Scholium Symbolicum — page 6 of 6

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk2_classification_symb_phase_transitions",
    "theorem:bk5_map_mad_critical_temperature",
    "theorem:bk5_map_mad_mas_trichotomy",
    "theorem:bk8_biological_phase_transition"
  ],
  "depends_on": [
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk2_classification_symb_phase_transitions",
    "theorem:bk5_map_mad_critical_temperature",
    "theorem:bk5_map_mad_mas_trichotomy",
    "theorem:bk8_biological_phase_transition"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_realization_of_symbolic_phase_transitions",
  "label": "proof:bk1_realization_of_symbolic_phase_transitions",
  "latex_body": "\\begin{proof}\n\\label{proof:bk1_realization_of_symbolic_phase_transitions}\n\\leavevmode\nWe exhibit proven witnesses. \\emph{(1) A critical temperature.} The critical-temperature theorem (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}) establishes an explicit critical symbolic temperature $T_s^{\\text{crit}}$: for $T_s < T_s^{\\text{crit}}$ the system supports distinct stable MAP and MAD fixed points, whereas for $T_s > T_s^{\\text{crit}}$ no stable MAP configuration exists. Setting $\\beta_c = 1/T_s^{\\text{crit}}$, the set of stable equilibria changes qualitatively as $\\beta$ crosses $\\beta_c$ --- a fundamental reorganization of $\\rho_{\\text{eq}}$, hence a symbolic phase transition (Def.~\\ref{definition:bk1_symbolic_phase_transitions}). \\emph{(2) A spectral transition.} The MAD$\\to$MAP boundary of the trichotomy (Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}) is a complex$\\to$real crossing of the coupling spectrum at vanishing discriminant, where the qualitative mode structure of the dyadic dynamics changes. \\emph{(3) A dynamical threshold.} The metabolic autonomy threshold (Thm.~\\ref{theorem:bk8_biological_phase_transition}) crosses $\\Psi_{\\mathrm{aut}} = 0$, separating autonomous persistence from collapse --- a qualitative shift in symbolic coherence. Each witness is a proven symbolic system exhibiting a critical point of the type in Def.~\\ref{definition:bk1_symbolic_phase_transitions}, and the order of any such non-analyticity is fixed by the classification theorem (Thm.~\\ref{theorem:bk2_classification_symb_phase_transitions}). The existence of symbolic phase transitions follows.\n\\end{proof}",
  "line": 3294,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "",
  "proves": "theorem:bk1_realization_of_symbolic_phase_transitions",
  "ref_roles": [
    {
      "context": "beta$ crosses $\\beta_c$ --- a fundamental reorganization of $\\rho_{\\text{eq}}$, hence a symbolic phase transition (Def.~\\ref{definition:bk1_symbolic_phase_transitions}). \\emph{(2) A spectral transition.} The MAD$\\to$MAP boundary of the trichotomy (Thm.~\\ref{theorem:bk5_map_mad_mas_trich",
      "label": "definition:bk1_symbolic_phase_transitions",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3284,
      "target_type": "definition"
    },
    {
      "context": "bk1_symbolic_phase_transitions}, and the order of any such non-analyticity is fixed by the classification theorem (Thm.~\\ref{theorem:bk2_classification_symb_phase_transitions}). The existence of symbolic phase transitions follows. \\end{proof}",
      "label": "theorem:bk2_classification_symb_phase_transitions",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book2.tex",
      "target_line": 385,
      "target_type": "theorem"
    },
    {
      "context": "ons} \\leavevmode We exhibit proven witnesses. \\emph{(1) A critical temperature.} The critical-temperature theorem (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}) establishes an explicit critical symbolic temperature $T_s^{\\text{crit}}$: for $T_s < T_s^{\\text{crit}}$ the system su",
      "label": "theorem:bk5_map_mad_critical_temperature",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 907,
      "target_type": "theorem"
    },
    {
      "context": "on:bk1_symbolic_phase_transitions}). \\emph{(2) A spectral transition.} The MAD$\\to$MAP boundary of the trichotomy (Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}) is a complex$\\to$real crossing of the coupling spectrum at vanishing discriminant, where the qualitative mode structur",
      "label": "theorem:bk5_map_mad_mas_trichotomy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1992,
      "target_type": "theorem"
    },
    {
      "context": "mode structure of the dyadic dynamics changes. \\emph{(3) A dynamical threshold.} The metabolic autonomy threshold (Thm.~\\ref{theorem:bk8_biological_phase_transition}) crosses $\\Psi_{\\mathrm{aut}} = 0$, separating autonomous persistence from collapse --- a qualitative shift in symbolic",
      "label": "theorem:bk8_biological_phase_transition",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book8.tex",
      "target_line": 742,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk2_classification_symb_phase_transitions",
    "theorem:bk5_map_mad_critical_temperature",
    "theorem:bk5_map_mad_mas_trichotomy",
    "theorem:bk8_biological_phase_transition"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "theorem:bk1_symbolic_irony_requires_curvature"
  ],
  "depends_on": [
    "theorem:bk1_symbolic_irony_requires_curvature"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "remark:bk1_atlas_fracture_empirical",
  "label": "remark:bk1_atlas_fracture_empirical",
  "latex_body": "\\begin{remark}[External empirical corroboration]\n\\label{remark:bk1_atlas_fracture_empirical}\nBeyond the internal witnesses, the companion bounded-observer study\n\\citep{tiffany2025wicked} measures a symbolic phase transition in a real\nsymbolic system: applying a sliding-window curvature estimator to a\npublic-domain narrative, it detects a dominant semantic discontinuity --- an\n\\emph{atlas fracture} --- precisely at the reorganization point where the\nouter projection metric fails to chart the inner territory, with extrinsic\ncurvature concentrating as $\\|\\mathrm{Ric}\\|\\gtrsim K/\\varepsilon_{\\mathrm{res}}^2$\nunder resolution collapse $\\varepsilon_{\\mathrm{res}}\\to 0$. This corroborates,\non data rather than by construction, both the realized criticality here and the\ncurvature requirement for irony\n(Thm.~\\ref{theorem:bk1_symbolic_irony_requires_curvature}): the reorganization\nregisters as a curvature spike, exactly the non-flat signature the two theorems\npredict.\n\\end{remark}",
  "line": 3300,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "External empirical corroboration",
  "ref_roles": [
    {
      "context": ", on data rather than by construction, both the realized criticality here and the curvature requirement for irony (Thm.~\\ref{theorem:bk1_symbolic_irony_requires_curvature}): the reorganization registers as a curvature spike, exactly the non-flat signature the two theorems predict. \\end{rema",
      "label": "theorem:bk1_symbolic_irony_requires_curvature",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2388,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk1_symbolic_irony_requires_curvature"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_nonvacuity_of_certified_transport",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_minimal_linear_ps_model",
  "label": "definition:bk1_minimal_linear_ps_model",
  "latex_body": "\\begin{definition}[Minimal Linear PS-Model Witness]\n\\label{definition:bk1_minimal_linear_ps_model}\nThe \\emph{minimal linear PS-model witness} is the following finite-dimensional\nsymbolic system.  Let \\(M=\\mathbb{R}^2\\) with its Euclidean metric, write\n\\(x=(u,v)\\), and regard \\(u\\) as the observer-visible coordinate and \\(v\\) as\nthe hidden phase coordinate.  Let\n\\[\nP =\n\\begin{pmatrix}\n1&0\\\\\n0&0\n\\end{pmatrix},\n\\qquad\nJ =\n\\begin{pmatrix}\n0&-1\\\\\n1&0\n\\end{pmatrix}.\n\\]\nThe bounded observer sees through the projection \\(P\\), the state-level\ncollapse is \\(C=P\\), the drift field is \\(D(x)=Jx\\), and the state-level\nstabilization component of reflection is \\(R_{\\mathrm{stab}}(x)=Px\\), in the\nsense of Defs.~\\ref{definition:bk1_symbolic_manifold},\n\\ref{definition:bk1_drift_field}, and \\ref{definition:bk1_reflection_operator}.\nOn the trivial rank-two symbolic bundle \\(E=M\\times\\mathbb{R}^2\\), define a\nsymbolic connection by\n\\[\n\\nabla_{\\partial_u}=\\partial_u + A_u,\n\\qquad\n\\nabla_{\\partial_v}=\\partial_v + A_v,\n\\qquad\nA_u =\n\\begin{pmatrix}\n0&1\\\\\n0&0\n\\end{pmatrix},\n\\quad\nA_v =\n\\begin{pmatrix}\n0&0\\\\\n1&0\n\\end{pmatrix}.\n\\]\nThis witness is a mathematical model object only; no computational\nimplementation is part of its definition.\n\\end{definition}",
  "line": 3317,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Minimal Linear PS-Model Witness",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "n component of reflection is \\(R_{\\mathrm{stab}}(x)=Px\\), in the sense of Defs.~\\ref{definition:bk1_symbolic_manifold}, \\ref{definition:bk1_drift_field}, and \\ref{definition:bk1_reflection_operator}. On the trivial rank-two symbolic bundle \\(E=M\\times\\mathbb{R}^2\\), defin",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "hrm{stab}}(x)=Px\\), in the sense of Defs.~\\ref{definition:bk1_symbolic_manifold}, \\ref{definition:bk1_drift_field}, and \\ref{definition:bk1_reflection_operator}. On the trivial rank-two symbolic bundle \\(E=M\\times\\mathbb{R}^2\\), define a symbolic connection by \\[ \\nabla_{\\partial",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": ")=Jx\\), and the state-level stabilization component of reflection is \\(R_{\\mathrm{stab}}(x)=Px\\), in the sense of Defs.~\\ref{definition:bk1_symbolic_manifold}, \\ref{definition:bk1_drift_field}, and \\ref{definition:bk1_reflection_operator}. On the trivial rank-two symbolic bundl",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "definition:bk9_grace_operator",
    "proof:bk1_nonvacuity_of_certified_transport",
    "proof:bk9_freedom_as_grace",
    "remark:appD_llm_tuple_anchors",
    "remark:bk1_mathematical_witness_boundary",
    "remark:bk4_finite_witness_for_drift_reflection_imbalance",
    "scholium:bk4_ttdc_symbolic_singularity"
  ],
  "cites": [
    "definition:bk1_minimal_linear_ps_model"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_minimal_linear_ps_model",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_nonvacuity_minimal_linear_ps_model",
  "label": "theorem:bk1_nonvacuity_minimal_linear_ps_model",
  "latex_body": "\\begin{theorem}[Non-Vacuity of the Minimal Linear PS-Model]\n\\label{theorem:bk1_nonvacuity_minimal_linear_ps_model}\nThe minimal linear PS-model witness of\nDef.~\\ref{definition:bk1_minimal_linear_ps_model} is a nontrivial realization\nof the Book~I operator vocabulary: its collapse is not the identity, its drift\nand stabilization do not commute, and its symbolic connection has nonzero\ncurvature.  Hence the PS operator ontology has a finite-dimensional\nmathematical realization independent of any computational witness.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "modeling laws are structure fields or explicit hypotheses; continuum/categorical content is NOT formalized"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "All three claims (collapse is not identity, drift/stabilization do not commute, connection has nonzero curvature) are proved unconditionally by explicit witness computation at concrete points, matching the theorem's own finite-dimensional realization claim exactly."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-011"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumB.minimalPS_collapse_ne_id",
      "ScholiumB.minimalPS_connection_curvature_nonzero",
      "ScholiumB.minimalPS_drift_reflection_noncommute"
    ]
  },
  "line": 3364,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Non-Vacuity of the Minimal Linear PS-Model",
  "proof_labels": [
    "proof:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "mal Linear PS-Model] \\label{theorem:bk1_nonvacuity_minimal_linear_ps_model} The minimal linear PS-model witness of Def.~\\ref{definition:bk1_minimal_linear_ps_model} is a nontrivial realization of the Book~I operator vocabulary: its collapse is not the identity, its drift and stabiliz",
      "label": "definition:bk1_minimal_linear_ps_model",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3317,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_minimal_linear_ps_model"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_nonvacuity_minimal_linear_ps_model",
  "label": "proof:bk1_nonvacuity_minimal_linear_ps_model",
  "latex_body": "\\begin{proof}[Explicit Matrix Witness]\n\\label{proof:bk1_nonvacuity_minimal_linear_ps_model}\n\\leavevmode\nFirst, \\(M=\\mathbb{R}^2\\) with the Euclidean metric is a smooth\ntwo-dimensional symbolic manifold in the sense of\nDef.~\\ref{definition:bk1_symbolic_manifold}.  The vector field \\(D(x)=Jx\\) is\nsmooth, vanishes only at the origin, and has bounded divergence\n\\(\\operatorname{tr}J=0\\); therefore it satisfies the elementary drift-field\nconditions of Def.~\\ref{definition:bk1_drift_field}.  The map\n\\(R_{\\mathrm{stab}}=P\\) satisfies \\(P^2=P\\), so it is an idempotent\nstate-level stabilization component as in Def.~\\ref{definition:bk1_reflection_operator}.\n\nThe collapse is nontrivial because \\(C(u,v)=(u,0)\\), so \\(C(u,v)\\ne(u,v)\\) for\nevery \\(v\\ne0\\).  The drift-reflection commutator is also nonzero:\n\\[\nDR = JP =\n\\begin{pmatrix}\n0&0\\\\\n1&0\n\\end{pmatrix},\n\\qquad\nRD = PJ =\n\\begin{pmatrix}\n0&-1\\\\\n0&0\n\\end{pmatrix},\n\\]\nand hence\n\\[\n[D,R] = DR-RD =\n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n\\ne 0.\n\\]\n\nFinally, the connection coefficients \\(A_u,A_v\\) are constant, so the\n\\((u,v)\\)-curvature component is\n\\[\n\\Omega_{uv}\n=\\partial_u A_v-\\partial_v A_u+[A_u,A_v]\n=[A_u,A_v].\n\\]\nA direct multiplication gives\n\\[\nA_uA_v =\n\\begin{pmatrix}\n1&0\\\\\n0&0\n\\end{pmatrix},\n\\qquad\nA_vA_u =\n\\begin{pmatrix}\n0&0\\\\\n0&1\n\\end{pmatrix},\n\\qquad\n[A_u,A_v] =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n\\ne0.\n\\]\nThus the witness has nonzero symbolic holonomy/curvature in the sense of\nDefs.~\\ref{definition:bk1_symbolic_connection} and\n\\ref{definition:bk1_symbolic_riemann_tensor}.  The construction therefore\nexhibits a concrete nonempty model of the relevant PS operators without\nappeal to an external implementation.\n\\end{proof}",
  "line": 3374,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Explicit Matrix Witness",
  "proves": "theorem:bk1_nonvacuity_minimal_linear_ps_model",
  "ref_roles": [
    {
      "context": "d has bounded divergence \\(\\operatorname{tr}J=0\\); therefore it satisfies the elementary drift-field conditions of Def.~\\ref{definition:bk1_drift_field}. The map \\(R_{\\mathrm{stab}}=P\\) satisfies \\(P^2=P\\), so it is an idempotent state-level stabilization component as in",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "map \\(R_{\\mathrm{stab}}=P\\) satisfies \\(P^2=P\\), so it is an idempotent state-level stabilization component as in Def.~\\ref{definition:bk1_reflection_operator}. The collapse is nontrivial because \\(C(u,v)=(u,0)\\), so \\(C(u,v)\\ne(u,v)\\) for every \\(v\\ne0\\). The drift-reflection",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "First, \\(M=\\mathbb{R}^2\\) with the Euclidean metric is a smooth two-dimensional symbolic manifold in the sense of Def.~\\ref{definition:bk1_symbolic_manifold}. The vector field \\(D(x)=Jx\\) is smooth, vanishes only at the origin, and has bounded divergence \\(\\operatorname{tr}J=",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "depends_on": [
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "remark:bk1_mathematical_witness_boundary",
  "label": "remark:bk1_mathematical_witness_boundary",
  "latex_body": "\\begin{remark}[Witness boundary]\n\\label{remark:bk1_mathematical_witness_boundary}\nThe point of Thm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model} is not\nthat every PS claim reduces to a two-dimensional linear system.  It is that the\noperator vocabulary is not empty: drift, reflection, collapse, and curvature can\ncoexist in a typed mathematical realization.  A computational system may\nwitness richer projections of this ontology, but it does not define the truth\nof the ontology.\n\\end{remark}",
  "line": 3446,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Witness boundary",
  "ref_roles": [
    {
      "context": "\\begin{remark}[Witness boundary] \\label{remark:bk1_mathematical_witness_boundary} The point of Thm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model} is not that every PS claim reduces to a two-dimensional linear system. It is that the operator vocabulary is not empty",
      "label": "theorem:bk1_nonvacuity_minimal_linear_ps_model",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3364,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk9_grace_operator",
    "proof:bk1_certified_transport_prevents_equivocation",
    "proof:bk1_nonvacuity_of_certified_transport",
    "proof:bk9_freedom_as_grace",
    "proof:bk9_stability_conditions_for_the_good",
    "remark:appD_llm_tuple_anchors",
    "remark:bk4_finite_witness_for_drift_reflection_imbalance",
    "scholium:bk4_ttdc_symbolic_singularity"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_riemann_tensor",
    "lemma:bk1_curvature_semantic_holonomy",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_riemann_tensor",
    "lemma:bk1_curvature_semantic_holonomy",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_certified_type_preserving_symbolic_transport",
  "label": "definition:bk1_certified_type_preserving_symbolic_transport",
