Exact LaTeX body
\begin{proposition}[Drift-Reflection Correspondence]
\label{proposition:bk6_drift_reflection_correspondence}
For any symbolic system $\mathcal{S}$ in reflective equilibrium, the drift field $D$ (Def.~\ref{definition:bk1_drift_field}) and reflection operator $R$ (Def.~\ref{definition:bk1_reflection_operator}) satisfy:
\begin{equation}
D = \frac{1}{2}(R - R^{-1}) + \mathcal{O}(\|R - \text{Id}\|_{\text{op}}^2)
\end{equation}
establishing a fundamental correspondence between reflective processes and symbolic drift.
See Prop.~\ref{proposition:bk6_reflective_mutation_inhibition} and Thm.~\ref{theorem:bk5_rift_reflection_balance_in_strategy_space}.
\begin{proof}[Drift Reflection Commutation Equilibrium]
\label{proof:bk6_drift_reflection_commutation_equilibrium}
\leavevmode
In reflective equilibrium, the symbolic system’s evolution is governed by a mutual commutation of drift and reflection. Formally, this is expressed as:
\[
R \circ \Phi_t = \Phi_t \circ R,
\]
where \( \Phi_t \) is the symbolic flow generated by the drift operator \( D \). This equality asserts that symbolic transformation under drift is structurally preserved by the reflection operator — a hallmark of equilibrium dynamics.
Differentiating both sides with respect to \( t \) at \( t = 0 \) yields:
\[
\left.\frac{d}{dt} R \circ \Phi_t \right|_{t=0} = \left. \frac{d}{dt} \Phi_t \circ R \right|_{t=0},
\]
which simplifies to the operator identity:
\[
DR = RD.
\]
This expresses extbf{infinitesimal commutativity}: at the level of symbolic generators, drift and reflection preserve each other’s action. This directly supports the balance condition described in the mutation-equilibrium proof (\ref{proof:bk6_mutation_equilibrium_entropy_balance}), where symbolic entropy production and dissipation reach parity.
Now assume that the reflection operator is extbf{near-identity}, i.e., \( R \approx \text{Id} \), as in the setting of stable coherence-preserving dynamics discussed in (\ref{proof:bk6_stable_reflective_submanifold}). Expanding \( R \) around identity as:
\[
R = \text{Id} + \epsilon A + \mathcal{O}(\epsilon^2),
\]
and applying the commutation condition, we find that:
\[
D \approx \frac{1}{2}(R - R^{-1}) + \mathcal{O}(\|R - \text{Id}\|_{\text{op}}^2),
\]
which characterizes drift as a extbf{symmetric deviation} from identity induced by reflection asymmetry. This interpretation reinforces the extbf{bifurcation boundary condition} established in (\ref{proof:bk6_symbolic_fokker_planck_bifurcation}) and maintains symbolic free energy (Def.~\ref{definition:bk2_symbolic_free_energy}) beneath the mutation threshold (\ref{proof:bk6_symbolic_mutation_threshold}) in the coherent regime.
Thus, in reflective equilibrium, symbolic drift arises as a geometric consequence of small reflective deviation, ensuring stability and coherence within symbolic dynamics.
\end{proof}
\end{proposition}
Complete structured record
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"latex_body": "\\begin{proposition}[Drift-Reflection Correspondence]\n\\label{proposition:bk6_drift_reflection_correspondence}\nFor any symbolic system $\\mathcal{S}$ in reflective equilibrium, the drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) and reflection operator $R$ (Def.~\\ref{definition:bk1_reflection_operator}) satisfy:\n\\begin{equation}\nD = \\frac{1}{2}(R - R^{-1}) + \\mathcal{O}(\\|R - \\text{Id}\\|_{\\text{op}}^2)\n\\end{equation}\nestablishing a fundamental correspondence between reflective processes and symbolic drift.\nSee Prop.~\\ref{proposition:bk6_reflective_mutation_inhibition} and Thm.~\\ref{theorem:bk5_rift_reflection_balance_in_strategy_space}.\n\\begin{proof}[Drift Reflection Commutation Equilibrium]\n\\label{proof:bk6_drift_reflection_commutation_equilibrium}\n\\leavevmode\n\nIn reflective equilibrium, the symbolic system’s evolution is governed by a mutual commutation of drift and reflection. Formally, this is expressed as:\n\\[\nR \\circ \\Phi_t = \\Phi_t \\circ R,\n\\]\nwhere \\( \\Phi_t \\) is the symbolic flow generated by the drift operator \\( D \\). This equality asserts that symbolic transformation under drift is structurally preserved by the reflection operator — a hallmark of equilibrium dynamics.\n\nDifferentiating both sides with respect to \\( t \\) at \\( t = 0 \\) yields:\n\\[\n\\left.\\frac{d}{dt} R \\circ \\Phi_t \\right|_{t=0} = \\left. \\frac{d}{dt} \\Phi_t \\circ R \\right|_{t=0},\n\\]\nwhich simplifies to the operator identity:\n\\[\nDR = RD.\n\\]\nThis expresses \textbf{infinitesimal commutativity}: at the level of symbolic generators, drift and reflection preserve each other’s action. This directly supports the balance condition described in the mutation-equilibrium proof (\\ref{proof:bk6_mutation_equilibrium_entropy_balance}), where symbolic entropy production and dissipation reach parity.\nNow assume that the reflection operator is \textbf{near-identity}, i.e., \\( R \\approx \\text{Id} \\), as in the setting of stable coherence-preserving dynamics discussed in (\\ref{proof:bk6_stable_reflective_submanifold}). Expanding \\( R \\) around identity as:\n\\[\nR = \\text{Id} + \\epsilon A + \\mathcal{O}(\\epsilon^2),\n\\]\nand applying the commutation condition, we find that:\n\\[\nD \\approx \\frac{1}{2}(R - R^{-1}) + \\mathcal{O}(\\|R - \\text{Id}\\|_{\\text{op}}^2),\n\\]\nwhich characterizes drift as a \textbf{symmetric deviation} from identity induced by reflection asymmetry. This interpretation reinforces the \textbf{bifurcation boundary condition} established in (\\ref{proof:bk6_symbolic_fokker_planck_bifurcation}) and maintains symbolic free energy (Def.~\\ref{definition:bk2_symbolic_free_energy}) beneath the mutation threshold (\\ref{proof:bk6_symbolic_mutation_threshold}) in the coherent regime.\nThus, in reflective equilibrium, symbolic drift arises as a geometric consequence of small reflective deviation, ensuring stability and coherence within symbolic dynamics.\n\\end{proof}\n\\end{proposition}",
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"context": "t_reflection_correspondence} For any symbolic system $\\mathcal{S}$ in reflective equilibrium, the drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) and reflection operator $R$ (Def.~\\ref{definition:bk1_reflection_operator}) satisfy: \\begin{equation} D = \\frac{1}{2}(",
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