Complete structured record
{
"book": "book4",
"cited_by": [
"proof:bk4_sketch_stokes",
"proof:bk5_coherence_through_dynamic_equilibriium",
"scholium:bk4_zero_is_idealized_in_boundedness"
],
"cites": [
"corollary:bk4_emergence_of_classical_ge",
"definition:bk4_fuzzy_divergence_operator",
"definition:bk4_induced_area",
"proposition:bk4_fuzzy_deriv_algebra",
"proposition:bk4_quantum_geometry",
"theorem:bk4_fuzzy_divergence",
"theorem:bk4_symbolic_stokes"
],
"depends_on": [
"corollary:bk4_emergence_of_classical_ge",
"definition:bk4_fuzzy_divergence_operator",
"definition:bk4_induced_area",
"proposition:bk4_fuzzy_deriv_algebra",
"proposition:bk4_quantum_geometry",
"theorem:bk4_fuzzy_divergence",
"theorem:bk4_symbolic_stokes"
],
"file": "book4.tex",
"id": "theorem:bk4_fuzzy_divergence_theorem",
"label": "theorem:bk4_fuzzy_divergence_theorem",
"latex_body": "\\begin{theorem}[Fuzzy Divergence Theorem]\n\\label{theorem:bk4_fuzzy_divergence_theorem}\nCombining Def.~\\ref{definition:bk4_fuzzy_divergence_operator}, the local divergence law in Thm.~\\ref{theorem:bk4_fuzzy_divergence}, derivative algebra from Prop.~\\ref{proposition:bk4_fuzzy_deriv_algebra}, observer-induced measure geometry (Def.~\\ref{definition:bk4_induced_area}), symbolic holonomy (with the optional certified quantum interpretation of Prop.~\\ref{proposition:bk4_quantum_geometry}), and the classical-limit correspondence in Cor.~\\ref{corollary:bk4_emergence_of_classical_ge}, this theorem is the flux-divergence counterpart of the Stokes relation in Thm.~\\ref{theorem:bk4_symbolic_stokes}.\nLet $\\Omega$ be a region in the fuzzy membrane $\\tilde{M}$ with boundary $\\partial\\Omega$. For a fuzzy vector field $\\vec{V}$, the fuzzy flux across the boundary is related to the fuzzy divergence within the volume by:\n\\[\n\\oint_{\\mathcal{O}}^{\\partial\\Omega} \\vec{V} \\cdot d\\vec{A}_{\\mathcal{O}} = \\iiint_{\\Omega} (\\text{div}_{\\mathcal{O}} \\vec{V}) \\, dV_{\\mathcal{O}} + \\mathcal{H}_{\\mathcal{O}}(\\Omega, \\vec{V})\n\\]\nwhere $\\oint_{\\mathcal{O}}$ is the fuzzy surface integral, $d\\vec{A}_{\\mathcal{O}}$ and $dV_{\\mathcal{O}}$ are observer-induced area and volume elements, and $\\mathcal{H}_{\\mathcal{O}}$ is a \\textbf{Boundary Holonomy Term} that accounts for information leakage or generation across the observer's fuzzy boundary.\n\\end{theorem}",
"lean_alignment": {
"conditions": [
"continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
"modeling laws are structure fields or explicit hypotheses"
],
"countermodels": [],
"full_record": "bib/principia_lean_alignment.json",
"kernel_certified": true,
"notes": [
"Instance: observerValue := the fuzzy boundary flux integral, classicalValue := the fuzzy volume-divergence integral, correction := H_O(Omega,V) (Boundary Holonomy Term)."