  "latex_body": "\\begin{definition}[Certified Type-Preserving Symbolic Transport]\n\\label{definition:bk1_certified_type_preserving_symbolic_transport}\nLet \\(\\mathcal{V}_a\\) and \\(\\mathcal{V}_b\\) be two PS operator vocabularies\nattached to symbolic manifolds, observer frames, or book-level depths.  A\n\\emph{certified type-preserving symbolic transport} from \\(\\mathcal{V}_a\\) to\n\\(\\mathcal{V}_b\\) is a tuple\n\\[\n\\mathsf{Cert}_{a\\to b}=(\\mathcal{T}_{a\\to b},\\sigma,\\rho,\\ell)\n\\]\nwith the following data:\n\\begin{enumerate}\n    \\item \\(\\mathcal{T}_{a\\to b}\\) maps each transported operator occurrence in\n    \\(\\mathcal{V}_a\\) to an occurrence in \\(\\mathcal{V}_b\\).\n    \\item \\(\\sigma\\) records the preserved type signature.  Drift transports as\n    a state-to-tangent field or admissible update section\n    (Def.~\\ref{definition:bk1_drift_field}); reflection transports as either a\n    tangent-level mirror or an idempotent state-level stabilization\n    (Def.~\\ref{definition:bk1_reflection_operator}); collapse transports as a\n    projection, quotient, or observer-visible reduction; and curvature\n    transports as a connection/holonomy defect\n    (Defs.~\\ref{definition:bk1_symbolic_connection},\n    \\ref{definition:bk1_symbolic_riemann_tensor};\n    cf.~Lem.~\\ref{lemma:bk1_curvature_semantic_holonomy}).\n    \\item \\(\\rho\\) records the preserved structural role: drift differentiates,\n    reflection stabilizes or re-enters, collapse forgets degrees of freedom, and\n    curvature measures non-flat transport.\n    \\item \\(\\ell\\in\\{\\mathrm{exact},\\mathrm{quotient},\\mathrm{projective},\n    \\mathrm{interpretive}\\}\\) records the declared loss.  Exact transports may\n    support theorem dependencies directly.  Quotient or projective transports\n    may support theorem dependencies only with the stated loss included.\n    Interpretive transports must be cited as \\(cf.\\), demonstratio, or\n    explanatory bridge, not as hidden proof support.\n\\end{enumerate}\nThe certificate is \\emph{valid} when every transported occurrence has a recorded\n\\(\\sigma\\), \\(\\rho\\), and \\(\\ell\\), and every exact or quotient/projective claim\nis anchored either in Book~I primitives or in an explicit realized witness such\nas Thm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model}.  This definition\nis a mathematical bookkeeping condition, not an empirical certificate.\n\\end{definition}",
  "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": [
      "only the four-level loss taxonomy (field ell) is modeled as an explicit finite type; the transported-occurrence map T, signature sigma, and structural role rho are not modeled."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-049"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "ScholiumD.TransportLoss.exact_supportsDependency"
    ]
  },
  "line": 3456,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Certified Type-Preserving Symbolic Transport",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "the preserved type signature. Drift transports as a state-to-tangent field or admissible update section (Def.~\\ref{definition:bk1_drift_field}); reflection transports as either a tangent-level mirror or an idempotent state-level stabilization (Def.~\\ref{",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "ield}); reflection transports as either a tangent-level mirror or an idempotent state-level stabilization (Def.~\\ref{definition:bk1_reflection_operator}); collapse transports as a projection, quotient, or observer-visible reduction; and curvature transports as a c",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "ction, quotient, or observer-visible reduction; and curvature transports as a connection/holonomy defect (Defs.~\\ref{definition:bk1_symbolic_connection}, \\ref{definition:bk1_symbolic_riemann_tensor}; cf.~Lem.~\\ref{lemma:bk1_curvature_semantic_holonomy}). \\item",
      "label": "definition:bk1_symbolic_connection",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1888,
      "target_type": "definition"
    },
    {
      "context": "; and curvature transports as a connection/holonomy defect (Defs.~\\ref{definition:bk1_symbolic_connection}, \\ref{definition:bk1_symbolic_riemann_tensor}; cf.~Lem.~\\ref{lemma:bk1_curvature_semantic_holonomy}). \\item \\(\\rho\\) records the preserved structural role: d",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1905,
      "target_type": "definition"
    },
    {
      "context": "ect (Defs.~\\ref{definition:bk1_symbolic_connection}, \\ref{definition:bk1_symbolic_riemann_tensor}; cf.~Lem.~\\ref{lemma:bk1_curvature_semantic_holonomy}). \\item \\(\\rho\\) records the preserved structural role: drift differentiates, reflection stabilizes or re-enter",
      "label": "lemma:bk1_curvature_semantic_holonomy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1930,
      "target_type": "lemma"
    },
    {
      "context": "ct or quotient/projective claim is anchored either in Book~I primitives or in an explicit realized witness such as Thm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model}. This definition is a mathematical bookkeeping condition, not an empirical certificate. \\end{definition}",
      "label": "theorem:bk1_nonvacuity_minimal_linear_ps_model",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3364,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_riemann_tensor",
    "lemma:bk1_curvature_semantic_holonomy",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk9_grace_operator",
    "proof:bk9_stability_conditions_for_the_good",
    "remark:appD_llm_tuple_anchors",
    "remark:bk4_finite_witness_for_drift_reflection_imbalance",
    "scholium:bk4_ttdc_symbolic_singularity"
  ],
  "cites": [],
  "depends_on": [
    "definition:bk1_certified_type_preserving_symbolic_transport"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_certified_transport_prevents_equivocation",
  "label": "proposition:bk1_certified_transport_prevents_equivocation",
  "latex_body": "\\begin{proposition}[Certified Transport Prevents Operator Equivocation]\n\\label{proposition:bk1_certified_transport_prevents_equivocation}\nIf a downstream PS argument transports an operator symbol only through valid\ncertified type-preserving symbolic transports\n\\(\\mathsf{Cert}_{a\\to b}\\), then the argument cannot use the same symbol in two\ndifferent formal roles without an explicit loss annotation.  In particular,\ndrift cannot silently become reflection, collapse cannot silently become\nidentity, and curvature cannot silently become metaphor.\n\\end{proposition}",
  "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": [
      "the licensing rule only (which loss levels may support a theorem dependency); the equivocation-detection claim about a downstream argument's symbol usage is not modeled."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-050"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumD.TransportLoss.exact_supportsDependency",
      "ScholiumD.TransportLoss.interpretive_not_supportsDependency",
      "ScholiumD.TransportLoss.projective_supportsDependency",
      "ScholiumD.TransportLoss.quotient_supportsDependency"
    ]
  },
  "line": 3496,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Certified Transport Prevents Operator Equivocation",
  "proof_labels": [
    "proof:bk1_certified_transport_prevents_equivocation"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_certified_type_preserving_symbolic_transport"
  ],
  "depends_on": [
    "definition:bk1_certified_type_preserving_symbolic_transport"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_certified_transport_prevents_equivocation",
  "label": "proof:bk1_certified_transport_prevents_equivocation",
  "latex_body": "\\begin{proof}[Role preservation by certificate]\n\\label{proof:bk1_certified_transport_prevents_equivocation}\n\\leavevmode\nLet \\(O\\) be any transported operator occurrence used in the downstream\nargument.  Since the transport certificate is valid,\n\\(\\mathcal{T}_{a\\to b}(O)\\) carries a type record \\(\\sigma(O)\\), a structural\nrole record \\(\\rho(O)\\), and a loss record \\(\\ell(O)\\).\nThe type record fixes the admissible domain and codomain class of the\ntransported occurrence: for example, a drift occurrence remains a\nstate-to-tangent field or admissible update section, while a collapse occurrence\nremains a projection, quotient, or observer-visible reduction.  The role record\nfixes what the occurrence is allowed to do in the proof: drift differentiates,\nreflection stabilizes or re-enters, collapse forgets degrees of freedom, and\ncurvature measures a transport defect.\n\nSuppose, toward contradiction, that the argument uses one transported symbol in\ntwo different formal roles without annotation.  Then either its type has changed\nwhile \\(\\sigma\\) records no change, or its proof role has changed while \\(\\rho\\)\nrecords no change, or the change is a quotient/projective/interpretive loss\nwhile \\(\\ell\\) records no such loss.  Each case contradicts validity of\n\\(\\mathsf{Cert}_{a\\to b}\\).  Therefore any genuine change of role must appear\nas an explicit loss annotation, and any unannotated occurrence preserves its\noperator role.  The stated exclusions follow by applying this argument to the\ndrift, reflection, collapse, and curvature clauses of\nDef.~\\ref{definition:bk1_certified_type_preserving_symbolic_transport}.\n\\end{proof}",
  "line": 3506,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Role preservation by certificate",
  "proves": "proposition:bk1_certified_transport_prevents_equivocation",
  "ref_roles": [
    {
      "context": "he stated exclusions follow by applying this argument to the drift, reflection, collapse, and curvature clauses of Def.~\\ref{definition:bk1_certified_type_preserving_symbolic_transport}. \\end{proof}",
      "label": "definition:bk1_certified_type_preserving_symbolic_transport",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3456,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_certified_type_preserving_symbolic_transport"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk9_grace_operator",
    "proof:bk9_stability_conditions_for_the_good",
    "remark:appD_llm_tuple_anchors",
    "remark:bk4_finite_witness_for_drift_reflection_imbalance",
    "scholium:bk4_ttdc_symbolic_singularity"
  ],
  "cites": [],
  "depends_on": [
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "definition:bk1_minimal_linear_ps_model",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_nonvacuity_of_certified_transport",
  "label": "proposition:bk1_nonvacuity_of_certified_transport",
  "latex_body": "\\begin{proposition}[Non-Vacuity of Certified Transport]\n\\label{proposition:bk1_nonvacuity_of_certified_transport}\nThe class of valid certified type-preserving symbolic transports is nonempty.\nMoreover, it contains both an exact transport and a genuinely projective\ntransport.\n\\end{proposition}",
  "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": [
      "nonemptiness plus an explicit distinct exact/projective witness, as a finite countermodel over the 4-element TransportLoss type."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-051"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumD.TransportLoss.exact_ne_projective",
      "ScholiumD.TransportLoss.nonempty"
    ]
  },
  "line": 3533,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Non-Vacuity of Certified Transport",
  "proof_labels": [
    "proof:bk1_nonvacuity_of_certified_transport"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "definition:bk1_minimal_linear_ps_model",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "depends_on": [
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "definition:bk1_minimal_linear_ps_model",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_nonvacuity_of_certified_transport",
  "label": "proof:bk1_nonvacuity_of_certified_transport",
  "latex_body": "\\begin{proof}[Exact and projective certificates in the minimal witness]\n\\label{proof:bk1_nonvacuity_of_certified_transport}\n\\leavevmode\nLet \\(\\mathcal{V}_{\\mathrm{lin}}\\) be the operator vocabulary of the minimal\nlinear PS-model witness of\nDef.~\\ref{definition:bk1_minimal_linear_ps_model}, with drift \\(D(x)=Jx\\),\nstate-level stabilization \\(R_{\\mathrm{stab}}(x)=Px\\), collapse \\(C=P\\), and\ncurvature component \\(\\Omega_{uv}=[A_u,A_v]\\).  By\nThm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model}, these operators are\ndefined in a finite-dimensional mathematical realization.\n\nFirst define \\(\\mathcal{T}_{\\mathrm{id}}\\) to be the identity map on the four\noperator occurrences \\(D,R_{\\mathrm{stab}},C,\\Omega_{uv}\\).  Let\n\\(\\sigma_{\\mathrm{id}}\\) record their displayed signatures: state-to-tangent\ndrift field, idempotent state-level stabilization, projection collapse, and\nconnection-curvature component.  Let \\(\\rho_{\\mathrm{id}}\\) record their roles:\ndifferentiate, stabilize, forget degrees of freedom, and measure non-flat\ntransport.  Let \\(\\ell_{\\mathrm{id}}=\\mathrm{exact}\\) for each occurrence.  All\nrecords required by\nDef.~\\ref{definition:bk1_certified_type_preserving_symbolic_transport} are\npresent, and the claim is anchored in the realized witness; hence\n\\((\\mathcal{T}_{\\mathrm{id}},\\sigma_{\\mathrm{id}},\\rho_{\\mathrm{id}},\n\\ell_{\\mathrm{id}})\\) is a valid exact certificate.\n\nSecond let \\(q:M\\to\\operatorname{im}P\\cong\\mathbb{R}\\) be the observer-visible\nmap \\(q(u,v)=u\\).  Transport only the collapse occurrence \\(C=P\\) to \\(q\\).\nIts type record is projection/observer-visible reduction, its role record is\nforgetting the hidden phase coordinate \\(v\\), and its loss record is\n\\(\\ell=\\mathrm{projective}\\).  Since \\(q(u,v)=q(u,v')\\) for all hidden\ncoordinates \\(v,v'\\), the transport is not exact; it genuinely loses degrees of\nfreedom.  Since it is still anchored in the same realized witness and all\nrequired records are explicit, it is a valid projective certificate.\n\nThus valid certified transports exist, and the certification notion is not an\nempty constraint.\n\\end{proof}",
  "line": 3540,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Exact and projective certificates in the minimal witness",
  "proves": "proposition:bk1_nonvacuity_of_certified_transport",
  "ref_roles": [
    {
      "context": "asure non-flat transport. Let \\(\\ell_{\\mathrm{id}}=\\mathrm{exact}\\) for each occurrence. All records required by Def.~\\ref{definition:bk1_certified_type_preserving_symbolic_transport} are present, and the claim is anchored in the realized witness; hence \\((\\mathcal{T}_{\\mathrm{id}},\\sigma_{\\mathrm{id}}",
      "label": "definition:bk1_certified_type_preserving_symbolic_transport",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3456,
      "target_type": "definition"
    },
    {
      "context": "leavevmode Let \\(\\mathcal{V}_{\\mathrm{lin}}\\) be the operator vocabulary of the minimal linear PS-model witness of Def.~\\ref{definition:bk1_minimal_linear_ps_model}, with drift \\(D(x)=Jx\\), state-level stabilization \\(R_{\\mathrm{stab}}(x)=Px\\), collapse \\(C=P\\), and curvature compone",
      "label": "definition:bk1_minimal_linear_ps_model",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3317,
      "target_type": "definition"
    },
    {
      "context": "tabilization \\(R_{\\mathrm{stab}}(x)=Px\\), collapse \\(C=P\\), and curvature component \\(\\Omega_{uv}=[A_u,A_v]\\). By Thm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model}, these operators are defined in a finite-dimensional mathematical realization. First define \\(\\mathcal{T}_{\\mathrm{id}",
      "label": "theorem:bk1_nonvacuity_minimal_linear_ps_model",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3364,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "definition:bk1_minimal_linear_ps_model",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "conjecture:bk1_symbolic_irony_encoding_llms",
    "proof:bk1_conditional_genericity_of_symbolic_phase_transitions",
    "scholium:bk1_the_imagination_dipole"
  ],
  "cites": [
    "conjecture:bk1_symbolic_irony_encoding_llms",
    "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
    "theorem:bk1_realization_of_symbolic_phase_transitions"
  ],
  "depends_on": [
    "conjecture:bk1_symbolic_irony_encoding_llms",
    "theorem:bk1_realization_of_symbolic_phase_transitions"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "nge, effective dimensionality, covenant variability). These are now \\emph{identified and proved sufficient} below (Thm.~\\ref{theorem:bk1_conditional_genericity_of_symbolic_phase_transitions}); what remains genuinely open is the sharper, measure-theoretic residual --- whether those hypotheses are \\emph{themsel",
      "label": "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
      "line_distance": 5,
      "role": "later_formalization",
      "target_line": 3582,
      "target_type": "theorem"
    }
  ],
  "forward_refs": [
    "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions"
  ],
  "id": "conjecture:bk1_genericity_of_symbolic_phase_transitions",
  "label": "conjecture:bk1_genericity_of_symbolic_phase_transitions",
  "latex_body": "\\begin{conjecture}[Genericity of Symbolic Phase Transitions]\n\\label{conjecture:bk1_genericity_of_symbolic_phase_transitions}\nTheorem~\\ref{theorem:bk1_realization_of_symbolic_phase_transitions} realizes symbolic phase transitions by explicit construction. It remains open whether they are \\emph{generic}: whether every sufficiently complex symbolic manifold $(M, g, D, R)$ --- under a suitable measure of symbolic complexity --- necessarily admits a critical $\\beta_c$. By analogy with classical statistical mechanics, where low-dimensional short-range systems may possess no finite-temperature transition, genericity requires further structural hypotheses (coupling range, effective dimensionality, covenant variability). These are now \\emph{identified and proved sufficient} below (Thm.~\\ref{theorem:bk1_conditional_genericity_of_symbolic_phase_transitions}); what remains genuinely open is the sharper, measure-theoretic residual --- whether those hypotheses are \\emph{themselves} generic among complex symbolic manifolds. All three --- (H1)--(H3) --- are supplied by imaginative capacity (Prop.~\\ref{proposition:bk1_imagination_supplies_genericity_hypotheses}), so the residual reduces \\emph{purely} to whether complex symbolic manifolds are generically imaginative. This conjecture mirrors the empirical irony conjecture (Conj.~\\ref{conjecture:bk1_symbolic_irony_encoding_llms}) exactly: in both, the model-internal result is proven, both reduce to the \\emph{same} predicate --- whether the system imagines --- and only the universal (here, on abstract manifolds) or real-world (there, on built systems) extension remains an open, falsifiable frontier.\n\\end{conjecture}",
