Book II — De Thermodynamica Symbolica — page 1 of 2

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk2_symbolic_temperature",
    "lemma:bk2_finiteness_of_symbolic_entropy",
    "scholium:bk1_epistemic_humility"
  ],
  "depends_on": [
    "scholium:bk1_epistemic_humility"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "",
      "label": "definition:bk2_symbolic_entropy",
      "line_distance": 112,
      "role": "navigation",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk2_symbolic_free_energy",
      "line_distance": 133,
      "role": "navigation",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk2_symbolic_phase_transitio",
      "line_distance": 375,
      "role": "navigation",
      "target_line": 377,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk2_symbolic_temperature",
      "line_distance": 146,
      "role": "navigation",
      "target_line": 148,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "lemma:bk2_finiteness_of_symbolic_entropy",
      "line_distance": 121,
      "role": "navigation",
      "target_line": 123,
      "target_type": "lemma"
    }
  ],
  "forward_refs": [
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk2_symbolic_temperature",
    "lemma:bk2_finiteness_of_symbolic_entropy"
  ],
  "id": "sec:bk2_foundations_symbolic_thermodynamics",
  "label": "sec:bk2_foundations_symbolic_thermodynamics",
  "latex_body": "",
  "line": 2,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Foundations of Symbolic Thermodynamics",
  "ref_roles": [
    {
      "context": "",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": false,
      "role": "forward_navigation",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": false,
      "role": "forward_navigation",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": false,
      "role": "forward_navigation",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk2_symbolic_temperature",
      "logical_support": false,
      "role": "forward_navigation",
      "target_file": "book2.tex",
      "target_line": 148,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "lemma:bk2_finiteness_of_symbolic_entropy",
      "logical_support": false,
      "role": "forward_navigation",
      "target_file": "book2.tex",
      "target_line": 123,
      "target_type": "lemma"
    },
    {
      "context": "",
      "label": "scholium:bk1_epistemic_humility",
      "logical_support": false,
      "role": "navigation",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 648,
      "target_type": "scholium"
    }
  ],
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book2.tex",
  "id": "subsec:bk2_symbolic_states_probability_measures",
  "label": "subsec:bk2_symbolic_states_probability_measures",
  "latex_body": "",
  "line": 20,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic States and Probability Measures",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_hamiltonian",
    "definition:bk3_integration_differentiation_pressures",
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_knowledge_structure",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk5_symbolic_operator_space",
    "lemma:bk2_wellposedness_symb_prob_space",
    "lemma:bk3_symbiotic_stability_conditions",
    "lemma:bk4_fragmentation_cascade",
    "proof:bk2_probability_structure_on_manifold",
    "proof:bk3_coupling_energy_symbolic_hamiltonian",
    "proof:bk3_differentiation_knowledge_structure",
    "proof:bk4_fragmentation_identity_stability",
    "proof:bk4_information_bottleneck_symbolic_filter",
    "proof:bk4_symbolic_curvature_fragmentation",
    "theorem:bk4_drift_reflection_imbalance"
  ],
  "cites": [
    "definition:bk1_symbolic_manifold_existence",
    "proof:bk2_probability_structure_on_manifold"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold_existence"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "el{definition:bk2_symbolic_probability_spa} The triple $(M, \\mathcal{B}, \\mu_g)$ forms a probability space ~(see proof~\\ref{proof:bk2_probability_structure_on_manifold}) where: \\begin{enumerate} \\item $M$ is the symbolic manifold (Def.~\\ref{definition:bk1_symbolic_manifold_existence}",
      "label": "proof:bk2_probability_structure_on_manifold",
      "line_distance": 30,
      "role": "proof_below",
      "target_line": 53,
      "target_type": "proof"
    }
  ],
  "forward_refs": [
    "proof:bk2_probability_structure_on_manifold"
  ],
  "id": "definition:bk2_symbolic_probability_spa",
  "label": "definition:bk2_symbolic_probability_spa",
  "latex_body": "\\begin{definition}[Symbolic Probability Space] \n\\label{definition:bk2_symbolic_probability_spa} \nThe triple $(M, \\mathcal{B}, \\mu_g)$ forms a probability space\n~(see proof~\\ref{proof:bk2_probability_structure_on_manifold})\nwhere:\n\\begin{enumerate}\n    \\item $M$ is the symbolic manifold (Def.~\\ref{definition:bk1_symbolic_manifold_existence});\n    \\item $\\mathcal{B}$ is the Borel $\\sigma$-algebra generated by the topology on $M$;\n    \\item $\\mu_g$ is the normalized Riemannian volume measure induced by the symbolic metric $g$, satisfying $\\mu_g(M) = 1$.\n\\end{enumerate}\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite probability space as density structure; the Borel/volume apparatus is not certified."
    ],
    "record_ids": [
      "MAP-BOOK2-001"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "Book2.gibbs_isDensity"
    ]
  },
  "line": 23,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Probability Space",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ref{proof:bk2_probability_structure_on_manifold}) where: \\begin{enumerate} \\item $M$ is the symbolic manifold (Def.~\\ref{definition:bk1_symbolic_manifold_existence}); \\item $\\mathcal{B}$ is the Borel $\\sigma$-algebra generated by the topology on $M$; \\item $\\mu_g$ is the norm",
      "label": "definition:bk1_symbolic_manifold_existence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2792,
      "target_type": "definition"
    },
    {
      "context": "el{definition:bk2_symbolic_probability_spa} The triple $(M, \\mathcal{B}, \\mu_g)$ forms a probability space ~(see proof~\\ref{proof:bk2_probability_structure_on_manifold}) where: \\begin{enumerate} \\item $M$ is the symbolic manifold (Def.~\\ref{definition:bk1_symbolic_manifold_existence}",
      "label": "proof:bk2_probability_structure_on_manifold",
      "logical_support": false,
      "role": "forward_proof_below",
      "target_file": "book2.tex",
      "target_line": 53,
      "target_type": "proof"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold_existence",
    "proof:bk2_probability_structure_on_manifold"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "axiom:bk2_symbolic_fokker_planck_equation",
    "definition:bk2_symbolic_energy",
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_wasserstein_met",
    "definition:bk3_integration_differentiation_pressures",
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_knowledge_structure",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk2_conservation_of_probability",
    "lemma:bk2_finiteness_of_symbolic_entropy",
    "lemma:bk3_symbiotic_stability_conditions",
    "proof:bk2_bounded_symbolic_entropy",
    "proof:bk2_fokker_planck_probability_conservation",
    "proof:bk2_global_local_temp_relation",
    "proof:bk2_symbolic_drift_equilibrium_yields_gibbs_measure",
    "proof:bk2_symbolic_free_energy_dissipation",
    "proposition:bk2_global_local_temp_relation",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "cites": [
    "definition:bk2_symbolic_probability_spa"
  ],
  "depends_on": [
    "definition:bk2_symbolic_probability_spa"
  ],
  "file": "book2.tex",
  "id": "definition:bk2__symbolic_probability_density",
  "label": "definition:bk2__symbolic_probability_density",
  "latex_body": "\\begin{definition}[Symbolic Probability Density] \n\\label{definition:bk2__symbolic_probability_density} \nA symbolic probability density at symbolic time $s$ is a measurable function $\\rho(\\cdot, s): M \\rightarrow \\mathbb{R}_{\\geq 0}$ satisfying (see def~\\ref{definition:bk2_symbolic_probability_spa}):\n\\begin{enumerate}\n    \\item Normalization: $\\int_M \\rho(x, s) \\, d\\mu_g(x) = 1$;\n    \\item Absolute continuity: $\\rho(\\cdot, s) \\ll \\mu_g$;\n    \\item Regularity: We restrict to the space\n    \\[\n    \\mathcal{P}(M) = \\left\\{ \\rho \\in C^\\infty(M) \\mid \\rho > 0,\\; \\int_M \\rho \\, d\\mu_g = 1 \\right\\}.\n    \\]\n\\end{enumerate}\n\\end{definition}",
  "line": 35,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Probability Density",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "y at symbolic time $s$ is a measurable function $\\rho(\\cdot, s): M \\rightarrow \\mathbb{R}_{\\geq 0}$ satisfying (see def~\\ref{definition:bk2_symbolic_probability_spa}): \\begin{enumerate} \\item Normalization: $\\int_M \\rho(x, s) \\, d\\mu_g(x) = 1$; \\item Absolute continuity: $\\rho",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_probability_spa"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk2_symbolic_probability_spa"
  ],
  "depends_on": [
    "axiom:bk1_topological_regularity",
    "definition:bk1_bounded_observer",
    "definition:bk2_symbolic_probability_spa",
    "lemma:bk1_local_stability_analysis",
    "theorem:bk1_manifold_emergence"
  ],
  "file": "book2.tex",
  "id": "lemma:bk2_wellposedness_symb_prob_space",
  "label": "lemma:bk2_wellposedness_symb_prob_space",
  "latex_body": "\\begin{lemma}[Well-posedness of Symbolic Probability Space] \n\\label{lemma:bk2_wellposedness_symb_prob_space} \nThe symbolic probability space $(M, \\mathcal{B}, \\mu_g)$ is well-defined for any bounded symbolic observer (see def~\\ref{definition:bk1_bounded_observer}) embedded within the system (see def~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite kernel: the Gibbs state is a genuine density (positive, sums to one)."
    ],
    "record_ids": [
      "MAP-BOOK2-002"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2.gibbs_isDensity"
    ]
  },
  "line": 48,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Well-posedness of Symbolic Probability Space",
  "proof_labels": [
    "proof:bk2_probability_structure_on_manifold"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "e} The symbolic probability space $(M, \\mathcal{B}, \\mu_g)$ is well-defined for any bounded symbolic observer (see def~\\ref{definition:bk1_bounded_observer}) embedded within the system (see def~\\ref{definition:bk2_symbolic_probability_spa}). \\end{lemma}",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "d for any bounded symbolic observer (see def~\\ref{definition:bk1_bounded_observer}) embedded within the system (see def~\\ref{definition:bk2_symbolic_probability_spa}). \\end{lemma}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk2_symbolic_probability_spa"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book2",
  "cited_by": [
    "definition:bk2_symbolic_probability_spa"
  ],
  "cites": [
    "axiom:bk1_topological_regularity",
    "definition:bk2_symbolic_probability_spa",
    "lemma:bk1_local_stability_analysis",
    "theorem:bk1_manifold_emergence"
  ],
  "depends_on": [
    "axiom:bk1_topological_regularity",
    "definition:bk2_symbolic_probability_spa",
    "lemma:bk1_local_stability_analysis",
    "theorem:bk1_manifold_emergence"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_probability_structure_on_manifold",
  "label": "proof:bk2_probability_structure_on_manifold",
  "latex_body": "\\begin{proof}[Symbolic Probability Structure on Emergent Manifold]\n\\label{proof:bk2_probability_structure_on_manifold}\n\\leavevmode\n\nBy Axiom~\\ref{axiom:bk1_topological_regularity}, the manifold $M$ is Hausdorff, second-countable, and paracompact, so the Borel $\\sigma$-algebra $\\mathcal{B}$ is well-defined.\nThe symbolic metric $g$ from Lemma~\\ref{lemma:bk1_local_stability_analysis} induces a Riemannian volume form $\\omega_g$ on $M$.\nSince $M$ emerges through the colimit process (Theorem~\\ref{theorem:bk1_manifold_emergence}) as connected and paracompact, it has finite total volume $V = \\int_M \\omega_g < \\infty$.\nNormalize to $\\mu_g = \\omega_g/V$ so that $\\mu_g(M)=1$. Hence $(M, \\mathcal{B}, \\mu_g)$ satisfies the probability-space axioms (see def~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{proof}",
  "line": 53,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Probability Structure on Emergent Manifold",
  "proves": "lemma:bk2_wellposedness_symb_prob_space",
  "ref_roles": [
    {
      "context": "c Probability Structure on Emergent Manifold] \\label{proof:bk2_probability_structure_on_manifold} \\leavevmode By Axiom~\\ref{axiom:bk1_topological_regularity}, the manifold $M$ is Hausdorff, second-countable, and paracompact, so the Borel $\\sigma$-algebra $\\mathcal{B}$ is well-",
      "label": "axiom:bk1_topological_regularity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2758,
      "target_type": "axiom"
    },
    {
      "context": "u_g = \\omega_g/V$ so that $\\mu_g(M)=1$. Hence $(M, \\mathcal{B}, \\mu_g)$ satisfies the probability-space axioms (see def~\\ref{definition:bk2_symbolic_probability_spa}). \\end{proof}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "table, and paracompact, so the Borel $\\sigma$-algebra $\\mathcal{B}$ is well-defined. The symbolic metric $g$ from Lemma~\\ref{lemma:bk1_local_stability_analysis} induces a Riemannian volume form $\\omega_g$ on $M$. Since $M$ emerges through the colimit process (Theorem~\\ref{theorem",
      "label": "lemma:bk1_local_stability_analysis",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3707,
      "target_type": "lemma"
    },
    {
      "context": "ty_analysis} induces a Riemannian volume form $\\omega_g$ on $M$. Since $M$ emerges through the colimit process (Theorem~\\ref{theorem:bk1_manifold_emergence}) as connected and paracompact, it has finite total volume $V = \\int_M \\omega_g < \\infty$. Normalize to $\\mu_g = \\omega_",
      "label": "theorem:bk1_manifold_emergence",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2771,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "axiom:bk1_topological_regularity",
    "definition:bk2_symbolic_probability_spa",
    "lemma:bk1_local_stability_analysis",
    "theorem:bk1_manifold_emergence"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "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": "book2.tex",
  "id": "subsec:bk2_core_thermodynamic_quantities",
  "label": "subsec:bk2_core_thermodynamic_quantities",
  "latex_body": "",
  "line": 63,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Core Thermodynamic Quantities",
  "ref_roles": [
    {
      "context": "",
      "label": "definition:bk1_drift_field",
      "logical_support": false,
      "role": "navigation",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk1_reflection_operator",
      "logical_support": false,
      "role": "navigation",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": false,
      "role": "navigation",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [
    "assumption:appB_srv_dissipativity",
    "axiom:bk2_gradient_structure_drift",
    "axiom:bk5_adaptation",
    "axiom:bk7_convergence_potential",
    "corollary:bk2_interpretative_framework",
    "definition:bk2_symbolic_energy",
    "definition:bk2_symbolic_partition_funct",
    "definition:bk3_membrane_thermodynamics",
    "lemma:appB_energy_contraction",
    "lemma:bk2_wellposedness_symb_hamiltonian",
    "proof:bk1_sketch_observed_consequences",
    "proof:bk2_coherence_of_symbolic_therm",
    "proof:bk2_interpretative_framework",
    "proof:bk3_coupling_energy_symbolic_hamiltonian",
    "remark:bk2_symbolic_hamiltonian",
    "sec:bk7_definitionnes_septimae_structures_of_convergence"
  ],
  "cites": [
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_probability_spa"
  ],
  "depends_on": [
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_probability_spa"
  ],
  "file": "book2.tex",
  "id": "definition:bk2_symbolic_hamiltonian",
  "label": "definition:bk2_symbolic_hamiltonian",
  "latex_body": "\\begin{definition}[Symbolic Hamiltonian] \n\\label{definition:bk2_symbolic_hamiltonian} \nThe symbolic Hamiltonian $H: M \\rightarrow \\mathbb{R}$ is defined as:\n\\[\nH(x) = \\frac{\\kappa}{\\|D(x)\\|_g + \\epsilon} + \\lambda \\cdot \\text{tr}(\\mathcal{L}_x)\n\\]\nwhere (see def~\\ref{definition:bk2_symbolic_probability_spa}; see also def~\\ref{definition:bk1_reflection_operator}):\n\\begin{enumerate}\n    \\item $\\kappa, \\lambda > 0$ are scaling constants;\n    \\item $\\|D(x)\\|_g$ denotes the norm of the drift vector at point $x$ with respect to the metric $g$;\n    \\item $\\epsilon > 0$ is a regularization constant ensuring well-definedness;\n    \\item $\\mathcal{L}_x = P_{R(x) \\leftarrow x} \\circ dR_x$ is the linearization of the reflection operator at $x$, where $dR_x$ is the differential of $R$ at $x$ and $P_{R(x) \\leftarrow x}$ denotes parallel transport from $x$ to $R(x)$ along the unique minimizing geodesic.\n\\end{enumerate}\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": false,
    "notes": [
      "Only the regularized denominator's positivity is modeled; the drift-norm and parallel-transported reflection-linearization trace terms are not."
