal-Reaching values are Quasimetrics). Quasimetric Reinforcement Learning studies goal-conditioned control through directed distances that compose over time (Wang et al., 2023). In reaching-cost problems, the optimal value function is V