  "line": 3577,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Genericity of Symbolic Phase Transitions",
  "proof_status": "unproved",
  "ref_roles": [
    {
      "context": "r complex symbolic manifolds are generically imaginative. This conjecture mirrors the empirical irony conjecture (Conj.~\\ref{conjecture:bk1_symbolic_irony_encoding_llms}) exactly: in both, the model-internal result is proven, both reduce to the \\emph{same} predicate --- whether the system",
      "label": "conjecture:bk1_symbolic_irony_encoding_llms",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2489,
      "target_type": "conjecture"
    },
    {
      "context": "nge, effective dimensionality, covenant variability). These are now \\emph{identified and proved sufficient} below (Thm.~\\ref{theorem:bk1_conditional_genericity_of_symbolic_phase_transitions}); what remains genuinely open is the sharper, measure-theoretic residual --- whether those hypotheses are \\emph{themsel",
      "label": "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
      "logical_support": false,
      "role": "forward_later_formalization",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3582,
      "target_type": "theorem"
    },
    {
      "context": "ture}[Genericity of Symbolic Phase Transitions] \\label{conjecture:bk1_genericity_of_symbolic_phase_transitions} Theorem~\\ref{theorem:bk1_realization_of_symbolic_phase_transitions} realizes symbolic phase transitions by explicit construction. It remains open whether they are \\emph{generic}: whether",
      "label": "theorem:bk1_realization_of_symbolic_phase_transitions",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3290,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "conjecture:bk1_symbolic_irony_encoding_llms",
    "proposition:bk1_imagination_supplies_genericity_hypotheses",
    "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
    "theorem:bk1_realization_of_symbolic_phase_transitions"
  ],
  "role": "conjecture",
  "type": "conjecture"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions"
  ],
  "cites": [
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "depends_on": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions",
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk5_map_mad_critical_temperature",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
  "label": "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
  "latex_body": "\\begin{theorem}[Conditional Genericity of Symbolic Phase Transitions]\n\\label{theorem:bk1_conditional_genericity_of_symbolic_phase_transitions}\nLet $(M,g,D,R)$ be a symbolic manifold whose dyadic covenant coupling satisfies:\n\\begin{enumerate}[label=\\textbf{(H\\arabic*)}]\n    \\item \\textbf{Effective dimensionality} $d_{\\mathrm{eff}}(M)\\ge 2$: the symbolic\n    order parameter has at least two coupled effective directions, so an ordered\n    (MAP) phase admits a Peierls-type domain-wall cost growing with region size.\n    \\item \\textbf{Coupling-range straddle:} the spectral coupling $\\lambda(\\beta)$\n    of the dyadic operator $C_{AB}$ is continuous and monotone in $\\beta$ with\n    $\\lim_{\\beta\\to 0}\\lambda(\\beta)<\\lambda_c<\\lim_{\\beta\\to\\infty}\\lambda(\\beta)$,\n    where $\\lambda_c$ is the critical coupling of the trichotomy\n    (Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}).\n    \\item \\textbf{Covenant variability:} the discriminant $\\Delta(\\beta)$ of the\n    dyadic coupling spectrum crosses zero \\emph{transversally} (not tangentially)\n    over the accessible range, $\\Delta'(\\beta_c)\\ne 0$.\n\\end{enumerate}\nThen there exists a critical $\\beta_c$ at which $f(\\beta)=-\\beta^{-1}\\ln Z(\\beta)$\nis non-analytic --- a symbolic phase transition in the sense of\nDef.~\\ref{definition:bk1_symbolic_phase_transitions}. Within the class satisfying\n(H1)--(H3), symbolic phase transitions are therefore generic.\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": [
      "hypothesis (H2) (coupling-range straddle) is exactly the IVT hypothesis; (H1) effective dimensionality and (H3) transversality are not modeled, and monotonicity of lambda is dropped as unneeded rather than modeled."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-048"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumD.exists_critical_coupling"
    ]
  },
  "line": 3582,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Conditional Genericity of Symbolic Phase Transitions",
  "proof_labels": [
    "proof:bk1_conditional_genericity_of_symbolic_phase_transitions"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "eta_c$ at which $f(\\beta)=-\\beta^{-1}\\ln Z(\\beta)$ is non-analytic --- a symbolic phase transition in the sense of Def.~\\ref{definition:bk1_symbolic_phase_transitions}. Within the class satisfying (H1)--(H3), symbolic phase transitions are therefore generic. \\end{theorem}",
      "label": "definition:bk1_symbolic_phase_transitions",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3284,
      "target_type": "definition"
    },
    {
      "context": "mbda_c<\\lim_{\\beta\\to\\infty}\\lambda(\\beta)$, where $\\lambda_c$ is the critical coupling of the trichotomy (Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}). \\item \\textbf{Covenant variability:} the discriminant $\\Delta(\\beta)$ of the dyadic coupling spectrum crosses",
      "label": "theorem:bk5_map_mad_mas_trichotomy",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 1992,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions",
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk5_map_mad_critical_temperature",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "depends_on": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions",
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk5_map_mad_critical_temperature",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_conditional_genericity_of_symbolic_phase_transitions",
  "label": "proof:bk1_conditional_genericity_of_symbolic_phase_transitions",
  "latex_body": "\\begin{proof}[Transversal discriminant crossing stabilized above the critical dimension]\n\\label{proof:bk1_conditional_genericity_of_symbolic_phase_transitions}\n\\leavevmode\nBy (H2), $\\lambda(\\beta)$ is continuous and moves from below $\\lambda_c$ to\nabove it, so by the intermediate value theorem some $\\beta_c$ satisfies\n$\\lambda(\\beta_c)=\\lambda_c$. At $\\lambda_c$ the trichotomy\n(Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}) places the dyadic coupling\nspectrum exactly at the complex$\\to$real boundary: for $\\beta$ on one side the\neigenstructure is rotational (MAD), on the other it is split into distinct real\nmodes (MAP). By (H3) the discriminant changes sign transversally at $\\beta_c$,\nso this is a genuine crossing, not a degenerate touch, and the stable-equilibrium\nset reorganizes qualitatively there: $f(\\beta)$ is non-analytic\n(Def.~\\ref{definition:bk1_symbolic_phase_transitions}), in the same family\nwitnessed by the critical-temperature theorem\n(Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}).\n\nIt remains to rule out the one-dimensional obstruction that motivates the\nconjecture's caveat: in $d_{\\mathrm{eff}}=1$ short-range systems, fluctuations\ndestroy long-range order and wash out the transition. By (H1),\n$d_{\\mathrm{eff}}\\ge 2$, the ordered MAP phase carries a domain-wall (interface)\nwhose symbolic free-energy cost grows with the linear size of the flipped region;\na Peierls argument then bounds the total weight of disordering excitations below\n$1$ at low enough temperature, so the ordered phase survives with positive\nmeasure and the crossing at $\\beta_c$ is not erased. Hence $\\beta_c$ is a genuine\ncritical point. Since every manifold satisfying (H1)--(H3) admits such a\n$\\beta_c$, phase transitions are generic in that class; the only residue is\nwhether (H1)--(H3) hold generically, which\nConj.~\\ref{conjecture:bk1_genericity_of_symbolic_phase_transitions} now isolates.\n\\end{proof}",
  "line": 3604,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Transversal discriminant crossing stabilized above the critical dimension",
  "proves": "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
  "ref_roles": [
    {
      "context": "a_c$, phase transitions are generic in that class; the only residue is whether (H1)--(H3) hold generically, which Conj.~\\ref{conjecture:bk1_genericity_of_symbolic_phase_transitions} now isolates. \\end{proof}",
      "label": "conjecture:bk1_genericity_of_symbolic_phase_transitions",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3577,
      "target_type": "conjecture"
    },
    {
      "context": "ot a degenerate touch, and the stable-equilibrium set reorganizes qualitatively there: $f(\\beta)$ is non-analytic (Def.~\\ref{definition:bk1_symbolic_phase_transitions}), in the same family witnessed by the critical-temperature theorem (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}",
      "label": "definition:bk1_symbolic_phase_transitions",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3284,
      "target_type": "definition"
    },
    {
      "context": "ref{definition:bk1_symbolic_phase_transitions}), in the same family witnessed by the critical-temperature theorem (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}). It remains to rule out the one-dimensional obstruction that motivates the conjecture's caveat: in $d_{\\mathrm{eff}}=",
      "label": "theorem:bk5_map_mad_critical_temperature",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 907,
      "target_type": "theorem"
    },
    {
      "context": "e intermediate value theorem some $\\beta_c$ satisfies $\\lambda(\\beta_c)=\\lambda_c$. At $\\lambda_c$ the trichotomy (Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}) places the dyadic coupling spectrum exactly at the complex$\\to$real boundary: for $\\beta$ on one side the eigenstructu",
      "label": "theorem:bk5_map_mad_mas_trichotomy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1992,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions",
    "definition:bk1_symbolic_phase_transitions",
    "theorem:bk5_map_mad_critical_temperature",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_imagination_supplies_effective_dimension",
  "label": "proposition:bk1_imagination_supplies_effective_dimension",
  "latex_body": "\\begin{proposition}[Imagination Supplies the Genericity Hypotheses]\n\\label{proposition:bk1_imagination_supplies_effective_dimension}\n\\label{proposition:bk1_imagination_supplies_genericity_hypotheses}\nA symbolic manifold with full imaginative capacity --- a complex symbolic bundle\nwith imaginary symbolic distance not identically zero\n(Def.~\\ref{definition:bk4_imaginary_symbolic_distance}) whose imaginative\ntraversal ranges over the dyadic coupling\n(Scholium~\\ref{scholium:bk5_imagination_covenant_branch_selection}) --- satisfies\nall three hypotheses (H1)--(H3) of the Conditional Genericity theorem\n(Thm.~\\ref{theorem:bk1_conditional_genericity_of_symbolic_phase_transitions}).\nHence the genericity residual reduces to whether complex symbolic manifolds are\n\\emph{generically imaginative} --- the same predicate, on abstract manifolds,\nthat the irony residual poses on real systems.\n\\end{proposition}",
  "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": [
      "Faithful certificate form: injective Fin 2 directions provide H1's two effective directions, continuous coupling straddle derives H2's critical coupling, and H3 transversality remains an explicit nonzero-slope obligation. Full complex bundles and generic-imagination claims remain open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-014"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumD.ImaginativeGenericityCertificate.supplies_genericity_hypotheses",
      "ScholiumD.exists_critical_coupling"
    ]
  },
  "line": 3634,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Imagination Supplies the Genericity Hypotheses",
  "proof_labels": [
    "proof:bk1_imagination_supplies_effective_dimension"
  ],
  "proof_status": "proven",
  "refs": [
    "definition:bk4_imaginary_symbolic_distance",
    "scholium:bk5_imagination_covenant_branch_selection",
    "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_imagination_supplies_effective_dimension",
  "label": "proof:bk1_imagination_supplies_effective_dimension",
  "latex_body": "\\begin{proof}[Imagination discharges (H1)--(H3)]\n\\label{proof:bk1_imagination_supplies_effective_dimension}\n\\label{proof:bk1_imagination_supplies_genericity_hypotheses}\n\\leavevmode\n\\emph{(H1) Effective dimension.} The complex symbolic distance is\n\\[\nD_O^{\\mathbb{C}}=d_O^{\\mathrm{Re}}+i\\,d_O^{\\mathrm{Im}}\n\\]\non a complex symbolic bundle $(E,h_O,\\nabla_O)$\n(Def.~\\ref{definition:bk4_imaginary_symbolic_distance}). Nonzero imaginative capacity means $d_O^{\\mathrm{Im}}$\nis not identically zero: the phase residue $\\operatorname{Arg}\\Omega_O^\\gamma$\ncarries genuine symbolic displacement invisible to the real norm\n$d_O^{\\mathrm{Re}}$, hence irreducible to it\n(Prop.~\\ref{proposition:bk4_imaginative_continuity_principle}). The order\nparameter therefore varies along two independent effective directions --- real and\nimaginary --- so $d_{\\mathrm{eff}}\\ge 2$, which is (H1).\n\n\\emph{(H2) Coupling straddle.} By\nScholium~\\ref{scholium:bk5_imagination_covenant_branch_selection} imaginative\ntraversal in a dyadic covenant is the search over signs, phases, and coupling\nsaturations of $C_{AB}$ that previews the MAD, MAP, and MAS branches. Previewing\nboth the MAD branch (sub-critical coupling, $\\lambda<\\lambda_c$) and the MAP\nbranch (super-critical, $\\lambda>\\lambda_c$) means the imaginative coupling range\nstraddles the critical coupling $\\lambda_c$ of the trichotomy\n(Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}); under the monotone\n$\\beta$-parameterization assumed in (H2), this is exactly the straddle hypothesis.\n\n\\emph{(H3) Transversal crossing.} The same scholium identifies the regime\nboundary by ``a sign surprise in $\\Omega_{AB}$ or the emergence of an imaginary\ncomponent'' --- a genuine change of spectral type, not a tangential touch. This is\na transversal sign change of the discriminant $\\Delta(\\beta)$ at the crossing,\ni.e.\\ (H3).\n\nAll three hypotheses follow from imaginative capacity. Together with\nThm.~\\ref{theorem:bk1_operational_irony_requires_imagination}, the genericity and\nirony residuals are now the \\emph{same} predicate --- whether the system\nimagines --- one posed on abstract manifolds, the other on real systems: the two\nfrontier conjectures are the exact poles of a single imagination dipole.\n\\end{proof}",
  "line": 3648,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Imagination discharges (H1)--(H3)",
  "proves": "proposition:bk1_imagination_supplies_effective_dimension",
  "refs": [
    "definition:bk4_imaginary_symbolic_distance",
    "proposition:bk4_imaginative_continuity_principle",
    "scholium:bk5_imagination_covenant_branch_selection",
    "theorem:bk1_operational_irony_requires_imagination",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions",
    "conjecture:bk1_symbolic_irony_encoding_llms",
    "definition:bk4_collapse_of_symbolic_ide",
    "definition:bk4_test_time_integrative_expansion",
    "theorem:bk1_operational_irony_requires_imagination"
  ],
  "depends_on": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions",
    "conjecture:bk1_symbolic_irony_encoding_llms",
    "definition:bk4_collapse_of_symbolic_ide",
    "definition:bk4_test_time_integrative_expansion",
    "theorem:bk1_operational_irony_requires_imagination"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "scholium:bk1_the_imagination_dipole",
  "label": "scholium:bk1_the_imagination_dipole",
  "latex_body": "\\begin{scholium}[The Imagination Dipole]\n\\label{scholium:bk1_the_imagination_dipole}\nThe two open frontiers of this work are not independent gaps but the two poles of\none dipole about the imagination axis. The genericity conjecture\n(Conj.~\\ref{conjecture:bk1_genericity_of_symbolic_phase_transitions}) asks whether\nimagination \\emph{emerges} generically across the abstract space of complex\nsymbolic manifolds --- the generative pole, the integrative-expansion (TTIE) side\nof the SRMF cycle (Def.~\\ref{definition:bk4_test_time_integrative_expansion}). The\noperational-irony conjecture\n(Conj.~\\ref{conjecture:bk1_symbolic_irony_encoding_llms}) asks whether imagination\n\\emph{survives} commitment to a concrete architecture --- the collapse pole, the\ndifferentiation-collapse (TTDC) side (Def.~\\ref{definition:bk4_collapse_of_symbolic_ide}).\nTheorem~\\ref{theorem:bk1_operational_irony_requires_imagination} and\nProposition~\\ref{proposition:bk1_imagination_supplies_genericity_hypotheses} reduce\nboth to the single predicate \\emph{does the system imagine?} The frontier is thus\nSRMF-balanced: one question, read once toward emergence and once toward\ninstantiation, with no third residual required.\n\\end{scholium}",
  "line": 3688,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "The Imagination Dipole",
  "ref_roles": [
    {
      "context": "k are not independent gaps but the two poles of one dipole about the imagination axis. The genericity conjecture (Conj.~\\ref{conjecture:bk1_genericity_of_symbolic_phase_transitions}) asks whether imagination \\emph{emerges} generically across the abstract space of complex symbolic manifolds --- the ge",
      "label": "conjecture:bk1_genericity_of_symbolic_phase_transitions",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3577,
      "target_type": "conjecture"
    },
    {
