Book V — De Vita Symbolica — page 2 of 6

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    {
      "context": "from differential survival rates in Thm.~\\ref{theorem:bk5__map_dominance} and evolutionary-game updates encoded in Def.~\\ref{definition:bk5_symbolic_replicator_dynamics}. Strategies with higher persistence probability (maintaining $F_s > 0$ under stronger drift) increase in population fre",
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  "id": "proof:bk5_symbolic_fitness_differentials",
  "label": "proof:bk5_symbolic_fitness_differentials",
  "latex_body": "\\begin{proof}[Survival Differentials and Symbolic Fitness]\n\\label{proof:bk5_symbolic_fitness_differentials}\n\\leavevmode\n\nThis follows from differential survival rates in\nThm.~\\ref{theorem:bk5__map_dominance} and evolutionary-game updates encoded in\nDef.~\\ref{definition:bk5_symbolic_replicator_dynamics}.\nStrategies with higher persistence probability\n(maintaining $F_s > 0$ under stronger drift) increase in population frequency\nover time.\n\\end{proof}",
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  "name": "Survival Differentials and Symbolic Fitness",
  "proves": "corollary:bk5_map_evolutionary_advantag",
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    {
      "context": "from differential survival rates in Thm.~\\ref{theorem:bk5__map_dominance} and evolutionary-game updates encoded in Def.~\\ref{definition:bk5_symbolic_replicator_dynamics}. Strategies with higher persistence probability (maintaining $F_s > 0$ under stronger drift) increase in population fre",
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      "context": "ss] \\label{proof:bk5_symbolic_fitness_differentials} \\leavevmode This follows from differential survival rates in Thm.~\\ref{theorem:bk5__map_dominance} and evolutionary-game updates encoded in Def.~\\ref{definition:bk5_symbolic_replicator_dynamics}. Strategies with higher",
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  "label": "sec:bk5_reflective_equilibrium_in_symbolic_systems",
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  "line": 496,
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  "line": 499,
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    "definition:bk7_operational_resolution_uncertainties",
    "lemma:bk7_involutive_dual_symmetry",
    "scholium:bk7_constrained_uncertainty_motivation",
    "sec:bk7_pisu_universal_symbolic_uncertainty",
    "subsec:bk7_pisu_axiom_statement",
    "subsec:bk7_pisu_regimes",
    "subsec:bk7_pisu_revisited_power_uncertainty",
    "subsec:bk7_sources_regimes_uncertainty",
    "theorem:bk5_reflective_stability_criterion"
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    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold"
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  "id": "definition:bk5_reflective_drift_coupling_tensor",
  "label": "definition:bk5_reflective_drift_coupling_tensor",
  "latex_body": "\\begin{definition}[Reflective-Drift Coupling Tensor] \\label{definition:bk5_reflective_drift_coupling_tensor}\nFor symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ with respective drift operators $\\drift_A, \\drift_B$ (derived from Def.~\\ref{definition:bk1_drift_field}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) and reflection operators $\\reflect_A, \\reflect_B$ (derived from Def.~\\ref{definition:bk1_reflection_operator}), their \\emph{reflective-drift coupling tensor} $\\mathcal{C}_{AB}$ is defined as:\n\\begin{equation}\n\\mathcal{C}_{AB} := \\drift_A \\circ \\reflect_B + \\drift_B \\circ \\reflect_A\n\\end{equation}\nThis tensor quantifies the net effect of each membrane's reflective capacity on the other's drift dynamics.\n\\end{definition}",
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  "name": "Reflective-Drift Coupling Tensor",
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  "ref_roles": [
    {
      "context": "bolic membranes $\\Membrane_A$ and $\\Membrane_B$ with respective drift operators $\\drift_A, \\drift_B$ (derived from Def.~\\ref{definition:bk1_drift_field}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) and reflection operators $\\reflect_A, \\refl",
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    {
