Complete structured record
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{
"context": "system $\\{P_\\lambda\\}_{\\lambda \\in \\Lambda}$ admits an observer-relative differentiable structure. \\\\ (see Theorem~\\ref{theorem:appB_metric_completion}) \\item The reflection operators $\\{R_\\lambda\\}$ induce $\\mathcal{O}$-differentiable stabilization fields on $\\tild",
"label": "theorem:appB_metric_completion",
"role": "appendix_teaser",
"target_file": "appendix_symbolic_reflexive_validation.tex",
"target_line": 182,
"target_type": "theorem"
}
],
"appendix_teaser_refs": [
"theorem:appB_metric_completion"
],
"book": "book4",
"cited_by": [
"corollary:bk4_emergence_of_classical_ge",
"corollary:bk4_smoothness_as_epistemic_phenomenon",
"proof:bk4_drift_reflection_field",
"proof:bk4_drift_reflection_summary",
"proof:bk4_emergence_of_classical_ge",
"proof:bk4_fuzzy_substitution_drift_smoothing",
"proof:bk4_observer_functor_induced_structure",
"proof:bk4_observer_relative_smoothness",
"remark:bk4_fuzzy",
"remark:bk4_fuzzy_notation",
"theorem:bk4_compatibility_drift_reflective_operations",
"theorem:bk4_restated_fuzzy_symbolic_geometry_theorem"
],
"cites": [
"axiom:bk2_gradient_structure_drift",
"definition:bk1_bounded_observer",
"definition:bk4_fuzzy_symbolic_substitution",
"definition:bk4_observer_differentiable_",
"definition:bk4_substituted_drift_field",
"theorem:appB_metric_completion"
],
"depends_on": [
"axiom:bk2_gradient_structure_drift",
"definition:bk1_bounded_observer",
"definition:bk4_fuzzy_symbolic_substitution",
"definition:bk4_observer_differentiable_",
"definition:bk4_substituted_drift_field",
"lemma:bk4_local_differentiability_substituted_drift",
"lemma:bk4_observer_relative_smoothness",
"theorem:bk2_coherence_of_symbolic_therm"
],
"file": "book4.tex",
"id": "theorem:bk4_fuzzy_symbolic_geometry_theorem",
"label": "theorem:bk4_fuzzy_symbolic_geometry_theorem",
"latex_body": "\\begin{theorem}[Fuzzy Symbolic Geometry Theorem]\n\\label{theorem:bk4_fuzzy_symbolic_geometry_theorem}\n\nLet $\\{P_\\lambda\\}_{\\lambda \\in \\Lambda}$ be a symbolic system with symbolic drift operators $\\{D_\\lambda\\}_{\\lambda \\in \\Lambda}$ and reflection operators $\\{R_\\lambda\\}_{\\lambda \\in \\Lambda}$, and let $\\mathcal{O} = (N_\\mathcal{O}, \\{\\delta^n_\\mathcal{O}\\}, \\epsilon_\\mathcal{O})$ be a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}).\n\nSuppose there exists a fuzzy symbolic substitution $u : \\bigcup_\\lambda P_\\lambda \\to \\tilde{M}$ (Def.~\\ref{definition:bk4_fuzzy_symbolic_substitution}) such that:\n\n\\begin{enumerate}\n \\item For all $\\lambda < \\mu \\in \\Lambda$, we have:\n \\[\n \\|\\delta^n_\\mathcal{O}(u(x_\\mu) - u(x_\\lambda))\\| < \\epsilon_\\mathcal{O}(x_\\lambda)\n \\quad \\text{whenever} \\quad \\|x_\\mu - x_\\lambda\\| < \\eta_{\\lambda\\mu}\n \\]\n for some $\\eta_{\\lambda\\mu} > 0$, with $x_\\lambda \\in P_\\lambda$, $x_\\mu \\in P_\\mu$, and $\\delta^n_\\mathcal{O}$ as in Def.~\\ref{definition:bk4_observer_differentiable_}. \n \n \\item The substituted drift fields \n \\[\n \\tilde{D}_\\lambda := u_*(D_\\lambda) = \\delta^1_\\mathcal{O}u \\circ D_\\lambda \\circ u^{-1}\n \\]\n (Def.~\\ref{definition:bk4_substituted_drift_field}) \n are $\\mathcal{O}$-differentiable (Def.~\\ref{definition:bk4_observer_differentiable_}) on domains $\\{u(U_\\lambda)\\}$ for some neighborhoods $\\{U_\\lambda \\subset P_\\lambda\\}$.\n (see Axiom~\\ref{axiom:bk2_gradient_structure_drift})\n \n \\item For each $\\lambda \\in \\Lambda$, there exists a local chart $(U_\\lambda, \\tilde{\\phi}_\\lambda)$ with \n \\[\n \\tilde{\\phi}_\\lambda: u(U_\\lambda) \\to V_\\lambda \\subset \\mathbb{R}^{d_\\lambda}\n \\]\n such that the chart representations \n \\[\n \\tilde{\\phi}_\\lambda \\circ \\tilde{D}_\\lambda \\circ \\tilde{\\phi}_\\lambda^{-1}\n \\]\n converge in the $C^k$ topology, where $k = \\min(N_\\mathcal{O}, N)$ for some $N \\geq 1$.\n\\end{enumerate}\n\nThen the following consequences hold:\n\n\\begin{enumerate}\n \\item The observer $\\mathcal{O}$ perceives $\\tilde{M}$ as a smooth manifold of symbolic emergence.\n\n \\item The original symbolic system $\\{P_\\lambda\\}_{\\lambda \\in \\Lambda}$ admits an observer-relative differentiable structure.\n\\\\\n (see Theorem~\\ref{theorem:appB_metric_completion})\n\n \\item The reflection operators $\\{R_\\lambda\\}$ induce $\\mathcal{O}$-differentiable stabilization fields on $\\tilde{M}$.\n\\end{enumerate}\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": false,
"notes": [
"Only consequence 2 (\"admits an observer-relative differentiable structure\") is covered, re-read over FracturedAtlas.ChartComplex with Glued as the named hypothesis consuming the source's chart-convergence condition. Consequences 1 and 3 (perceiving a smooth manifold; reflection-induced stabilization fields) are not modeled."