    ],
    "record_ids": [
      "MAP-BOOK2-017"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book7B.hamiltonian_denom_pos"
    ]
  },
  "line": 67,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Hamiltonian",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "+ \\lambda \\cdot \\text{tr}(\\mathcal{L}_x) \\] where (see def~\\ref{definition:bk2_symbolic_probability_spa}; see also def~\\ref{definition:bk1_reflection_operator}): \\begin{enumerate} \\item $\\kappa, \\lambda > 0$ are scaling constants; \\item $\\|D(x)\\|_g$ denotes the norm of t",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "s defined as: \\[ H(x) = \\frac{\\kappa}{\\|D(x)\\|_g + \\epsilon} + \\lambda \\cdot \\text{tr}(\\mathcal{L}_x) \\] where (see def~\\ref{definition:bk2_symbolic_probability_spa}; see also def~\\ref{definition:bk1_reflection_operator}): \\begin{enumerate} \\item $\\kappa, \\lambda > 0$ are scaling",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_probability_spa"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_hamiltonian",
    "subsec:appB_srv_trace8",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "definition:bk2_symbolic_hamiltonian",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "zation. \\item \\textbf{Equilibrium compatibility:} The Gibbs measure $\\rho_{\\mathrm{eq}} \\propto e^{-\\beta H}$ (Thm.~\\ref{theorem:bk2_equilibrium_distribution}) must be the unique equilibrium of the Fokker--Planck dynamics (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank",
      "label": "theorem:bk2_equilibrium_distribution",
      "line_distance": 134,
      "role": "teaser",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "forward_refs": [
    "theorem:bk2_equilibrium_distribution"
  ],
  "id": "remark:bk2_symbolic_hamiltonian",
  "label": "remark:bk2_symbolic_hamiltonian",
  "latex_body": "\\begin{remark}[Motivating the Canonical Symbolic Hamiltonian]\n\\label{remark:bk2_symbolic_hamiltonian}\nThe form of $H$ in Def.~\\ref{definition:bk2_symbolic_hamiltonian} is fixed by three requirements:\n\\begin{enumerate}\n    \\item \\textbf{Bounded below, smooth:} $H \\in C^\\infty(M)$ and $H > 0$ everywhere (ensured by the $\\epsilon$-regularization in the denominator and the positivity of the trace term).\n    \\item \\textbf{Drift--reflection balance:} $H$ must encode the tension between drift magnitude $\\|D(x)\\|_g$ and reflective stabilization $\\mathrm{tr}(\\mathcal{L}_x)$. High drift decreases $H(x)$, signaling instability; strong reflection increases it, signaling stabilization.\n    \\item \\textbf{Equilibrium compatibility:} The Gibbs measure $\\rho_{\\mathrm{eq}} \\propto e^{-\\beta H}$ (Thm.~\\ref{theorem:bk2_equilibrium_distribution}) must be the unique equilibrium of the Fokker--Planck dynamics (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}).\n\\end{enumerate}\nGiven these constraints, $H$ is canonical up to the gauge choices $\\kappa, \\lambda > 0$ (which set the relative weighting of drift and reflection) and $\\epsilon > 0$ (which regularizes the drift singularity). The structural correspondence between this Hamiltonian and the Operatio's pre-parametric skeleton is demonstrated in SRV Trace~8 (\\S\\ref{subsec:appB_srv_trace8}).\n\\end{remark}",
  "line": 82,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Motivating the Canonical Symbolic Hamiltonian",
  "ref_roles": [
    {
      "context": "{remark}[Motivating the Canonical Symbolic Hamiltonian] \\label{remark:bk2_symbolic_hamiltonian} The form of $H$ in Def.~\\ref{definition:bk2_symbolic_hamiltonian} is fixed by three requirements: \\begin{enumerate} \\item \\textbf{Bounded below, smooth:} $H \\in C^\\infty(M)$ and $H",
      "label": "definition:bk2_symbolic_hamiltonian",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 67,
      "target_type": "definition"
    },
    {
      "context": "al correspondence between this Hamiltonian and the Operatio's pre-parametric skeleton is demonstrated in SRV Trace~8 (\\S\\ref{subsec:appB_srv_trace8}). \\end{remark}",
      "label": "subsec:appB_srv_trace8",
      "logical_support": false,
      "role": "navigation",
      "target_file": "trace8.tex",
      "target_line": 1,
      "target_type": "section"
    },
    {
      "context": "$ (Thm.~\\ref{theorem:bk2_equilibrium_distribution}) must be the unique equilibrium of the Fokker--Planck dynamics (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). \\end{enumerate} Given these constraints, $H$ is canonical up to the gauge choices $\\kappa, \\lambda > 0$ (which set th",
      "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": "zation. \\item \\textbf{Equilibrium compatibility:} The Gibbs measure $\\rho_{\\mathrm{eq}} \\propto e^{-\\beta H}$ (Thm.~\\ref{theorem:bk2_equilibrium_distribution}) must be the unique equilibrium of the Fokker--Planck dynamics (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_hamiltonian",
    "subsec:appB_srv_trace8",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_hamiltonian",
    "proof:bk2_smoothness_symbolic_hamiltonian"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_hamiltonian",
    "theorem:bk1_emergence_of_drift_field"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "tonian $H$ (defined in def~\\ref{definition:bk2_symbolic_hamiltonian}) is well-defined and smooth on $M$ (see also proof~\\ref{proof:bk2_smoothness_symbolic_hamiltonian}). \\end{lemma}",
      "label": "proof:bk2_smoothness_symbolic_hamiltonian",
      "line_distance": 5,
      "role": "proof_below",
      "target_line": 98,
      "target_type": "proof"
    }
  ],
  "forward_refs": [
    "proof:bk2_smoothness_symbolic_hamiltonian"
  ],
  "id": "lemma:bk2_wellposedness_symb_hamiltonian",
  "label": "lemma:bk2_wellposedness_symb_hamiltonian",
  "latex_body": "\\begin{lemma}[Well-posedness of Symbolic Hamiltonian] \n\\label{lemma:bk2_wellposedness_symb_hamiltonian} \nThe symbolic Hamiltonian $H$ (defined in def~\\ref{definition:bk2_symbolic_hamiltonian}) is well-defined and smooth on $M$ (see also proof~\\ref{proof:bk2_smoothness_symbolic_hamiltonian}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Only the real-analytic non-vanishing-denominator core of well-posedness is modeled; smoothness on the manifold M is not."
    ],
    "record_ids": [
      "MAP-BOOK2-018"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book7B.hamiltonian_denom_pos"
    ]
  },
  "line": 93,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Well-posedness of Symbolic Hamiltonian",
  "proof_labels": [
    "proof:bk2_smoothness_symbolic_hamiltonian"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "f Symbolic Hamiltonian] \\label{lemma:bk2_wellposedness_symb_hamiltonian} The symbolic Hamiltonian $H$ (defined in def~\\ref{definition:bk2_symbolic_hamiltonian}) is well-defined and smooth on $M$ (see also proof~\\ref{proof:bk2_smoothness_symbolic_hamiltonian}). \\end{lemma}",
      "label": "definition:bk2_symbolic_hamiltonian",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 67,
      "target_type": "definition"
    },
    {
      "context": "tonian $H$ (defined in def~\\ref{definition:bk2_symbolic_hamiltonian}) is well-defined and smooth on $M$ (see also proof~\\ref{proof:bk2_smoothness_symbolic_hamiltonian}). \\end{lemma}",
      "label": "proof:bk2_smoothness_symbolic_hamiltonian",
      "logical_support": false,
      "role": "forward_proof_below",
      "target_file": "book2.tex",
      "target_line": 98,
      "target_type": "proof"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_hamiltonian",
    "proof:bk2_smoothness_symbolic_hamiltonian"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book2",
  "cited_by": [
    "lemma:bk2_wellposedness_symb_hamiltonian"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_emergence_of_drift_field"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_emergence_of_drift_field"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_smoothness_symbolic_hamiltonian",
  "label": "proof:bk2_smoothness_symbolic_hamiltonian",
  "latex_body": "\\begin{proof}[Smoothness of Symbolic Hamiltonian]\n\\label{proof:bk2_smoothness_symbolic_hamiltonian}\n\\leavevmode\n\nThe drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) is smooth on $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) by Theorem~\\ref{theorem:bk1_emergence_of_drift_field}, so $\\|D(x)\\|_g$ is smooth and positive. The regularization term $\\epsilon > 0$ ensures the denominator never vanishes. The reflection operator $R$ is smooth (Def.~\\ref{definition:bk1_reflection_operator}), so its differential $dR_x$ exists and varies smoothly with $x$. For each $x \\in M$, the geodesic distance $d_g(x, R(x))$ is finite due to the completeness of $(M, g)$, and the parallel transport $P_{R(x) \\leftarrow x}$ is well-defined along the unique minimizing geodesic. The parallel transport operator varies smoothly with its endpoints in a neighborhood where the exponential map is a diffeomorphism. The trace operation preserves smoothness. Therefore, $H \\in C^\\infty(M)$.\n\\end{proof}",
  "line": 98,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Smoothness of Symbolic Hamiltonian",
  "proves": "lemma:bk2_wellposedness_symb_hamiltonian",
  "ref_roles": [
    {
      "context": "hness of Symbolic Hamiltonian] \\label{proof:bk2_smoothness_symbolic_hamiltonian} \\leavevmode The drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) is smooth on $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) by Theorem~\\ref{theorem:bk1_emergence_of_drift_field},",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "regularization term $\\epsilon > 0$ ensures the denominator never vanishes. The reflection operator $R$ is smooth (Def.~\\ref{definition:bk1_reflection_operator}), so its differential $dR_x$ exists and varies smoothly with $x$. For each $x \\in M$, the geodesic distance $d_g(x, R(x",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "s_symbolic_hamiltonian} \\leavevmode The drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) is smooth on $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) by Theorem~\\ref{theorem:bk1_emergence_of_drift_field}, so $\\|D(x)\\|_g$ is smooth and positive. The regularization term",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "d $D$ (Def.~\\ref{definition:bk1_drift_field}) is smooth on $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) by Theorem~\\ref{theorem:bk1_emergence_of_drift_field}, so $\\|D(x)\\|_g$ is smooth and positive. The regularization term $\\epsilon > 0$ ensures the denominator never vanishes.",
      "label": "theorem:bk1_emergence_of_drift_field",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2856,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_emergence_of_drift_field"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [
    "axiom:bk5_energy_conservation",
    "axiom:bk5_metabolic_persistence",
    "axiom:bk7_convergence_potential",
    "definition:bk2_symbolic_temperature",
    "definition:bk3_membrane_thermodynamics",
    "definition:bk5_symbolic_energy",
    "demonstratio:bk7_free_energy_balance_equilibrium",
    "proof:bk5_coherence_through_dynamic_equilibriium",
    "proof:bk7_reflective_convergence_to_stable_identity",
    "subsec:bk5_symbolic_free_energy_and_stability",
    "subsubsec:bk7_formal_definition_of_symbolic_confidence_ch_i"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_hamiltonian"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_hamiltonian"
  ],
  "file": "book2.tex",
  "id": "definition:bk2_symbolic_energy",
  "label": "definition:bk2_symbolic_energy",
  "latex_body": "\\begin{definition}[Symbolic Energy] \n\\label{definition:bk2_symbolic_energy} \nThe symbolic energy at symbolic time $s$ is defined as:\n\\[\nE_s = \\int_M \\rho(x,s) H(x) \\, d\\mu_g(x)\n\\]\nrepresenting the expectation value of the Hamiltonian (see def~\\ref{definition:bk2_symbolic_hamiltonian}) with respect to the probability density $\\rho(\\cdot,s)$ (see def~\\ref{definition:bk2__symbolic_probability_density}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The bridge identity: Gibbs mean energy = -d/dbeta log Z."