      "context": "$M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) and reflection operators $\\reflect_A, \\reflect_B$ (derived from Def.~\\ref{definition:bk1_reflection_operator}), their \\emph{reflective-drift coupling tensor} $\\mathcal{C}_{AB}$ is defined as: \\begin{equation} \\mathcal{C}_{AB} :=",
      "label": "definition:bk1_reflection_operator",
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      "role": "definition_anchor",
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    {
      "context": "operators $\\drift_A, \\drift_B$ (derived from Def.~\\ref{definition:bk1_drift_field}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) and reflection operators $\\reflect_A, \\reflect_B$ (derived from Def.~\\ref{definition:bk1_reflection_operator}), their",
      "label": "definition:bk1_symbolic_manifold",
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    "definition:bk1_drift_field",
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      "context": "is the scalar control parameter for Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}, used immediately in Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}. The spectral radius of the reflective–drift coupling tensor \\( \\mathcal{C}_{AB} \\), denoted \\( \\rho(\\mathcal{C}_{AB})",
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  "latex_body": "\\begin{definition}[Spectral Radius of Coupling Tensor] \\label{definition:bk5_spectral_radius_of_coupl}\nThis is the scalar control parameter for Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}, used immediately in Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}.\nThe spectral radius of the reflective–drift coupling tensor \\( \\mathcal{C}_{AB} \\), denoted \\( \\rho(\\mathcal{C}_{AB}) \\), is defined as:\n\\begin{equation}\n\\rho(\\mathcal{C}_{AB}) := \\max\\{|\\lambda| : \\lambda \\in \\sigma(\\mathcal{C}_{AB})\\}\n\\end{equation}\nWhere $\\sigma(\\mathcal{C}_{AB})$ denotes the spectrum (set of eigenvalues) of $\\mathcal{C}_{AB}$ when viewed as a linear operator on the combined state space $\\Membrane_A \\otimes \\Membrane_B$.\n\\end{definition}",
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  "name": "Spectral Radius of Coupling Tensor",
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  "ref_roles": [
    {
      "context": "is the scalar control parameter for Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}, used immediately in Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}. The spectral radius of the reflective–drift coupling tensor \\( \\mathcal{C}_{AB} \\), denoted \\( \\rho(\\mathcal{C}_{AB})",
      "label": "axiom:bk5_reflective_equilibrium_stability_flux",
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      "context": "adius of Coupling Tensor] \\label{definition:bk5_spectral_radius_of_coupl} This is the scalar control parameter for Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}, used immediately in Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}. The spectral radius of the reflective–d",
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  "label": "axiom:bk5_reflective_equilibrium_stability_flux",
  "latex_body": "\\begin{axiom}[Reflective Equilibrium Stability]\n\\label{axiom:bk5_reflective_equilibrium_stability_flux}\nIt refines the MAP condition (Thm.~\\ref{theorem:bk5_map_equilibrium}) into a spectral criterion tied to symbolic temperature (Def.~\\ref{definition:bk2_symbolic_temperature}).\nA symbolic system attains reflective equilibrium with another system if their coupled reflective-drift tensor $\\mathcal{C}_{AB}$ exhibits a bounded spectral radius relative to a critical stability threshold. Specifically:\n\\begin{equation}\n\\rho(\\mathcal{C}_{AB}) < \\lambda_{\\text{crit}}\n\\end{equation}\nWhere $\\lambda_{\\text{crit}}$ is the critical spectral radius threshold given by:\n\\begin{equation}\n\\lambda_{\\text{crit}} = \\frac{T_s \\cdot \\min\\{\\eta_A, \\eta_B\\}}{\\max\\{\\|\\drift_A\\|, \\|\\drift_B\\|\\}} \n\\end{equation}\nWith $T_s$ representing symbolic temperature, $\\eta_A$ and $\\eta_B$ the symbolic coherence densities of the respective membranes, and $\\|\\drift_i\\|$ the operator norm of the drift operator.\nThis condition ensures stable inter-membrane viability and mutually sustained symbolic free energy over time.\n\\end{axiom}",
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  "name": "Reflective Equilibrium Stability",