],
"record_ids": [
"MAP-BOOK4A-061"
],
"statuses": [
"open_bridge"
],
"witnesses": [
"Book4D.chart_geometry_exists_iff_glued",
"Book4D.chart_glued_yields_single_geometry"
]
},
"line": 3726,
"macros_used": [],
"matter_region": "mainmatter",
"matter_role": "canonical_book",
"name": "Fuzzy Symbolic Geometry Theorem",
"proof_labels": [
"proof:bk4_fuzzy_substitution_drift_smoothing"
],
"proof_status": "proven",
"ref_roles": [
{
"context": "ifferentiable_}) on domains $\\{u(U_\\lambda)\\}$ for some neighborhoods $\\{U_\\lambda \\subset P_\\lambda\\}$. (see Axiom~\\ref{axiom:bk2_gradient_structure_drift}) \\item For each $\\lambda \\in \\Lambda$, there exists a local chart $(U_\\lambda, \\tilde{\\phi}_\\lambda)$ with",
"label": "axiom:bk2_gradient_structure_drift",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book2.tex",
"target_line": 162,
"target_type": "axiom"
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{
"context": "}$, and let $\\mathcal{O} = (N_\\mathcal{O}, \\{\\delta^n_\\mathcal{O}\\}, \\epsilon_\\mathcal{O})$ be a bounded observer (Def.~\\ref{definition:bk1_bounded_observer}). Suppose there exists a fuzzy symbolic substitution $u : \\bigcup_\\lambda P_\\lambda \\to \\tilde{M}$ (Def.~\\ref{definiti",
"label": "definition:bk1_bounded_observer",
"logical_support": true,
"role": "definition_anchor",
"target_file": "scholium_symbolicum.tex",
"target_line": 27,
"target_type": "definition"
},
{
"context": "ded_observer}). Suppose there exists a fuzzy symbolic substitution $u : \\bigcup_\\lambda P_\\lambda \\to \\tilde{M}$ (Def.~\\ref{definition:bk4_fuzzy_symbolic_substitution}) such that: \\begin{enumerate} \\item For all $\\lambda < \\mu \\in \\Lambda$, we have: \\[ \\|\\delta^n_\\mathcal{O",
"label": "definition:bk4_fuzzy_symbolic_substitution",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book4.tex",
"target_line": 3294,
"target_type": "definition"
},
{
"context": "some $\\eta_{\\lambda\\mu} > 0$, with $x_\\lambda \\in P_\\lambda$, $x_\\mu \\in P_\\mu$, and $\\delta^n_\\mathcal{O}$ as in Def.~\\ref{definition:bk4_observer_differentiable_}. \\item The substituted drift fields \\[ \\tilde{D}_\\lambda := u_*(D_\\lambda) = \\delta^1_\\mathcal{O}u \\",
"label": "definition:bk4_observer_differentiable_",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book4.tex",
"target_line": 3306,
"target_type": "definition"
},
{
"context": "s \\[ \\tilde{D}_\\lambda := u_*(D_\\lambda) = \\delta^1_\\mathcal{O}u \\circ D_\\lambda \\circ u^{-1} \\] (Def.~\\ref{definition:bk4_substituted_drift_field}) are $\\mathcal{O}$-differentiable (Def.~\\ref{definition:bk4_observer_differentiable_}) on domains $\\{u(U_\\lambda)\\",
"label": "definition:bk4_substituted_drift_field",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book4.tex",
"target_line": 3314,
"target_type": "definition"
},
{
"context": "system $\\{P_\\lambda\\}_{\\lambda \\in \\Lambda}$ admits an observer-relative differentiable structure. \\\\ (see Theorem~\\ref{theorem:appB_metric_completion}) \\item The reflection operators $\\{R_\\lambda\\}$ induce $\\mathcal{O}$-differentiable stabilization fields on $\\tild",
"label": "theorem:appB_metric_completion",
"logical_support": false,
"role": "appendix_teaser",
"target_file": "appendix_symbolic_reflexive_validation.tex",
"target_line": 182,
"target_type": "theorem"
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"refs": [
"axiom:bk2_gradient_structure_drift",
"definition:bk1_bounded_observer",
"definition:bk4_fuzzy_symbolic_substitution",
"definition:bk4_observer_differentiable_",
"definition:bk4_substituted_drift_field",
"theorem:appB_metric_completion"
],
"role": "theorem",
"type": "theorem"
}