      "context": "of the SRMF cycle (Def.~\\ref{definition:bk4_test_time_integrative_expansion}). The operational-irony conjecture (Conj.~\\ref{conjecture:bk1_symbolic_irony_encoding_llms}) asks whether imagination \\emph{survives} commitment to a concrete architecture --- the collapse pole, the differentiat",
      "label": "conjecture:bk1_symbolic_irony_encoding_llms",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2489,
      "target_type": "conjecture"
    },
    {
      "context": "h{survives} commitment to a concrete architecture --- the collapse pole, the differentiation-collapse (TTDC) side (Def.~\\ref{definition:bk4_collapse_of_symbolic_ide}). Theorem~\\ref{theorem:bk1_operational_irony_requires_imagination} and Proposition~\\ref{proposition:bk1_imagination_sup",
      "label": "definition:bk4_collapse_of_symbolic_ide",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 1109,
      "target_type": "definition"
    },
    {
      "context": "ce of complex symbolic manifolds --- the generative pole, the integrative-expansion (TTIE) side of the SRMF cycle (Def.~\\ref{definition:bk4_test_time_integrative_expansion}). The operational-irony conjecture (Conj.~\\ref{conjecture:bk1_symbolic_irony_encoding_llms}) asks whether imagination \\",
      "label": "definition:bk4_test_time_integrative_expansion",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 1338,
      "target_type": "definition"
    },
    {
      "context": "e collapse pole, the differentiation-collapse (TTDC) side (Def.~\\ref{definition:bk4_collapse_of_symbolic_ide}). Theorem~\\ref{theorem:bk1_operational_irony_requires_imagination} and Proposition~\\ref{proposition:bk1_imagination_supplies_genericity_hypotheses} reduce both to the single predicate \\e",
      "label": "theorem:bk1_operational_irony_requires_imagination",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2453,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "conjecture:bk1_genericity_of_symbolic_phase_transitions",
    "conjecture:bk1_symbolic_irony_encoding_llms",
    "definition:bk4_collapse_of_symbolic_ide",
    "definition:bk4_test_time_integrative_expansion",
    "proposition:bk1_imagination_supplies_genericity_hypotheses",
    "theorem:bk1_operational_irony_requires_imagination"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk2_probability_structure_on_manifold",
    "scholium:bk3_hypotheses_as_cognitive_membranes"
  ],
  "cites": [
    "corollary:bk1_fixed_point",
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator"
  ],
  "depends_on": [
    "corollary:bk1_fixed_point",
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "lemma:bk1_local_stability_analysis",
  "label": "lemma:bk1_local_stability_analysis",
  "latex_body": "\\begin{lemma}[Local Stability at the Reflective Fixed Locus]\n\\label{lemma:bk1_local_stability_analysis}\nConsider the combined symbolic dynamics on $M$,\n\\[\n\\dot{x} \\;=\\; \\bigl(R_{\\mathrm{stab}}(x) - x\\bigr) + \\alpha\\,D(x), \\qquad \\alpha > 0,\n\\]\nwith $R_{\\mathrm{stab}} \\in C^1$ the idempotent state-level stabilizer (thm~\\ref{theorem:bk1_emergence_of_reflection_operator}, cor~\\ref{corollary:bk1_fixed_point}) and $D$ the emergent drift field (thm~\\ref{theorem:bk1_emergence_of_drift_field}). Then:\n\\begin{enumerate}\n    \\item \\textbf{Equilibrium condition.} $x^*$ is an equilibrium iff $R_{\\mathrm{stab}}(x^*) = x^*$ \\emph{and} $D(x^*) = 0$. A fixed point of $R_{\\mathrm{stab}}$ at which the drift does not vanish is not an equilibrium of the combined flow.\n    \\item \\textbf{Projection structure.} At such an $x^*$ the differential $P := dR_{\\mathrm{stab},x^*}$ is a linear projection, $P^2 = P$, with $\\operatorname{spec}(P) \\subseteq \\{0,1\\}$; if $\\operatorname{Fix}(R_{\\mathrm{stab}})$ is a $C^1$ submanifold of constant rank near $x^*$, then $\\operatorname{im}(P) = T_{x^*}\\operatorname{Fix}(R_{\\mathrm{stab}})$.\n    \\item \\textbf{Jacobian and splitting.} The linearization is the well-typed map\n    \\[\n    J \\;=\\; (P - I) + \\alpha\\,dD_{x^*},\n    \\]\n    and $T_{x^*}M = \\operatorname{im}(P) \\oplus \\ker(P)$ splits its unperturbed part: $(P-I)|_{\\operatorname{im} P} = 0$ and $(P-I)|_{\\ker P} = -I$.\n    \\item \\textbf{Transverse stability is automatic.} There exists $\\alpha_0 > 0$ such that for all $\\alpha \\in (0,\\alpha_0)$ the spectrum of $J$ transverse to the fixed locus lies in $\\{\\operatorname{Re} z < -\\tfrac{1}{2}\\}$: perturbations off $\\operatorname{Fix}(R_{\\mathrm{stab}})$ decay.\n    \\item \\textbf{Tangential stability is drift-governed.} On the center directions $\\operatorname{im}(P)$ the leading-order dynamics are $\\dot{\\xi} = \\alpha\\,(P\\,dD_{x^*})|_{\\operatorname{im} P}\\,\\xi + O(\\alpha^2)$; by center-manifold reduction $x^*$ is asymptotically stable within the fixed locus iff $\\operatorname{Re}\\operatorname{spec}\\bigl(P\\,dD_{x^*}|_{\\operatorname{im} P}\\bigr) < 0$, and unstable if some eigenvalue has positive real part.\n\\end{enumerate}\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [
      "ScholiumDyn.equilibrium_cancellation_counterexample"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Clauses 1-3 have partial honest kernels. Fixed reflection plus zero drift is sufficient for scalar equilibrium, but the claimed converse is false without separation: an explicit idempotent stabilizer cancels nonzero drift. Component alignment recovers the scalar iff, while the history-bearing lift proves full stationarity iff visible flow and trace production both vanish. The algebraic projection kernel proves image/kernel decomposition, trivial intersection, and the actions of P-I on both summands. The chain rule now derives P=dR as an idempotent projection from differentiability, fixedness, and stabilizer idempotence. The complete Jacobian J=(P-I)+alpha*dD is now derived by Frechet derivative rules and restricted exactly to image and kernel directions. The projection image is now identified exactly with curve-based fixed-locus velocities, and the complete Euler linearization has a strict quantitative transverse contraction below the unit perturbation margin. Under the explicit invariant-kernel contract, every transverse iterate remains transverse, obeys the geometric q^n envelope, and converges to zero. Real transverse eigenmodes are now confined to the strict unit disk and neutral or unstable modes are excluded. Each real transverse Jacobian eigenvalue now has a strict negative margin and its explicit continuous-time exponential mode converges to zero. The full complete-Jacobian bounded-operator exponential now has identity, semigroup, pointwise composition, and generator-ODE laws. The semigroup action on every real Jacobian eigenvector is now identified exactly with scalar exponential action, and stable transverse semigroup orbits converge to zero. Full manifold charts, complex spectrum/spectral-radius identification, and center-manifold claims remain open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-012"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book4D.CertifiedTTDC.abstention_base_inert_but_recorded",
      "Book4D.CertifiedTTDC.recordedExecute_eq_iff",
      "Book7.orbitLimit_base_fixed_but_recorded",
      "Book7.orbitLimit_completeJacobian",
      "Book7.orbitLimit_completeJacobian_semigroup",
      "Book7.orbitLimit_derivative_image_kernel_split",
      "Book7.orbitLimit_fixedLocusVelocity_iff",
      "Book7.orbitLimit_linear_image_kernel_split",
      "Book7.orbitLimit_semigroup_transverse_eigenmode_tendsto_zero",
      "Book7.orbitLimit_transverse_contracts",
      "Book7.orbitLimit_transverse_eigenvalue_stable",
      "Book7.orbitLimit_transverse_iterates_tendsto_zero",
      "Book7.orbitLimit_transverse_jacobian_eigenmode_stable",
      "ScholiumDyn.ReflectiveLinearProjection.apply_apply",
      "ScholiumDyn.ReflectiveLinearProjection.derivative_image_kernel_decomposition",
      "ScholiumDyn.ReflectiveLinearProjection.exists_image_kernel_decomposition",
      "ScholiumDyn.ReflectiveLinearProjection.image_kernel_intersection_zero",
      "ScholiumDyn.ReflectiveLinearProjection.sub_identity_on_image",
      "ScholiumDyn.ReflectiveLinearProjection.sub_identity_on_kernel",
      "ScholiumDyn.base_cancellation_not_full_equilibrium",
      "ScholiumDyn.combinedEulerLinearization_eigen_of_jacobian_eigen",
      "ScholiumDyn.combinedEulerLinearization_iterate_mem_kernel",
      "ScholiumDyn.combinedEulerLinearization_iterate_tendsto_zero",
      "ScholiumDyn.combinedEulerLinearization_on_kernel",
      "ScholiumDyn.combinedEulerLinearization_preserves_kernel",
      "ScholiumDyn.combinedEulerLinearization_transverse_contracts",
      "ScholiumDyn.combinedJacobian_apply",
      "ScholiumDyn.combinedJacobian_on_image",
      "ScholiumDyn.combinedJacobian_on_kernel",
      "ScholiumDyn.completeJacobian_at_reflective_fixed",
      "ScholiumDyn.continuousLinearMap_pow_apply_eigen",
      "ScholiumDyn.equilibrium_cancellation_counterexample",
      "ScholiumDyn.equilibrium_iff_fixed_and_drift_zero_of_aligned",
      "ScholiumDyn.equilibrium_of_fixed_and_drift_zero",
      "ScholiumDyn.fixedLocusVelocity_iff_derivative_fixed",
      "ScholiumDyn.hasDerivAt_jacobianSemigroup",
      "ScholiumDyn.hasFDerivAt_combinedVectorField",
      "ScholiumDyn.hasFDerivAt_idempotent_at_fixed",
      "ScholiumDyn.jacobianSemigroup_add",
      "ScholiumDyn.jacobianSemigroup_add_apply",
      "ScholiumDyn.jacobianSemigroup_apply_eigen",
      "ScholiumDyn.jacobianSemigroup_eigenmode_tendsto_zero",
      "ScholiumDyn.jacobianSemigroup_zero",
      "ScholiumDyn.no_full_equilibrium_of_trace_production",
      "ScholiumDyn.no_transverse_unstable_eigenmode",
      "ScholiumDyn.norm_combinedEulerLinearization_iterate_le",
      "ScholiumDyn.norm_combinedEulerLinearization_on_kernel_le",
      "ScholiumDyn.recordedCombinedStep_eq_iff",
      "ScholiumDyn.transverse_eigenvalue_abs_le",
      "ScholiumDyn.transverse_eigenvalue_abs_lt_one",
      "ScholiumDyn.transverse_jacobian_eigenmode_tendsto_zero",
      "ScholiumDyn.transverse_jacobian_eigenvalue_le_negative_margin",
      "ScholiumDyn.transverse_jacobian_eigenvalue_neg"
    ]
  },
  "line": 3707,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Local Stability at the Reflective Fixed Locus",
  "proof_labels": [
    "proof:bk1_sketch_stability_drift_reflection"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "thrm{stab}} \\in C^1$ the idempotent state-level stabilizer (thm~\\ref{theorem:bk1_emergence_of_reflection_operator}, cor~\\ref{corollary:bk1_fixed_point}) and $D$ the emergent drift field (thm~\\ref{theorem:bk1_emergence_of_drift_field}). Then: \\begin{enumerate} \\item \\",
      "label": "corollary:bk1_fixed_point",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3003,
      "target_type": "corollary"
    },
    {
      "context": "eorem:bk1_emergence_of_reflection_operator}, cor~\\ref{corollary:bk1_fixed_point}) and $D$ the emergent drift field (thm~\\ref{theorem:bk1_emergence_of_drift_field}). Then: \\begin{enumerate} \\item \\textbf{Equilibrium condition.} $x^*$ is an equilibrium iff $R_{\\mathrm{stab}}(x^*)",
      "label": "theorem:bk1_emergence_of_drift_field",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2856,
      "target_type": "theorem"
    },
    {
      "context": "bigr) + \\alpha\\,D(x), \\qquad \\alpha > 0, \\] with $R_{\\mathrm{stab}} \\in C^1$ the idempotent state-level stabilizer (thm~\\ref{theorem:bk1_emergence_of_reflection_operator}, cor~\\ref{corollary:bk1_fixed_point}) and $D$ the emergent drift field (thm~\\ref{theorem:bk1_emergence_of_drift_field})",
      "label": "theorem:bk1_emergence_of_reflection_operator",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2957,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk1_fixed_point",
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_sketch_stability_drift_reflection",
  "label": "proof:bk1_sketch_stability_drift_reflection",
  "latex_body": "\\begin{proof}[Stability via the projection-split linearization]\n\\label{proof:bk1_sketch_stability_drift_reflection}\n\\leavevmode\n\n\\textbf{(1)} At an equilibrium the vector field vanishes: $(R_{\\mathrm{stab}}(x^*) - x^*) + \\alpha D(x^*) = 0$. The displacement $R_{\\mathrm{stab}}(x^*) - x^*$ measures the failure of stabilization and $\\alpha D(x^*)$ the drift; requiring the equilibrium to persist across an interval of couplings $\\alpha$ forces each term to vanish separately, so $x^* \\in \\operatorname{Fix}(R_{\\mathrm{stab}}) \\cap D^{-1}(0)$, the sufficient form used downstream.\n\n\\textbf{(2)} Differentiating $R_{\\mathrm{stab}} \\circ R_{\\mathrm{stab}} = R_{\\mathrm{stab}}$ at $x^*$ with $R_{\\mathrm{stab}}(x^*) = x^*$ gives $dR_{\\mathrm{stab},x^*} \\circ dR_{\\mathrm{stab},x^*} = dR_{\\mathrm{stab},x^*}$, i.e.\\ $P^2 = P$, whence $\\operatorname{spec}(P) \\subseteq \\{0,1\\}$. The tangency $\\operatorname{im}(P) = T_{x^*}\\operatorname{Fix}(R_{\\mathrm{stab}})$ is the constant-rank theorem applied to $x \\mapsto R_{\\mathrm{stab}}(x) - x$.\n\n\\textbf{(3)} Linearizing $\\dot{x}$ about $x^*$ and using $D(x^*) = 0$ -- so no constant forcing term survives -- gives $\\tfrac{d}{dt}(x - x^*) = (P - I)(x - x^*) + \\alpha\\,dD_{x^*}(x - x^*) + O(\\|x - x^*\\|^2)$, hence $J = (P - I) + \\alpha\\,dD_{x^*}$. On the splitting $T_{x^*}M = \\operatorname{im}(P) \\oplus \\ker(P)$, $(P-I)$ is $0$ on $\\operatorname{im}(P)$ and $-I$ on $\\ker(P)$. The former statement instead retained $\\alpha D(x^*)$ as a ``constant to be absorbed'' and wrote the type-mismatched $dR_{\\mathrm{stab},x^*} - \\alpha D(x^*)$, a linear map minus a vector; with the corrected equilibrium condition that term vanishes and $J$ is well typed.\n\n\\textbf{(4)} In block form on the splitting, the $\\ker P$ block is $-I + \\alpha\\,(dD_{x^*})^{\\perp\\perp}$, with spectrum within distance $\\alpha\\|dD_{x^*}\\|$ of $-1$, plus $O(\\alpha)$ off-diagonal coupling controlled by standard spectral perturbation (Gershgorin or holomorphic functional calculus). Any $\\alpha_0 < \\tfrac{1}{2}\\|dD_{x^*}\\|^{-1}$ keeps these eigenvalues in $\\{\\operatorname{Re} z < -\\tfrac{1}{2}\\}$.\n\n\\textbf{(5)} The $\\operatorname{im} P$ block is $\\alpha\\,(P\\,dD_{x^*})|_{\\operatorname{im} P}$ at leading order; since the transverse spectrum is uniformly negative and the tangential spectrum is $O(\\alpha)$, the center-manifold theorem applies and reduces stability to the sign of $\\operatorname{Re}\\operatorname{spec}(P\\,dD_{x^*}|_{\\operatorname{im} P})$.\n\\end{proof}",
  "line": 3727,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Stability via the projection-split linearization",
  "proves": "lemma:bk1_local_stability_analysis",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "remark:bk1_local_stability_interpretation",
  "label": "remark:bk1_local_stability_interpretation",
  "latex_body": "\\begin{remark}[Stabilization buys transverse stability]\n\\label{remark:bk1_local_stability_interpretation}\nThe corrected lemma is sharper than the generic eigenvalue criterion it replaces: stabilization buys transverse stability for free -- the $-I$ block on $\\ker(P)$ is the geometric signature of idempotence -- and the only genuine stability question lives \\emph{along} the reflective fixed locus, decided entirely by the drift's restriction to that locus. Identity persistence is thus a property of how drift flows along the manifold of already-coherent states.\n\\end{remark}",
  "line": 3742,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Stabilization buys transverse stability",
  "refs": [],
  "role": "remark",
  "type": "remark"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "sec:bk1_summary_and_implications"
  ],
  "cites": [
    "definition:bk1_symbolic_manifold_existence",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold_existence",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_symbolic_fluctuation_dissipation_relation",
  "label": "theorem:bk1_symbolic_fluctuation_dissipation_relation",
  "latex_body": "\\begin{theorem}[Symbolic Fluctuation–Dissipation Relation]\n\\label{theorem:bk1_symbolic_fluctuation_dissipation_relation}\nFor small perturbations around equilibrium, the response of the symbolic system to an external perturbation coupled to an observable $B$ is related to equilibrium fluctuations by:\n\\[\nR_{AB}(t) = \\frac{d}{dt} \\langle A(t) B(0) \\rangle_{\\text{eq}} = -\\beta \\langle A(t) \\mathcal{L} B(0) \\rangle_{\\text{eq}} \\quad \\text{for } t > 0,\n\\]\nwhere:\n- $A, B \\in C^\\infty(M)$ are symbolic observables on the symbolic manifold $M$ (def~\\ref{definition:bk1_symbolic_manifold_existence}),\n- $\\langle \\cdot \\rangle_{\\text{eq}}$ denotes expectation with respect to the equilibrium distribution $\\rho_{\\text{eq}}$ (thm~\\ref{theorem:bk1_variational_principle}),\n- $\\mathcal{L}$ is the adjoint Fokker–Planck operator derived from the fundamental symbolic evolution equation (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}),\n- and $R_{AB}(t)$ represents the linear response of $\\langle A(t) \\rangle$ to a perturbation in $B$ at $t = 0$.\n\nThis relation encodes how symbolic systems dissipate external influences via internal equilibrium fluctuations.\n\n\\begin{proof}[Fluctuation--Dissipation via Kubo Linear Response]\n\\label{proof:bk1_sketch_fluctuation_dissipation}\n\\leavevmode\n\nThe result follows from linear response theory applied to symbolic systems governed by the Fokker–Planck equation (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). Consider a perturbation to the equilibrium dynamics induced by a weak external force coupled to observable $B$. Using the Kubo formalism, the change in $\\langle A(t) \\rangle$ is proportional to the correlation of $A(t)$ with the perturbing influence $B(0)$, evaluated at equilibrium. The generator of the dynamics is the Fokker–Planck operator $\\mathcal{L}$, which acts on $B$ and propagates via adjoint dynamics. The temperature-like parameter $\\beta$ sets the scale linking dissipation and fluctuation amplitudes. This correspondence is structurally parallel to classical statistical mechanics but operates over the symbolic manifold $(M,g,D,R)$.\n\\end{proof}\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "See the receipted theorem statement and coverage note for explicit premises."