    ],
    "record_ids": [
      "MAP-BOOK2-014"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book2.energy_eq_neg_deriv_log_partition"
    ]
  },
  "line": 105,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Energy",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "an (see def~\\ref{definition:bk2_symbolic_hamiltonian}) with respect to the probability density $\\rho(\\cdot,s)$ (see def~\\ref{definition:bk2__symbolic_probability_density}). \\end{definition}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "fined as: \\[ E_s = \\int_M \\rho(x,s) H(x) \\, d\\mu_g(x) \\] representing the expectation value of the Hamiltonian (see def~\\ref{definition:bk2_symbolic_hamiltonian}) with respect to the probability density $\\rho(\\cdot,s)$ (see def~\\ref{definition:bk2__symbolic_probability_density}).",
      "label": "definition:bk2_symbolic_hamiltonian",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 67,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_hamiltonian"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "axiom:bk5_energy_conservation",
    "axiom:bk5_metabolic_persistence",
    "axiom:bk7_convergence_potential",
    "axiom:bk9_emergent_autonomy",
    "corollary:bk2_interpretative_framework",
    "corollary:bk8_translation_limit",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_temperature",
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_reflexive_encoding",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_symbolic_freedom_measure",
    "definition:bk5_entropy_inflection_point",
    "definition:bk8_entropy_shift",
    "definition:bk8_metabolic_programming_cycle",
    "definition:bk8_refinement_objective",
    "definition:bk8_symbolic_interface",
    "demonstratio:bk5_negative_reflection_instability",
    "demonstratio:bk7_free_energy_balance_equilibrium",
    "lemma:bk2_finiteness_of_symbolic_entropy",
    "lemma:bk2_thermodynamic_consistency_hypothesis_manifolds",
    "proof:bk2_bounded_symbolic_entropy",
    "proof:bk2_interpretative_framework",
    "proof:bk4_emergence_conditions",
    "proof:bk5_energy_conservation_under_reflective_coupling",
    "proof:bk5_entropy_increase_from_drift",
    "proof:bk5_symbolic_free_energy_stability_condition",
    "proof:bk7_reflective_convergence_to_stable_identity",
    "proof:bk8_translation_limit",
    "remark:bk3_toward_symbolic_evolution",
    "scholium:bk2_on_hypotheses_as_thermodyn",
    "scholium:bk4_fuzzy_exponential_growth",
    "scholium:bk5_metabolic_cost_of_cognition",
    "sec:bk2_foundations_symbolic_thermodynamics",
    "sec:bk5_funadmenta_symbolicae_vitae",
    "sec:bk7_pisu_universal_symbolic_uncertainty",
    "subsec:appD_core_resonance",
    "subsec:bk3_preamble_to_symbiosis",
    "subsec:bk5_symbolic_free_energy_and_stability",
    "subsec:bk6_structural_requirements_for_regulation",
    "subsec:bk7_pisu_formula",
    "subsec:bk7_pisu_implications",
    "subsubsec:bk7_formal_definition_of_symbolic_loss_loss",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk7_reflective_convergence_to_stable_identity",
    "theorem:bk8_holographic_surface_entropy"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "lemma:bk2_finiteness_of_symbolic_entropy"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "entropy to the continuous manifold setting (see def~\\ref{definition:bk2__symbolic_probability_density}; see also lemma~\\ref{lemma:bk2_finiteness_of_symbolic_entropy}). \\end{definition}",
      "label": "lemma:bk2_finiteness_of_symbolic_entropy",
      "line_distance": 9,
      "role": "teaser",
      "target_line": 123,
      "target_type": "lemma"
    }
  ],
  "forward_refs": [
    "lemma:bk2_finiteness_of_symbolic_entropy"
  ],
  "id": "definition:bk2_symbolic_entropy",
  "label": "definition:bk2_symbolic_entropy",
  "latex_body": "\\begin{definition}[Symbolic Entropy] \n\\label{definition:bk2_symbolic_entropy} \nThe symbolic entropy at symbolic time $s$ is defined as:\n\\[\nS_s = -\\int_M \\rho(x,s) \\log\\rho(x,s) \\, d\\mu_g(x)\n\\]\nThis generalizes the Shannon entropy to the continuous manifold setting (see def~\\ref{definition:bk2__symbolic_probability_density}; see also lemma~\\ref{lemma:bk2_finiteness_of_symbolic_entropy}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite entropy with the 0 log 0 = 0 convention."
    ],
    "record_ids": [
      "MAP-BOOK2-003"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "Book2.entropy_nonneg"
    ]
  },
  "line": 114,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Entropy",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "rho(x,s) \\log\\rho(x,s) \\, d\\mu_g(x) \\] This generalizes the Shannon entropy to the continuous manifold setting (see def~\\ref{definition:bk2__symbolic_probability_density}; see also lemma~\\ref{lemma:bk2_finiteness_of_symbolic_entropy}). \\end{definition}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "entropy to the continuous manifold setting (see def~\\ref{definition:bk2__symbolic_probability_density}; see also lemma~\\ref{lemma:bk2_finiteness_of_symbolic_entropy}). \\end{definition}",
      "label": "lemma:bk2_finiteness_of_symbolic_entropy",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "book2.tex",
      "target_line": 123,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "lemma:bk2_finiteness_of_symbolic_entropy"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "definition:bk2_symbolic_entropy",
    "sec:bk2_foundations_symbolic_thermodynamics"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy"
  ],
  "file": "book2.tex",
  "id": "lemma:bk2_finiteness_of_symbolic_entropy",
  "label": "lemma:bk2_finiteness_of_symbolic_entropy",
  "latex_body": "\\begin{lemma}[Finiteness of Symbolic Entropy] \n\\label{lemma:bk2_finiteness_of_symbolic_entropy} \nFor any density $\\rho \\in \\mathcal{P}(M)$, the symbolic entropy $S_s$ (see def~\\ref{definition:bk2_symbolic_entropy}) is finite (see def~\\ref{definition:bk2__symbolic_probability_density}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Two-sided quantitative bounds 0 <= S <= log n."
    ],
    "record_ids": [
      "MAP-BOOK2-004"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book2.entropy_le_log_card",
      "Book2.entropy_nonneg"
    ]
  },
  "line": 123,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Finiteness of Symbolic Entropy",
  "proof_labels": [
    "proof:bk2_bounded_symbolic_entropy"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "\\rho \\in \\mathcal{P}(M)$, the symbolic entropy $S_s$ (see def~\\ref{definition:bk2_symbolic_entropy}) is finite (see def~\\ref{definition:bk2__symbolic_probability_density}). \\end{lemma}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "mma:bk2_finiteness_of_symbolic_entropy} For any density $\\rho \\in \\mathcal{P}(M)$, the symbolic entropy $S_s$ (see def~\\ref{definition:bk2_symbolic_entropy}) is finite (see def~\\ref{definition:bk2__symbolic_probability_density}). \\end{lemma}",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_bounded_symbolic_entropy",
  "label": "proof:bk2_bounded_symbolic_entropy",
  "latex_body": "\\begin{proof}[Boundedness of Symbolic Entropy on Compact Manifold]\n\\label{proof:bk2_bounded_symbolic_entropy}\n\\leavevmode\n\nSince $M$ is compact and $\\rho \\in \\mathcal{P}(M)$ is smooth and strictly positive, there exist constants $0 < m \\leq \\rho(x) \\leq M < \\infty$ for all $x \\in M$. Therefore, $|\\rho(x) \\log\\rho(x)| \\leq M|\\log m|$ is bounded, and the integral $S_s = -\\int_M \\rho \\log\\rho \\, d\\mu_g$ (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2__symbolic_probability_density}) converges to a finite value.\n\\end{proof}",
  "line": 128,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Boundedness of Symbolic Entropy on Compact Manifold",
  "proves": "lemma:bk2_finiteness_of_symbolic_entropy",
  "ref_roles": [
    {
      "context": ", and the integral $S_s = -\\int_M \\rho \\log\\rho \\, d\\mu_g$ (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2__symbolic_probability_density}) converges to a finite value. \\end{proof}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "e, $|\\rho(x) \\log\\rho(x)| \\leq M|\\log m|$ is bounded, and the integral $S_s = -\\int_M \\rho \\log\\rho \\, d\\mu_g$ (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2__symbolic_probability_density}) converges to a finite value. \\end{proof}",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [
    "axiom:appC_axiom_of_memory",
    "axiom:bk5_adaptation",
    "axiom:bk5_positive_free_energy",
    "axiom:bk8_coherence_horizon",
    "axiom:bk8_mutation_phase_shift",
    "axiom:bk9_bounded_liberation_principle",
    "axiom:bk9_emergent_autonomy",
    "corollary:bk2_interpretative_framework",
    "corollary:bk9_freedomentropy_complementarity",
    "definition:appC_bounded_reflexive_emergence",
    "definition:bk1_bounded_reflexive_emergence",
    "definition:bk3_autophagic_drift",
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk4_collapse_of_symbolic_ide",
    "definition:bk4_individuation_path",
    "definition:bk4_meta_stable_symbolic_str",
    "definition:bk4_symbolic_spinor_bundle",
    "definition:bk4_symbolic_transition_rate",
    "definition:bk4_symbolic_work_functional",
    "definition:bk4_test_time_coherent_sampling",
    "definition:bk5_complexity_stability_maintenance",
    "definition:bk5_metabolic_capacity_mc_",
    "definition:bk5_mutually_assured_progress",
    "definition:bk5_process_free_energy",
    "definition:bk5_symbolic_fitness",
    "definition:bk5_symbolic_strategy",
    "definition:bk5_viability_domain",
    "definition:bk6_mutation_threshold",
    "definition:bk6_symbolic_free_energy_functional",
    "definition:bk6_symbolic_regulatory_cycle",
    "definition:bk7_convergent_symbolic_identity",
    "definition:bk7_symbolic_free_energy",
    "definition:bk8_metabolic_programming_cycle",
    "definition:bk8_projective_compression_operator",
    "definition:bk8_structural_regulators",
    "definition:bk8_symbolic_adjacency",
    "definition:bk8_symbolic_stress_tensor",
    "definition:bk8_translation_loss",
    "definition:bk9_frame_selection_reflection",
    "definition:bk9_symbolic_thermodynamic_stress",
    "demonstratio:bk4_symbolic_thermodynamics",
    "demonstratio:bk8_symbolic_unkotting",
    "lemma:bk2_thermodynamic_consistency_hypothesis_manifolds",
    "lemma:bk7_coarsegrained_convexity",
    "proof:bk2_interpretative_framework",
    "proof:bk2_sketch_wasserstein_gradient_flow",
    "proof:bk2_symbolic_free_energy_dissipation",
    "proof:bk2_symbolic_h_theorem",
    "proof:bk4_lipschitz_continuity_symbolic_drift",
    "proof:bk4_sketch_observer_resolution_floor",
    "proof:bk5_map_invasion_dynamics",
    "proof:bk5_map_perturbation_robustness",
    "proof:bk5_map_resistance_to_drift",
    "proof:bk5_metabolic_capacity_non_decreasing",
    "proof:bk5_operator_convergence",
    "proof:bk5_symbolic_free_energy_stability_condition",
    "proof:bk5_symbolic_temperature_threshold",
    "proof:bk6_drift_reflection_commutation_equilibrium",
    "proof:bk6_stable_reflective_submanifold",
    "proof:bk6_symbolic_mutation_threshold",
    "proof:bk9_freedomentropy_complementarity",
    "proof:bk9_pathologies_of_coherence",
    "proof:bk9_symbolic_masking_and_unmasking",
    "proposition:bk5_golden_ratio_thermodynamic_optimum",
    "proposition:bk5_symbolic_ess_via_map_observability_variant",
    "proposition:bk5_symbolic_life_criterion",
    "remark:bk3_toward_symbolic_evolution",
    "remark:bk4_ttpr_entropy",
    "remark:bk8_inference_principle_over_confidence_loss_tradeoff",
    "remark:bk9_gauge_theoretic_perspective",
    "scholium:bk2_on_hypotheses_as_thermodyn",
    "scholium:bk3_hypotheses_as_cognitive_membranes",
    "scholium:bk4_fuzzy_logarithmic_resolution",
    "scholium:bk4_symbolic_entanglement",
    "scholium:bk4_symbolic_interference",
    "scholium:bk4_ttcs_stochastic_operator",
    "scholium:bk5_life_on_edge_of_chaos",
    "scholium:bk5_map_as_fundamental_organizational_principle",
    "scholium:bk5_metabolic_cost_of_cognition",
    "scholium:bk6_hypotheses_as_regulatory_mutation_manifolds",
    "scholium:bk7_reflective_selection_as_principled_convergence",
    "scholium:bk8_telephone_game",
    "sec:bk2_foundations_symbolic_thermodynamics",
    "sec:bk5_funadmenta_symbolicae_vitae",
    "sec:bk7_axiomata_septima_the_laws_of_convergence",
    "sec:bk7_pisu_universal_symbolic_uncertainty",
    "sec:bk8_axiomata_octava",
    "subsec:appC_born_interpretation_ps",
    "subsec:appD_core_resonance",
    "subsec:bk2_symbolic_phase_transitions",
    "subsec:bk3_preamble_to_symbiosis",
    "subsec:bk4_ttie_operator_algebra",
    "subsec:bk6_structural_requirements_for_regulation",
    "subsec:bk7_formalizing_reflective_selection_confidence_loss_and_symbolic_",
    "subsec:bk7_pisu_formula",
    "subsec:bk7_pisu_implications",
    "subsec:bk9_limits_of_repair",
    "theorem:appC_fundamental_irreversibility_final",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk2_wasserstein_gradient_flow",
    "theorem:bk4_symbolic_identity_continuit",
    "theorem:bk5_operator_convergence",
    "theorem:bk5_reflective_equilibrium_conservation",
    "theorem:bk7_reflective_convergence_to_stable_identity",
    "theorem:bk8_biological_phase_transition",
    "theorem:bk8_sr_convergence",
    "theorem:bk8_threshold_of_metabolic_autonomy",
    "theorem:bk9_good_as_lyapunov_basin"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy",
    "theorem:bk2_h_theorem_for_symbolic_evol"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "rho(x) \\left(H(x) + \\beta^{-1}\\log\\rho(x)\\right) d\\mu_g(x) \\] This quantity decreases under symbolic evolution (see thm~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}). \\end{definition}",
      "label": "theorem:bk2_h_theorem_for_symbolic_evol",
      "line_distance": 120,
      "role": "teaser",
      "target_line": 255,
      "target_type": "theorem"
    }
  ],
  "forward_refs": [
    "theorem:bk2_h_theorem_for_symbolic_evol"
  ],
  "id": "definition:bk2_symbolic_free_energy",
  "label": "definition:bk2_symbolic_free_energy",
  "latex_body": "\\begin{definition}[Symbolic Free Energy] \n\\label{definition:bk2_symbolic_free_energy} \nThe symbolic free energy functional $F_\\beta: \\mathcal{P}(M) \\rightarrow \\mathbb{R}$ is defined for inverse temperature parameter $\\beta > 0$ as:\n\\[\nF_\\beta[\\rho] = \\int_M \\rho(x) H(x) \\, d\\mu_g(x) - \\beta^{-1} S[\\rho]\n\\]\nwhere $S[\\rho] = -\\int_M \\rho(x) \\log\\rho(x) \\, d\\mu_g(x)$ is the entropy functional (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2__symbolic_probability_density}). This can be rewritten as:\n\\[\nF_\\beta[\\rho] = \\int_M \\rho(x) \\left(H(x) + \\beta^{-1}\\log\\rho(x)\\right) d\\mu_g(x)\n\\]\nThis quantity decreases under symbolic evolution (see thm~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite free-energy functional; its decrease along the flow is a named open item."
