The Geometry of Memory: How Actions Leave Traces in Your Body

October 22, 2025

marcel blattner | October, 2025

Tangential Action Spaces (TAS) offer a principled geometric framework for understanding a fundamental constraint on embodied agency: the trade-off between energetic efficiency and experiential memory . By representing agents as hierarchies of abstract spaces connected by projection maps, TAS reveals when memory is possible, what memory costs, how learning and action interact, and why social roles can emerge from local energetic trade-offs .

The framework synthesizes insights from geometric mechanics, embodied cognition, optimal control, and predictive processing into a coherent mathematical picture . Perhaps most importantly, TAS demonstrates that memory and experience are not abstract, disembodied phenomena but are physical processes with measurable energetic costs. Every time you learn a new skill or form a habit, your body is traversing paths through high-dimensional state spaces. The traces left by those paths, the holonomy they accumulate, are the geometric substrate of embodied memory.

The framework is predictive, not just descriptive. It makes quantitative claims about energy costs and memory magnitudes that are testable with current technologies . The extension to non-Abelian structure groups ensures applicability to realistic systems with complex rotational kinematics. For robotics and AI, TAS provides clear design principles . For biology, it offers testable hypotheses about motor control and the metabolic costs of learning .

Understanding this geometry may ultimately help us design better robots, treat motor disorders, and build more robust learning systems. The cost-memory duality is not a limitation but a fundamental feature of embodied existence, a constraint that shapes what agents can remember, how they must move, and what strategies they can adopt for navigating an uncertain world.

To understand how this works, let's explore the core concepts of the TAS framework.

The Problem: Bridging Bodies, Minds, and Goals

Living systems face a remarkable computational challenge. At any given moment, your body occupies some physical state. From this high-dimensional reality, your nervous system somehow extracts a much simpler cognitive representation, which in turn is used to form abstract intentions.

This creates a natural hierarchy with three levels:

• Physical Space (P) encompasses the complete, high-definition state of your body: potentially millions of degrees of freedom, including every muscle fiber activation, joint position, metabolic variable, and neural firing pattern.

• Cognitive Space (C) contains task-relevant representations. This isn't just simplified physical data (like hand position or velocity), but also more abstract concepts, such as "learned motor schemas," "world models," and "cognitive strategies" that guide action selection .

• Intentional Space (I) holds the most abstract goals and objectives, such as "I want coffee" or "I should maintain balance," representing what you want to achieve, independent of how you'll achieve it.

It is important to clarify that these "spaces" are not hidden dimensions in spacetime. They are abstract representational states organized in a hierarchy. The framework is a mathematical way to model an agent's state, orchestrated by the interaction between these abstract levels.

The critical operations happen when you move between these levels. Perception flows "up" from physical to cognitive to intentional (these are the "projections"). Action flows "down" from intentions through cognitive plans to concrete physical movements (these are the "lifts"). The TAS framework formalizes this hierarchy using differential geometry, treating each level as a smooth mathematical space connected by these projection and lift maps .

The Core Insight: The Space of Choices

Here's where things get interesting. When you want to move from one cognitive state to another (say, from "hand at rest" to "hand grasping cup") there are generally many different ways your physical body could achieve this. Different muscle activation patterns, different trajectories, or different joint angle sequences might all result in the same perceived outcome.

In mathematical terms, the cognitive-to-physical mapping is underdetermined. For any desired change in cognitive space, there's typically an entire subspace of possible physical motions that would work. The question becomes: which one should your body choose?

Two Fundamental Cases

The TAS framework reveals that the answer depends critically on the geometry of the perception map, or how physical states (P) project onto cognitive representations (C) . There are two structurally different cases:

Case One: One-to-One Mappings (Diffeomorphisms)
Sometimes, each cognitive state corresponds to exactly one physical state, at least within a small local region. In this case, there's a unique "geometric lift" that simply inverts the perception map . This lift minimizes instantaneous energy expenditure, produces no path-dependent memory, and creates closed loops: if you return to the same cognitive state, your physical state returns exactly to where it started .

Think of this like following a well-lit path through a forest. There's one clear route, and retracing your steps brings you exactly back to the start.

Case Two: Many-to-One Mappings (Fibrations)
More commonly, multiple physical states map to the same cognitive state. For example, you might perceive "arm extended forward" while having many different internal muscle tensions or metabolic states that don't affect the perceived position. These "hidden" abstract or internal dimensions form what mathematicians call fibres.

In this case, there's a special lift called the metric lift that minimizes energy among all possible lifts . This lift is uniquely determined by the physical body's "metric" (a function that encodes the energetic cost of motion in each direction). Importantly, this minimal-energy lift may or may not produce memory, depending on the "curvature" of the connection.