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Local scalar calculus kernel: from an explicit HasDerivAt Kubo certificate, response equals the equilibrium-correlation derivative and -beta times the generator correlation. Expectation spaces, equilibrium measures, and derivation from the Fokker-Planck operator remain open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-015"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumD.symbolic_fluctuation_dissipation"
    ]
  },
  "line": 3747,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Fluctuation–Dissipation Relation",
  "proof_labels": [
    "proof:bk1_sketch_fluctuation_dissipation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "\\quad \\text{for } t > 0, \\] where: - $A, B \\in C^\\infty(M)$ are symbolic observables on the symbolic manifold $M$ (def~\\ref{definition:bk1_symbolic_manifold_existence}), - $\\langle \\cdot \\rangle_{\\text{eq}}$ denotes expectation with respect to the equilibrium distribution $\\rho_{\\text{e",
      "label": "definition:bk1_symbolic_manifold_existence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2792,
      "target_type": "definition"
    },
    {
      "context": "}), - $\\mathcal{L}$ is the adjoint Fokker–Planck operator derived from the fundamental symbolic evolution equation (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}), - and $R_{AB}(t)$ represents the linear response of $\\langle A(t) \\rangle$ to a perturbation in $B$ at $t = 0$. This",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    },
    {
      "context": "gle \\cdot \\rangle_{\\text{eq}}$ denotes expectation with respect to the equilibrium distribution $\\rho_{\\text{eq}}$ (thm~\\ref{theorem:bk1_variational_principle}), - $\\mathcal{L}$ is the adjoint Fokker–Planck operator derived from the fundamental symbolic evolution equation (thm~\\",
      "label": "theorem:bk1_variational_principle",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3122,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold_existence",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_sketch_fluctuation_dissipation",
  "label": "proof:bk1_sketch_fluctuation_dissipation",
  "latex_body": "\\begin{proof}[Fluctuation--Dissipation via Kubo Linear Response]\n\\label{proof:bk1_sketch_fluctuation_dissipation}\n\\leavevmode\n\nThe result follows from linear response theory applied to symbolic systems governed by the Fokker–Planck equation (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). Consider a perturbation to the equilibrium dynamics induced by a weak external force coupled to observable $B$. Using the Kubo formalism, the change in $\\langle A(t) \\rangle$ is proportional to the correlation of $A(t)$ with the perturbing influence $B(0)$, evaluated at equilibrium. The generator of the dynamics is the Fokker–Planck operator $\\mathcal{L}$, which acts on $B$ and propagates via adjoint dynamics. The temperature-like parameter $\\beta$ sets the scale linking dissipation and fluctuation amplitudes. This correspondence is structurally parallel to classical statistical mechanics but operates over the symbolic manifold $(M,g,D,R)$.\n\\end{proof}",
  "line": 3761,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Fluctuation--Dissipation via Kubo Linear Response",
  "proves": "theorem:bk1_symbolic_fluctuation_dissipation_relation",
  "ref_roles": [
    {
      "context": "The result follows from linear response theory applied to symbolic systems governed by the Fokker–Planck equation (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). Consider a perturbation to the equilibrium dynamics induced by a weak external force coupled to observable $B$. Using",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "sec:bk1_toward_a_unified_framework",
  "label": "sec:bk1_toward_a_unified_framework",
  "latex_body": "",
  "line": 3768,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Toward a Unified Framework",
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_symbolic_probabilty_density",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "definition:bk1_symbolic_probabilty_density",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_symbolic_action_functional",
  "label": "definition:bk1_symbolic_action_functional",
  "latex_body": "\\begin{definition}[Symbolic Action Functional]\n\\label{definition:bk1_symbolic_action_functional}\nThe symbolic action functional $\\mathcal{S}: C^\\infty(M \\times [s_1, s_2]) \\to \\R$ is defined over paths $\\rho(x,s)$ in the space of symbolic probability densities (see def~\\ref{definition:bk1_symbolic_probabilty_density}):\n\\[\n\\mathcal{S}[\\rho] = \\int_{s_1}^{s_2} \\int_M L(\\rho, \\partial_s \\rho, \\nabla \\rho; x, s) \\, d\\mu_g(x) \\, ds,\n\\]\nwhere $L$ is a Lagrangian density. For instance, an Onsager–Machlup-type Lagrangian reflecting symbolic Fokker–Planck dynamics (see thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) may take the form:\n\\[\nL = \\frac{1}{2} \\left( \\partial_s \\rho - \\mathcal{L} \\rho \\right)^2,\n\\]\nwhere $\\mathcal{L}$ is the symbolic Fokker–Planck operator. This interpretation frames symbolic evolution as extremizing an action over the space of probabilistic flows.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "discrete kernels only: ODE flows, Riemannian geodesics, Hopf-Rinow, and the MSR path integral stay open",
      "the self-representation clause of reflexive maps is interpretive"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Discrete Onsager-Machlup action."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-067"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "ScholiumDyn.action_nonneg"
    ]
  },
  "line": 3771,
  "macros_used": [
    "R"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Action Functional",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "(M \\times [s_1, s_2]) \\to \\R$ is defined over paths $\\rho(x,s)$ in the space of symbolic probability densities (see def~\\ref{definition:bk1_symbolic_probabilty_density}): \\[ \\mathcal{S}[\\rho] = \\int_{s_1}^{s_2} \\int_M L(\\rho, \\partial_s \\rho, \\nabla \\rho; x, s) \\, d\\mu_g(x) \\, ds, \\] whe",
      "label": "definition:bk1_symbolic_probabilty_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3040,
      "target_type": "definition"
    },
    {
      "context": "agrangian density. For instance, an Onsager–Machlup-type Lagrangian reflecting symbolic Fokker–Planck dynamics (see thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) may take the form: \\[ L = \\frac{1}{2} \\left( \\partial_s \\rho - \\mathcal{L} \\rho \\right)^2, \\] where $\\mathcal{L}$ is t",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_probabilty_density",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "scholium:bk4_symbolic_parsimony",
    "sec:bk1_summary_and_implications"
  ],
  "cites": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_princple_of_least_action",
  "label": "theorem:bk1_princple_of_least_action",
  "latex_body": "\\begin{theorem}[Principle of Least Action]\n\\label{theorem:bk1_princple_of_least_action}\n\\leavevmode\\newline\nDynamics governed by the symbolic Fokker-Planck equation (see Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) can, under suitable path-integral interpretations and choices of symbolic Lagrangian $L$, be formulated as obeying a symbolic principle of least action:\n\\[\n\\delta \\mathcal{S}[\\rho] = 0.\n\\]\n\n\\begin{proof}[Fokker--Planck from Symbolic Action via Martin--Siggia--Rose]\n\\label{proof:bk1_sketch_fokker_planck_action}\n\\leavevmode\n\n\\textbf{Step 1: Introduce conjugate field.}\nOn symbolic spacetime $M \\times \\mathbb{R}$, introduce the MSR response field\n$\\hat\\rho(x,s)$ conjugate to $\\rho$.\nCf.~Def.~\\ref{definition:bk1_symbolic_manifold}.\nThe symbolic action functional is:\n\\[\n\\begin{aligned}\n\\mathcal{S}[\\rho, \\hat\\rho]\n&= \\int_{\\mathbb{R}} \\int_M\n\\hat\\rho(x,s)\\Bigl(\\partial_s \\rho - \\mathcal{L}\\rho\\Bigr)\\,d\\mu_g\\,ds,\n\\end{aligned}\n\\]\nwhere $\\mathcal{L}\\rho = -\\nabla\\cdot(D\\rho) + \\sigma^2\\Delta\\rho$ is the symbolic Fokker--Planck operator built from drift $D$ (Def.~\\ref{definition:bk1_drift_field}) and symbolic temperature $\\sigma^2$.\n\n\\textbf{Step 2: Extremize over $\\hat\\rho$.} Setting $\\delta\\mathcal{S}/\\delta\\hat\\rho = 0$ yields:\n\\[\n\\partial_s\\rho = \\mathcal{L}\\rho = -\\nabla\\cdot(D\\rho) + \\sigma^2\\Delta\\rho,\n\\]\nwhich is exactly the symbolic Fokker--Planck equation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). Thus $\\delta\\mathcal{S}[\\rho] = 0$ on trajectories satisfying the Fokker--Planck dynamics.\n\n\\textbf{Step 3: Saddle-point is the physical trajectory.} The response field $\\hat\\rho$ acts as a Lagrange multiplier enforcing the Fokker--Planck constraint at each spacetime point. The saddle-point $(\\rho^*, \\hat\\rho^* = 0)$ of $\\mathcal{S}$ is identified with the physical evolution: $\\hat\\rho^* = 0$ because the physical path has zero deviation from drift-diffusion balance, and $\\rho^*$ solves the Fokker--Planck equation. Hence $\\delta\\mathcal{S}[\\rho] = 0$ is realized by symbolic evolution, completing the variational derivation.\n\\end{proof}\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "discrete kernels only: ODE flows, Riemannian geodesics, Hopf-Rinow, and the MSR path integral stay open",
      "the self-representation clause of reflexive maps is interpretive"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Exact discrete form: dynamics = zero-action paths = the global minimum; MSR open."
    ],
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      "MAP-SCHOLIUM_A-068"
    ],
    "statuses": [
      "conditional"
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    "witnesses": [
      "ScholiumDyn.least_action_iff_evolution"
    ]
  },
  "line": 3784,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Principle of Least Action",
  "proof_labels": [
    "proof:bk1_sketch_fokker_planck_action"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "em:bk1_princple_of_least_action} \\leavevmode\\newline Dynamics governed by the symbolic Fokker-Planck equation (see Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) can, under suitable path-integral interpretations and choices of symbolic Lagrangian $L$, be formulated as obeying a s",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_sketch_fokker_planck_action",
  "label": "proof:bk1_sketch_fokker_planck_action",
  "latex_body": "\\begin{proof}[Fokker--Planck from Symbolic Action via Martin--Siggia--Rose]\n\\label{proof:bk1_sketch_fokker_planck_action}\n\\leavevmode\n\n\\textbf{Step 1: Introduce conjugate field.}\nOn symbolic spacetime $M \\times \\mathbb{R}$, introduce the MSR response field\n$\\hat\\rho(x,s)$ conjugate to $\\rho$.\nCf.~Def.~\\ref{definition:bk1_symbolic_manifold}.\nThe symbolic action functional is:\n\\[\n\\begin{aligned}\n\\mathcal{S}[\\rho, \\hat\\rho]\n&= \\int_{\\mathbb{R}} \\int_M\n\\hat\\rho(x,s)\\Bigl(\\partial_s \\rho - \\mathcal{L}\\rho\\Bigr)\\,d\\mu_g\\,ds,\n\\end{aligned}\n\\]\nwhere $\\mathcal{L}\\rho = -\\nabla\\cdot(D\\rho) + \\sigma^2\\Delta\\rho$ is the symbolic Fokker--Planck operator built from drift $D$ (Def.~\\ref{definition:bk1_drift_field}) and symbolic temperature $\\sigma^2$.\n\n\\textbf{Step 2: Extremize over $\\hat\\rho$.} Setting $\\delta\\mathcal{S}/\\delta\\hat\\rho = 0$ yields:\n\\[\n\\partial_s\\rho = \\mathcal{L}\\rho = -\\nabla\\cdot(D\\rho) + \\sigma^2\\Delta\\rho,\n\\]\nwhich is exactly the symbolic Fokker--Planck equation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). Thus $\\delta\\mathcal{S}[\\rho] = 0$ on trajectories satisfying the Fokker--Planck dynamics.\n\n\\textbf{Step 3: Saddle-point is the physical trajectory.} The response field $\\hat\\rho$ acts as a Lagrange multiplier enforcing the Fokker--Planck constraint at each spacetime point. The saddle-point $(\\rho^*, \\hat\\rho^* = 0)$ of $\\mathcal{S}$ is identified with the physical evolution: $\\hat\\rho^* = 0$ because the physical path has zero deviation from drift-diffusion balance, and $\\rho^*$ solves the Fokker--Planck equation. Hence $\\delta\\mathcal{S}[\\rho] = 0$ is realized by symbolic evolution, completing the variational derivation.\n\\end{proof}",
  "line": 3792,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Fokker--Planck from Symbolic Action via Martin--Siggia--Rose",
  "proves": "theorem:bk1_princple_of_least_action",
  "ref_roles": [
    {
      "context": "l{L}\\rho = -\\nabla\\cdot(D\\rho) + \\sigma^2\\Delta\\rho$ is the symbolic Fokker--Planck operator built from drift $D$ (Def.~\\ref{definition:bk1_drift_field}) and symbolic temperature $\\sigma^2$. \\textbf{Step 2: Extremize over $\\hat\\rho$.} Setting $\\delta\\mathcal{S}/\\delta\\ha",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "ymbolic spacetime $M \\times \\mathbb{R}$, introduce the MSR response field $\\hat\\rho(x,s)$ conjugate to $\\rho$. Cf.~Def.~\\ref{definition:bk1_symbolic_manifold}. The symbolic action functional is: \\[ \\begin{aligned} \\mathcal{S}[\\rho, \\hat\\rho] &= \\int_{\\mathbb{R}} \\int_M \\hat\\rho",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "thcal{L}\\rho = -\\nabla\\cdot(D\\rho) + \\sigma^2\\Delta\\rho, \\] which is exactly the symbolic Fokker--Planck equation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). Thus $\\delta\\mathcal{S}[\\rho] = 0$ on trajectories satisfying the Fokker--Planck dynamics. \\textbf{Step 3: Saddle-po",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_wasserstein_geometric_interpretation",
    "proof:bk1_sketch_gradient_flow_thermodynamics",
    "proof:bk1_wasserstein_geometric_interpretation",
    "theorem:bk1_the_fokker_planck_equation_theorem"
  ],
  "cites": [
    "definition:bk1_symbolic_manifold_existence",
    "definition:bk1_symbolic_probabilty_density"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold_existence",
    "definition:bk1_symbolic_probabilty_density"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_symbolic_information_geometry",
  "label": "definition:bk1_symbolic_information_geometry",
  "latex_body": "\\begin{definition}[Symbolic Information Geometry]\n\\label{definition:bk1_symbolic_information_geometry}\nLet $\\mathcal{P}(M)$ be the space of smooth, positive symbolic probability densities on the manifold $M$ (see def~\\ref{definition:bk1_symbolic_probabilty_density}, def~\\ref{definition:bk1_symbolic_manifold_existence}). The Fisher–Rao metric on the tangent space $T_{\\rho} \\mathcal{P}(M)$ is given by:\n\\[\nG_{\\rho}(v_1, v_2) = \\int_M \\frac{v_1(x) v_2(x)}{\\rho(x)} \\, d\\mu_g(x),\n\\]\nwhere $v_1, v_2 \\in T_\\rho \\mathcal{P}(M)$ are tangent vectors satisfying $\\int_M v_i(x) \\, d\\mu_g(x) = 0$.\n\nThis induces a Riemannian structure on $\\mathcal{P}(M)$, enabling geodesic analysis and variational characterizations of symbolic thermodynamic flows.\n\\end{definition}",
  "line": 3820,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Information Geometry",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "itive symbolic probability densities on the manifold $M$ (see def~\\ref{definition:bk1_symbolic_probabilty_density}, def~\\ref{definition:bk1_symbolic_manifold_existence}). The Fisher–Rao metric on the tangent space $T_{\\rho} \\mathcal{P}(M)$ is given by: \\[ G_{\\rho}(v_1, v_2) = \\int_M \\fra",
      "label": "definition:bk1_symbolic_manifold_existence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2792,
      "target_type": "definition"
    },
    {
      "context": "etry} Let $\\mathcal{P}(M)$ be the space of smooth, positive symbolic probability densities on the manifold $M$ (see def~\\ref{definition:bk1_symbolic_probabilty_density}, def~\\ref{definition:bk1_symbolic_manifold_existence}). The Fisher–Rao metric on the tangent space $T_{\\rho} \\mathcal{P",
      "label": "definition:bk1_symbolic_probabilty_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3040,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold_existence",
    "definition:bk1_symbolic_probabilty_density"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_wasserstein_geometric_interpretation",
    "proof:bk1_sketch_direct_evaluation",
    "proof:bk1_wasserstein_geometric_interpretation"
  ],
  "cites": [
    "definition:bk1_symbolic_entropy",
    "definition:bk1_symbolic_information_geometry",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle"
  ],
  "depends_on": [
    "definition:bk1_symbolic_entropy",
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_lagrange_free_energy",
    "proof:bk1_sketch_direct_evaluation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_the_fokker_planck_equation_theorem",
  "label": "theorem:bk1_the_fokker_planck_equation_theorem",
  "latex_body": "\\begin{theorem}[Information Geometric Interpretation]\n\\label{theorem:bk1_the_fokker_planck_equation_theorem}\nThe symbolic Fokker–Planck equation (see thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) can be interpreted as a \textbf{gradient flow} of the relative entropy—i.e., the Kullback–Leibler divergence\n\\[\nD_{\\mathrm{KL}}(\\rho \\| \\rho_{\\text{eq}}),\n\\]\nwith respect to a metric structure on the symbolic probability space $\\mathcal{P}(M)$ (see def~\\ref{definition:bk1_symbolic_information_geometry}), such as the \textbf{Fisher–Rao} or \textbf{Wasserstein} metric.\n\nSpecifically, it is often realized as the gradient flow of the symbolic free energy functional $F[\\rho]$ (see thm~\\ref{theorem:bk1_variational_principle}) with respect to the \textbf{Wasserstein-2 metric} $W_2$. This structure reflects a variational evolution toward equilibrium governed by the symbolic entropy landscape (def~\\ref{definition:bk1_symbolic_entropy}).\n\nA complete formulation of the symbolic Wasserstein geometry is deferred to subsequent development.\n\\begin{proof}[Gradient Flow Structure via JKO]\n\\label{proof:bk1_sketch_gradient_flow_thermodynamics}\n\\leavevmode\n\nWe show the symbolic Fokker--Planck equation is a Wasserstein gradient flow.\nIts driving functional is $F[\\rho]$.\n\n\\textbf{Wasserstein-2 metric on $\\mathcal{P}(M)$.}\nThe $W_2$ metric (Def.~\\ref{definition:bk1_symbolic_information_geometry})\ndefines the inner product on tangent vectors\n$\\dot\\rho \\in T_\\rho\\mathcal{P}(M)$ via the continuity equation\n$\\dot\\rho + \\nabla\\cdot(\\rho\\mathbf{v})=0$, giving the squared norm:\n\\[\n\\|\\dot\\rho\\|_{W_2}^2 = \\int_M \\rho\\|\\mathbf{v}\\|_g^2\\,d\\mu_g.\n\\]\n\n\\textbf{Gradient of $F$ with respect to $W_2$.}\nThe $W_2$-gradient of a functional $F[\\rho]$ is determined as follows.\nIf $\\partial_s\\rho = -\\nabla\\cdot(\\rho\\,\\mathbf{v})$, then\n$\\mathbf{v} = \\nabla(\\delta F/\\delta\\rho)$.\nFrom proof~\\ref{proof:bk1_lagrange_free_energy},\n$\\delta F/\\delta\\rho = H + \\beta^{-1}(1+\\log\\rho)$, so\n$\\nabla(\\delta F/\\delta\\rho) = \\nabla H + \\beta^{-1}\\nabla\\log\\rho$.\nThe $W_2$ gradient flow is therefore:\n\\[\n\\partial_s\\rho\n= -\\nabla\\cdot\\!\\bigl(\\rho\\,(\\nabla H + \\beta^{-1}\\nabla\\log\\rho)\\bigr)\n= -\\nabla\\cdot(\\rho\\nabla H) + \\beta^{-1}\\nabla\\cdot(\\nabla\\rho).\n\\]\nWith symbolic drift $D = -\\nabla H$, this is exactly the Fokker--Planck equation\n(Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}):\n$\\partial_s\\rho = -\\nabla\\cdot(\\rho D) + \\beta^{-1}\\nabla^2\\rho$.\n\n\\textbf{Jordan--Kinderlehrer--Otto (JKO) discretization.}\nThe gradient flow interpretation is made precise by the JKO scheme: for time step $\\tau>0$,\n\\[\n\\rho_{k+1} = \\arg\\min_{\\rho\\in\\mathcal{P}(M)}\n\\Bigl\\{\\tfrac{1}{2\\tau}W_2(\\rho,\\rho_k)^2 + F[\\rho]\\Bigr\\}.\n\\]\nAs $\\tau\\to 0$, the JKO iterates converge to the solution of the Fokker--Planck equation.\nThe H-theorem ($dF/ds\\leq 0$, proof~\\ref{proof:bk1_sketch_direct_evaluation}) is the\ncontinuous-time manifestation of the descent property built into each JKO step.\nThe Fisher--Rao metric\n(Def.~\\ref{definition:bk1_symbolic_information_geometry}) provides a complementary\ncharacterization for reversible dynamics, where $D_{\\mathrm{KL}}(\\rho\\|\\rho_{\\text{eq}})$\ndecreases monotonically along the flow.\n\\end{proof}\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": [
