Tangential Action Spaces: Memory and Cost

• Tangential Action Spaces are a differential‐geometric framework that models embodied agents as hierarchical fibre bundles linking physical, cognitive, and intentional spaces.
• The framework uses connection curvature and holonomy to quantify intrinsic geometric memory effects, where nonzero curvature incurs an energetic penalty.
• TAS establish a cost-memory duality by demonstrating that excess energy cost is proportional to the square of the fibre holonomy, guiding efficient controller design.

Tangential Action Spaces (TAS) are a differential-geometric framework linking the geometry of action in embodied agents to fundamental trade-offs between memory and energetic cost. Developed to characterize how memory arises and is priced in physical and cognitive systems, TAS provide a unified treatment of holonomic and nonholonomic dynamics, geometric mechanics, and the design of efficient controllers. Central to the framework is the modeling of agents as hierarchical fibre bundles, with projections mapping physical, cognitive, and intentional spaces. The geometry of these projections governs whether memory mechanisms must be engineered (via vertical dynamics) or arise intrinsically through the curvature and holonomy of the connection structure. Tangential Action Spaces thereby reveal universal energetic costs for path-dependent, memory-carrying behavior and establish rigorous design principles for both biological and artificial agents (Blattner, 2 Sep 2025).

1. Geometric Structure and Hierarchical Manifold Model

At the core of TAS is a hierarchical decomposition of agent state:

• The physical manifold $(P,G)$ is an $m$-dimensional Riemannian manifold representing "body + actuators," with metric $G$ capturing kinetic cost or energy.
• The cognitive manifold $C$ is an $n$-dimensional smooth manifold, $n\leq m$, encoding task-relevant variables as perceived or abstracted from $P$.
• The intentional manifold $I$ of dimension $k\leq n$ encapsulates agent-level goals or plans.

The interactions are encoded via two surjective submersions of constant rank: $$\Phi: P \longrightarrow C, \qquad \Psi: C \longrightarrow I,$$ each admitting local trivializations—so both $m$0 and $m$1 are smooth fibre bundles.

• $m$2 ("perception") is typically a many-to-one mapping effecting a coarse-graining of the physical state, crucial in embodied contexts.
• $m$3 ("intention") maps many cognitive states to the same intentional goal.

Given $m$4, a lift $m$5 selects a physical velocity $m$6 obeying $m$7. The structure and freedom of these lifts underpin memory and energy cost in TAS (Blattner, 2 Sep 2025).

2. Connections, Curvature, and Holonomy

The geometric machinery of TAS centers on connections and their curvature:

• Ehresmann Connection: At each $m$8, the tangent space splits as $m$9, with $G$0 (the vertical subspace) and $G$1 (horizontal distribution) chosen so that $G$2 is an isomorphism.
• The unique horizontal lift of $G$3 is the element $G$4 with $G$5.

A connection 1-form $G$6 with values in the (here, Abelian) structure group defines the curvature 2-form: $G$7 For a loop $G$8 bounding a surface $G$9, the holonomy is given by

$C$0

Nonzero $C$1 captures the presence of geometric memory: horizontal transport around $C$2 does not close, marking "failure to return" in physical state due to traversed cognitive loop. Thus, curvature $C$3 intrinsically quantifies memory (Blattner, 2 Sep 2025).

3. Classification of Tangential Action Spaces

A principal theorem classifies TAS systems depending on the nature of the projection $C$4:

1. Local Diffeomorphism ($C$5):
   • $C$6 invertible everywhere; the geometric lift is unique:

   $C$7

• No intrinsic connection curvature or holonomy arises. Any memory effect must be engineered by prescribing vertical dynamics in a hidden fibre $C$8 orthogonal to $C$9 (i.e., additional state variables not visible to $n$0).

1. Genuine Fibration ($n$1):
   • $n$2; many horizontal distributions are possible.
   • Nontrivial curvature $n$3 in the chosen connection yields intrinsic geometric memory: closed loops in $n$4 can lift to open paths in $n$5 ("holonomy"). The memory mechanism is thus embedded in the geometry of the action space.

This classification demarcates when memory is implemented by system design (engineered, diffeomorphic case) versus when it arises naturally from geometry (fibration, curved connection) (Blattner, 2 Sep 2025).