    ],
    "record_ids": [
      "MAP-BOOK2-005"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "Book2.freeEnergy_gibbs"
    ]
  },
  "line": 135,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Free Energy",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "rho(x) \\log\\rho(x) \\, d\\mu_g(x)$ is the entropy functional (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2__symbolic_probability_density}). This can be rewritten as: \\[ F_\\beta[\\rho] = \\int_M \\rho(x) \\left(H(x) + \\beta^{-1}\\log\\rho(x)\\right) d\\mu_g(x) \\] Th",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": ") - \\beta^{-1} S[\\rho] \\] where $S[\\rho] = -\\int_M \\rho(x) \\log\\rho(x) \\, d\\mu_g(x)$ is the entropy functional (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2__symbolic_probability_density}). This can be rewritten as: \\[ F_\\beta[\\rho] = \\int_M",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "rho(x) \\left(H(x) + \\beta^{-1}\\log\\rho(x)\\right) d\\mu_g(x) \\] This quantity decreases under symbolic evolution (see thm~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}). \\end{definition}",
      "label": "theorem:bk2_h_theorem_for_symbolic_evol",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "book2.tex",
      "target_line": 255,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_entropy",
    "theorem:bk2_h_theorem_for_symbolic_evol"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "axiom:bk5_energy_conservation",
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "axiom:bk7_convergence_potential",
    "corollary:bk2_interpretative_framework",
    "corollary:bk8_emergent_cognitive_scaffold",
    "definition:bk2_local_symbolic_temperature",
    "definition:bk3_membrane_thermodynamics",
    "definition:bk4_symbolic_transition_rate",
    "definition:bk4_test_time_coherent_sampling",
    "definition:bk5_process_free_energy",
    "definition:bk5_reflective_coupling_stab",
    "definition:bk7_frame_temperature_quotient",
    "definition:bk8_temperature_freedom",
    "demonstratio:bk4_ising_model_covenant",
    "demonstratio:bk7_free_energy_balance_equilibrium",
    "lemma:bk2_thermodynamic_consistency_hypothesis_manifolds",
    "proof:bk2_global_local_temp_relation",
    "proof:bk2_interpretative_framework",
    "proof:bk5_symbolic_temperature_threshold",
    "proposition:bk2_global_local_temp_relation",
    "scholium:bk2_on_hypotheses_as_thermodyn",
    "sec:bk2_foundations_symbolic_thermodynamics",
    "subsec:bk5_symbolic_free_energy_and_stability",
    "theorem:bk5_reflective_stability_criterion",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "cites": [
    "definition:bk2_symbolic_energy",
    "definition:bk2_symbolic_entropy"
  ],
  "depends_on": [
    "definition:bk2_symbolic_energy",
    "definition:bk2_symbolic_entropy"
  ],
  "file": "book2.tex",
  "id": "definition:bk2_symbolic_temperature",
  "label": "definition:bk2_symbolic_temperature",
  "latex_body": "\\begin{definition}[Symbolic Temperature] \n\\label{definition:bk2_symbolic_temperature} \nThe global symbolic temperature $T_s$ at symbolic time $s$ is defined thermodynamically as:\n\\[\nT_s^{-1} = \\frac{\\partial S_s}{\\partial E_s}\n\\]\nwhen the relationship between $S_s$ and $E_s$ is differentiable (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2_symbolic_energy}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "finite alphabet; strict suboptimality/unique minimizer hypotheses explicit; limits along atTop in beta",
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "The derivative bridge plus the full trichotomy of the temperature knob: uniform at beta 0, variational balance at finite beta, freezing/concentration at beta to infinity; the dS/dE form itself is not formalized."
    ],
    "record_ids": [
      "MAP-BOOK2-015"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book2.energy_eq_neg_deriv_log_partition",
      "Book2.gibbs_concentrates",
      "Book2.gibbs_freezes"
    ]
  },
  "line": 148,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Temperature",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "the relationship between $S_s$ and $E_s$ is differentiable (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2_symbolic_energy}). \\end{definition}",
      "label": "definition:bk2_symbolic_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 105,
      "target_type": "definition"
    },
    {
      "context": "_s^{-1} = \\frac{\\partial S_s}{\\partial E_s} \\] when the relationship between $S_s$ and $E_s$ is differentiable (see def~\\ref{definition:bk2_symbolic_entropy}; see also def~\\ref{definition:bk2_symbolic_energy}). \\end{definition}",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_energy",
    "definition:bk2_symbolic_entropy"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book2.tex",
  "id": "subsec:bk2_evolution_equations",
  "label": "subsec:bk2_evolution_equations",
  "latex_body": "",
  "line": 157,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Evolution Equations",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [
    "axiom:bk2_symbolic_fokker_planck_equation",
    "proof:bk2_coherence_of_symbolic_therm",
    "proof:bk2_symbolic_fluctuation_dissipation_relation",
    "proof:bk4_drift_reflection_summary",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk4_fuzzy_symbolic_geometry_theorem"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk2_symbolic_hamiltonian"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk2_symbolic_hamiltonian"
  ],
  "file": "book2.tex",
  "id": "axiom:bk2_gradient_structure_drift",
  "label": "axiom:bk2_gradient_structure_drift",
  "latex_body": "\\begin{axiom}[Gradient Structure of Symbolic Drift]\n\\label{axiom:bk2_gradient_structure_drift}\nThe symbolic drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) is related to the symbolic Hamiltonian $H$ (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) by:\n\\[\nD(x) = -\\nabla_g H(x) + \\xi(x)\n\\]\nwhere $\\nabla_g$ is the gradient with respect to the metric $g$, and $\\xi(x)$ is a solenoidal field (i.e., $\\nabla_g \\cdot \\xi = 0$) representing non-conservative components of the symbolic dynamics.\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Detailed balance is the finite shadow of the gradient condition; the 3-cycle exhibits the solenoidal component."
    ],
    "record_ids": [
      "MAP-BOOK2-011"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2.cycle_stationary_not_reversible",
      "Book2.detailedBalance_stationary"
    ]
  },
  "line": 162,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Gradient Structure of Symbolic Drift",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "om}[Gradient Structure of Symbolic Drift] \\label{axiom:bk2_gradient_structure_drift} The symbolic drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) is related to the symbolic Hamiltonian $H$ (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) by: \\[ D(x) = -\\nabla_g H(",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "} The symbolic drift field $D$ (Def.~\\ref{definition:bk1_drift_field}) is related to the symbolic Hamiltonian $H$ (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) by: \\[ D(x) = -\\nabla_g H(x) + \\xi(x) \\] where $\\nabla_g$ is the gradient with respect to the metric $g$, and $\\xi(x)$",
      "label": "definition:bk2_symbolic_hamiltonian",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 67,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk2_symbolic_hamiltonian"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk9_pathologies_of_coherence",
    "proof:bk9_symbolic_masking_and_unmasking"
  ],
  "cites": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2__symbolic_probability_density",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2__symbolic_probability_density",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "book2.tex",
  "id": "axiom:bk2_symbolic_fokker_planck_equation",
  "label": "axiom:bk2_symbolic_fokker_planck_equation",
  "latex_body": "\\begin{axiom}[Symbolic Fokker-Planck Equation]\n\\label{axiom:bk2_symbolic_fokker_planck_equation}\nIn continuity with the Book I symbolic evolution law (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) and the drift decomposition in axiom~\\ref{axiom:bk2_gradient_structure_drift}, the evolution of the symbolic probability density $\\rho$ (def~\\ref{definition:bk2__symbolic_probability_density}) is governed by:\n\\[\n\\frac{\\partial \\rho}{\\partial s} = -\\nabla_g \\cdot (\\rho D) + \\sigma^2 \\nabla_g^2 \\rho\n\\]\nwhere:\n\\begin{enumerate}\n    \\item $\\nabla_g \\cdot$ is the divergence operator with respect to the metric $g$;\n    \\item $\\nabla_g^2$ is the Laplace-Beltrami operator on $(M,g)$;\n    \\item $\\sigma^2 > 0$ is the symbolic diffusion coefficient, related to the inverse temperature by $\\sigma^2 = \\beta^{-1}$.\n\\end{enumerate}\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite skeleton only: evolution by a stochastic kernel; the manifold PDE is not certified."
    ],
    "record_ids": [
      "MAP-BOOK2-010"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2.evolve_conserves"
    ]
  },
  "line": 171,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Fokker-Planck Equation",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "c evolution law (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) and the drift decomposition in axiom~\\ref{axiom:bk2_gradient_structure_drift}, the evolution of the symbolic probability density $\\rho$ (def~\\ref{definition:bk2__symbolic_probability_density}) is g",
      "label": "axiom:bk2_gradient_structure_drift",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 162,
      "target_type": "axiom"
    },
    {
      "context": "sition in axiom~\\ref{axiom:bk2_gradient_structure_drift}, the evolution of the symbolic probability density $\\rho$ (def~\\ref{definition:bk2__symbolic_probability_density}) is governed by: \\[ \\frac{\\partial \\rho}{\\partial s} = -\\nabla_g \\cdot (\\rho D) + \\sigma^2 \\nabla_g^2 \\rho \\] where: \\b",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "k Equation] \\label{axiom:bk2_symbolic_fokker_planck_equation} In continuity with the Book I symbolic evolution law (thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) and the drift decomposition in axiom~\\ref{axiom:bk2_gradient_structure_drift}, the evolution of the symbolic probabili",
      "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": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2__symbolic_probability_density",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk2_fokker_planck_probability_conservation"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "proof:bk2_fokker_planck_probability_conservation"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "Fokker-Planck equation preserves the total probability: \\[ \\frac{d}{ds}\\int_M \\rho(x,s) \\, d\\mu_g(x) = 0 \\] (see proof~\\ref{proof:bk2_fokker_planck_probability_conservation}; see also def~\\ref{definition:bk2__symbolic_probability_density}). \\end{lemma}",
      "label": "proof:bk2_fokker_planck_probability_conservation",
      "line_distance": 9,
      "role": "proof_below",
      "target_line": 194,
      "target_type": "proof"
    }
  ],
  "forward_refs": [
    "proof:bk2_fokker_planck_probability_conservation"
  ],
  "id": "lemma:bk2_conservation_of_probability",
  "label": "lemma:bk2_conservation_of_probability",
  "latex_body": "\\begin{lemma}[Conservation of Probability] \n\\label{lemma:bk2_conservation_of_probability} \nThe symbolic Fokker-Planck equation preserves the total probability: \n\\[\n\\frac{d}{ds}\\int_M \\rho(x,s) \\, d\\mu_g(x) = 0\n\\]\n(see proof~\\ref{proof:bk2_fokker_planck_probability_conservation}; see also def~\\ref{definition:bk2__symbolic_probability_density}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite Markov skeleton: row-stochastic evolution conserves total mass."