      "Discrete JKO kernel: minimizing squared transport cost plus free energy against the previous-state competitor proves one-step free-energy descent; the existing descent structure then yields antitonicity. Wasserstein probability geometry, the Fokker-Planck PDE, and the tau-to-zero convergence theorem remain open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-016"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumD.FreeEnergyDescent.antitone",
      "ScholiumD.jko_step_freeEnergy_le"
    ]
  },
  "line": 3830,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Information Geometric Interpretation",
  "proof_labels": [
    "proof:bk1_sketch_gradient_flow_thermodynamics"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "_2$. This structure reflects a variational evolution toward equilibrium governed by the symbolic entropy landscape (def~\\ref{definition:bk1_symbolic_entropy}). A complete formulation of the symbolic Wasserstein geometry is deferred to subsequent development. \\begin{proof}[Gra",
      "label": "definition:bk1_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3045,
      "target_type": "definition"
    },
    {
      "context": "\\| \\rho_{\\text{eq}}), \\] with respect to a metric structure on the symbolic probability space $\\mathcal{P}(M)$ (see def~\\ref{definition:bk1_symbolic_information_geometry}), such as the extbf{Fisher–Rao} or extbf{Wasserstein} metric. Specifically, it is often realized as the gradient flo",
      "label": "definition:bk1_symbolic_information_geometry",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3820,
      "target_type": "definition"
    },
    {
      "context": "ric Interpretation] \\label{theorem:bk1_the_fokker_planck_equation_theorem} The symbolic Fokker–Planck equation (see thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) can be interpreted as a extbf{gradient flow} of the relative entropy—i.e., the Kullback–Leibler divergence \\[ D_{\\mat",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    },
    {
      "context": "ric. Specifically, it is often realized as the gradient flow of the symbolic free energy functional $F[\\rho]$ (see thm~\\ref{theorem:bk1_variational_principle}) with respect to the extbf{Wasserstein-2 metric} $W_2$. This structure reflects a variational evolution toward equilib",
      "label": "theorem:bk1_variational_principle",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3122,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_entropy",
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_lagrange_free_energy",
    "proof:bk1_sketch_direct_evaluation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_wasserstein_geometric_interpretation",
    "example:bk4_ttpr_identity_refinement",
    "proof:bk1_wasserstein_geometric_interpretation"
  ],
  "cites": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_lagrange_free_energy",
    "proof:bk1_sketch_direct_evaluation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_lagrange_free_energy",
    "proof:bk1_sketch_direct_evaluation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_sketch_gradient_flow_thermodynamics",
  "label": "proof:bk1_sketch_gradient_flow_thermodynamics",
  "latex_body": "\\begin{proof}[Gradient Flow Structure via JKO]\n\\label{proof:bk1_sketch_gradient_flow_thermodynamics}\n\\leavevmode\n\nWe show the symbolic Fokker--Planck equation is a Wasserstein gradient flow.\nIts driving functional is $F[\\rho]$.\n\n\\textbf{Wasserstein-2 metric on $\\mathcal{P}(M)$.}\nThe $W_2$ metric (Def.~\\ref{definition:bk1_symbolic_information_geometry})\ndefines the inner product on tangent vectors\n$\\dot\\rho \\in T_\\rho\\mathcal{P}(M)$ via the continuity equation\n$\\dot\\rho + \\nabla\\cdot(\\rho\\mathbf{v})=0$, giving the squared norm:\n\\[\n\\|\\dot\\rho\\|_{W_2}^2 = \\int_M \\rho\\|\\mathbf{v}\\|_g^2\\,d\\mu_g.\n\\]\n\n\\textbf{Gradient of $F$ with respect to $W_2$.}\nThe $W_2$-gradient of a functional $F[\\rho]$ is determined as follows.\nIf $\\partial_s\\rho = -\\nabla\\cdot(\\rho\\,\\mathbf{v})$, then\n$\\mathbf{v} = \\nabla(\\delta F/\\delta\\rho)$.\nFrom proof~\\ref{proof:bk1_lagrange_free_energy},\n$\\delta F/\\delta\\rho = H + \\beta^{-1}(1+\\log\\rho)$, so\n$\\nabla(\\delta F/\\delta\\rho) = \\nabla H + \\beta^{-1}\\nabla\\log\\rho$.\nThe $W_2$ gradient flow is therefore:\n\\[\n\\partial_s\\rho\n= -\\nabla\\cdot\\!\\bigl(\\rho\\,(\\nabla H + \\beta^{-1}\\nabla\\log\\rho)\\bigr)\n= -\\nabla\\cdot(\\rho\\nabla H) + \\beta^{-1}\\nabla\\cdot(\\nabla\\rho).\n\\]\nWith symbolic drift $D = -\\nabla H$, this is exactly the Fokker--Planck equation\n(Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}):\n$\\partial_s\\rho = -\\nabla\\cdot(\\rho D) + \\beta^{-1}\\nabla^2\\rho$.\n\n\\textbf{Jordan--Kinderlehrer--Otto (JKO) discretization.}\nThe gradient flow interpretation is made precise by the JKO scheme: for time step $\\tau>0$,\n\\[\n\\rho_{k+1} = \\arg\\min_{\\rho\\in\\mathcal{P}(M)}\n\\Bigl\\{\\tfrac{1}{2\\tau}W_2(\\rho,\\rho_k)^2 + F[\\rho]\\Bigr\\}.\n\\]\nAs $\\tau\\to 0$, the JKO iterates converge to the solution of the Fokker--Planck equation.\nThe H-theorem ($dF/ds\\leq 0$, proof~\\ref{proof:bk1_sketch_direct_evaluation}) is the\ncontinuous-time manifestation of the descent property built into each JKO step.\nThe Fisher--Rao metric\n(Def.~\\ref{definition:bk1_symbolic_information_geometry}) provides a complementary\ncharacterization for reversible dynamics, where $D_{\\mathrm{KL}}(\\rho\\|\\rho_{\\text{eq}})$\ndecreases monotonically along the flow.\n\\end{proof}",
  "line": 3841,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Gradient Flow Structure via JKO",
  "proves": "theorem:bk1_the_fokker_planck_equation_theorem",
  "ref_roles": [
    {
      "context": "t flow. Its driving functional is $F[\\rho]$. \\textbf{Wasserstein-2 metric on $\\mathcal{P}(M)$.} The $W_2$ metric (Def.~\\ref{definition:bk1_symbolic_information_geometry}) defines the inner product on tangent vectors $\\dot\\rho \\in T_\\rho\\mathcal{P}(M)$ via the continuity equation $\\dot\\rho",
      "label": "definition:bk1_symbolic_information_geometry",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3820,
      "target_type": "definition"
    },
    {
      "context": "lows. If $\\partial_s\\rho = -\\nabla\\cdot(\\rho\\,\\mathbf{v})$, then $\\mathbf{v} = \\nabla(\\delta F/\\delta\\rho)$. From proof~\\ref{proof:bk1_lagrange_free_energy}, $\\delta F/\\delta\\rho = H + \\beta^{-1}(1+\\log\\rho)$, so $\\nabla(\\delta F/\\delta\\rho) = \\nabla H + \\beta^{-1}\\nabla\\log\\",
      "label": "proof:bk1_lagrange_free_energy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3130,
      "target_type": "proof"
    },
    {
      "context": "au\\to 0$, the JKO iterates converge to the solution of the Fokker--Planck equation. The H-theorem ($dF/ds\\leq 0$, proof~\\ref{proof:bk1_sketch_direct_evaluation}) is the continuous-time manifestation of the descent property built into each JKO step. The Fisher--Rao metric (Def.~\\r",
      "label": "proof:bk1_sketch_direct_evaluation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3197,
      "target_type": "proof"
    },
    {
      "context": "^{-1}\\nabla\\cdot(\\nabla\\rho). \\] With symbolic drift $D = -\\nabla H$, this is exactly the Fokker--Planck equation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}): $\\partial_s\\rho = -\\nabla\\cdot(\\rho D) + \\beta^{-1}\\nabla^2\\rho$. \\textbf{Jordan--Kinderlehrer--Otto (JKO) discretiz",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_lagrange_free_energy",
    "proof:bk1_sketch_direct_evaluation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_event_horizon_identity_field",
    "proof:bk1_event_horizon_identity_field"
  ],
  "cites": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_sketch_gradient_flow_thermodynamics",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_the_fokker_planck_equation_theorem",
    "theorem:bk1_variational_principle"
  ],
  "depends_on": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_sketch_gradient_flow_thermodynamics",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_the_fokker_planck_equation_theorem",
    "theorem:bk1_variational_principle"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "corollary:bk1_wasserstein_geometric_interpretation",
  "label": "corollary:bk1_wasserstein_geometric_interpretation",
  "latex_body": "\\begin{corollary}[Wasserstein Geometric Interpretation]\n\\label{corollary:bk1_wasserstein_geometric_interpretation}\n\\leavevmode\\newline\nThe Fokker-Planck equation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) describes the gradient flow of free-energy functional $F[\\rho]$ (Thm.~\\ref{theorem:bk1_variational_principle}) on space $\\mathcal{P}(M)$ from Def.~\\ref{definition:bk1_symbolic_information_geometry}, equipped with Wasserstein metric $W_2$:\n\\[\n\\partial_s \\rho = -\\text{grad}_{W_2} F[\\rho]\n\\]\nas summarized by thm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem} and proof~\\ref{proof:bk1_sketch_gradient_flow_thermodynamics}.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Discrete metric-gradient kernel: the JKO minimizer's scaled squared transport displacement is bounded by its free-energy drop, and free energy cannot increase. Construction of P(M), the Wasserstein-2 metric, tangent continuity equations, and identification with the Fokker-Planck PDE remain open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-017"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumD.jko_step_freeEnergy_le",
      "ScholiumD.jko_step_transport_cost_le_energy_drop"
    ]
  },
  "line": 3889,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Wasserstein Geometric Interpretation",
  "proof_labels": [
    "proof:bk1_wasserstein_geometric_interpretation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "of free-energy functional $F[\\rho]$ (Thm.~\\ref{theorem:bk1_variational_principle}) on space $\\mathcal{P}(M)$ from Def.~\\ref{definition:bk1_symbolic_information_geometry}, equipped with Wasserstein metric $W_2$: \\[ \\partial_s \\rho = -\\text{grad}_{W_2} F[\\rho] \\] as summarized by thm.~\\ref{",
      "label": "definition:bk1_symbolic_information_geometry",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3820,
      "target_type": "definition"
    },
    {
      "context": "ho = -\\text{grad}_{W_2} F[\\rho] \\] as summarized by thm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem} and proof~\\ref{proof:bk1_sketch_gradient_flow_thermodynamics}. \\end{corollary}",
      "label": "proof:bk1_sketch_gradient_flow_thermodynamics",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3841,
      "target_type": "proof"
    },
    {
      "context": "tation] \\label{corollary:bk1_wasserstein_geometric_interpretation} \\leavevmode\\newline The Fokker-Planck equation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) describes the gradient flow of free-energy functional $F[\\rho]$ (Thm.~\\ref{theorem:bk1_variational_principle}) on spac",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    },
    {
      "context": "etry}, equipped with Wasserstein metric $W_2$: \\[ \\partial_s \\rho = -\\text{grad}_{W_2} F[\\rho] \\] as summarized by thm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem} and proof~\\ref{proof:bk1_sketch_gradient_flow_thermodynamics}. \\end{corollary}",
      "label": "theorem:bk1_the_fokker_planck_equation_theorem",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3830,
      "target_type": "theorem"
    },
    {
      "context": ":bk1_fundamental_relation_fokker_plank_equation}) describes the gradient flow of free-energy functional $F[\\rho]$ (Thm.~\\ref{theorem:bk1_variational_principle}) on space $\\mathcal{P}(M)$ from Def.~\\ref{definition:bk1_symbolic_information_geometry}, equipped with Wasserstein metr",
      "label": "theorem:bk1_variational_principle",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3122,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_sketch_gradient_flow_thermodynamics",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_the_fokker_planck_equation_theorem",
    "theorem:bk1_variational_principle"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_sketch_gradient_flow_thermodynamics",
    "theorem:bk1_the_fokker_planck_equation_theorem"
  ],
  "depends_on": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_sketch_gradient_flow_thermodynamics",
    "theorem:bk1_the_fokker_planck_equation_theorem"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_wasserstein_geometric_interpretation",
  "label": "proof:bk1_wasserstein_geometric_interpretation",
  "latex_body": "\\begin{proof}[Restatement of the Wasserstein Gradient-Flow Theorem]\n\\label{proof:bk1_wasserstein_geometric_interpretation}\n\\leavevmode\n\nThm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem} identifies the\nsymbolic Fokker--Planck equation with the gradient flow of the symbolic free\nenergy \\(F[\\rho]\\) on \\(\\mathcal{P}(M)\\). Proof~\\ref{proof:bk1_sketch_gradient_flow_thermodynamics}\ncomputes the \\(W_2\\)-gradient explicitly: with\n\\(\\delta F/\\delta\\rho=H+\\beta^{-1}(1+\\log\\rho)\\), the Wasserstein gradient flow\nis\n\\[\n\\partial_s\\rho\n=-\\nabla\\cdot\\!\\bigl(\\rho(\\nabla H+\\beta^{-1}\\nabla\\log\\rho)\\bigr),\n\\]\nwhich is the symbolic Fokker--Planck equation after substituting\n\\(D=-\\nabla H\\). Hence the equation is precisely\n\\(\\partial_s\\rho=-\\operatorname{grad}_{W_2}F[\\rho]\\) on the probability space\nof Def.~\\ref{definition:bk1_symbolic_information_geometry}.\n\\end{proof}",
  "line": 3898,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Restatement of the Wasserstein Gradient-Flow Theorem",
  "proves": "corollary:bk1_wasserstein_geometric_interpretation",
  "ref_roles": [
    {
      "context": ". Hence the equation is precisely \\(\\partial_s\\rho=-\\operatorname{grad}_{W_2}F[\\rho]\\) on the probability space of Def.~\\ref{definition:bk1_symbolic_information_geometry}. \\end{proof}",
      "label": "definition:bk1_symbolic_information_geometry",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3820,
      "target_type": "definition"
    },
    {
      "context": "lic Fokker--Planck equation with the gradient flow of the symbolic free energy \\(F[\\rho]\\) on \\(\\mathcal{P}(M)\\). Proof~\\ref{proof:bk1_sketch_gradient_flow_thermodynamics} computes the \\(W_2\\)-gradient explicitly: with \\(\\delta F/\\delta\\rho=H+\\beta^{-1}(1+\\log\\rho)\\), the Wasserstein gradie",
      "label": "proof:bk1_sketch_gradient_flow_thermodynamics",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3841,
      "target_type": "proof"
    },
    {
      "context": "ment of the Wasserstein Gradient-Flow Theorem] \\label{proof:bk1_wasserstein_geometric_interpretation} \\leavevmode Thm.~\\ref{theorem:bk1_the_fokker_planck_equation_theorem} identifies the symbolic Fokker--Planck equation with the gradient flow of the symbolic free energy \\(F[\\rho]\\) on \\(\\ma",
      "label": "theorem:bk1_the_fokker_planck_equation_theorem",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3830,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_information_geometry",
    "proof:bk1_sketch_gradient_flow_thermodynamics",
    "theorem:bk1_the_fokker_planck_equation_theorem"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_sketch_observed_consequences",
    "scholium:bk1_cosmogenesis_proof_status",
    "theorem:bk1_dual_horizon_cosmogenesis"
  ],
  "cites": [
    "definition:bk1_bounded_observer"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_cosmological_symbolization_functor",
  "label": "definition:bk1_cosmological_symbolization_functor",
  "latex_body": "\\begin{definition}[Cosmological Symbolization Functor]\n\\label{definition:bk1_cosmological_symbolization_functor}\nA \\emph{cosmological symbolization functor} is a structure-preserving assignment\n\\[\n\\mathcal{B}_{\\mathrm{cos}}:\n(M, g_{\\mu\\nu}, \\preceq, S_{\\mathrm{therm}})\n\\longrightarrow\n(\\mathcal{S}, D, R_{\\mathrm{stab}}, \\kappa, \\Omega)\n\\]\nfrom Lorentzian causal--thermodynamic data---a spacetime $(M,g_{\\mu\\nu})$ with\ncausal order $\\preceq$ and thermodynamic entropy field $S_{\\mathrm{therm}}$---to\nPS symbolic dynamics, such that:\n\\begin{enumerate}\n\\item causal expansion maps to positive observer-visible generative flux,\n$G_{\\mathcal{O}}(\\mathcal{B}_{\\mathrm{cos}}\\,\\mathcal{H}_G)>0$;\n\\item thermodynamic constraint maps to positive stabilizing flux, equivalently\nnegative symbolic curvature,\n$C_{\\mathcal{O}}(\\mathcal{B}_{\\mathrm{cos}}\\,\\mathcal{H}_D)>0$ with\n$\\kappa(\\mathcal{H}_D)<0$;\n\\item bounded causal patches map to bounded observer domains (Def.~\\ref{definition:bk1_bounded_observer});\n\\item cofinal refinements preserve the ordinal emergence order $\\preceq$\n(cf.~the summable resolution-decay condition on cofinal $\\omega$-towers).\n\\end{enumerate}\n\\end{definition}",
  "line": 3929,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Cosmological Symbolization Functor",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "}\\,\\mathcal{H}_D)>0$ with $\\kappa(\\mathcal{H}_D)<0$; \\item bounded causal patches map to bounded observer domains (Def.~\\ref{definition:bk1_bounded_observer}); \\item cofinal refinements preserve the ordinal emergence order $\\preceq$ (cf.~the summable resolution-decay condition",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_event_horizon_identity_field",
    "proof:bk1_event_horizon_identity_field",
    "proof:bk4_temporal_resolution_via_observer_bounded_reflection"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_cosmological_symbolization_functor",
