variational_dynamics_deterministic
plain-language theorem explainer
Variational trajectories in the Recognition Science ledger are uniquely determined by their initial configuration. Researchers modeling deterministic ledger evolution would cite this to justify unique futures from initial conditions plus the update rule. The proof proceeds by induction on discrete time, using the induction hypothesis to match log charges and then invoking uniqueness of the variational step.
Claim. Let $N > 0$. Suppose two trajectories $traj_1, traj_2 : ℕ → Configuration_N$ are both variational, i.e., each successive pair satisfies the variational successor condition (constrained global minimization of total defect subject to log-charge conservation). If the entries of $traj_1(0)$ equal those of $traj_2(0)$, then the entries of $traj_1(t)$ equal those of $traj_2(t)$ for every $t$.
background
In the Variational Dynamics module a Trajectory is the type ℕ → Configuration N. A trajectory is variational when every step satisfies IsVariationalSuccessor, which encodes the update rule state(t+1) = argmin_{s ∈ Feasible(state(t))} TotalDefect(s) while preserving the total log-ratio charge. The module sits atop LawOfExistence (defect equals J), InitialCondition (zero-defect start), TimeEmergence (non-increasing defect), and Determinism (unique minimizers from strict convexity of J). Upstream results supply the defect functional and successor arithmetic used to construct compatible steps.
proof idea
Proof by induction on t. Base case t=0 is immediate from the initial-entries hypothesis. For the successor step the induction hypothesis yields equal log charges at time n; this is used to build a compatibility witness showing traj₂(n+1) is a variational successor of traj₁(n). The conclusion follows by applying the uniqueness lemma variational_step_unique to the pair of successors.
why it matters
The theorem supplies the equation-of-motion analogue of Laplacian determinism inside F-008, closing the gap between the energy landscape and the explicit update rule. It guarantees unique paths under the constrained J-cost minimization that defines the ledger dynamics and thereby supports downstream claims about monotone defect along trajectories. The result sits within the Recognition forcing chain by ensuring that the discrete-time evolution respects the uniqueness forced by J-convexity.
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