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      "context": "the MAP condition (Thm.~\\ref{theorem:bk5_map_equilibrium}) into a spectral criterion tied to symbolic temperature (Def.~\\ref{definition:bk2_symbolic_temperature}). A symbolic system attains reflective equilibrium with another system if their coupled reflective-drift tensor $\\mathc",
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      "target_file": "book2.tex",
      "target_line": 148,
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    },
    {
      "context": "ctive Equilibrium Stability] \\label{axiom:bk5_reflective_equilibrium_stability_flux} It refines the MAP condition (Thm.~\\ref{theorem:bk5_map_equilibrium}) into a spectral criterion tied to symbolic temperature (Def.~\\ref{definition:bk2_symbolic_temperature}). A symbolic sy",
      "label": "theorem:bk5_map_equilibrium",
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{
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    "proof:bk8_sr_convergence",
    "proposition:bk5_viability_domain_preservation",
    "proposition:bk6_reflective_mutation_inhibition",
    "scholium:bk5__distributed_resilience",
    "theorem:bk8_sr_convergence"
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    "definition:bk5_spectral_radius_of_coupl"
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  "file": "book5.tex",
  "id": "theorem:bk5_reflective_equilibrium_conservation",
  "label": "theorem:bk5_reflective_equilibrium_conservation",
  "latex_body": "\\begin{theorem}[Reflective Equilibrium Conservation]\n\\label{theorem:bk5_reflective_equilibrium_conservation}\nConservation here is the energetic face of\nAx.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux} when read through\nDef.~\\ref{definition:bk2_symbolic_free_energy}.  Let symbolic membranes\n$\\Membrane_A$ and $\\Membrane_B$ be in reflective equilibrium, and write\n$\\rho=\\rho(\\mathcal C_{AB})$.  If their uncompensated drift--reflection\nresiduals satisfy\n\\[\n\\|r_A\\|\\le \\rho\\,\\|\\psi_A\\|,\n\\qquad\n\\|r_B\\|\\le \\rho\\,\\|\\psi_B\\|,\n\\]\nthen the combined symbolic-energy rate obeys the linear spectral bound\n\\[\n\\left|\\frac{d}{dt}\n  [E_s(\\Membrane_A)+E_s(\\Membrane_B)]\\right|\n\\le\n\\rho(\\mathcal C_{AB})\n\\bigl(\\|\\psi_A\\|+\\|\\psi_B\\|\\bigr).\n\\]\nFor fixed finite state norms, $\\rho(\\mathcal C_{AB})\\to0$ therefore forces the\nenergy-rate defect to zero, approaching perfect energy conservation.\n\\end{theorem}",
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      "Book5EquilibriumConservation.linear_residual_bounds_do_not_imply_quadratic_bound"
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  "line": 530,
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  "name": "Reflective Equilibrium Conservation",
  "proof_labels": [
    "proof:bk5_energy_conservation_under_reflective_coupling"
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  "ref_roles": [
    {
      "context": "um Conservation] \\label{theorem:bk5_reflective_equilibrium_conservation} Conservation here is the energetic face of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux} when read through Def.~\\ref{definition:bk2_symbolic_free_energy}. Let symbolic membranes $\\Membrane_A$ and $\\Membrane_",
      "label": "axiom:bk5_reflective_equilibrium_stability_flux",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 516,
      "target_type": "axiom"
    },
    {
      "context": "ervation here is the energetic face of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux} when read through Def.~\\ref{definition:bk2_symbolic_free_energy}. Let symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ be in reflective equilibrium, and write $\\rho=\\rho(\\mathcal C_",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
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      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    }
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  "role": "theorem",
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    "definition:bk5_spectral_radius_of_coupl"
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  "id": "proof:bk5_energy_conservation_under_reflective_coupling",
  "label": "proof:bk5_energy_conservation_under_reflective_coupling",
