remarkmainmatter
Contraction is sufficient, not necessary: the descent route
remark:bk4_ttpr_descent_route
Exact LaTeX body
\begin{remark}[Contraction is sufficient, not necessary: the descent route]
\label{remark:bk4_ttpr_descent_route}
Axiom~\ref{axiom:bk4_refinement_contraction} posits a strict contraction, which is a
strong hypothesis: a bounded-observer refinement operator need not be globally
Lipschitz with constant below one. It suffices that refinement \emph{descend} the
observer-relative symbolic entropy -- the ``annealing'' of the preceding remark --
forming a free-energy descent pair in the sense of
Thm.~\ref{theorem:bk7_reflective_convergence_to_stable_identity}: if there is a
bounded-below, lower semicontinuous potential $\Phi$ with
$d_{\mathcal{O}}(s, \mathcal{R}(s)) \le \Phi(s) - \Phi(\mathcal{R}(s))$, the refinement
orbit has summable increments and converges to a fixed point by the telescoping
argument, with no contraction constant required. The contraction of
Axiom~\ref{axiom:bk4_refinement_contraction} is then the special case in which $\Phi$
is comparable to $d_{\mathcal{O}}(\cdot, s^*)$, and it additionally certifies the
geometric rate $\kappa^{n}/(1-\kappa)$ derived above. Stating the weaker descent
condition keeps TTPR convergence from resting on a contraction the bounded observer
may not supply.
\end{remark}Depends on
Cites
Reference roles
| Target | Role | Logical support |
|---|---|---|
axiom:bk4_refinement_contraction | definition_anchor | yes |
theorem:bk7_reflective_convergence_to_stable_identity | formal_dependency | yes |
Complete structured record
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"id": "remark:bk4_ttpr_descent_route",
"label": "remark:bk4_ttpr_descent_route",
"latex_body": "\\begin{remark}[Contraction is sufficient, not necessary: the descent route]\n\\label{remark:bk4_ttpr_descent_route}\nAxiom~\\ref{axiom:bk4_refinement_contraction} posits a strict contraction, which is a\nstrong hypothesis: a bounded-observer refinement operator need not be globally\nLipschitz with constant below one. It suffices that refinement \\emph{descend} the\nobserver-relative symbolic entropy -- the ``annealing'' of the preceding remark --\nforming a free-energy descent pair in the sense of\nThm.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}: if there is a\nbounded-below, lower semicontinuous potential $\\Phi$ with\n$d_{\\mathcal{O}}(s, \\mathcal{R}(s)) \\le \\Phi(s) - \\Phi(\\mathcal{R}(s))$, the refinement\norbit has summable increments and converges to a fixed point by the telescoping\nargument, with no contraction constant required. The contraction of\nAxiom~\\ref{axiom:bk4_refinement_contraction} is then the special case in which $\\Phi$\nis comparable to $d_{\\mathcal{O}}(\\cdot, s^*)$, and it additionally certifies the\ngeometric rate $\\kappa^{n}/(1-\\kappa)$ derived above. Stating the weaker descent\ncondition keeps TTPR convergence from resting on a contraction the bounded observer\nmay not supply.\n\\end{remark}",
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"context": "\\begin{remark}[Contraction is sufficient, not necessary: the descent route] \\label{remark:bk4_ttpr_descent_route} Axiom~\\ref{axiom:bk4_refinement_contraction} posits a strict contraction, which is a strong hypothesis: a bounded-observer refinement operator need not be globally",
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"context": "ymbolic entropy -- the ``annealing'' of the preceding remark -- forming a free-energy descent pair in the sense of Thm.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}: if there is a bounded-below, lower semicontinuous potential $\\Phi$ with $d_{\\mathcal{O}}(s, \\mathcal{R}(s)) \\le \\Phi(s",
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