This is more like hiking through mountains where multiple trails lead to the same viewpoint. The most energy-efficient route might not be the one you take if you're also trying to explore or remember the terrain.

The Cost-Memory Duality: Memory Requires Extra Energy

The central finding of TAS is elegant and precise: any lift that creates path-dependent memory must expend strictly more energy than the metric lift.

Let's unpack this carefully. The metric lift represents the energy-optimal baseline: the most efficient way to achieve a given cognitive change while respecting the geometry of your body . Path-dependent memory, technically called holonomy, occurs when traversing a closed loop in cognitive space leaves your physical state in a different configuration than where you started . The "gap" between start and end points in physical space is the "memory" of the path taken.

The excess energy required to create this memory (beyond the metric-lift baseline) scales in a specific way: it grows quadratically with the magnitude of the memory for small loops. This means doubling the amount of memory you want to store requires quadrupling the energy expenditure. This quadratic relationship is universal and appears whether memory arises from natural geometric curvature or from deliberately engineered dynamics.

A Concrete Example:

To make this concrete, consider a simplified model from the paper: the strip-sine system. Imagine a robot with "visible" coordinates that we can observe (like its hand position) and a "hidden" internal state that we cannot directly measure (like a motor's internal counter or a metabolic variable) . The perception map, which builds the cognitive state, depends only on the visible coordinates. This means the hidden state lives in the fibre; it's part of the physical reality but invisible to the task-level representation.

Now, let's address what "tracing a circle in cognitive space" means. This simply means the robot executes a repetitive, cyclical task where its task-relevant variables return perfectly to the start.

For example, the cognitive plan is "move the hand in a circle." This means the perceived (x, y) coordinates of the hand end up exactly where they began. The plan is a closed loop. The crucial question is: when the plan completes this loop, does the robot's full physical state (including its hidden internal state) also return to the start?

Case 1: The "Lazy" Path (Metric Lift)
If the robot executes this circular plan using the metric lift (the most energy-efficient path), it does so without engaging the hidden state. The hidden state remains constant . The hand returns to the start, and the hidden state is unchanged. The result is zero memory and minimal energy.

Case 2: The "Memory" Path (Prescribed Dynamic)
But what if we want the robot to remember how many loops it has traced? We can add a prescribed dynamic: a specific rule that forces the hidden state to change as the robot moves. For example, this rule makes the hidden state accumulate the signed area enclosed by the path .

Now we have memory. After one counterclockwise loop, the cognitive plan is back where it started, but the full physical state is not. The hidden state has a new value (e.g., "1 loop"). This "gap" between the physical start and end is the holonomy, and it functions as memory.

But this memory comes at a cost. To activate that hidden state, the robot had to deviate from the "lazy" path, expending instantaneous excess power. When we integrate this over the full trajectory, we find the total excess energy scales quadratically with the accumulated memory. So doubling the memory (e.g., tracing a loop with twice the area) quadruples the excess energy cost .

This quadratic relationship is a universal feature of small-loop memory formation in TAS systems. It doesn't matter whether the memory comes from natural geometric curvature or from engineered prescribed dynamics; the cost-memory trade-off follows the same mathematical law.

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Classification: Four Types of Path-Dependent Systems

The TAS framework naturally classifies embodied systems based on how path-dependence can arise. This taxonomy provides a principled way to understand when and why biological or robotic systems exhibit history-dependence.

Intrinsically Conservative Systems These have one-to-one mappings between cognitive and physical states and use the geometric lift. There's no intrinsic memory mechanism. Holonomy is zero for all loops, and energy is always at the minimum defined by the geometric lift. A simple example would be a point mass moving in ordinary space with direct position sensing; wherever you go, coming back to the same position leaves you in exactly the same state.

Conditionally Conservative Systems These have fibre structure (multiple physical states per cognitive state) but with a flat connection, meaning zero curvature. They produce no memory for small contractible loops. Energy is at a minimum when using the metric lift. An example is a cylindrical space with a flat connection. The cylinder has nontrivial topology (you can wind around it) but locally there's no curvature, so small loops don't accumulate memory .

Geometrically Nonconservative Systems These have fibre structure with a curved connection, meaning the curvature is nonzero. Memory arises from this connection curvature. For small loops, holonomy grows linearly with the loop's area . Energy depends on which connection you choose; the metric connection minimizes it. A helical fibration provides a clean example . Imagine a structure where moving in circles in the base space naturally induces a vertical twist. The constant vertical twist creates predictable memory that grows linearly with area. This is purely geometric.

