Exact LaTeX body
\begin{theorem}[Bounded-increment parameter lift]
\label{theorem:appD_bounded_increment_parameter_lift}
Let \(M\) be a finite-dimensional symbolic manifold equipped with an
observer-relative norm \(\|\cdot\|_{\Obs}\), and let
\[
r:M\longrightarrow \mathbb{R}^{k}
\]
be a \(C^{1}\) residual map encoding \(k\) simultaneous symbolic constraints
near \(x\in M\). Work in a normal neighborhood of \(x\), write
\(J_x = Dr_x:T_xM\to\mathbb{R}^k\), and suppose the bounded observer can make
only increments \(v\in T_xM\) with \(\|v\|_{\Obs}\leq B\), up to local
linearization error \(O(\kappa\|v\|_{\Obs}^{2})\) from symbolic curvature
(cf.~Def.~\ref{definition:bk4_symbolic_curvature} and
Def.~\ref{definition:bk6_symbolic_curvature_tensor}). Define the first-order
simultaneous satisfaction cost
\[
d_M(x)
=
\inf\{\|v\|_{\Obs}: J_xv=-r(x)\}.
\]
If \(d_M(x)>B\), then no bounded first-order increment satisfies all
constraints at \(x\), apart from the stated curvature-scale correction.
Now let \(\Lambda\) be a parameter manifold and extend the residual to
\[
R:M\times\Lambda\longrightarrow \mathbb{R}^{k},
\qquad R(x,\lambda_0)=r(x),
\]
with derivative
\[
D R_{(x,\lambda_0)}(v,\mu)=J_xv+J_{\lambda}\mu .
\]
For any positive parameter weight \(\alpha\), define
\[
d_{M\times\Lambda}(x,\lambda_0)
=
\inf\left\{
\bigl(\|v\|_{\Obs}^{2}+\alpha^{2}\|\mu\|^{2}\bigr)^{1/2}
:
J_xv+J_{\lambda}\mu=-r(x)
\right\}.
\]
Then \(d_{M\times\Lambda}(x,\lambda_0)\leq d_M(x)\). The inequality is strict
exactly when the introduced parameter direction contributes a non-redundant
constraint-canceling component: equivalently, the least weighted-norm solution
of \(J_xv+J_{\lambda}\mu=-r(x)\) has \(\mu\neq0\). Consequently, a new
parameter relieves a bounded-increment obstruction precisely when it enlarges
the accessible tangent cone in a direction relevant to the residual.
\end{theorem}
Complete structured record
{
"book": "appendix_symbolic_framing",
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"definition:bk4_symbolic_curvature",
"definition:bk6_symbolic_curvature_tensor"
],
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"definition:appD_llm_observer_tuple",
"definition:bk4_symbolic_curvature",
"definition:bk6_symbolic_curvature_tensor",
"definition:bk7_symbolic_reflexive_validation_srv",
"theorem:bk4_test_time_differentiation_c"
],
"file": "appendix_symbolic_framing.tex",
"id": "theorem:appD_bounded_increment_parameter_lift",
"label": "theorem:appD_bounded_increment_parameter_lift",
"latex_body": "\\begin{theorem}[Bounded-increment parameter lift]\n\\label{theorem:appD_bounded_increment_parameter_lift}\nLet \\(M\\) be a finite-dimensional symbolic manifold equipped with an\nobserver-relative norm \\(\\|\\cdot\\|_{\\Obs}\\), and let\n\\[\n r:M\\longrightarrow \\mathbb{R}^{k}\n\\]\nbe a \\(C^{1}\\) residual map encoding \\(k\\) simultaneous symbolic constraints\nnear \\(x\\in M\\). Work in a normal neighborhood of \\(x\\), write\n\\(J_x = Dr_x:T_xM\\to\\mathbb{R}^k\\), and suppose the bounded observer can make\nonly increments \\(v\\in T_xM\\) with \\(\\|v\\|_{\\Obs}\\leq B\\), up to local\nlinearization error \\(O(\\kappa\\|v\\|_{\\Obs}^{2})\\) from symbolic curvature\n(cf.~Def.~\\ref{definition:bk4_symbolic_curvature} and\nDef.