theoremprovenmainmatter
Regularity of Symbolic Curvature: Continuity and Differentiability
theorem:bk4_curvature_continuity
Exact LaTeX body
\begin{theorem}[Regularity of Symbolic Curvature: Continuity and Differentiability]
\label{theorem:bk4_curvature_continuity}
\label{theorem:bk4_curvature_differentiability}
The symbolic curvature $\kappa_O$ (Def.~\ref{definition:bk4_symbolic_curvature}) inherits the regularity of the operators that generate it:
\begin{enumerate}
\item (\textbf{Continuity}) if $\delta_O$ and $K_O$ are continuous, then $\kappa_O : \mathcal{S} \to \mathbb{R}^+$ is continuous on the proto-symbolic space of Def.~\ref{definition:bk4_proto_symbolic_space};
\item (\textbf{Differentiability}) if $R_\lambda$ (Def.~\ref{definition:bk4_reflexive_operator}) and $K_O$ are $C^2$ smooth, then $\kappa_O$ is twice differentiable.
\end{enumerate}
This is a consequence of curvature arising as the interaction product of drift ($D$) and reflection ($R$) rather than as a primitive (cf.~Corollary~\ref{corollary:bk1_dimensional_bounds_emergence} on rank bounds of emergent curvature).
\end{theorem}Complete structured record
{
"book": "book4",
"cited_by": [],
"cites": [],
"depends_on": [],
"file": "book4.tex",
"id": "theorem:bk4_curvature_continuity",
"label": "theorem:bk4_curvature_continuity",
"latex_body": "\\begin{theorem}[Regularity of Symbolic Curvature: Continuity and Differentiability]\n\\label{theorem:bk4_curvature_continuity}\n\\label{theorem:bk4_curvature_differentiability}\nThe symbolic curvature $\\kappa_O$ (Def.~\\ref{definition:bk4_symbolic_curvature}) inherits the regularity of the operators that generate it:\n\\begin{enumerate}\n \\item (\\textbf{Continuity}) if $\\delta_O$ and $K_O$ are continuous, then $\\kappa_O : \\mathcal{S} \\to \\mathbb{R}^+$ is continuous on the proto-symbolic space of Def.~\\ref{definition:bk4_proto_symbolic_space};\n \\item (\\textbf{Differentiability}) if $R_\\lambda$ (Def.~\\ref{definition:bk4_reflexive_operator}) and $K_O$ are $C^2$ smooth, then $\\kappa_O$ is twice differentiable.\n\\end{enumerate}\nThis is a consequence of curvature arising as the interaction product of drift ($D$) and reflection ($R$) rather than as a primitive (cf.~Corollary~\\ref{corollary:bk1_dimensional_bounds_emergence} on rank bounds of emergent curvature).\n\\end{theorem}",
"lean_alignment": {
"conditions": [
"contraction constant is the modeling hypothesis for TTPR stability; the differentiable-manifold constraint space, recursion-depth dynamics, and differentiability clause stay open"
],
"countermodels": [],
"full_record": "bib/principia_lean_alignment.json",
"kernel_certified": true,
"notes": [
"Curvature inherits continuity from its generating operators; the differentiability/manifold clauses stay open."
],
"record_ids": [
"MAP-BOOK4A-089"
],
"statuses": [
"conditional"
],
"witnesses": [
"Book4Ref.curvature_inherits_continuity"
]
},
"line": 498,
"macros_used": [],
"matter_region": "mainmatter",
"matter_role": "canonical_book",
"name": "Regularity of Symbolic Curvature: Continuity and Differentiability",
"proof_labels": [
"proof:bk4_curvature_continuity"
],
"proof_status": "proven",
"refs": [
"corollary:bk1_dimensional_bounds_emergence",
"definition:bk4_proto_symbolic_space",
"definition:bk4_reflexive_operator",
"definition:bk4_symbolic_curvature"
],
"role": "theorem",
"type": "theorem"
}