Dynamically Nonconservative Systems These have one-to-one mappings but use prescribed dynamics that deliberately deviate from the geometric lift. Memory is engineered through these rules . Holonomy depends on the prescribed dynamics, and there's always excess energy relative to the metric lift. The strip-sine system described earlier fits here. The visible coordinates have a one-to-one relationship with the cognitive space, but we've added a hidden variable with dynamics that integrate the area, deliberately creating memory where none would exist naturally .

Where TAS Fits: Context in Current Research

The TAS framework builds on and connects several established research traditions. Understanding where it fits helps clarify both what's new and what problems it addresses.

Geometric Mechanics and Robotics
Differential geometry has long been applied to robotic systems and locomotion studies . Concepts like fibre bundles, connections, and geometric phases have appeared in analyses of parallel parking problems, snake robots, and gauge theories of locomotion . However, this prior work focused largely on the mechanics and control of movement itself, without incorporating cognitive states or explicitly addressing memory. TAS extends this geometric tradition by introducing cognitive and intentional layers on top of the physical manifold, asking new questions about the energetic price of internal memory .

Dynamical Systems and Enactivism
The enactive approach to cognition emphasizes that cognition arises through dynamic interaction between an organism and its environment. Dynamical systems theory models agents as coupled differential equations evolving over time. These approaches compellingly illustrate sensorimotor contingencies, but they often lack explicit geometric structure distinguishing "physical" from "cognitive" coordinates. TAS contributes a formal geometric scaffolding to this perspective. The projection from physical to cognitive space is a concrete realization of structural coupling . TAS can quantify qualitative enactive concepts, like how an agent's history alters its state, as path-dependent parallel transport yielding measurable holonomy.

Optimal Motor Control
Biomechanics and robotics have extensively studied energy-efficient movement, using models like minimum-jerk or minimum metabolic energy to explain motor patterns . These approaches, however, typically do not address path-dependent memory or hysteresis . There's no notion that taking different routes to the same destination could leave an agent in different internal states. TAS bridges this gap by establishing the metric lift as the instantaneous energetic baseline and showing how any memory-encoding deviation incurs measurable excess cost .

Predictive Processing and Active Inference
The Free Energy Principle and Active Inference frameworks propose that intelligent agents minimize prediction error or variational free energy . These theories provide a high-level normative target for adaptive behavior. However, they usually abstract away detailed geometry, lumping physical interactions into generic probabilistic mappings . TAS can provide a geometric grounding for these ideas. It introduces the idea that there's an energy-optimal (geodesic) way to realize any prediction, namely the metric lift, and that deviating from this geodesic corresponds to encoding additional information as holonomic memory. This suggests a refinement: agents must balance minimizing surprise (an informational goal) against the physical imperative to conserve energy .

Beyond Single Agents: Reflective TAS and Social Dynamics

A powerful extension addresses self-modification: agents that can update their own perceptual or action models . This leads to Reflective TAS (rTAS), which adds a model manifold representing learnable parameters. The perception map now depends on this model state.

A block metric prices both physical effort and model change. The parameter lambda ($\lambda$) controls the effort-learning trade-off: large lambda makes model updates expensive, pushing the system toward rigid behavior, while small lambda allows cheap adaptation but risks instability.

Cross-Curvature and Re-Entry: In Reflective TAS, the connection becomes block-valued . There's physical holonomy and also meta-memory (holonomy in model space) . Most interesting are the cross-curvature terms that implement re-entry: model changes can induce physical holonomy, and physical motion can induce meta-holonomy . These cross-terms provide a geometric mechanism for phenomena like motor learning (where physical practice induces model updates) and mental practice (where imagined movements alter subsequent performance).

Multi-Agent Interactions: When multiple reflective agents interact, remarkably rich behaviors emerge from simple local rules.

• Emergent role specialization occurs when two agents have different learning costs. The agent with cheaper learning spontaneously takes on more adaptation, while the partner with expensive learning becomes a stable anchor. A leader-follower split arises purely from local energetic trade-offs.

• Resonance catastrophe can happen when both agents are highly adaptive (small ) and strongly coupled. Positive feedback loops can arise, leading to runaway model activity and energy expenditure .

• Phase transitions appear as coupling strength varies. The system can undergo discontinuous changes between adaptive and rigid regimes, with optimal strategies shifting abruptly at critical coupling values .

These examples illustrate how rTAS provides a bottom-up mechanism for grounding social dynamics in individual energetic constraints.

Generalizations: Non-Abelian Structure Groups

The framework described so far applies most directly to systems where the hidden fibre dimensions behave independently (technically, where the structure group is "Abelian") . However, many real-world systems involve non-commutative structure. Consider a robot with rotational degrees of freedom: rotating first around one axis then another produces a different result than rotating in the opposite order. This non-commutativity fundamentally changes the geometric structure.

