proofmainmatter
proof:bk8_threshold_of_metabolic_autonomy
proof:bk8_threshold_of_metabolic_autonomy
Exact LaTeX body
\begin{proof}
\label{proof:bk8_threshold_of_metabolic_autonomy}
\leavevmode
This is the autonomy functional of the Threshold of Autonomy theorem (Thm.~\ref{theorem:bk8_biological_phase_transition}) expressed for the metabolic-programming dynamics, and the argument transfers verbatim. By the Metabolic Sufficiency Criterion (Axiom~\ref{axiom:bk8_mutation_phase_shift}) each cycle dissipates a fixed quantum $\delta_F>0$ of knot free energy, so $\Psi_{\mathrm{aut}}$ is the long-run average dissipation rate; balance of production against repair gives metabolic autonomy iff $\Psi_{\mathrm{aut}}\ge 0$. When $\Psi_{\mathrm{aut}}>0$, $\freeenergy^{\text{knot}}$ decays while identity stability $\identitystability$ (Def.~\ref{definition:bk8_identitystability}), rising as torsion is resolved, converges by the Grönwall estimate to $\identitystability^{(\infty)}\ge 1-2e^{-\gamma t}$, $\gamma>0$ (cf.~Thm.~\ref{theorem:bk8_thermodynamic_necessity_of_symbolic_metabolism}). Thus $\Psi_{\mathrm{aut}}\ge 0$ is exactly the threshold of metabolic autonomy.
\end{proof}Depends on
Cites
Reference roles
| Target | Role | Logical support |
|---|---|---|
axiom:bk8_mutation_phase_shift | definition_anchor | yes |
definition:bk8_identitystability | definition_anchor | yes |
theorem:bk8_biological_phase_transition | proof_support | yes |
theorem:bk8_thermodynamic_necessity_of_symbolic_metabolism | cf_near_match | yes |
Complete structured record
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"latex_body": "\\begin{proof}\n\\label{proof:bk8_threshold_of_metabolic_autonomy}\n\\leavevmode\nThis is the autonomy functional of the Threshold of Autonomy theorem (Thm.~\\ref{theorem:bk8_biological_phase_transition}) expressed for the metabolic-programming dynamics, and the argument transfers verbatim. By the Metabolic Sufficiency Criterion (Axiom~\\ref{axiom:bk8_mutation_phase_shift}) each cycle dissipates a fixed quantum $\\delta_F>0$ of knot free energy, so $\\Psi_{\\mathrm{aut}}$ is the long-run average dissipation rate; balance of production against repair gives metabolic autonomy iff $\\Psi_{\\mathrm{aut}}\\ge 0$. When $\\Psi_{\\mathrm{aut}}>0$, $\\freeenergy^{\\text{knot}}$ decays while identity stability $\\identitystability$ (Def.~\\ref{definition:bk8_identitystability}), rising as torsion is resolved, converges by the Grönwall estimate to $\\identitystability^{(\\infty)}\\ge 1-2e^{-\\gamma t}$, $\\gamma>0$ (cf.~Thm.~\\ref{theorem:bk8_thermodynamic_necessity_of_symbolic_metabolism}). Thus $\\Psi_{\\mathrm{aut}}\\ge 0$ is exactly the threshold of metabolic autonomy.\n\\end{proof}",
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"context": "the metabolic-programming dynamics, and the argument transfers verbatim. By the Metabolic Sufficiency Criterion (Axiom~\\ref{axiom:bk8_mutation_phase_shift}) each cycle dissipates a fixed quantum $\\delta_F>0$ of knot free energy, so $\\Psi_{\\mathrm{aut}}$ is the long-run avera",
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"context": "threshold_of_metabolic_autonomy} \\leavevmode This is the autonomy functional of the Threshold of Autonomy theorem (Thm.~\\ref{theorem:bk8_biological_phase_transition}) expressed for the metabolic-programming dynamics, and the argument transfers verbatim. By the Metabolic Sufficiency Cr",
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