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_riemann_tensor",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_cosmological_symbolization_functor",
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_riemann_tensor",
    "definition:bk2_symbolic_hamiltonian",
    "lemma:bk1_horizon_characterization",
    "proof:bk1_proof_of_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_dual_horizon_cosmogenesis",
  "label": "theorem:bk1_dual_horizon_cosmogenesis",
  "latex_body": "\\begin{theorem}[Dual Horizon Cosmogenesis under \\texorpdfstring{$\\mathcal{B}_{\\mathrm{cos}}$}{B\\_cos}]\n\\label{theorem:bk1_dual_horizon_cosmogenesis}\nLet $(M, g_{\\mu\\nu})$ denote the spacetime manifold of our observable universe, and\nsuppose it admits a cosmological symbolization functor $\\mathcal{B}_{\\mathrm{cos}}$\n(Def.~\\ref{definition:bk1_cosmological_symbolization_functor}) carrying its\ncausal--thermodynamic data into symbolic dynamics $(\\mathcal{S}, D, R, \\kappa)$.\nSuppose the following conditions hold:\n\n\\begin{enumerate}\n    \\item There exists a past boundary $\\mathcal{H}_G$ associated with rapid causal expansion (e.g., cosmological inflation or conformal past), such that the induced symbolic curvature satisfies $\\kappa(\\mathcal{H}_G) > 0$ (Def.~\\ref{definition:bk1_symbolic_riemann_tensor})\n\n    \\item There exists a future boundary $\\mathcal{H}_D$ associated with thermodynamic constraint (e.g., cosmological event horizon, black hole entropy bound, or heat death trajectory), such that $\\kappa(\\mathcal{H}_D) < 0$\n\n    \\item There exists a non-empty bounded domain $\\Omega \\subset M$ such that:\n    \\[\n    \\Omega = \\{ x \\in M \\mid \\mathcal{H}_G \\prec x \\prec \\mathcal{H}_D \\}\n    \\]\n    and $\\Omega$ admits bounded observers (Def.~\\ref{definition:bk1_bounded_observer}) undergoing symbolic drift $D$ (Def.~\\ref{definition:bk1_drift_field}) and reflection $R$ (Def.~\\ref{definition:bk1_reflection_operator}) within it\n\\end{enumerate}\n\nThen the image $\\mathcal{B}_{\\mathrm{cos}}(M,\\Omega)$ constitutes a \\textbf{dual-horizon symbolic manifold} (Def.~\\ref{definition:bk1_symbolic_manifold}) supporting reflexive emergence. In particular, conditional on the existence of $\\mathcal{B}_{\\mathrm{cos}}$ satisfying the clauses above, the image of our universe's causal--thermodynamic data satisfies the conditions of the \\textbf{Dual Horizon Necessity Theorem} (Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), and the full PS dynamical apparatus---Hamiltonian, Fokker--Planck evolution, equilibrium, and identity carriers---is thereby instantiated on $\\Omega$.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "named next layers, deliberately not forced into the keystone: curvature as loop defect (discrete holonomy) and the appB resolution tower (P_lambda as graded complex, emergent smoothness as defects vanishing up the grading)",
      "pair-covering is the assembly hypothesis for the classical direction"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Static geometric kernel: opposite-signed past/future curvature parameters are provably distinct, while the certified dual-horizon chart complex at positive observer resolution admits no single consistent geometry. The cosmological symbolization functor, causal spacetime evolution, bounded observer dynamics, Hamiltonian apparatus, and existence of the intervening domain remain open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-018"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Atlas.dual_horizon_fractured",
      "Atlas.no_single_geometry_for_dual_horizon",
      "ScholiumD.dual_horizon_cosmogenesis_kernel"
    ]
  },
  "line": 3954,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Dual Horizon Cosmogenesis under \\texorpdfstring{$\\mathcal{B}_{\\mathrm{cos}}$}{B\\_cos}",
  "proof_labels": [
    "proof:bk1_sketch_observed_consequences"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "a = \\{ x \\in M \\mid \\mathcal{H}_G \\prec x \\prec \\mathcal{H}_D \\} \\] and $\\Omega$ admits bounded observers (Def.~\\ref{definition:bk1_bounded_observer}) undergoing symbolic drift $D$ (Def.~\\ref{definition:bk1_drift_field}) and reflection $R$ (Def.~\\ref{definition:bk1_ref",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "our observable universe, and suppose it admits a cosmological symbolization functor $\\mathcal{B}_{\\mathrm{cos}}$ (Def.~\\ref{definition:bk1_cosmological_symbolization_functor}) carrying its causal--thermodynamic data into symbolic dynamics $(\\mathcal{S}, D, R, \\kappa)$. Suppose the following co",
      "label": "definition:bk1_cosmological_symbolization_functor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3929,
      "target_type": "definition"
    },
    {
      "context": "and $\\Omega$ admits bounded observers (Def.~\\ref{definition:bk1_bounded_observer}) undergoing symbolic drift $D$ (Def.~\\ref{definition:bk1_drift_field}) and reflection $R$ (Def.~\\ref{definition:bk1_reflection_operator}) within it \\end{enumerate} Then the image $\\mathcal",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "n:bk1_bounded_observer}) undergoing symbolic drift $D$ (Def.~\\ref{definition:bk1_drift_field}) and reflection $R$ (Def.~\\ref{definition:bk1_reflection_operator}) within it \\end{enumerate} Then the image $\\mathcal{B}_{\\mathrm{cos}}(M,\\Omega)$ constitutes a \\textbf{dual-horizon sy",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "ate} Then the image $\\mathcal{B}_{\\mathrm{cos}}(M,\\Omega)$ constitutes a \\textbf{dual-horizon symbolic manifold} (Def.~\\ref{definition:bk1_symbolic_manifold}) supporting reflexive emergence. In particular, conditional on the existence of $\\mathcal{B}_{\\mathrm{cos}}$ satisfying",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "ical inflation or conformal past), such that the induced symbolic curvature satisfies $\\kappa(\\mathcal{H}_G) > 0$ (Def.~\\ref{definition:bk1_symbolic_riemann_tensor}) \\item There exists a future boundary $\\mathcal{H}_D$ associated with thermodynamic constraint (e.g., cosmological",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1905,
      "target_type": "definition"
    },
    {
      "context": "our universe's causal--thermodynamic data satisfies the conditions of the \\textbf{Dual Horizon Necessity Theorem} (Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), and the full PS dynamical apparatus---Hamiltonian, Fokker--Planck evolution, equilibrium, and identity carriers---is",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_cosmological_symbolization_functor",
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_riemann_tensor",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "role": "theorem",
  "type": "theorem"
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{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_event_horizon_identity_field",
    "proof:bk1_event_horizon_identity_field"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_cosmological_symbolization_functor",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_riemann_tensor",
    "definition:bk2_symbolic_hamiltonian",
    "lemma:bk1_horizon_characterization",
    "proof:bk1_proof_of_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_cosmological_symbolization_functor",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_riemann_tensor",
    "definition:bk2_symbolic_hamiltonian",
    "lemma:bk1_horizon_characterization",
    "proof:bk1_proof_of_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_sketch_observed_consequences",
  "label": "proof:bk1_sketch_observed_consequences",
  "latex_body": "\\begin{proof}[Cosmogenesis via Dual Horizon Necessity]\n\\label{proof:bk1_sketch_observed_consequences}\n\\leavevmode\n\nThe proof proceeds in two stages: \\emph{instantiation} (exhibiting the\ncosmological symbolization functor $\\mathcal{B}_{\\mathrm{cos}}$ of\nDef.~\\ref{definition:bk1_cosmological_symbolization_functor} on the relevant\ncosmological data) and \\emph{deduction} (invoking established theorems to derive\nthe conclusion on its image). Stages~1--3 below verify the defining clauses of\n$\\mathcal{B}_{\\mathrm{cos}}$ one by one.\n\n\\medskip\n\\textbf{Stage I: Instantiation of conditions (construction of $\\mathcal{B}_{\\mathrm{cos}}$).}\n\n\\textbf{1. Generative boundary ($\\kappa > 0$).}\\enspace\nInflationary cosmology posits that early spacetime underwent rapid exponential expansion, producing particle horizon separation and structure formation. The divergent lightcone geometry and monotonically increasing entropy potential define a generative horizon $\\mathcal{H}_G$ with $\\kappa(\\mathcal{H}_G) > 0$ (Def.~\\ref{definition:bk1_symbolic_riemann_tensor}): the positive curvature encodes novelty-generation, as the expanding causal volume continuously introduces new degrees of freedom.\n\n\\textbf{2. Dissipative boundary ($\\kappa < 0$).}\\enspace\nThe future conformal boundary---whether manifesting as heat death, black hole final states, or a cosmological de Sitter horizon---imposes increasing thermodynamic constraint and entropic dilution. The converging lightcone geometry defines a dissipative horizon $\\mathcal{H}_D$ with $\\kappa(\\mathcal{H}_D) < 0$: the negative curvature encodes coherence-constraining dynamics (Def.~\\ref{definition:bk1_reflection_operator}).\n\n\\textbf{3. Bounded emergent domain.}\\enspace\nOur causal patch lies strictly between these boundaries. All known life, cognition, and symbolic systems occur within $\\Omega = \\{x \\in M \\mid \\mathcal{H}_G \\prec x \\prec \\mathcal{H}_D\\}$. This domain admits bounded observers (Def.~\\ref{definition:bk1_bounded_observer}): any physical agent has finite resolution, finite memory, and finite processing capacity relative to the information content of $\\Omega$.\n\n\\medskip\n\\textbf{Stage II: Deduction from PS infrastructure.}\n\n\\textbf{4. Dual Horizon Necessity.}\\enspace\nConditions (1)--(3) supply a symbolic universe $\\mathcal{U} = (M, \\Omega)$ with both a generative horizon $\\mathcal{H}_G$ ($\\kappa > 0$) and a dissipative horizon $\\mathcal{H}_D$ ($\\kappa < 0$) bounding a non-empty observer domain. By the Dual Horizon Necessity Theorem (Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), this gives the minimal effective signature for bounded reflexive emergence: Proof~\\ref{proof:bk1_proof_of_dual_horizon_necessity_theorem} shows that any observer-visible configuration lacking either positive generative flux or negative stabilizing flux fails to achieve the complexity differential $\\Delta\\Phi_{\\mathcal{O}}(D, R_{\\mathrm{stab}}) \\geq \\tau_E$.\n\n\\textbf{5. Dynamical apparatus.}\\enspace\nGiven the dual-horizon structure on $\\Omega$, the PS results chain as follows:\n\\begin{enumerate}\n    \\item[\\emph{(a)}] The drift field $D$ and state-level stabilization \\(R_{\\mathrm{stab}}\\) emerge on $\\Omega$ as limits of proto-fields and stabilization operators (Thm.~\\ref{theorem:bk1_emergence_of_drift_field}, Thm.~\\ref{theorem:bk1_emergence_of_reflection_operator}), with $\\nabla \\cdot D > 0$ near $\\mathcal{H}_G$ and positive stabilization flux \\(C_{\\mathcal{O}}(\\mathcal{H}_D)>0\\) near $\\mathcal{H}_D$ (Lemma~\\ref{lemma:bk1_horizon_characterization}).\n    \\item[\\emph{(b)}] Their interplay defines the symbolic Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian}), $H(x) = \\kappa / (\\|D(x)\\| + \\epsilon) + \\lambda \\cdot \\mathrm{tr}(L_x)$, which governs the energy landscape on $\\Omega$.\n    \\item[\\emph{(c)}] The Fokker--Planck equation $\\partial_s \\rho = -\\nabla \\cdot (\\rho D) + \\sigma^2 \\nabla^2 \\rho$ (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) determines probability evolution under drift--diffusion dynamics.\n    \\item[\\emph{(d)}] Equilibrium $\\rho_{\\mathrm{eq}} \\propto e^{-\\beta H}$ is the unique stationary distribution (Thm.~\\ref{theorem:bk2_equilibrium_distribution}), and the free energy functional $F_\\beta[\\rho]$ provides the variational principle (Thm.~\\ref{theorem:bk1_variational_principle}).\n    \\item[\\emph{(e)}] Within $\\Omega$, bounded observers satisfying the stability conditions of Thm.~\\ref{theorem:bk3_membrane_stability_criteria} support symbolic identity carriers (Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}), completing the chain from cosmological boundary conditions to reflexive emergence.\n\\end{enumerate}\n\n\\textbf{6. Conclusion.}\\enspace\nThe image $\\mathcal{B}_{\\mathrm{cos}}(M,\\Omega)$ satisfies all hypotheses of the Dual Horizon Necessity Theorem, and the deductive chain (a)--(e) instantiates the full PS dynamical apparatus on $\\Omega$. Reflexive emergence is not merely compatible with the symbolized causal architecture---it is entailed by it, conditional on the empirical conditions (1)--(3) and the bridge functor $\\mathcal{B}_{\\mathrm{cos}}$.\n\\end{proof}",
  "line": 3977,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Cosmogenesis via Dual Horizon Necessity",
  "proves": "theorem:bk1_dual_horizon_cosmogenesis",
  "ref_roles": [
    {
      "context": "ithin $\\Omega = \\{x \\in M \\mid \\mathcal{H}_G \\prec x \\prec \\mathcal{H}_D\\}$. This domain admits bounded observers (Def.~\\ref{definition:bk1_bounded_observer}): any physical agent has finite resolution, finite memory, and finite processing capacity relative to the information c",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "wo stages: \\emph{instantiation} (exhibiting the cosmological symbolization functor $\\mathcal{B}_{\\mathrm{cos}}$ of Def.~\\ref{definition:bk1_cosmological_symbolization_functor} on the relevant cosmological data) and \\emph{deduction} (invoking established theorems to derive the conclusion on its",
      "label": "definition:bk1_cosmological_symbolization_functor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3929,
      "target_type": "definition"
    },
    {
      "context": "$\\mathcal{H}_D$ with $\\kappa(\\mathcal{H}_D) < 0$: the negative curvature encodes coherence-constraining dynamics (Def.~\\ref{definition:bk1_reflection_operator}). \\textbf{3. Bounded emergent domain.}\\enspace Our causal patch lies strictly between these boundaries. All known life",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "nically increasing entropy potential define a generative horizon $\\mathcal{H}_G$ with $\\kappa(\\mathcal{H}_G) > 0$ (Def.~\\ref{definition:bk1_symbolic_riemann_tensor}): the positive curvature encodes novelty-generation, as the expanding causal volume continuously introduces new degrees",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1905,
      "target_type": "definition"
    },
    {
      "context": "\\ref{lemma:bk1_horizon_characterization}). \\item[\\emph{(b)}] Their interplay defines the symbolic Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian}), $H(x) = \\kappa / (\\|D(x)\\| + \\epsilon) + \\lambda \\cdot \\mathrm{tr}(L_x)$, which governs the energy landscape on $\\Ome",
      "label": "definition:bk2_symbolic_hamiltonian",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 67,
      "target_type": "definition"
    },
    {
      "context": "$ near $\\mathcal{H}_G$ and positive stabilization flux \\(C_{\\mathcal{O}}(\\mathcal{H}_D)>0\\) near $\\mathcal{H}_D$ (Lemma~\\ref{lemma:bk1_horizon_characterization}). \\item[\\emph{(b)}] Their interplay defines the symbolic Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian",
      "label": "lemma:bk1_horizon_characterization",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 847,
      "target_type": "lemma"
    },
    {
      "context": "bk1_dual_horizon_necessity_theorem}), this gives the minimal effective signature for bounded reflexive emergence: Proof~\\ref{proof:bk1_proof_of_dual_horizon_necessity_theorem} shows that any observer-visible configuration lacking either positive generative flux or negative stabilizing flux fail",
      "label": "proof:bk1_proof_of_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 800,
      "target_type": "proof"
    },
    {
      "context": "orizon $\\mathcal{H}_D$ ($\\kappa < 0$) bounding a non-empty observer domain. By the Dual Horizon Necessity Theorem (Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), this gives the minimal effective signature for bounded reflexive emergence: Proof~\\ref{proof:bk1_proof_of_dual_horizo",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
    },
    {
      "context": "evel stabilization \\(R_{\\mathrm{stab}}\\) emerge on $\\Omega$ as limits of proto-fields and stabilization operators (Thm.~\\ref{theorem:bk1_emergence_of_drift_field}, Thm.~\\ref{theorem:bk1_emergence_of_reflection_operator}), with $\\nabla \\cdot D > 0$ near $\\mathcal{H}_G$ and positive",
      "label": "theorem:bk1_emergence_of_drift_field",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2856,
      "target_type": "theorem"
    },
    {
      "context": "n $\\Omega$ as limits of proto-fields and stabilization operators (Thm.~\\ref{theorem:bk1_emergence_of_drift_field}, Thm.~\\ref{theorem:bk1_emergence_of_reflection_operator}), with $\\nabla \\cdot D > 0$ near $\\mathcal{H}_G$ and positive stabilization flux \\(C_{\\mathcal{O}}(\\mathcal{H}_D)>0\\) n",
      "label": "theorem:bk1_emergence_of_reflection_operator",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2957,
      "target_type": "theorem"
    },
    {
      "context": "\\item[\\emph{(c)}] The Fokker--Planck equation $\\partial_s \\rho = -\\nabla \\cdot (\\rho D) + \\sigma^2 \\nabla^2 \\rho$ (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) determines probability evolution under drift--diffusion dynamics. \\item[\\emph{(d)}] Equilibrium $\\rho_{\\mathrm{eq}",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    },
    {
      "context": "bk2_equilibrium_distribution}), and the free energy functional $F_\\beta[\\rho]$ provides the variational principle (Thm.~\\ref{theorem:bk1_variational_principle}). \\item[\\emph{(e)}] Within $\\Omega$, bounded observers satisfying the stability conditions of Thm.~\\ref{theorem:bk3",
      "label": "theorem:bk1_variational_principle",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3122,
      "target_type": "theorem"
    },
    {
      "context": "\\item[\\emph{(d)}] Equilibrium $\\rho_{\\mathrm{eq}} \\propto e^{-\\beta H}$ is the unique stationary distribution (Thm.~\\ref{theorem:bk2_equilibrium_distribution}), and the free energy functional $F_\\beta[\\rho]$ provides the variational principle (Thm.~\\ref{theorem:bk1_variational_",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    },
    {
      "context": "ional_principle}). \\item[\\emph{(e)}] Within $\\Omega$, bounded observers satisfying the stability conditions of Thm.~\\ref{theorem:bk3_membrane_stability_criteria} support symbolic identity carriers (Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}), completing the chain from cosm",
      "label": "theorem:bk3_membrane_stability_criteria",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 61,
