Complete structured record
{
"book": "book4",
"cited_by": [],
"cites": [
"definition:bk4_tilda_substitution",
"lemma:bk4_convergence_of_symbolic_drift",
"theorem:bk4_fuzzy_chain_rule",
"theorem:bk4_fuzzy_product_rule",
"theorem:bk4_fuzzy_sum_rule"
],
"depends_on": [
"definition:bk4_tilda_substitution",
"lemma:bk4_convergence_of_symbolic_drift"
],
"file": "book4.tex",
"forward_ref_roles": [
{
"context": "the fuzzy sum, product, and chain rules (Thms.~\\ref{theorem:bk4_fuzzy_sum_rule}, \\ref{theorem:bk4_fuzzy_product_rule}, \\ref{theorem:bk4_fuzzy_chain_rule}). ($\\Rightarrow$) Conversely, if $\\delta^1_O \\widetilde{D}$ exists in $\\mathcal{L}(T\\widetilde{M}, T\\widetilde{M})$, ex",
"label": "theorem:bk4_fuzzy_chain_rule",
"line_distance": 58,
"role": "teaser",
"target_line": 4286,
"target_type": "theorem"
},
{
"context": "E}_L, \\mathcal{E}_P, \\mathcal{E}_C$ by the fuzzy sum, product, and chain rules (Thms.~\\ref{theorem:bk4_fuzzy_sum_rule}, \\ref{theorem:bk4_fuzzy_product_rule}, \\ref{theorem:bk4_fuzzy_chain_rule}). ($\\Rightarrow$) Conversely, if $\\delta^1_O \\widetilde{D}$ exists in $\\mathcal{L}(",
"label": "theorem:bk4_fuzzy_product_rule",
"line_distance": 209,
"role": "teaser",
"target_line": 4437,
"target_type": "theorem"
},
{
"context": "lon_O$-bounded residues $\\mathcal{E}_L, \\mathcal{E}_P, \\mathcal{E}_C$ by the fuzzy sum, product, and chain rules (Thms.~\\ref{theorem:bk4_fuzzy_sum_rule}, \\ref{theorem:bk4_fuzzy_product_rule}, \\ref{theorem:bk4_fuzzy_chain_rule}). ($\\Rightarrow$) Conversely, if $\\delta^1_O",
"label": "theorem:bk4_fuzzy_sum_rule",
"line_distance": 605,
"role": "teaser",
"target_line": 4833,
"target_type": "theorem"
}
],
"forward_refs": [
"theorem:bk4_fuzzy_chain_rule",
"theorem:bk4_fuzzy_product_rule",
"theorem:bk4_fuzzy_sum_rule"
],
"id": "proof:bk4_existence_observer_valid_derivatives",
"label": "proof:bk4_existence_observer_valid_derivatives",
"latex_body": "\\begin{proof}\n\\label{proof:bk4_existence_observer_valid_derivatives}\n\\leavevmode\n($\\Leftarrow$) Conditions (1)--(3) are exactly the hypotheses under which Lemma~\\ref{lemma:bk4_convergence_of_symbolic_drift} applies: (1) is the convergence rate controlling the second-order differences against the first-order ones, (3) is the reflective stabilization of all differences up to order $N_O$, and (2) is the ratio-compatibility of the tilda-substitution (Def.~\\ref{definition:bk4_tilda_substitution}). By that lemma the transfinite drift sequence is Cauchy in the observer operator norm, so the limit $\\delta^1_O \\widetilde{D} = \\lim_{\\lambda} \\delta^1_O(D_{\\lambda+1} - D_\\lambda)$ exists in $\\mathcal{L}(T\\widetilde{M}, T\\widetilde{M})$. The derivative then obeys linearity, the product law, and the chain law up to the $\\epsilon_O$-bounded residues $\\mathcal{E}_L, \\mathcal{E}_P, \\mathcal{E}_C$ by the fuzzy sum, product, and chain rules (Thms.~\\ref{theorem:bk4_fuzzy_sum_rule}, \\ref{theorem:bk4_fuzzy_product_rule}, \\ref{theorem:bk4_fuzzy_chain_rule}).\n($\\Rightarrow$) Conversely, if $\\delta^1_O \\widetilde{D}$ exists in $\\mathcal{L}(T\\widetilde{M}, T\\widetilde{M})$, existence of the operator-norm limit forces the second-order increments to be sub-dominant to the first-order increments---otherwise the difference quotients do not converge---which is condition~(1); boundedness of the limit operator at the resolution scale $\\epsilon_O$ forces the reflective stabilization~(3) and the substitution ratio bound~(2). The conditions are therefore necessary as well as sufficient.\n\\end{proof}",