    ],
    "record_ids": [
      "MAP-BOOK2-009"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2.evolve_conserves"
    ]
  },
  "line": 185,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conservation of Probability",
  "proof_labels": [
    "proof:bk2_fokker_planck_probability_conservation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ds}\\int_M \\rho(x,s) \\, d\\mu_g(x) = 0 \\] (see proof~\\ref{proof:bk2_fokker_planck_probability_conservation}; see also def~\\ref{definition:bk2__symbolic_probability_density}). \\end{lemma}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "Fokker-Planck equation preserves the total probability: \\[ \\frac{d}{ds}\\int_M \\rho(x,s) \\, d\\mu_g(x) = 0 \\] (see proof~\\ref{proof:bk2_fokker_planck_probability_conservation}; see also def~\\ref{definition:bk2__symbolic_probability_density}). \\end{lemma}",
      "label": "proof:bk2_fokker_planck_probability_conservation",
      "logical_support": false,
      "role": "forward_proof_below",
      "target_file": "book2.tex",
      "target_line": 194,
      "target_type": "proof"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "proof:bk2_fokker_planck_probability_conservation"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book2",
  "cited_by": [
    "lemma:bk2_conservation_of_probability"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "lemma:bk2_conservation_of_probability"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "lemma:bk2_conservation_of_probability"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_fokker_planck_probability_conservation",
  "label": "proof:bk2_fokker_planck_probability_conservation",
  "latex_body": "\\begin{proof}[Probability Conservation in Symbolic Fokker–Planck Equation]\n\\label{proof:bk2_fokker_planck_probability_conservation}\n\\leavevmode\n\nIntegrate the Fokker-Planck equation over $M$\n(see Def.~\\ref{definition:bk2__symbolic_probability_density} and\nLem.~\\ref{lemma:bk2_conservation_of_probability}):\n\\[\n\\int_M \\frac{\\partial \\rho}{\\partial s} \\, d\\mu_g \n= -\\int_M \\nabla_g \\cdot (\\rho D) \\, d\\mu_g \n+ \\sigma^2 \\int_M \\nabla_g^2 \\rho \\, d\\mu_g\n\\]\nBy the divergence theorem on the compact manifold $M$ (which has no boundary), both integrals on the right-hand side vanish:\n\\[\n\\int_M \\nabla_g \\cdot (\\rho D) \\, d\\mu_g = \\int_{\\partial M} (\\rho D) \\cdot \\mathbf{n} \\, d\\sigma = 0\n\\]\nand similarly for the Laplacian term. Therefore:\n\\[\n\\frac{d}{ds} \\int_M \\rho \\, d\\mu_g = 0\n\\]\n\\end{proof}",
  "line": 194,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Probability Conservation in Symbolic Fokker–Planck Equation",
  "proves": "lemma:bk2_conservation_of_probability",
  "ref_roles": [
    {
      "context": "{proof:bk2_fokker_planck_probability_conservation} \\leavevmode Integrate the Fokker-Planck equation over $M$ (see Def.~\\ref{definition:bk2__symbolic_probability_density} and Lem.~\\ref{lemma:bk2_conservation_of_probability}): \\[ \\int_M \\frac{\\partial \\rho}{\\partial s} \\, d\\mu_g = -\\int_M",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "de Integrate the Fokker-Planck equation over $M$ (see Def.~\\ref{definition:bk2__symbolic_probability_density} and Lem.~\\ref{lemma:bk2_conservation_of_probability}): \\[ \\int_M \\frac{\\partial \\rho}{\\partial s} \\, d\\mu_g = -\\int_M \\nabla_g \\cdot (\\rho D) \\, d\\mu_g + \\sigma^2 \\int_M",
      "label": "lemma:bk2_conservation_of_probability",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book2.tex",
      "target_line": 185,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "lemma:bk2_conservation_of_probability"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [
    "definition:bk2_symbolic_partition_funct",
    "definition:bk2_symbolic_response_functi",
    "proof:bk1_sketch_observed_consequences",
    "proof:bk2_coherence_of_symbolic_therm",
    "proof:bk2_sketch_wasserstein_gradient_flow",
    "proof:bk2_symbolic_drift_equilibrium_yields_gibbs_measure",
    "proof:bk2_symbolic_fluctuation_dissipation_relation",
    "proof:bk2_symbolic_free_energy_dissipation",
    "proof:bk2_symbolic_h_theorem",
    "proof:bk5_entropy_increase_from_drift",
    "proof:bk5_operator_convergence",
    "remark:bk2_symbolic_hamiltonian",
    "subsec:bk2_symbolic_phase_transitions",
    "theorem:bk2_coherence_of_symbolic_therm",
    "theorem:bk2_emergence_structure_symb_thermo",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk2_symbolic_fluctuation_dissipation_relation",
    "theorem:bk2_wasserstein_gradient_flow"
  ],
  "cites": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2_symbolic_partition_funct"
  ],
  "depends_on": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2__symbolic_probability_density"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "re $\\beta = \\sigma^{-2}$ and the partition function is: \\[ Z = \\int_M e^{-\\beta H(x)} \\, d\\mu_g(x) \\quad \\text{(see def~\\ref{definition:bk2_symbolic_partition_funct})} \\] \\end{theorem}",
      "label": "definition:bk2_symbolic_partition_funct",
      "line_distance": 153,
      "role": "teaser",
      "target_line": 369,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk2_symbolic_partition_funct"
  ],
  "id": "theorem:bk2_equilibrium_distribution",
  "label": "theorem:bk2_equilibrium_distribution",
  "latex_body": "\\begin{theorem}[Equilibrium Distribution] \n\\label{theorem:bk2_equilibrium_distribution} \nUnder the gradient condition $D = -\\nabla_g H$ (i.e., when the solenoidal component $\\xi = 0$ in Axiom~\\ref{axiom:bk2_gradient_structure_drift}), the unique equilibrium distribution $\\rho_{eq}$ satisfying $\\partial \\rho / \\partial s = 0$ for the symbolic Fokker-Planck equation is given by:\n\\[\n\\rho_{eq}(x) = Z^{-1} e^{-\\beta H(x)}\n\\]\nwhere $\\beta = \\sigma^{-2}$ and the partition function is:\n\\[\nZ = \\int_M e^{-\\beta H(x)} \\, d\\mu_g(x) \\quad \\text{(see def~\\ref{definition:bk2_symbolic_partition_funct})}\n\\]\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite kernel: stationarity under detailed balance plus unique variational characterization; the PDE derivation is not certified."
    ],
    "record_ids": [
      "MAP-BOOK2-007"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2.detailedBalance_stationary",
      "Book2.gibbs_minimizes",
      "Book2.gibbs_unique_minimizer"
    ]
  },
  "line": 216,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Equilibrium Distribution",
  "proof_labels": [
    "proof:bk2_symbolic_drift_equilibrium_yields_gibbs_measure"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "m_distribution} Under the gradient condition $D = -\\nabla_g H$ (i.e., when the solenoidal component $\\xi = 0$ in Axiom~\\ref{axiom:bk2_gradient_structure_drift}), the unique equilibrium distribution $\\rho_{eq}$ satisfying $\\partial \\rho / \\partial s = 0$ for the symbolic Fokker-P",
      "label": "axiom:bk2_gradient_structure_drift",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 162,
      "target_type": "axiom"
    },
    {
      "context": "re $\\beta = \\sigma^{-2}$ and the partition function is: \\[ Z = \\int_M e^{-\\beta H(x)} \\, d\\mu_g(x) \\quad \\text{(see def~\\ref{definition:bk2_symbolic_partition_funct})} \\] \\end{theorem}",
      "label": "definition:bk2_symbolic_partition_funct",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "book2.tex",
      "target_line": 369,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2_symbolic_partition_funct"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_symbolic_drift_equilibrium_yields_gibbs_measure",
  "label": "proof:bk2_symbolic_drift_equilibrium_yields_gibbs_measure",
  "latex_body": "\\begin{proof}[Proof: Symbolic Drift Equilibrium Yields Gibbs Measure]\n\\label{proof:bk2_symbolic_drift_equilibrium_yields_gibbs_measure}\n\\leavevmode\n\nAt equilibrium, we require $\\partial \\rho / \\partial s = 0$, which gives:\n\\[\n\\nabla_g \\cdot (\\rho D) = \\sigma^2 \\nabla_g^2 \\rho\n\\]\nDefine the probability current $J = \\rho D - \\sigma^2 \\nabla_g \\rho$. Then the equilibrium condition becomes $\\nabla_g \\cdot J = 0$. For a simply connected manifold, this admits the solution $J = 0$, giving:\n\\[\n\\rho D = \\sigma^2 \\nabla_g \\rho\n\\]\nSubstituting $D = -\\nabla_g H$:\n\\[\n-\\rho \\nabla_g H = \\sigma^2 \\nabla_g \\rho\n\\]\nDividing by $\\rho > 0$:\n\\[\n\\nabla_g \\log \\rho = -\\sigma^{-2} \\nabla_g H = -\\beta \\nabla_g H\n\\]\nThis integrates to give:\n\\[\n\\log \\rho = -\\beta H + C\n\\]\nfor some constant $C$. The normalization condition $\\int_M \\rho \\, d\\mu_g = 1$ determines $C = \\log Z^{-1}$, yielding the Gibbs-Boltzmann distribution (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; see also def~\\ref{definition:bk2__symbolic_probability_density}).\n\\end{proof}",
  "line": 228,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Proof: Symbolic Drift Equilibrium Yields Gibbs Measure",
  "proves": "theorem:bk2_equilibrium_distribution",
  "ref_roles": [
    {
      "context": "^{-1}$, yielding the Gibbs-Boltzmann distribution (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; see also def~\\ref{definition:bk2__symbolic_probability_density}). \\end{proof}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "dition $\\int_M \\rho \\, d\\mu_g = 1$ determines $C = \\log Z^{-1}$, yielding the Gibbs-Boltzmann distribution (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; see also def~\\ref{definition:bk2__symbolic_probability_density}). \\end{proof}",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [
    "axiom:bk5_positive_free_energy",
    "definition:bk2_symbolic_free_energy",
    "lemma:bk5_map_fitness_advantage",
    "proof:bk2_coherence_of_symbolic_therm",
    "proof:bk2_interpretative_framework",
    "proof:bk2_symbolic_free_energy_dissipation",
    "proof:bk2_symbolic_h_theorem",
    "proof:bk3_membrane_stability_energy_permeability",
    "proof:bk3_sketch_necessity_for_continuous_operation",
    "proof:bk5_entropy_increase_from_drift",
    "proof:bk5_map_resistance_to_drift",
    "proof:bk5_max_sustainable_drift",
    "proof:bk5_operator_convergence",
    "proof:bk8_sr_convergence",
    "proof:bk9_framework_functional_identity",
    "sec:bk7_scholium_convergence_as_symbolic_inhalation",
    "subsec:bk7_pisu_implications",
    "theorem:bk2_coherence_of_symbolic_therm",
    "theorem:bk2_emergence_structure_symb_thermo",
    "theorem:bk8_sr_convergence"
  ],
  "cites": [
    "definition:bk2_symbolic_free_energy",
    "proof:bk2_symbolic_free_energy_dissipation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_free_energy",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "ion, satisfying: \\[ \\frac{dF_\\beta[\\rho]}{ds} \\leq 0 \\] with equality if and only if $\\rho = \\rho_{eq}$ (see also proof~\\ref{proof:bk2_symbolic_free_energy_dissipation}). \\end{theorem}",
      "label": "proof:bk2_symbolic_free_energy_dissipation",
      "line_distance": 13,
      "role": "proof_below",
      "target_line": 268,
      "target_type": "proof"
    }
  ],
  "forward_refs": [
    "proof:bk2_symbolic_free_energy_dissipation"
  ],
  "id": "theorem:bk2_h_theorem_for_symbolic_evol",
  "label": "theorem:bk2_h_theorem_for_symbolic_evol",
  "latex_body": "\\begin{theorem}[H-Theorem for Symbolic Evolution] \n\\label{theorem:bk2_h_theorem_for_symbolic_evol} \nUnder the gradient condition $D = -\\nabla_g H$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}), the symbolic free energy functional\n\\[\nF_\\beta[\\rho] = \\int_M \\rho \\left( H + \\beta^{-1} \\log \\rho \\right) \\, d\\mu_g\n\\]\n(see def~\\ref{definition:bk2_symbolic_free_energy}) is a Lyapunov functional for the symbolic Fokker-Planck evolution, satisfying:\n\\[\n\\frac{dF_\\beta[\\rho]}{ds} \\leq 0\n\\]\nwith equality if and only if $\\rho = \\rho_{eq}$ (see also proof~\\ref{proof:bk2_symbolic_free_energy_dissipation}).\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Both halves now certified finitely: unique variational endpoint (Book2) and monotone decrease under any detailed-balance stochastic step via finite data-processing (Book2H); the continuum Fokker-Planck flow is not certified."