4. Energy-Cost and Memory Duality

A fundamental TAS result links energetic cost with memory:

• The energy of a trajectory $n$6 in $n$7 is

$n$8

• For specified $n$9 in $n\leq m$0, the metric lift is the minimum-energy lift:

$n\leq m$1

• Cost-Memory Theorem: For any loop $n\leq m$2, the excess cost incurred over the optimal lift is proportional to the square of the fibre holonomy:

$n\leq m$3

In Abelian cases, for holonomy coordinate $n\leq m$4:

$n\leq m$5

This cost-memory duality establishes a universal quantitative principle: any nonzero geometric memory (measurable by fibre holonomy) exacts an energetic penalty quadratic in holonomy, regardless of the physical or robotic context (Blattner, 2 Sep 2025).

5. Concrete Models and Illustrative Systems

TAS principles are validated across several explicit systems, illustrating both engineered and geometric memory:

System	Projection Structure	Memory Mechanism	Energy Penalty
Strip-Sine	Diffeomorphism + hidden	Prescribed fibre dynamics	$n\leq m$6
Helical fibration	Fibration, const. curvature	Geometric (via curvature)	$n\leq m$7
Twisted fibration	Fibration, variable curv.	Geometric (varied)	Nonlinear in holonomy
Flat fibration	Fibration, flat	None (zero curvature)	$n\leq m$8
Cylindrical	Flat, non-simply connected	None (zero curvature)	$n\leq m$9

• Strip–Sine (engineered memory): Explicit dependence on a hidden variable $P$0 controlled by $P$1, with holonomy $P$2, excess cost quadratic in $P$3.
• Helical/Twisted fibrations: Nontrivial connection 1-forms induce fibre holonomies $P$4 proportional to the area enclosed in $P$5; energetic penalties again quadratic in holonomy.
• Flat/cylindrical fibrations: Zero curvature yields zero holonomy and no extra energy cost for loops.

These detailed systems confirm that connection curvature, not base topology, is the decisive factor for geometric memory and energetic penalty (Blattner, 2 Sep 2025).

6. Implications for Biological and Robotic Systems

TAS bridges geometric mechanics with embodied cognition and optimal control, providing explanatory power across both theoretical and engineering domains:

• Biological motor control: The source of diversity in animal movements and "laziness" in energy expenditure follows directly from the geometry of projection morphologies and connection curvature.
• Robotic design: The classification and cost-memory duality yield explicit design rules:
  1. Morphology $P$6 should minimize unnecessary curvature when memory is undesired.
  2. Natural fibre structures with modest, task-aligned curvature enable built-in, low-cost memory.
  3. Connection or hidden-fibre parameters may be tuned to adjust the trade-off between memory capacity and energetic efficiency.
  4. Integrating energetic penalties into intentional dynamics through feedback (using e.g., $P$7) produces strategies balancing goal achievement and cost minimization.

A plausible implication is that TAS illuminate why certain "lazily optimal" behavioral strategies are observed in both natural and artificial agents, and provide actionable principles for the synthesis of efficient controllers (Blattner, 2 Sep 2025).

7. Tangential Action Spaces in Broader Geometric and Topological Contexts

The concept of tangential action and tangential symmetric spaces also appears in the representation theory and topology of homogeneous spaces, as exemplified by the study of tangential homogeneous spaces $P$8 constructed via Cartan involution in the context of Lie group actions (Tojo, 2021):

• Here, the tangential homogeneous space $P$9 is formed by taking the Cartan decomposition $I$0 and defining a "θ-twist" real form $I$1.

• The resulting tangential space, diffeomorphic to a vector bundle over a compact quotient, features prominently in classification and in questions of existence of compact Clifford–Klein forms.
• Classification distinguishes irreducible classical tangential symmetric spaces by types (AI, AII, etc.), with their topological and geometric complexity influencing possible quotients and the existence of invariant measures.

While these tangential symmetric spaces are not identical to TAS as developed in embodied control, the shared focus on fibre bundle geometry, projection-induced structures, and their effect on dynamical or topological properties reflects a broad thematic unity (Tojo, 2021).

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Tangential Action Spaces thus form a powerful and unifying language for the analysis of memory, cost, and geometry in physical and computational agents. Their rigorous classification and quantitative cost-memory duality underlie both explanatory and prescriptive advances in optimal control, robotics, and embodied cognition (Blattner, 2 Sep 2025).