the agent's state `s = (q, p)` exists on the cotangent bundle `tm` of a semantic manifold `m`.
the base point `q ∈ m` represents the agent's semantic position.
the covector `p ∈ tm` represents its semantic momentum or intent.
the topology of `m` defines the relational structure of meaning.

the system's dynamics are governed by the hamiltonian `h(q, p) = (1/2) pᵀ g(q)⁻¹ p + v(q) + c(q, ψ)`.
the kinetic term uses a metric tensor `g(q)` to define the cost of changing state.
the potential `v(q)` is an internal landscape of semantic value.
the coupling term `c(q, ψ)` models interaction with an ambient semantic field `ψ`.

the agent's trajectory evolves according to hamilton's equations with dissipation:
`dq/dt = ∂h/∂p`
`dp/dt = -∂h/∂q - γp`

learning updates the potential to minimize error:
`v_{t+1}(q) = v_t(q) - η ∇l`.

the embodiment operator `π: tm → a` projects the agent's state into an interaction substrate:
`a_t = π(q_t, p_t)`.