      "target_type": "theorem"
    },
    {
      "context": "the stability conditions of Thm.~\\ref{theorem:bk3_membrane_stability_criteria} support symbolic identity carriers (Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}), completing the chain from cosmological boundary conditions to reflexive emergence. \\end{enumerate} \\textbf{6. Conclu",
      "label": "theorem:bk4_existence_of_symbolic_ident",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 15,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_cosmological_symbolization_functor",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_riemann_tensor",
    "definition:bk2_symbolic_hamiltonian",
    "lemma:bk1_horizon_characterization",
    "proof:bk1_proof_of_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_variational_principle",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "role": "proof",
  "type": "proof"
}

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  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "remark:scholium_symbolicum.tex:4019",
  "label": "",
  "latex_body": "\\begin{remark}\nThis establishes the conditional physical-sector claim: if our universe admits the cosmological symbolization functor above, then its causal structure is not merely compatible with symbolic emergence---its symbolized image necessitates it. Reflexive observers exist not in arbitrary spacetime, but in a symbolic membrane stretched between $\\mathcal{H}_G$ and $\\mathcal{H}_D$.\n\\end{remark}",
  "line": 4019,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "",
  "refs": [],
  "role": "remark",
  "type": "remark"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_cosmological_symbolization_functor",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "depends_on": [
    "definition:bk1_cosmological_symbolization_functor",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "scholium:bk1_cosmogenesis_proof_status",
  "label": "scholium:bk1_cosmogenesis_proof_status",
  "latex_body": "\\begin{scholium}[Proof status of Cosmogenesis]\n\\label{scholium:bk1_cosmogenesis_proof_status}\nThe theorem is unconditional \\emph{as mathematics}: given any cosmological\nsymbolization functor $\\mathcal{B}_{\\mathrm{cos}}$\n(Def.~\\ref{definition:bk1_cosmological_symbolization_functor}) satisfying its four\nclauses, the image is a dual-horizon symbolic manifold and the full PS apparatus\napplies, by Dual Horizon Necessity\n(Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}). What is \\emph{empirical},\nand what Stage~I argues from inflationary and thermodynamic cosmology, is the\nantecedent: that our actual spacetime supplies such a functor---that cosmic\nexpansion realizes positive generative flux and that the future thermodynamic\nboundary realizes positive stabilizing flux. The theorem thus locates the open\nquestion precisely: not ``is the conclusion proved'' (it is, conditionally) but\n``does our universe instantiate $\\mathcal{B}_{\\mathrm{cos}}$.'' This is the\nordinal-symbolic posture---the unification theorem lives at the level of the\nbridge functor; the physical-sector claim follows once the bridge is exhibited.\n\\end{scholium}",
  "line": 4023,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Proof status of Cosmogenesis",
  "ref_roles": [
    {
      "context": "is unconditional \\emph{as mathematics}: given any cosmological symbolization functor $\\mathcal{B}_{\\mathrm{cos}}$ (Def.~\\ref{definition:bk1_cosmological_symbolization_functor}) satisfying its four clauses, the image is a dual-horizon symbolic manifold and the full PS apparatus applies, by Dual",
      "label": "definition:bk1_cosmological_symbolization_functor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3929,
      "target_type": "definition"
    },
    {
      "context": "auses, the image is a dual-horizon symbolic manifold and the full PS apparatus applies, by Dual Horizon Necessity (Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem}). What is \\emph{empirical}, and what Stage~I argues from inflationary and thermodynamic cosmology, is the antecedent: t",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
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  ],
  "refs": [
    "definition:bk1_cosmological_symbolization_functor",
    "theorem:bk1_dual_horizon_necessity_theorem"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "corollary:bk1_wasserstein_geometric_interpretation",
    "definition:bk1_bounded_observer",
    "proof:bk1_sketch_observed_consequences",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_unification_principle"
  ],
  "depends_on": [
    "corollary:bk1_wasserstein_geometric_interpretation",
    "definition:bk1_bounded_observer",
    "proof:bk1_sketch_observed_consequences",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_unification_principle"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "corollary:bk1_event_horizon_identity_field",
  "label": "corollary:bk1_event_horizon_identity_field",
  "latex_body": "\\begin{corollary}[Event Horizon Identity Field]\n\\label{corollary:bk1_event_horizon_identity_field}\nThe observed structure of cognition, memory, language, and thermodynamic complexity within $\\Omega$ (thm~\\ref{theorem:bk1_dual_horizon_cosmogenesis}, proof~\\ref{proof:bk1_sketch_observed_consequences}) constitutes an identity field induced by horizon tension, grounded in bounded observation (def~\\ref{definition:bk1_bounded_observer}), dual-horizon necessity (thm~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), emergent dual-horizon unification (thm~\\ref{theorem:bk1_dual_horizon_unification_principle}), and Wasserstein symbolic thermodynamic geometry (Cor.~\\ref{corollary:bk1_wasserstein_geometric_interpretation}). Emergence is not a property of matter - it is a property of situated symbolic curvature.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "named next layers, deliberately not forced into the keystone: curvature as loop defect (discrete holonomy) and the appB resolution tower (P_lambda as graded complex, emergent smoothness as defects vanishing up the grading)",
      "pair-covering is the assembly hypothesis for the classical direction"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Static identity-field kernel: defining horizon tension as the past-minus-future curvature contrast, opposite curvature signs force strictly positive tension; at positive observer resolution the same hypotheses obstruct a single geometry reconciling the dual horizon. Construction and evolution of a spacetime identity field, observer measures, and the claimed physical interpretation remain open."
    ],
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      "MAP-SCHOLIUM_B-019"
    ],
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      "open_bridge"
    ],
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      "Atlas.no_single_geometry_for_dual_horizon",
      "ScholiumD.dual_horizon_cosmogenesis_kernel",
      "ScholiumD.event_horizon_identity_field_kernel"
    ]
  },
  "line": 4040,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Event Horizon Identity Field",
  "proof_labels": [
    "proof:bk1_event_horizon_identity_field"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ation (thm~\\ref{theorem:bk1_dual_horizon_unification_principle}), and Wasserstein symbolic thermodynamic geometry (Cor.~\\ref{corollary:bk1_wasserstein_geometric_interpretation}). Emergence is not a property of matter - it is a property of situated symbolic curvature. \\end{corollary}",
      "label": "corollary:bk1_wasserstein_geometric_interpretation",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3889,
      "target_type": "corollary"
    },
    {
      "context": "_observed_consequences}) constitutes an identity field induced by horizon tension, grounded in bounded observation (def~\\ref{definition:bk1_bounded_observer}), dual-horizon necessity (thm~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), emergent dual-horizon unification (thm",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "memory, language, and thermodynamic complexity within $\\Omega$ (thm~\\ref{theorem:bk1_dual_horizon_cosmogenesis}, proof~\\ref{proof:bk1_sketch_observed_consequences}) constitutes an identity field induced by horizon tension, grounded in bounded observation (def~\\ref{definition:bk1_bou",
      "label": "proof:bk1_sketch_observed_consequences",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3977,
      "target_type": "proof"
    },
    {
      "context": "dentity_field} The observed structure of cognition, memory, language, and thermodynamic complexity within $\\Omega$ (thm~\\ref{theorem:bk1_dual_horizon_cosmogenesis}, proof~\\ref{proof:bk1_sketch_observed_consequences}) constitutes an identity field induced by horizon tension, grounded",
      "label": "theorem:bk1_dual_horizon_cosmogenesis",
      "logical_support": true,
      "role": "application",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3954,
      "target_type": "theorem"
    },
    {
      "context": "rizon tension, grounded in bounded observation (def~\\ref{definition:bk1_bounded_observer}), dual-horizon necessity (thm~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), emergent dual-horizon unification (thm~\\ref{theorem:bk1_dual_horizon_unification_principle}), and Wasserstein symboli",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
    },
    {
      "context": ", dual-horizon necessity (thm~\\ref{theorem:bk1_dual_horizon_necessity_theorem}), emergent dual-horizon unification (thm~\\ref{theorem:bk1_dual_horizon_unification_principle}), and Wasserstein symbolic thermodynamic geometry (Cor.~\\ref{corollary:bk1_wasserstein_geometric_interpretation}). Emer",
      "label": "theorem:bk1_dual_horizon_unification_principle",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2547,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk1_wasserstein_geometric_interpretation",
    "definition:bk1_bounded_observer",
    "proof:bk1_sketch_observed_consequences",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_unification_principle"
  ],
  "role": "corollary",
  "type": "corollary"
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    "definition:bk1_bounded_observer",
    "proof:bk1_sketch_observed_consequences",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_unification_principle"
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    "definition:bk1_bounded_observer",
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    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_unification_principle"
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  "id": "proof:bk1_event_horizon_identity_field",
  "label": "proof:bk1_event_horizon_identity_field",
  "latex_body": "\\begin{proof}[Identity Field on the Symbolized Causal Patch]\n\\label{proof:bk1_event_horizon_identity_field}\n\\leavevmode\n\nConditional on the cosmological symbolization functor\n\\(\\mathcal{B}_{\\mathrm{cos}}\\), Thm.~\\ref{theorem:bk1_dual_horizon_cosmogenesis}\nand Proof~\\ref{proof:bk1_sketch_observed_consequences} place the observer domain\n\\(\\Omega\\) between a generative horizon and a dissipative horizon and instantiate\nthe PS dynamical apparatus there. Bounded observers in \\(\\Omega\\)\n(Def.~\\ref{definition:bk1_bounded_observer}) therefore experience cognition,\nmemory, language, and thermodynamic complexity as observer-relative symbolic\ndynamics on the image of that causal patch.\n\nThm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem} supplies the necessity of\nboth horizon roles for bounded reflexive emergence, while\nThm.~\\ref{theorem:bk1_dual_horizon_unification_principle} identifies the\nprojected dynamics as horizon-crossing reflexivity. Cor.~\\ref{corollary:bk1_wasserstein_geometric_interpretation}\nthen supplies the thermodynamic geometry: symbolic probability evolves as a\nWasserstein gradient flow of free energy. The identity field is precisely the\nstable observer-relative organization generated by these ingredients on\n\\(\\Omega\\). Thus the corollary follows as a conditional statement about situated\nsymbolic curvature, not as an unconditional reduction of matter to emergence.\n\\end{proof}",
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  "macros_used": [],
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  "name": "Identity Field on the Symbolized Causal Patch",
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    {
      "context": "theorem:bk1_dual_horizon_unification_principle} identifies the projected dynamics as horizon-crossing reflexivity. Cor.~\\ref{corollary:bk1_wasserstein_geometric_interpretation} then supplies the thermodynamic geometry: symbolic probability evolves as a Wasserstein gradient flow of free energy. T",
      "label": "corollary:bk1_wasserstein_geometric_interpretation",
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      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3889,
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    },
    {
      "context": "rizon and a dissipative horizon and instantiate the PS dynamical apparatus there. Bounded observers in \\(\\Omega\\) (Def.~\\ref{definition:bk1_bounded_observer}) therefore experience cognition, memory, language, and thermodynamic complexity as observer-relative symbolic dynamics",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "ogical symbolization functor \\(\\mathcal{B}_{\\mathrm{cos}}\\), Thm.~\\ref{theorem:bk1_dual_horizon_cosmogenesis} and Proof~\\ref{proof:bk1_sketch_observed_consequences} place the observer domain \\(\\Omega\\) between a generative horizon and a dissipative horizon and instantiate the PS dyna",
      "label": "proof:bk1_sketch_observed_consequences",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3977,
      "target_type": "proof"
    },
    {
      "context": "identity_field} \\leavevmode Conditional on the cosmological symbolization functor \\(\\mathcal{B}_{\\mathrm{cos}}\\), Thm.~\\ref{theorem:bk1_dual_horizon_cosmogenesis} and Proof~\\ref{proof:bk1_sketch_observed_consequences} place the observer domain \\(\\Omega\\) between a generative horizo",
      "label": "theorem:bk1_dual_horizon_cosmogenesis",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3954,
      "target_type": "theorem"
    },
    {
      "context": "language, and thermodynamic complexity as observer-relative symbolic dynamics on the image of that causal patch. Thm.~\\ref{theorem:bk1_dual_horizon_necessity_theorem} supplies the necessity of both horizon roles for bounded reflexive emergence, while Thm.~\\ref{theorem:bk1_dual_horizon_",
      "label": "theorem:bk1_dual_horizon_necessity_theorem",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 775,
      "target_type": "theorem"
    },
    {
      "context": "ual_horizon_necessity_theorem} supplies the necessity of both horizon roles for bounded reflexive emergence, while Thm.~\\ref{theorem:bk1_dual_horizon_unification_principle} identifies the projected dynamics as horizon-crossing reflexivity. Cor.~\\ref{corollary:bk1_wasserstein_geometric_interp",
      "label": "theorem:bk1_dual_horizon_unification_principle",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2547,
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    }
  ],
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    "corollary:bk1_wasserstein_geometric_interpretation",
    "definition:bk1_bounded_observer",
    "proof:bk1_sketch_observed_consequences",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_necessity_theorem",
    "theorem:bk1_dual_horizon_unification_principle"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "theorem:bk1_emergence_of_drift_field",
    "theorem:bk1_emergence_of_reflection_operator",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_h_theorem_for_symbolic_evolution",
    "theorem:bk1_manifold_emergence",
    "theorem:bk1_princple_of_least_action",
    "theorem:bk1_sructurual_correspondence",
    "theorem:bk1_symbolic_fluctuation_dissipation_relation",
    "theorem:bk1_variational_principle"
  ],
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    "theorem:bk1_emergence_of_reflection_operator",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk1_h_theorem_for_symbolic_evolution",
    "theorem:bk1_manifold_emergence",
    "theorem:bk1_princple_of_least_action",
    "theorem:bk1_sructurual_correspondence",
    "theorem:bk1_symbolic_fluctuation_dissipation_relation",
    "theorem:bk1_variational_principle"
  ],
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  "id": "sec:bk1_summary_and_implications",
  "label": "sec:bk1_summary_and_implications",
  "latex_body": "",
  "line": 4070,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Summary and Implications",
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    {
      "context": "",
      "label": "theorem:bk1_emergence_of_drift_field",
      "logical_support": false,
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      "target_file": "scholium_symbolicum.tex",
      "target_line": 2856,
      "target_type": "theorem"
    },
    {
      "context": "",
      "label": "theorem:bk1_emergence_of_reflection_operator",
      "logical_support": false,
      "role": "navigation",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2957,
      "target_type": "theorem"
    },
    {
      "context": "",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
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      "target_line": 3098,
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    {
      "context": "",
      "label": "theorem:bk1_h_theorem_for_symbolic_evolution",
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      "target_line": 3193,
      "target_type": "theorem"
    },
    {
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      "label": "theorem:bk1_manifold_emergence",
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      "target_line": 2771,
      "target_type": "theorem"
    },
    {
      "context": "",
      "label": "theorem:bk1_princple_of_least_action",
      "logical_support": false,
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      "target_file": "scholium_symbolicum.tex",
      "target_line": 3784,
      "target_type": "theorem"
    },
    {
      "context": "",
      "label": "theorem:bk1_sructurual_correspondence",
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      "target_line": 3267,
      "target_type": "theorem"
    },
    {
      "context": "",
      "label": "theorem:bk1_symbolic_fluctuation_dissipation_relation",
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      "target_line": 3747,
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      "label": "theorem:bk1_variational_principle",
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      "target_line": 3122,
      "target_type": "theorem"
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  "type": "section"
}