    ],
    "record_ids": [
      "MAP-BOOK2-008"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2.gibbs_minimizes",
      "Book2.gibbs_unique_minimizer",
      "Book2H.dataProcessing_kl",
      "Book2H.h_theorem"
    ]
  },
  "line": 255,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "H-Theorem for Symbolic Evolution",
  "proof_labels": [
    "proof:bk2_symbolic_free_energy_dissipation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "lic free energy functional \\[ F_\\beta[\\rho] = \\int_M \\rho \\left( H + \\beta^{-1} \\log \\rho \\right) \\, d\\mu_g \\] (see def~\\ref{definition:bk2_symbolic_free_energy}) is a Lyapunov functional for the symbolic Fokker-Planck evolution, satisfying: \\[ \\frac{dF_\\beta[\\rho]}{ds} \\leq 0 \\]",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "ion, satisfying: \\[ \\frac{dF_\\beta[\\rho]}{ds} \\leq 0 \\] with equality if and only if $\\rho = \\rho_{eq}$ (see also proof~\\ref{proof:bk2_symbolic_free_energy_dissipation}). \\end{theorem}",
      "label": "proof:bk2_symbolic_free_energy_dissipation",
      "logical_support": false,
      "role": "forward_proof_below",
      "target_file": "book2.tex",
      "target_line": 268,
      "target_type": "proof"
    },
    {
      "context": "volution] \\label{theorem:bk2_h_theorem_for_symbolic_evol} Under the gradient condition $D = -\\nabla_g H$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}), the symbolic free energy functional \\[ F_\\beta[\\rho] = \\int_M \\rho \\left( H + \\beta^{-1} \\log \\rho \\right) \\, d\\mu_g",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_free_energy",
    "proof:bk2_symbolic_free_energy_dissipation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book2",
  "cited_by": [
    "definition:bk4_meta_stable_symbolic_str",
    "proof:bk2_sketch_wasserstein_gradient_flow",
    "theorem:bk2_h_theorem_for_symbolic_evol"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_free_energy",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk2_h_theorem_for_symbolic_evol"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_free_energy",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk2_h_theorem_for_symbolic_evol"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_symbolic_free_energy_dissipation",
  "label": "proof:bk2_symbolic_free_energy_dissipation",
  "latex_body": "\\begin{proof}[Symbolic Free Energy Dissipation Principle]\n\\label{proof:bk2_symbolic_free_energy_dissipation}\n\\leavevmode\n\nDefine the symbolic chemical potential:\n\\[\n\\mu := \\frac{\\delta F_\\beta}{\\delta \\rho} = H + \\beta^{-1}(1 + \\log \\rho)\n\\]\nThe time derivative of the free energy (see def~\\ref{definition:bk2_symbolic_free_energy}) is:\n\\[\n\\frac{dF_\\beta[\\rho]}{ds} = \\int_M \\frac{\\partial \\rho}{\\partial s} \\mu \\, d\\mu_g\n\\]\nFrom the Fokker-Planck equation and integration by parts:\n\\[\n\\frac{dF_\\beta[\\rho]}{ds} = \\int_M [\\nabla_g \\cdot (\\rho D) - \\sigma^2 \\nabla_g^2 \\rho] \\mu \\, d\\mu_g = \\int_M [\\rho D - \\sigma^2 \\nabla_g \\rho] \\cdot \\nabla_g \\mu \\, d\\mu_g\n\\]\nUnder the gradient condition $D = -\\nabla_g H$, we have:\n\\[\n\\nabla_g \\mu = \\nabla_g H + \\beta^{-1} \\rho^{-1} \\nabla_g \\rho\n\\]\nTherefore:\n\\[\n\\rho D - \\sigma^2 \\nabla_g \\rho = -\\rho \\nabla_g H - \\beta^{-1} \\nabla_g \\rho = -\\rho \\left( \\nabla_g H + \\beta^{-1} \\rho^{-1} \\nabla_g \\rho \\right) = -\\rho \\nabla_g \\mu\n\\]\nThis gives:\n\\[\n\\frac{dF_\\beta[\\rho]}{ds} = -\\int_M \\rho \\|\\nabla_g \\mu\\|_g^2 \\, d\\mu_g \\leq 0\n\\]\nEquality holds if and only if $\\nabla_g \\mu = 0$, which implies $\\mu$ is constant on the support of $\\rho$, corresponding to the equilibrium distribution $\\rho_{eq}$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; cf. theorem~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}; see also def~\\ref{definition:bk2__symbolic_probability_density}).\n\\end{proof}",
  "line": 268,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Free Energy Dissipation Principle",
  "proves": "theorem:bk2_h_theorem_for_symbolic_evol",
  "ref_roles": [
    {
      "context": "rem~\\ref{theorem:bk2_equilibrium_distribution}; cf. theorem~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}; see also def~\\ref{definition:bk2__symbolic_probability_density}). \\end{proof}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "= \\frac{\\delta F_\\beta}{\\delta \\rho} = H + \\beta^{-1}(1 + \\log \\rho) \\] The time derivative of the free energy (see def~\\ref{definition:bk2_symbolic_free_energy}) is: \\[ \\frac{dF_\\beta[\\rho]}{ds} = \\int_M \\frac{\\partial \\rho}{\\partial s} \\mu \\, d\\mu_g \\] From the Fokker-Planck equ",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "lies $\\mu$ is constant on the support of $\\rho$, corresponding to the equilibrium distribution $\\rho_{eq}$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; cf. theorem~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}; see also def~\\ref{definition:bk2__symbolic_probability_dens",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    },
    {
      "context": "onding to the equilibrium distribution $\\rho_{eq}$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; cf. theorem~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}; see also def~\\ref{definition:bk2__symbolic_probability_density}). \\end{proof}",
      "label": "theorem:bk2_h_theorem_for_symbolic_evol",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 255,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_free_energy",
    "theorem:bk2_equilibrium_distribution",
    "theorem:bk2_h_theorem_for_symbolic_evol"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book2.tex",
  "id": "subsec:bk2_wasserstein_geometry",
  "label": "subsec:bk2_wasserstein_geometry",
  "latex_body": "",
  "line": 299,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Wasserstein Geometry and Gradient Flow Structure",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk2_sketch_wasserstein_gradient_flow",
    "proof:bk7_reflective_convergence_to_stable_identity",
    "theorem:bk2_wasserstein_gradient_flow",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density"
  ],
  "file": "book2.tex",
  "id": "definition:bk2_symbolic_wasserstein_met",
  "label": "definition:bk2_symbolic_wasserstein_met",
  "latex_body": "\\begin{definition}[Symbolic Wasserstein Metric] \n\\label{definition:bk2_symbolic_wasserstein_met} \nThe symbolic Wasserstein-2 metric $W_2$ on the space $\\mathcal{P}(M)$ of probability densities (see def~\\ref{definition:bk2__symbolic_probability_density}) is defined as:\n\\[\nW_2(\\rho_1, \\rho_2)^2 = \\inf_{\\pi \\in \\Pi(\\rho_1, \\rho_2)} \\int_{M \\times M} d_g(x,y)^2 \\, d\\pi(x,y)\n\\]\nwhere:\n\\begin{enumerate}\n    \\item $\\Pi(\\rho_1, \\rho_2)$ is the set of all couplings (joint probability measures) with marginals $\\rho_1 d\\mu_g$ and $\\rho_2 d\\mu_g$;\n    \\item $d_g$ is the geodesic distance on $(M,g)$.\n\\end{enumerate}\n\\end{definition}",
  "line": 302,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Wasserstein Metric",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "sserstein_met} The symbolic Wasserstein-2 metric $W_2$ on the space $\\mathcal{P}(M)$ of probability densities (see def~\\ref{definition:bk2__symbolic_probability_density}) is defined as: \\[ W_2(\\rho_1, \\rho_2)^2 = \\inf_{\\pi \\in \\Pi(\\rho_1, \\rho_2)} \\int_{M \\times M} d_g(x,y)^2 \\, d\\pi(x,y)",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk2_coherence_of_symbolic_therm",
    "proof:bk5_operator_convergence",
    "proof:bk8_sr_convergence",
    "proof:bk9_framework_functional_identity",
    "scholium:bk4_ttcs_potential_field",
    "sec:bk5_srmf_for_symbolic_operators_and_processes",
    "subsec:bk5_srmf_core_axioms",
    "subsec:bk7_pisu_formula",
    "theorem:bk5_operator_convergence",
    "theorem:bk8_sr_convergence"
  ],
  "cites": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_wasserstein_met",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_wasserstein_met",
    "proof:bk2_symbolic_free_energy_dissipation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "id": "theorem:bk2_wasserstein_gradient_flow",
  "label": "theorem:bk2_wasserstein_gradient_flow",
  "latex_body": "\\begin{theorem}[Wasserstein Gradient Flow] \n\\label{theorem:bk2_wasserstein_gradient_flow} \nUnder the gradient condition $D = -\\nabla_g H$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}), the symbolic Fokker-Planck equation can be interpreted as the gradient flow of the free energy functional $F_\\beta[\\rho]$ (see def~\\ref{definition:bk2_symbolic_free_energy}) with respect to the symbolic Wasserstein metric (see def~\\ref{definition:bk2_symbolic_wasserstein_met}):\n\\[\n\\frac{\\partial \\rho}{\\partial s} = -\\text{grad}_{W_2} F_\\beta[\\rho]\n\\]\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": [
      "Finite discrete gradient-flow kernel: every repeated detailed-balance step preserves density and the complete free-energy trajectory is antitone. No Wasserstein metric or continuum Fokker-Planck identification is asserted."
    ],
    "record_ids": [
      "MAP-BOOK2-020"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2H.freeEnergy_trajectory_antitone",
      "Book2H.trajectory_isDensity"
    ]
  },
  "line": 315,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Wasserstein Gradient Flow",
  "proof_labels": [
    "proof:bk2_sketch_wasserstein_gradient_flow"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "c Fokker-Planck equation can be interpreted as the gradient flow of the free energy functional $F_\\beta[\\rho]$ (see def~\\ref{definition:bk2_symbolic_free_energy}) with respect to the symbolic Wasserstein metric (see def~\\ref{definition:bk2_symbolic_wasserstein_met}): \\[ \\frac{\\par",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "eta[\\rho]$ (see def~\\ref{definition:bk2_symbolic_free_energy}) with respect to the symbolic Wasserstein metric (see def~\\ref{definition:bk2_symbolic_wasserstein_met}): \\[ \\frac{\\partial \\rho}{\\partial s} = -\\text{grad}_{W_2} F_\\beta[\\rho] \\] \\end{theorem}",
      "label": "definition:bk2_symbolic_wasserstein_met",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 302,
      "target_type": "definition"
    },
    {
      "context": "dient Flow] \\label{theorem:bk2_wasserstein_gradient_flow} Under the gradient condition $D = -\\nabla_g H$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}), the symbolic Fokker-Planck equation can be interpreted as the gradient flow of the free energy functional $F_\\beta[\\r",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_wasserstein_met",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_wasserstein_met",
    "proof:bk2_symbolic_free_energy_dissipation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_wasserstein_met",
    "proof:bk2_symbolic_free_energy_dissipation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_sketch_wasserstein_gradient_flow",
  "label": "proof:bk2_sketch_wasserstein_gradient_flow",
  "latex_body": "\\begin{proof}[Wasserstein Gradient Flow via Jordan--Kinderlehrer--Otto]\n\\label{proof:bk2_sketch_wasserstein_gradient_flow}\n\\leavevmode\n\n\\textbf{Wasserstein-2 gradient.}\nOn the space $\\mathcal{P}(M)$ equipped with the Wasserstein-2 metric $W_2$\n(Def.~\\ref{definition:bk2_symbolic_wasserstein_met}), the gradient of a functional\n$F[\\rho]$ is characterized as follows: if $\\partial_s\\rho + \\nabla_g\\cdot(\\rho v) = 0$\n(continuity equation), then $v = -\\nabla_g(\\delta F_\\beta/\\delta\\rho)$ defines\nthe $W_2$-gradient direction.\n\n\\textbf{Computing $\\delta F_\\beta/\\delta\\rho$.}\nFrom Def.~\\ref{definition:bk2_symbolic_free_energy},\n$F_\\beta[\\rho] = \\int_M \\rho H\\,d\\mu_g + \\beta^{-1}\\int_M\\rho\\log\\rho\\,d\\mu_g$.\nTaking the functional derivative:\n\\[\n\\frac{\\delta F_\\beta}{\\delta\\rho} = H(x) + \\beta^{-1}(1 + \\log\\rho).\n\\]\nTherefore the $W_2$-gradient velocity field is:\n\\[\nv = -\\nabla_g\\!\\left(H + \\beta^{-1}\\log\\rho\\right)\n  = -\\nabla_g H - \\beta^{-1}\\rho^{-1}\\nabla_g\\rho.\n\\]\nUnder the condition $D = -\\nabla_g H$\n(Thm.~\\ref{theorem:bk2_equilibrium_distribution}), this becomes\n$v = D - \\beta^{-1}\\rho^{-1}\\nabla_g\\rho$.\n\n\\textbf{Recovery of Fokker--Planck.}\nSubstituting into the continuity equation\n$\\partial_s\\rho + \\nabla_g\\cdot(\\rho v) = 0$:\n\\[\n\\frac{\\partial\\rho}{\\partial s}\n= -\\nabla_g\\cdot(\\rho v)\n= -\\nabla_g\\cdot(\\rho D) + \\beta^{-1}\\nabla_g\\cdot(\\nabla_g\\rho)\n= -\\nabla_g\\cdot(\\rho D) + \\beta^{-1}\\nabla_g^2\\rho,\n\\]\nwhich is exactly the symbolic Fokker--Planck equation\n(cf.~Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation},\nproof~\\ref{proof:bk2_symbolic_free_energy_dissipation}).\nHence $\\partial_s\\rho = -\\mathrm{grad}_{W_2}F_\\beta[\\rho]$.\n\\end{proof}",
  "line": 323,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Wasserstein Gradient Flow via Jordan--Kinderlehrer--Otto",
  "proves": "theorem:bk2_wasserstein_gradient_flow",
  "ref_roles": [
    {
      "context": "a F_\\beta/\\delta\\rho)$ defines the $W_2$-gradient direction. \\textbf{Computing $\\delta F_\\beta/\\delta\\rho$.} From Def.~\\ref{definition:bk2_symbolic_free_energy}, $F_\\beta[\\rho] = \\int_M \\rho H\\,d\\mu_g + \\beta^{-1}\\int_M\\rho\\log\\rho\\,d\\mu_g$. Taking the functional derivative: \\[ \\",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "mode \\textbf{Wasserstein-2 gradient.} On the space $\\mathcal{P}(M)$ equipped with the Wasserstein-2 metric $W_2$ (Def.~\\ref{definition:bk2_symbolic_wasserstein_met}), the gradient of a functional $F[\\rho]$ is characterized as follows: if $\\partial_s\\rho + \\nabla_g\\cdot(\\rho v) = 0$ (",
      "label": "definition:bk2_symbolic_wasserstein_met",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 302,
      "target_type": "definition"
    },
    {
      "context": "ctly the symbolic Fokker--Planck equation (cf.~Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}, proof~\\ref{proof:bk2_symbolic_free_energy_dissipation}). Hence $\\partial_s\\rho = -\\mathrm{grad}_{W_2}F_\\beta[\\rho]$. \\end{proof}",
      "label": "proof:bk2_symbolic_free_energy_dissipation",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 268,
      "target_type": "proof"
    },
    {
      "context": "= -\\nabla_g\\cdot(\\rho D) + \\beta^{-1}\\nabla_g^2\\rho, \\] which is exactly the symbolic Fokker--Planck equation (cf.~Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}, proof~\\ref{proof:bk2_symbolic_free_energy_dissipation}). Hence $\\partial_s\\rho = -\\mathrm{grad}_{W_2}F_\\beta[\\rho]$. \\",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    },
    {
      "context": "a^{-1}\\log\\rho\\right) = -\\nabla_g H - \\beta^{-1}\\rho^{-1}\\nabla_g\\rho. \\] Under the condition $D = -\\nabla_g H$ (Thm.~\\ref{theorem:bk2_equilibrium_distribution}), this becomes $v = D - \\beta^{-1}\\rho^{-1}\\nabla_g\\rho$. \\textbf{Recovery of Fokker--Planck.} Substituting into the c",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_wasserstein_met",
    "proof:bk2_symbolic_free_energy_dissipation",
    "theorem:bk1_fundamental_relation_fokker_plank_equation",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "id": "subsec:bk2_symbolic_phase_transitions",
  "label": "subsec:bk2_symbolic_phase_transitions",
  "latex_body": "",
  "line": 365,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Phase Transitions",
  "ref_roles": [
    {
      "context": "",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": false,
      "role": "navigation",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": false,
      "role": "navigation",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [
    "definition:bk2_symbolic_phase_transitio",
    "proof:bk2_symbolic_h_theorem",
    "theorem:bk2_equilibrium_distribution"
  ],
  "cites": [
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_hamiltonian",
    "definition:bk2_symbolic_phase_transitio",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_hamiltonian",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "forward_ref_roles": [
    {
      "context": "ion_funct} The symbolic partition function $Z(\\beta)$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; cf. def~\\ref{definition:bk2_symbolic_phase_transitio}), integrating the Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) over the symbolic manifold $M$ (Def.~\\re",
      "label": "definition:bk2_symbolic_phase_transitio",
      "line_distance": 8,
      "role": "interpretive_bridge",
      "target_line": 377,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk2_symbolic_phase_transitio"
  ],
  "id": "definition:bk2_symbolic_partition_funct",
  "label": "definition:bk2_symbolic_partition_funct",
  "latex_body": "\\begin{definition}[Symbolic Partition Function] \n\\label{definition:bk2_symbolic_partition_funct} \nThe symbolic partition function $Z(\\beta)$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; cf. def~\\ref{definition:bk2_symbolic_phase_transitio}), integrating the Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) over the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}), is defined as:\n\\[\nZ(\\beta) = \\int_M e^{-\\beta H(x)} \\, d\\mu_g(x)\n\\]\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite partition function: positivity and the equilibrium value -1/beta log Z."
