Complete structured record
{
"book": "appendix_symbolic_reflexive_validation",
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"cites": [
"definition:appB_symbolic_state_space",
"lemma:appB_energy_contraction",
"scholium:bk1_resolution_of_continuum_disjunction",
"theorem:appB_metric_completion",
"theorem:appB_smooth_atlas",
"theorem:appB_smoothness_emergence",
"theorem:appB_srv_cauchy"
],
"depends_on": [
"definition:appB_symbolic_state_space",
"lemma:appB_energy_contraction",
"scholium:bk1_resolution_of_continuum_disjunction",
"theorem:appB_metric_completion",
"theorem:appB_smooth_atlas",
"theorem:appB_smoothness_emergence",
"theorem:appB_srv_cauchy"
],
"file": "appendix_symbolic_reflexive_validation.tex",
"id": "proof:appB_resolution_of_smoothness",
"label": "proof:appB_resolution_of_smoothness",
"latex_body": "\\begin{proof}\n\\label{proof:appB_resolution_of_smoothness}\n\\leavevmode\nScholium~\\ref{scholium:bk1_resolution_of_continuum_disjunction} poses the disjunction between a discrete symbolic substrate and a continuous, smooth manifold. The construction of this appendix dissolves it constructively: from the discrete, finite-complexity symbolic tower $\\mathcal{P}=\\bigcup_\\lambda P_\\lambda$ (Def.~\\ref{definition:appB_symbolic_state_space}), the SRV dynamics are dissipative (Lemma~\\ref{lemma:appB_energy_contraction}) and their trajectories Cauchy (Thm.~\\ref{theorem:appB_srv_cauchy}); metric completion yields a separable complete space (Thm.~\\ref{theorem:appB_metric_completion}) carrying a smooth, paracompact manifold structure (Thm.~\\ref{theorem:appB_smooth_atlas}, Thm.~\\ref{theorem:appB_smoothness_emergence}). Smoothness therefore arises \\emph{from} the discrete layers under bounded observer resolution rather than being postulated beside them, which is precisely the resolution the Scholium calls for.\n\\end{proof}",
"line": 257,
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"name": "",
"proves": "corollary:appB_resolution_of_smoothness",
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{
"context": "es it constructively: from the discrete, finite-complexity symbolic tower $\\mathcal{P}=\\bigcup_\\lambda P_\\lambda$ (Def.~\\ref{definition:appB_symbolic_state_space}), the SRV dynamics are dissipative (Lemma~\\ref{lemma:appB_energy_contraction}) and their trajectories Cauchy (Thm.~\\ref",
"label": "definition:appB_symbolic_state_space",
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"role": "definition_anchor",
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"context": "}=\\bigcup_\\lambda P_\\lambda$ (Def.~\\ref{definition:appB_symbolic_state_space}), the SRV dynamics are dissipative (Lemma~\\ref{lemma:appB_energy_contraction}) and their trajectories Cauchy (Thm.~\\ref{theorem:appB_srv_cauchy}); metric completion yields a separable complete spac",
"label": "lemma:appB_energy_contraction",
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"context": "\\begin{proof} \\label{proof:appB_resolution_of_smoothness} \\leavevmode Scholium~\\ref{scholium:bk1_resolution_of_continuum_disjunction} poses the disjunction between a discrete symbolic substrate and a continuous, smooth manifold. The construction of this",
"label": "scholium:bk1_resolution_of_continuum_disjunction",
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"role": "proof_support",
"target_file": "scholium_symbolicum.tex",
"target_line": 2709,
"target_type": "scholium"
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{
"context": "eir trajectories Cauchy (Thm.~\\ref{theorem:appB_srv_cauchy}); metric completion yields a separable complete space (Thm.~\\ref{theorem:appB_metric_completion}) carrying a smooth, paracompact manifold structure (Thm.~\\ref{theorem:appB_smooth_atlas}, Thm.~\\ref{theorem:appB_smooth",
"label": "theorem:appB_metric_completion",
"logical_support": true,
"role": "proof_support",
"target_file": "appendix_symbolic_reflexive_validation.tex",
"target_line": 182,
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"context": "able complete space (Thm.~\\ref{theorem:appB_metric_completion}) carrying a smooth, paracompact manifold structure (Thm.~\\ref{theorem:appB_smooth_atlas}, Thm.~\\ref{theorem:appB_smoothness_emergence}). Smoothness therefore arises \\emph{from} the discrete layers under bound",
"label": "theorem:appB_smooth_atlas",
"logical_support": true,
"role": "proof_support",
"target_file": "appendix_symbolic_reflexive_validation.tex",
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"context": ":appB_metric_completion}) carrying a smooth, paracompact manifold structure (Thm.~\\ref{theorem:appB_smooth_atlas}, Thm.~\\ref{theorem:appB_smoothness_emergence}). Smoothness therefore arises \\emph{from} the discrete layers under bounded observer resolution rather than being postu",
"label": "theorem:appB_smoothness_emergence",
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"target_file": "appendix_symbolic_reflexive_validation.tex",
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"context": "ace}), the SRV dynamics are dissipative (Lemma~\\ref{lemma:appB_energy_contraction}) and their trajectories Cauchy (Thm.~\\ref{theorem:appB_srv_cauchy}); metric completion yields a separable complete space (Thm.~\\ref{theorem:appB_metric_completion}) carrying a smooth, pa",
"label": "theorem:appB_srv_cauchy",
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"lemma:appB_energy_contraction",
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"theorem:appB_smooth_atlas",
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