The TAS framework extends naturally to these non-Abelian structure groups, though with important modifications. In the non-Abelian case, holonomy becomes "group-valued" (like a rotation matrix) rather than a simple displacement vector. The cost-memory relationship still holds, but must be formulated using relative holonomy with respect to the metric lift .

Non-Abelian Examples and Diagnostics Two concrete examples illustrate this extension:

1. Rotations in three dimensions (SO(3)): A system where the physical space includes rotational orientation . For small rectangular loops in cognitive space, the accumulated rotation is proportional to loop area, but additional higher-order terms appear that depend on loop shape and the order of traversal, which are signatures of non-Abelian structure .

2. Rotation-translation coupling (SE(2)): This describes simultaneous rotations and translations in the plane . A classic nonholonomic effect emerges: rotating while translating produces a net lateral displacement proportional to the loop area . This is the mathematical essence of parallel parking or snake-like locomotion, now understood as group-valued holonomy.

A powerful diagnostic for non-Abelian effects uses commutator loops: paths that traverse a sequence and then reverse it in opposite order. In Abelian systems, these produce zero holonomy to leading order. In non-Abelian systems, commutator loops reveal next-order effects.

Practical Implications The non-Abelian extension is not merely a mathematical generalization; it's essential for modeling realistic robotic systems with rotational degrees of freedom, understanding human motor control involving joint rotations, and analyzing multi-body systems. The cost-memory duality remains intact: any non-Abelian lift that creates path-dependent memory incurs excess energy, scaling quadratically with the magnitude of the group-valued holonomy for small loops.

Testable Predictions and Practical Implications

The TAS framework makes concrete, falsifiable predictions that distinguish it from purely descriptive or qualitative theories.

For Biology

• First prediction: Stereotyped, well-practiced motor tasks should correspond to policies approximating the flat, energy-minimal metric lift.

• Second prediction: Tasks requiring adaptation should engage pathways that produce measurable holonomy at a metabolic cost.

• Third prediction: This excess cost should scale quadratically with the magnitude of accumulated memory for small repeated movements.

• Fourth prediction: In movements involving complex rotations (where non-Abelian structure is relevant), order-dependent effects should appear.

These predictions could be tested using metabolic measurements (like oxygen consumption) combined with kinematic analysis to estimate the metric-lift baseline and measure deviations .

For Robotics and AI

• Design principle one: For high-efficiency tasks, co-design robot morphology and controllers to make the perception-action map as "flat" as possible (minimize curvature).

• Design principle two: For tasks requiring memory, explicitly budget energy for holonomy using the cost-memory law.

• Design principle three: In multi-robot systems, assign different learning costs () to agents to create "scouts" (cheap learning) or "anchors" (expensive learning) to foster role specialization.

• Design principle four: To avoid runaway co-adaptation, ensure sufficient damping through either larger values or bounded coupling strength, as informed by phase-transition analysis .

• Design principle five: For robots with significant rotational degrees of freedom, explicitly account for non-Abelian structure and use commutator-loop diagnostics to verify behavior.

Operational Measurement

The framework is not just theoretical; it provides an operational recipe for empirical testing. Given time-series data from motion capture, one can estimate the differential of the perception map via local linear regression, construct the metric lift using the weighted pseudoinverse formula, compute instantaneous excess power as the squared difference between observed and optimal velocities, integrate over closed cognitive loops to get excess energy, and measure physical holonomy as the closure gap . One can then test the predicted quadratic relationship between excess energy and holonomy magnitude.

Current Limitations and Future Directions

The present formulation of TAS rests on simplifying assumptions that limit its immediate applicability but suggest clear extensions. These include relaxing the assumptions of time-invariant structures (to model development, aging, or injury recovery), a quadratic energy functional (to model impacts or explosive movements), and static manifolds.

Promising extensions include Stochastic TAS (to model how noise interacts with geometric memory), Evolutionary TAS (where learning costs and coupling strengths become evolvable traits), and Neuroscience applications (mapping TAS components onto specific neural circuits, like the cerebellum or basal ganglia) .

A Final Perspective

Tangential Action Spaces provide a new language for describing the deep connection between an agent's physical body, its internal models, and the energy it must spend to act and learn. It shows that memory is not an abstract file, but a physical, geometric trace etched into the state of a system by the path it has taken. The trade-off between efficiency and memory appears to be a fundamental and quantifiable feature of any embodied agent, shaping the very strategies available for it to navigate and understand its world.