    ],
    "record_ids": [
      "MAP-BOOK2-006"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book2.freeEnergy_gibbs",
      "Book2.gibbs_isDensity"
    ]
  },
  "line": 369,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Partition Function",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "io}), integrating the Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) over the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}), is defined as: \\[ Z(\\beta) = \\int_M e^{-\\beta H(x)} \\, d\\mu_g(x) \\] \\end{definition}",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "k2_equilibrium_distribution}; cf. def~\\ref{definition:bk2_symbolic_phase_transitio}), integrating the Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) over the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}), is defined as: \\[ Z(\\beta) = \\int_M e^{-",
      "label": "definition:bk2_symbolic_hamiltonian",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 67,
      "target_type": "definition"
    },
    {
      "context": "ion_funct} The symbolic partition function $Z(\\beta)$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; cf. def~\\ref{definition:bk2_symbolic_phase_transitio}), integrating the Hamiltonian (Def.~\\ref{definition:bk2_symbolic_hamiltonian}) over the symbolic manifold $M$ (Def.~\\re",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": false,
      "role": "forward_interpretive_bridge",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    },
    {
      "context": "ion Function] \\label{definition:bk2_symbolic_partition_funct} The symbolic partition function $Z(\\beta)$ (see theorem~\\ref{theorem:bk2_equilibrium_distribution}; cf. def~\\ref{definition:bk2_symbolic_phase_transitio}), integrating the Hamiltonian (Def.~\\ref{definition:bk2_symbolic",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_hamiltonian",
    "definition:bk2_symbolic_phase_transitio",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "definition:bk2_symbolic_partition_funct",
    "proof:bk2_classification_symb_phase_transitions",
    "proof:bk2_coherence_of_symbolic_therm",
    "proof:bk5_map_mad_mas_trichotomy",
    "proof:bk7_hilbert_banach_bridge",
    "sec:bk2_foundations_symbolic_thermodynamics",
    "theorem:bk2_classification_symb_phase_transitions",
    "theorem:bk5_map_mad_mas_trichotomy",
    "theorem:bk7_hilbert_banach_bridge"
  ],
  "cites": [
    "definition:bk2_symbolic_partition_funct"
  ],
  "depends_on": [
    "definition:bk2_symbolic_partition_funct"
  ],
  "file": "book2.tex",
  "id": "definition:bk2_symbolic_phase_transitio",
  "label": "definition:bk2_symbolic_phase_transitio",
  "latex_body": "\\begin{definition}[Symbolic Phase Transition] \n\\label{definition:bk2_symbolic_phase_transitio} \nA symbolic phase transition (see def~\\ref{definition:bk2_symbolic_partition_funct}) occurs at inverse temperature $\\beta_c$ if the free energy\n\\[\nf(\\beta) = -\\beta^{-1} \\ln Z(\\beta)\n\\]\nor its derivatives exhibit non-analytic behavior at $\\beta = \\beta_c$.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "The definition's non-analyticity is proved impossible at finite alphabet."
    ],
    "record_ids": [
      "MAP-BOOK2-012"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book2.no_finite_phase_transition"
    ]
  },
  "line": 377,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Phase Transition",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "tion}[Symbolic Phase Transition] \\label{definition:bk2_symbolic_phase_transitio} A symbolic phase transition (see def~\\ref{definition:bk2_symbolic_partition_funct}) occurs at inverse temperature $\\beta_c$ if the free energy \\[ f(\\beta) = -\\beta^{-1} \\ln Z(\\beta) \\] or its derivative",
      "label": "definition:bk2_symbolic_partition_funct",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 369,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_partition_funct"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk1_realization_of_symbolic_phase_transitions",
    "proof:bk2_coherence_of_symbolic_therm",
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "cites": [
    "definition:bk2_symbolic_phase_transitio"
  ],
  "depends_on": [
    "definition:bk2_symbolic_phase_transitio"
  ],
  "file": "book2.tex",
  "id": "theorem:bk2_classification_symb_phase_transitions",
  "label": "theorem:bk2_classification_symb_phase_transitions",
  "latex_body": "\\begin{theorem}[Classification of Symbolic Phase Transitions] \n\\label{theorem:bk2_classification_symb_phase_transitions} \nSymbolic phase transitions (see Def.~\\ref{definition:bk2_symbolic_phase_transitio}) are classified\nby the order of the first non-analytic derivative of the free energy $f(\\beta)$ at $\\beta_c$:\n\\textbf{first-order} transitions exhibit a discontinuity in $f'(\\beta)$ (energy discontinuity);\n\\textbf{second-order} transitions exhibit a discontinuity in $f''(\\beta)$ (heat capacity discontinuity);\nand \\textbf{higher-order} transitions exhibit discontinuities in derivatives of order $n \\geq 3$.\nThe order of the transition determines its thermodynamic signature and governs observable behavior near $\\beta_c$.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty symbolic alphabet (NeZero n)",
      "positive beta for the variational principle; nonzero beta for the equilibrium value",
      "the stochastic-kernel evolution law and detailed balance are named structures, not derived from the PDE"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Negative-space kernel: f differentiable at every beta > 0 finitely; the order classification is not certified."
    ],
    "record_ids": [
      "MAP-BOOK2-013"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book2.no_finite_phase_transition"
    ]
  },
  "line": 385,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Classification of Symbolic Phase Transitions",
  "proof_labels": [
    "proof:bk2_classification_symb_phase_transitions"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "lic Phase Transitions] \\label{theorem:bk2_classification_symb_phase_transitions} Symbolic phase transitions (see Def.~\\ref{definition:bk2_symbolic_phase_transitio}) are classified by the order of the first non-analytic derivative of the free energy $f(\\beta)$ at $\\beta_c$: \\textbf{f",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_phase_transitio"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_phase_transitio"
  ],
  "depends_on": [
    "definition:bk2_symbolic_phase_transitio"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_classification_symb_phase_transitions",
  "label": "proof:bk2_classification_symb_phase_transitions",
  "latex_body": "\\begin{proof}\n\\label{proof:bk2_classification_symb_phase_transitions}\n\\leavevmode\n\nBy Def.~\\ref{definition:bk2_symbolic_phase_transitio}, the datum that makes a\nsymbolic phase transition visible is precisely a non-analyticity of\n$f(\\beta)=-\\beta^{-1}\\ln Z(\\beta)$, or of one of its derivatives, at the critical\ninverse temperature $\\beta_c$. Let $m$ be the least derivative order for which\n$f^{(m)}$ fails to extend analytically through $\\beta_c$. Minimality of $m$\nimplies that all lower derivatives carry the same analytic germ on the two sides\nof $\\beta_c$, so the first failed derivative is well-defined.\n\nWhen $m=1$, the first thermodynamic response obtained from $f$ changes\ndiscontinuously; in the symbolic thermodynamic normalization this is the energy\nresponse. When $m=2$, the first derivative remains continuous while the next\nresponse, the heat-capacity response, is discontinuous. If $m\\geq 3$, the first\ntwo responses remain regular and the singularity is deferred to a higher\nderivative. These three mutually exclusive cases exhaust the possible least\norders $m$, so the classification follows from the definition of the transition.\n\\end{proof}",
  "line": 395,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "theorem:bk2_classification_symb_phase_transitions",
  "ref_roles": [
    {
      "context": "\\begin{proof} \\label{proof:bk2_classification_symb_phase_transitions} \\leavevmode By Def.~\\ref{definition:bk2_symbolic_phase_transitio}, the datum that makes a symbolic phase transition visible is precisely a non-analyticity of $f(\\beta)=-\\beta^{-1}\\ln Z(",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_phase_transitio"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book2.tex",
  "id": "subsec:bk2_symbolic_fluctuation_dissipation_relations",
  "label": "subsec:bk2_symbolic_fluctuation_dissipation_relations",
  "latex_body": "",
  "line": 416,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Fluctuation-Dissipation Relations",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk2_symbolic_fluctuation_dissipation_relation",
    "theorem:bk2_symbolic_fluctuation_dissipation_relation"
  ],
  "cites": [
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "id": "definition:bk2_symbolic_response_functi",
  "label": "definition:bk2_symbolic_response_functi",
  "latex_body": "\\begin{definition}[Symbolic Response Function] \n\\label{definition:bk2_symbolic_response_functi} \nFor observables $A, B: M \\to \\mathbb{R}$, the linear response function $\\chi_{AB}(t)$ is defined by (see thm~\\ref{theorem:bk2_equilibrium_distribution}):\n\\[\n\\langle A(s+t) \\rangle_h - \\langle A \\rangle_{eq} = \\int_0^t \\chi_{AB}(t-\\tau) h(\\tau) \\, d\\tau + O(h^2)\n\\]\nwhere $\\langle \\cdot \\rangle_h$ denotes expectation under the perturbed Hamiltonian $H' = H - hB$ and $\\langle \\cdot \\rangle_{eq}$ is the equilibrium expectation.\n\\end{definition}",
  "line": 419,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Response Function",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "e_functi} For observables $A, B: M \\to \\mathbb{R}$, the linear response function $\\chi_{AB}(t)$ is defined by (see thm~\\ref{theorem:bk2_equilibrium_distribution}): \\[ \\langle A(s+t) \\rangle_h - \\langle A \\rangle_{eq} = \\int_0^t \\chi_{AB}(t-\\tau) h(\\tau) \\, d\\tau + O(h^2) \\] where",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk2_coherence_of_symbolic_therm",
    "theorem:bk2_coherence_of_symbolic_therm"
  ],
  "cites": [
    "definition:bk2_symbolic_response_functi",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2_symbolic_response_functi",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "id": "theorem:bk2_symbolic_fluctuation_dissipation_relation",
  "label": "theorem:bk2_symbolic_fluctuation_dissipation_relation",
  "latex_body": "\\begin{theorem}[Symbolic Fluctuation-Dissipation Relation] \n\\label{theorem:bk2_symbolic_fluctuation_dissipation_relation} \nFor the symbolic Fokker-Planck dynamics in equilibrium (see thm~\\ref{theorem:bk2_equilibrium_distribution}), the response function (see def~\\ref{definition:bk2_symbolic_response_functi}) is related to the equilibrium correlation function by:\n\\[\n\\chi_{AB}(t) = \\beta \\frac{d}{dt}\\langle A(t) B(0) \\rangle_{eq} \\quad \\text{for } t > 0\n\\]\nwhere $A(t)$ evolves under the unperturbed dynamics.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "see per-anchor coverage-map notes for the exact scope of each conditional/partial grade"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite static Kubo kernel: under H_h = H - hB, the derivative at h=0 of the Gibbs expectation of A equals beta times the equilibrium covariance of A and B. The continuous-time correlation derivative requires a differentiable equilibrium semigroup and remains outside this kernel."