],
"record_ids": [
"MAP-BOOK4A-080"
],
"statuses": [
"conditional"
],
"witnesses": [
"Book4D.observer_correction_zero_iff_classical"
]
},
"line": 6341,
"macros_used": [],
"matter_region": "mainmatter",
"matter_role": "canonical_book",
"name": "Fuzzy Divergence Theorem",
"proof_labels": [
"proof:bk4_fuzzy_divergence_theorem"
],
"proof_status": "proven",
"ref_roles": [
{
"context": "quantum interpretation of Prop.~\\ref{proposition:bk4_quantum_geometry}), and the classical-limit correspondence in Cor.~\\ref{corollary:bk4_emergence_of_classical_ge}, this theorem is the flux-divergence counterpart of the Stokes relation in Thm.~\\ref{theorem:bk4_symbolic_stokes}. Let",
"label": "corollary:bk4_emergence_of_classical_ge",
"logical_support": true,
"role": "interpretive_bridge",
"target_file": "book4.tex",
"target_line": 4116,
"target_type": "corollary"
},
{
"context": "\\begin{theorem}[Fuzzy Divergence Theorem] \\label{theorem:bk4_fuzzy_divergence_theorem} Combining Def.~\\ref{definition:bk4_fuzzy_divergence_operator}, the local divergence law in Thm.~\\ref{theorem:bk4_fuzzy_divergence}, derivative algebra from Prop.~\\ref{proposition:bk",
"label": "definition:bk4_fuzzy_divergence_operator",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book4.tex",
"target_line": 6305,
"target_type": "definition"
},
{
"context": "ence}, derivative algebra from Prop.~\\ref{proposition:bk4_fuzzy_deriv_algebra}, observer-induced measure geometry (Def.~\\ref{definition:bk4_induced_area}), symbolic holonomy (with the optional certified quantum interpretation of Prop.~\\ref{proposition:bk4_quantum_geometry}",
"label": "definition:bk4_induced_area",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book4.tex",
"target_line": 5922,
"target_type": "definition"
},
{
"context": "ivergence_operator}, the local divergence law in Thm.~\\ref{theorem:bk4_fuzzy_divergence}, derivative algebra from Prop.~\\ref{proposition:bk4_fuzzy_deriv_algebra}, observer-induced measure geometry (Def.~\\ref{definition:bk4_induced_area}), symbolic holonomy (with the optional certi",
"label": "proposition:bk4_fuzzy_deriv_algebra",
"logical_support": true,
"role": "formal_dependency",
"target_file": "book4.tex",
"target_line": 4986,
"target_type": "proposition"
},
{
"context": "Def.~\\ref{definition:bk4_induced_area}), symbolic holonomy (with the optional certified quantum interpretation of Prop.~\\ref{proposition:bk4_quantum_geometry}), and the classical-limit correspondence in Cor.~\\ref{corollary:bk4_emergence_of_classical_ge}, this theorem is the flu",
"label": "proposition:bk4_quantum_geometry",
"logical_support": true,
"role": "interpretive_bridge",
"target_file": "book4.tex",
"target_line": 6194,
"target_type": "proposition"
},
{
"context": "zzy_divergence_theorem} Combining Def.~\\ref{definition:bk4_fuzzy_divergence_operator}, the local divergence law in Thm.~\\ref{theorem:bk4_fuzzy_divergence}, derivative algebra from Prop.~\\ref{proposition:bk4_fuzzy_deriv_algebra}, observer-induced measure geometry (Def.~\\ref{",
"label": "theorem:bk4_fuzzy_divergence",
"logical_support": true,
"role": "formal_dependency",
"target_file": "book4.tex",
"target_line": 5458,
"target_type": "theorem"
},
{
"context": "orollary:bk4_emergence_of_classical_ge}, this theorem is the flux-divergence counterpart of the Stokes relation in Thm.~\\ref{theorem:bk4_symbolic_stokes}. Let $\\Omega$ be a region in the fuzzy membrane $\\tilde{M}$ with boundary $\\partial\\Omega$. For a fuzzy vector field $\\",
"label": "theorem:bk4_symbolic_stokes",
"logical_support": true,
"role": "formal_dependency",
"target_file": "book4.tex",
"target_line": 5937,
"target_type": "theorem"
}
],
"refs": [
"corollary:bk4_emergence_of_classical_ge",
"definition:bk4_fuzzy_divergence_operator",
"definition:bk4_induced_area",
"proposition:bk4_fuzzy_deriv_algebra",
"proposition:bk4_quantum_geometry",
"theorem:bk4_fuzzy_divergence",
"theorem:bk4_symbolic_stokes"
],
"role": "theorem",
"type": "theorem"
}