"line": 4228,
"macros_used": [],
"matter_region": "mainmatter",
"matter_role": "canonical_book",
"name": "",
"proves": "theorem:bk4_existence_observer_valid_derivatives",
"ref_roles": [
{
"context": "stabilization of all differences up to order $N_O$, and (2) is the ratio-compatibility of the tilda-substitution (Def.~\\ref{definition:bk4_tilda_substitution}). By that lemma the transfinite drift sequence is Cauchy in the observer operator norm, so the limit $\\delta^1_O \\widet",
"label": "definition:bk4_tilda_substitution",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book4.tex",
"target_line": 4193,
"target_type": "definition"
},
{
"context": "observer_valid_derivatives} \\leavevmode ($\\Leftarrow$) Conditions (1)--(3) are exactly the hypotheses under which Lemma~\\ref{lemma:bk4_convergence_of_symbolic_drift} applies: (1) is the convergence rate controlling the second-order differences against the first-order ones, (3) is the",
"label": "lemma:bk4_convergence_of_symbolic_drift",
"logical_support": true,
"role": "proof_support",
"target_file": "book4.tex",
"target_line": 4159,
"target_type": "lemma"
},
{
"context": "the fuzzy sum, product, and chain rules (Thms.~\\ref{theorem:bk4_fuzzy_sum_rule}, \\ref{theorem:bk4_fuzzy_product_rule}, \\ref{theorem:bk4_fuzzy_chain_rule}). ($\\Rightarrow$) Conversely, if $\\delta^1_O \\widetilde{D}$ exists in $\\mathcal{L}(T\\widetilde{M}, T\\widetilde{M})$, ex",
"label": "theorem:bk4_fuzzy_chain_rule",
"logical_support": false,
"role": "forward_teaser",
"target_file": "book4.tex",
"target_line": 4286,
"target_type": "theorem"
},
{
"context": "E}_L, \\mathcal{E}_P, \\mathcal{E}_C$ by the fuzzy sum, product, and chain rules (Thms.~\\ref{theorem:bk4_fuzzy_sum_rule}, \\ref{theorem:bk4_fuzzy_product_rule}, \\ref{theorem:bk4_fuzzy_chain_rule}). ($\\Rightarrow$) Conversely, if $\\delta^1_O \\widetilde{D}$ exists in $\\mathcal{L}(",
"label": "theorem:bk4_fuzzy_product_rule",
"logical_support": false,
"role": "forward_teaser",
"target_file": "book4.tex",
"target_line": 4437,
"target_type": "theorem"
},
{
"context": "lon_O$-bounded residues $\\mathcal{E}_L, \\mathcal{E}_P, \\mathcal{E}_C$ by the fuzzy sum, product, and chain rules (Thms.~\\ref{theorem:bk4_fuzzy_sum_rule}, \\ref{theorem:bk4_fuzzy_product_rule}, \\ref{theorem:bk4_fuzzy_chain_rule}). ($\\Rightarrow$) Conversely, if $\\delta^1_O",
"label": "theorem:bk4_fuzzy_sum_rule",
"logical_support": false,
"role": "forward_teaser",
"target_file": "book4.tex",
"target_line": 4833,
"target_type": "theorem"
}
],
"refs": [
"definition:bk4_tilda_substitution",
"lemma:bk4_convergence_of_symbolic_drift",
"theorem:bk4_fuzzy_chain_rule",
"theorem:bk4_fuzzy_product_rule",
"theorem:bk4_fuzzy_sum_rule"
],
"role": "proof",
"type": "proof"
}