    ],
    "record_ids": [
      "MAP-BOOK2-021"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book2Response.fluctuation_response_hasDerivAt"
    ]
  },
  "line": 428,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Fluctuation-Dissipation Relation",
  "proof_labels": [
    "proof:bk2_symbolic_fluctuation_dissipation_relation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ker-Planck dynamics in equilibrium (see thm~\\ref{theorem:bk2_equilibrium_distribution}), the response function (see def~\\ref{definition:bk2_symbolic_response_functi}) is related to the equilibrium correlation function by: \\[ \\chi_{AB}(t) = \\beta \\frac{d}{dt}\\langle A(t) B(0) \\rangle_{",
      "label": "definition:bk2_symbolic_response_functi",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 419,
      "target_type": "definition"
    },
    {
      "context": "theorem:bk2_symbolic_fluctuation_dissipation_relation} For the symbolic Fokker-Planck dynamics in equilibrium (see thm~\\ref{theorem:bk2_equilibrium_distribution}), the response function (see def~\\ref{definition:bk2_symbolic_response_functi}) is related to the equilibrium correlati",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_response_functi",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2_symbolic_response_functi",
    "theorem:bk2_equilibrium_distribution"
  ],
  "depends_on": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2_symbolic_response_functi",
    "theorem:bk2_equilibrium_distribution"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_symbolic_fluctuation_dissipation_relation",
  "label": "proof:bk2_symbolic_fluctuation_dissipation_relation",
  "latex_body": "\\begin{proof}\n\\label{proof:bk2_symbolic_fluctuation_dissipation_relation}\n\\leavevmode\n\nBy Thm.~\\ref{theorem:bk2_equilibrium_distribution} the unperturbed stationary state is the Gibbs density $\\rho_{eq} = Z^{-1}e^{-\\beta H}$. Following the protocol of Def.~\\ref{definition:bk2_symbolic_response_functi}, prepare the system in the equilibrium of the perturbed Hamiltonian $H' = H - hB$ and release the field at $t=0$. Expanding the perturbed weight $\\propto e^{-\\beta(H-hB)}$ to first order in $h$ and renormalizing,\n\\[\n\\rho_h = \\rho_{eq}\\bigl[\\,1 + \\beta h\\,(B - \\langle B\\rangle_{eq})\\,\\bigr] + O(h^2),\n\\]\nthe correction integrating to zero as required. Evolving $A$ for $t>0$ under the unperturbed symbolic Fokker--Planck flow and using stationarity of $\\rho_{eq}$,\n\\[\n\\langle A(t)\\rangle_h - \\langle A\\rangle_{eq}\n   = \\beta h\\,\\bigl[\\langle A(t)B(0)\\rangle_{eq} - \\langle A\\rangle_{eq}\\langle B\\rangle_{eq}\\bigr] + O(h^2).\n\\tag{$\\star$}\n\\]\nBecause the symbolic drift is a gradient, $D = -\\nabla_g H$ (the condition under which $\\rho_{eq}$ is stationary, Ax.~\\ref{axiom:bk2_gradient_structure_drift}, Thm.~\\ref{theorem:bk2_equilibrium_distribution}), the dynamics satisfy detailed balance; the equilibrium correlation is therefore differentiable in $t$, and the disconnected term $\\langle A\\rangle_{eq}\\langle B\\rangle_{eq}$ is constant, so it is annihilated by $d/dt$. Identifying $(\\star)$ with the linear-response kernel of Def.~\\ref{definition:bk2_symbolic_response_functi} and differentiating the switch protocol then gives, with the sign fixed by the convention $H'=H-hB$,\n\\[\n\\chi_{AB}(t) = \\beta \\frac{d}{dt}\\langle A(t)B(0)\\rangle_{eq}, \\qquad t>0 .\n\\]\nEquilibrium fluctuations thus determine the dissipative response: the symbolic Kubo identity.\n\\end{proof}",
  "line": 437,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "theorem:bk2_symbolic_fluctuation_dissipation_relation",
  "ref_roles": [
    {
      "context": "] Because the symbolic drift is a gradient, $D = -\\nabla_g H$ (the condition under which $\\rho_{eq}$ is stationary, Ax.~\\ref{axiom:bk2_gradient_structure_drift}, Thm.~\\ref{theorem:bk2_equilibrium_distribution}), the dynamics satisfy detailed balance; the equilibrium correlation i",
      "label": "axiom:bk2_gradient_structure_drift",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 162,
      "target_type": "axiom"
    },
    {
      "context": "the unperturbed stationary state is the Gibbs density $\\rho_{eq} = Z^{-1}e^{-\\beta H}$. Following the protocol of Def.~\\ref{definition:bk2_symbolic_response_functi}, prepare the system in the equilibrium of the perturbed Hamiltonian $H' = H - hB$ and release the field at $t=0$. Expan",
      "label": "definition:bk2_symbolic_response_functi",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 419,
      "target_type": "definition"
    },
    {
      "context": "\\begin{proof} \\label{proof:bk2_symbolic_fluctuation_dissipation_relation} \\leavevmode By Thm.~\\ref{theorem:bk2_equilibrium_distribution} the unperturbed stationary state is the Gibbs density $\\rho_{eq} = Z^{-1}e^{-\\beta H}$. Following the protocol of Def.~",
      "label": "theorem:bk2_equilibrium_distribution",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book2.tex",
      "target_line": 216,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "axiom:bk2_gradient_structure_drift",
    "definition:bk2_symbolic_response_functi",
    "theorem:bk2_equilibrium_distribution"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book2.tex",
  "id": "subsec:bk2_local_temperature_geometry",
  "label": "subsec:bk2_local_temperature_geometry",
  "latex_body": "",
  "line": 458,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Local Temperature and Geometric Relations",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book2",
  "cited_by": [
    "proof:bk2_global_local_temp_relation",
    "proposition:bk2_global_local_temp_relation"
  ],
  "cites": [
    "definition:bk2_symbolic_temperature"
  ],
  "depends_on": [
    "definition:bk2_symbolic_temperature"
  ],
  "file": "book2.tex",
  "id": "definition:bk2_local_symbolic_temperature",
  "label": "definition:bk2_local_symbolic_temperature",
  "latex_body": "\\begin{definition}[Local Symbolic Temperature] \n\\label{definition:bk2_local_symbolic_temperature}\nThe local symbolic temperature (see def~\\ref{definition:bk2_symbolic_temperature}) at point $x \\in M$ and time $s$ is defined as:\n\\[\nT(x,s) = \\alpha \\left( \\|\\nabla_g \\cdot D(x)\\|_g + \\gamma \\|D(x)\\|_g \\right)^{-1}\n\\]\nwhere $\\alpha, \\gamma > 0$ are scaling constants, and $\\nabla_g \\cdot D$ is the divergence of the drift field.\n\\end{definition}",
  "line": 461,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Local Symbolic Temperature",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "[Local Symbolic Temperature] \\label{definition:bk2_local_symbolic_temperature} The local symbolic temperature (see def~\\ref{definition:bk2_symbolic_temperature}) at point $x \\in M$ and time $s$ is defined as: \\[ T(x,s) = \\alpha \\left( \\|\\nabla_g \\cdot D(x)\\|_g + \\gamma \\|D(x)\\|_g",
      "label": "definition:bk2_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 148,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_temperature"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book2",
  "cited_by": [
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_local_symbolic_temperature",
    "definition:bk2_symbolic_temperature"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_local_symbolic_temperature",
    "definition:bk2_symbolic_temperature"
  ],
  "file": "book2.tex",
  "id": "proposition:bk2_global_local_temp_relation",
  "label": "proposition:bk2_global_local_temp_relation",
  "latex_body": "\\begin{proposition}[Global-Local Temperature Relation] \n\\label{proposition:bk2_global_local_temp_relation} \nUnder local equilibrium conditions, the global symbolic temperature $T_s$ (def~\\ref{definition:bk2_symbolic_temperature}) relates to the local symbolic temperature $T(x,s)$ (def~\\ref{definition:bk2_local_symbolic_temperature}) through:\n\\[\nT_s^{-1} = \\int_M \\rho(x,s) T(x,s)^{-1} \\, d\\mu_g(x)\n\\]\nwhere $\\rho(\\cdot,s)$ is the symbolic probability density from def~\\ref{definition:bk2__symbolic_probability_density}.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "manifold measure form, specific masking free-energy functional, and Hilbert decoherence operator stay open per row notes"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Global inverse temperature = density-weighted mean of local inverse temps, bounded by the extremes; the manifold integral stays open."
    ],
    "record_ids": [
      "MAP-BOOK2-019"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ThermoRes.global_beta_between"
    ]
  },
  "line": 470,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Global-Local Temperature Relation",
  "proof_labels": [
    "proof:bk2_global_local_temp_relation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "^{-1} = \\int_M \\rho(x,s) T(x,s)^{-1} \\, d\\mu_g(x) \\] where $\\rho(\\cdot,s)$ is the symbolic probability density from def~\\ref{definition:bk2__symbolic_probability_density}. \\end{proposition}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "mperature $T_s$ (def~\\ref{definition:bk2_symbolic_temperature}) relates to the local symbolic temperature $T(x,s)$ (def~\\ref{definition:bk2_local_symbolic_temperature}) through: \\[ T_s^{-1} = \\int_M \\rho(x,s) T(x,s)^{-1} \\, d\\mu_g(x) \\] where $\\rho(\\cdot,s)$ is the symbolic probability",
      "label": "definition:bk2_local_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 461,
      "target_type": "definition"
    },
    {
      "context": "osition:bk2_global_local_temp_relation} Under local equilibrium conditions, the global symbolic temperature $T_s$ (def~\\ref{definition:bk2_symbolic_temperature}) relates to the local symbolic temperature $T(x,s)$ (def~\\ref{definition:bk2_local_symbolic_temperature}) through: \\[ T",
      "label": "definition:bk2_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 148,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_local_symbolic_temperature",
    "definition:bk2_symbolic_temperature"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_local_symbolic_temperature",
    "definition:bk2_symbolic_temperature"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_local_symbolic_temperature",
    "definition:bk2_symbolic_temperature"
  ],
  "file": "book2.tex",
  "id": "proof:bk2_global_local_temp_relation",
  "label": "proof:bk2_global_local_temp_relation",
  "latex_body": "\\begin{proof}\n\\label{proof:bk2_global_local_temp_relation}\n\\leavevmode\n\n\\begin{assumption}[Local equilibrium averaging]\nThe phrase ``under local equilibrium conditions'' means that a global\nquasistatic symbolic-energy variation decomposes into uniform local energy\nincrements, while entropy variations add with respect to the symbolic\nprobability density $\\rho(\\cdot,s)$ of Def.~\\ref{definition:bk2__symbolic_probability_density}.\n\\end{assumption}\n\nFor a local cell at $x$, Def.~\\ref{definition:bk2_local_symbolic_temperature}\nidentifies $T(x,s)^{-1}$ as the local entropy response per unit symbolic-energy\nincrement. Hence a quasistatic increment $\\delta E$ contributes\n$T(x,s)^{-1}\\delta E$ to the local entropy variation. Additivity under the\nlocal-equilibrium averaging assumption gives\n\\[\n\\delta S_s\n  = \\int_M \\rho(x,s)\\,T(x,s)^{-1}\\delta E\\,d\\mu_g(x).\n\\]\nDividing by the common increment $\\delta E$ and using the thermodynamic\ndefinition $T_s^{-1}=\\delta S_s/\\delta E$ from Def.~\\ref{definition:bk2_symbolic_temperature}\nyields\n\\[\nT_s^{-1} = \\int_M \\rho(x,s) T(x,s)^{-1}\\,d\\mu_g(x),\n\\]\nas claimed.\n\\end{proof}",
  "line": 479,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "proposition:bk2_global_local_temp_relation",
  "ref_roles": [
    {
      "context": "nergy increments, while entropy variations add with respect to the symbolic probability density $\\rho(\\cdot,s)$ of Def.~\\ref{definition:bk2__symbolic_probability_density}. \\end{assumption} For a local cell at $x$, Def.~\\ref{definition:bk2_local_symbolic_temperature} identifies $T(x,s)^{-1",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "o(\\cdot,s)$ of Def.~\\ref{definition:bk2__symbolic_probability_density}. \\end{assumption} For a local cell at $x$, Def.~\\ref{definition:bk2_local_symbolic_temperature} identifies $T(x,s)^{-1}$ as the local entropy response per unit symbolic-energy increment. Hence a quasistatic incremen",
      "label": "definition:bk2_local_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 461,
      "target_type": "definition"
    },
    {
      "context": "ding by the common increment $\\delta E$ and using the thermodynamic definition $T_s^{-1}=\\delta S_s/\\delta E$ from Def.~\\ref{definition:bk2_symbolic_temperature} yields \\[ T_s^{-1} = \\int_M \\rho(x,s) T(x,s)^{-1}\\,d\\mu_g(x), \\] as claimed. \\end{proof}",
      "label": "definition:bk2_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 148,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_local_symbolic_temperature",
    "definition:bk2_symbolic_temperature"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book2.tex",
  "id": "assumption:book2.tex:483",
  "label": "",
  "latex_body": "\\begin{assumption}[Local equilibrium averaging]\nThe phrase ``under local equilibrium conditions'' means that a global\nquasistatic symbolic-energy variation decomposes into uniform local energy\nincrements, while entropy variations add with respect to the symbolic\nprobability density $\\rho(\\cdot,s)$ of Def.~\\ref{definition:bk2__symbolic_probability_density}.\n\\end{assumption}",
  "line": 483,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Local equilibrium averaging",
  "proof_status": "definitional",
  "refs": [
    "definition:bk2__symbolic_probability_density"
  ],
  "role": "assumption",
  "type": "assumption"
}

{
  "book": "book2",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book2.tex",
  "id": "subsec:bk2_hypotheses_thermodynamic_surfaces",
  "label": "subsec:bk2_hypotheses_thermodynamic_surfaces",
  "latex_body": "",
  "line": 508,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Hypotheses as Thermodynamic Surfaces",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}