Complete structured record
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"definition:bk4_symbolic_curvature_formulations",
"definition:bk4_symbolic_space",
"definition:bk6_symbolic_curvature_tensor",
"definition:bk6_symbolic_density_evolution",
"definition:bk7_operational_resolution_uncertainties",
"definition:bk7_symbolic_norm",
"definition:bk8_sr_renormalization_group",
"definition:bk8_symbolic_hypothesis_manifold",
"lemma:bk7_involutive_dual_symmetry",
"proof:bk4_symbolic_curvature_properties",
"proof:bk8_curvature_entanglement_equivalence",
"proof:bk8_optimal_projection_path",
"proof:bk8_skech_via_euler_lagrange_flow_yields_geodesic",
"proof:bk9_curvature_resilience_bound",
"proposition:bk8_operator_curvature_flux",
"proposition:bk8_optimal_projection_path",
"proposition:bk9_curvature_resilience_bound",
"proposition:bk9_mechanisms_of_recognition",
"scholium:bk4_o_boundedness_unifying_principle",
"scholium:bk7_constrained_uncertainty_motivation",
"subsec:bk7_pisu_motivation",
"subsec:bk7_pisu_revisited_power_uncertainty",
"subsec:bk7_sources_regimes_uncertainty",
"theorem:appD_bounded_increment_parameter_lift",
"theorem:bk4_symbolic_curvature_properties",
"theorem:bk5_golden_ratio_curvature_scalar"
],
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"definition:bk1_symbolic_connection",
"definition:bk1_symbolic_field_curvature_tensor",
"definition:bk1_symbolic_manifold",
"lemma:bk2_thermodynamic_consistency_hypothesis_manifolds",
"proposition:bk4_geodesic_failure",
"sec:bk4_fuzzy_symbolic_geometry_observer_relative_smoothness"
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"definition:bk1_symbolic_field_curvature_tensor",
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"lemma:bk2_thermodynamic_consistency_hypothesis_manifolds"
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"id": "definition:bk4_symbolic_curvature",
"label": "definition:bk4_symbolic_curvature",
"latex_body": "\\begin{definition}[Symbolic Curvature]\n\\label{definition:bk4_symbolic_curvature}\nGiven $s \\in \\mathcal{S}_n$ on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) and observer $O = (K_O, \\delta_O, \\mathcal{B}_O)$, the \\emph{symbolic curvature} of $s$ relative to $O$ is (extending the bk1 curvature notion of Def.~\\ref{definition:bk1_symbolic_connection}, Def.~\\ref{definition:bk1_symbolic_field_curvature_tensor}):\n\\[\n\\kappa_O(s) := \\left\\| \\delta_O^2(R_\\lambda(s) - s) \\right\\|_{K_O}^2\n= \\big\\langle\\, \\delta_O^2(R_\\lambda(s) - s),\\; K_O\\,\\delta_O^2(R_\\lambda(s) - s) \\,\\big\\rangle\n\\]\nwhere:\n\\begin{itemize}\n \\item $\\delta_O^2 = \\delta_O \\circ \\delta_O$ is second-order observer derivation;\n \\item $\\|f\\|_{K_O}^2 := \\langle f, K_O f \\rangle$ is the kernel quadratic energy.\n\\end{itemize}\nSymbolic curvature is the kernel \\emph{energy} (degree two in the symbolic argument), not its square root, so that it scales as $|\\alpha|^2$ and matches the second-order character of curvature.\nWhen $\\kappa_O$ is bounded above by an observer-relative threshold $K_O$, the thermodynamic consistency of the observer's hypothesis manifold is guaranteed (cf.~Lem.~\\ref{lemma:bk2_thermodynamic_consistency_hypothesis_manifolds}).\n\\end{definition}",
"lean_alignment": {
"conditions": [
"continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
"modeling laws are structure fields or explicit hypotheses"
],
"countermodels": [],
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"notes": [
"Curvature modeled concretely as kappa K R lam s := K * (R lam s - s)^2, honestly degree-two in the symbolic argument per the definition's own stipulation. The second-order observer derivation delta_O^2 is not modeled, only its residual."
],
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"name": "Symbolic Curvature",
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{
"context": "O, \\mathcal{B}_O)$, the \\emph{symbolic curvature} of $s$ relative to $O$ is (extending the bk1 curvature notion of Def.~\\ref{definition:bk1_symbolic_connection}, Def.~\\ref{definition:bk1_symbolic_field_curvature_tensor}): \\[ \\kappa_O(s) := \\left\\| \\delta_O^2(R_\\lambda(s) - s) \\ri",
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{
"context": "e} of $s$ relative to $O$ is (extending the bk1 curvature notion of Def.~\\ref{definition:bk1_symbolic_connection}, Def.~\\ref{definition:bk1_symbolic_field_curvature_tensor}): \\[ \\kappa_O(s) := \\left\\| \\delta_O^2(R_\\lambda(s) - s) \\right\\|_{K_O}^2 = \\big\\langle\\, \\delta_O^2(R_\\lambda(s) - s),",
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"context": "lic Curvature] \\label{definition:bk4_symbolic_curvature} Given $s \\in \\mathcal{S}_n$ on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) and observer $O = (K_O, \\delta_O, \\mathcal{B}_O)$, the \\emph{symbolic curvature} of $s$ relative to $O$ is (extending",
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{
"context": "r-relative threshold $K_O$, the thermodynamic consistency of the observer's hypothesis manifold is guaranteed (cf.~Lem.~\\ref{lemma:bk2_thermodynamic_consistency_hypothesis_manifolds}). \\end{definition}",
"label": "lemma:bk2_thermodynamic_consistency_hypothesis_manifolds",
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