Explanation of totalEnergy in IndisputableMonolith.Action.Hamiltonian

(1) In plain English: The definition computes the total mechanical energy E(t) of a trajectory γ at time t. It evaluates the standard Hamiltonian H(q, p) = p²/(2m) + V(q) at the current position q = γ(t) and the conjugate momentum p = m · γ'(t), yielding kinetic plus potential energy.

(2) In Recognition Science this matters because the module derives the Hamiltonian formulation from the J-action via the Legendre transform in the quadratic small-strain limit QuadraticLimit.standardEL. The totalEnergy definition supplies the concrete object whose conservation follows from the Euler-Lagrange equation, linking classical mechanics to the recognition-cost framework.

(3) How to read the formal statement: It is a noncomputable definition

noncomputable def totalEnergy (m : ℝ) (V : ℝ → ℝ) (γ : ℝ → ℝ) (t : ℝ) : ℝ := standardHamiltonian m V (γ t) (conjugateMomentum m γ t)

The parameters are mass m, potential V, trajectory γ, and time t; the body composes the two helper definitions already present in the same module.

(4) Visible dependencies and certificates: Directly depends on standardHamiltonian and conjugateMomentum. It is the central object in the proof of energy_conservation, which also cites hamilton_equations_from_EL and the differentiability hypotheses. The module reports zero sorry and zero axiom.

(5) What this declaration does not prove: It only defines the energy functional; conservation itself is proved separately in energy_conservation under extra hypotheses (differentiability of V and γ, the factored derivative identity h_dE_eq_factored, and the Euler-Lagrange equation). It does not address the full non-quadratic J-action, recognition-specific costs, or any forcing-chain theorems outside this module.