~\\ref{definition:bk6_symbolic_curvature_tensor}). Define the first-order\nsimultaneous satisfaction cost\n\\[\n d_M(x)\n =\n \\inf\\{\\|v\\|_{\\Obs}: J_xv=-r(x)\\}.\n\\]\nIf \\(d_M(x)>B\\), then no bounded first-order increment satisfies all\nconstraints at \\(x\\), apart from the stated curvature-scale correction.\n\nNow let \\(\\Lambda\\) be a parameter manifold and extend the residual to\n\\[\n R:M\\times\\Lambda\\longrightarrow \\mathbb{R}^{k},\n \\qquad R(x,\\lambda_0)=r(x),\n\\]\nwith derivative\n\\[\n D R_{(x,\\lambda_0)}(v,\\mu)=J_xv+J_{\\lambda}\\mu .\n\\]\nFor any positive parameter weight \\(\\alpha\\), define\n\\[\n d_{M\\times\\Lambda}(x,\\lambda_0)\n =\n \\inf\\left\\{\n \\bigl(\\|v\\|_{\\Obs}^{2}+\\alpha^{2}\\|\\mu\\|^{2}\\bigr)^{1/2}\n :\n J_xv+J_{\\lambda}\\mu=-r(x)\n \\right\\}.\n\\]\nThen \\(d_{M\\times\\Lambda}(x,\\lambda_0)\\leq d_M(x)\\). The inequality is strict\nexactly when the introduced parameter direction contributes a non-redundant\nconstraint-canceling component: equivalently, the least weighted-norm solution\nof \\(J_xv+J_{\\lambda}\\mu=-r(x)\\) has \\(\\mu\\neq0\\). Consequently, a new\nparameter relieves a bounded-increment obstruction precisely when it enlarges\nthe accessible tangent cone in a direction relevant to the residual.\n\\end{theorem}",
"lean_alignment": {
"conditions": [
"modeling laws are structure fields or explicit hypotheses; continuum/categorical content is NOT formalized"
],
"countermodels": [],
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"kernel_certified": false,
"notes": [
"Erases the manifold/linear-algebra content (the residual map, its derivative J_x, tangent spaces, the curvature correction) and keeps only its abstract order-theoretic core: enlarging a feasible real-valued constraint set can only lower sInf, and strictly lowers it exactly when the enlarged set contains a witness below the old infimum -- the honest kernel of 'a new parameter relieves the obstruction precisely when it contributes a non-redundant direction.'"
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]
},
"line": 233,
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],
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"matter_role": "appendix_expansion",
"name": "Bounded-increment parameter lift",
"proof_labels": [
"proof:appD_bounded_increment_parameter_lift"
],
"proof_status": "proven",
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"context": "\\(\\|v\\|_{\\Obs}\\leq B\\), up to local linearization error \\(O(\\kappa\\|v\\|_{\\Obs}^{2})\\) from symbolic curvature (cf.~Def.~\\ref{definition:bk4_symbolic_curvature} and Def.~\\ref{definition:bk6_symbolic_curvature_tensor}). Define the first-order simultaneous satisfaction cost \\[",
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"context": "error \\(O(\\kappa\\|v\\|_{\\Obs}^{2})\\) from symbolic curvature (cf.~Def.~\\ref{definition:bk4_symbolic_curvature} and Def.~\\ref{definition:bk6_symbolic_curvature_tensor}). Define the first-order simultaneous satisfaction cost \\[ d_M(x) = \\inf\\{\\|v\\|_{\\Obs}: J_xv=-r(x)\\}. \\] If",
"label": "definition:bk6_symbolic_curvature_tensor",
"logical_support": true,
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