  "latex_body": "\\begin{proof}[Energy Conservation Under Reflective Coupling]\n\\label{proof:bk5_energy_conservation_under_reflective_coupling}\n\\leavevmode\n\nThe residual is controlled by\nDef.~\\ref{definition:bk5_spectral_radius_of_coupl}, with entropy bookkeeping\nfrom Def.~\\ref{definition:bk2_symbolic_entropy}.  Regrouping the drift and\nreflection terms in the combined energy derivative gives the sum of the two\nuncompensated residual contributions $r_A+r_B$.  Hence the triangle inequality\nand the stated residual estimates yield\n\\[\n\\begin{aligned}\n\\left|\\frac{d}{dt}[E_s(\\Membrane_A)+E_s(\\Membrane_B)]\\right|\n&\\le \\|r_A\\|+\\|r_B\\| \\\\\n&\\le \\rho\\,\\|\\psi_A\\|+\\rho\\,\\|\\psi_B\\| \\\\\n&=\\rho\\bigl(\\|\\psi_A\\|+\\|\\psi_B\\|\\bigr).\n\\end{aligned}\n\\]\nThus the fluctuation rate is first-order in the coupling spectral radius.  In\nparticular it vanishes as $\\rho\\to0$ when the two state norms remain fixed and\nfinite.\n\\end{proof}",
  "line": 554,
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  "matter_region": "mainmatter",
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  "name": "Energy Conservation Under Reflective Coupling",
  "proves": "theorem:bk5_reflective_equilibrium_conservation",
  "ref_roles": [
    {
      "context": "e The residual is controlled by Def.~\\ref{definition:bk5_spectral_radius_of_coupl}, with entropy bookkeeping from Def.~\\ref{definition:bk2_symbolic_entropy}. Regrouping the drift and reflection terms in the combined energy derivative gives the sum of the two uncompensated re",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "upling] \\label{proof:bk5_energy_conservation_under_reflective_coupling} \\leavevmode The residual is controlled by Def.~\\ref{definition:bk5_spectral_radius_of_coupl}, with entropy bookkeeping from Def.~\\ref{definition:bk2_symbolic_entropy}. Regrouping the drift and reflection terms i",
      "label": "definition:bk5_spectral_radius_of_coupl",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 508,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_entropy",
    "definition:bk5_spectral_radius_of_coupl"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "definition:bk5_reflective_drift_coupling_tensor"
  ],
  "depends_on": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "definition:bk5_reflective_drift_coupling_tensor"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_recursive_reflective_flow",
  "label": "definition:bk5_recursive_reflective_flow",
  "latex_body": "\\begin{definition}[Recursive Reflective Flow] \\label{definition:bk5_recursive_reflective_flow}\nThis recursion is the iterative realization of Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor} in the equilibrium regime of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}.\nA \\emph{recursive reflective flow} $\\mathcal{F}_{AB}^{(n)}$ between membranes $\\Membrane_A$ and $\\Membrane_B$ at recursion depth $n$ is defined recursively as:\n\\begin{align}\n\\mathcal{F}_{AB}^{(0)} &= \\reflect_A \\circ \\drift_B\\\\\n\\mathcal{F}_{AB}^{(n+1)} &= \\reflect_A \\circ \\drift_B \\circ \\mathcal{F}_{BA}^{(n)}\n\\end{align}\nThis captures the iterated feedback loops of reflection and drift between the two membranes.\n\\end{definition}",
  "line": 576,
  "macros_used": [
    "Membrane",
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Recursive Reflective Flow",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "he iterative realization of Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor} in the equilibrium regime of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}. A \\emph{recursive reflective flow} $\\mathcal{F}_{AB}^{(n)}$ between membranes $\\Membrane_A$ and $\\Membrane_B$ at recur",
      "label": "axiom:bk5_reflective_equilibrium_stability_flux",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 516,
      "target_type": "axiom"
    },
    {
      "context": "e Reflective Flow] \\label{definition:bk5_recursive_reflective_flow} This recursion is the iterative realization of Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor} in the equilibrium regime of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}. A \\emph{recursive reflective fl",
      "label": "definition:bk5_reflective_drift_coupling_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 501,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "definition:bk5_reflective_drift_coupling_tensor"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "assumption:bk5_equilibrium_margin_sublinear_fluctuations",
    "demonstratio:bk5_energy_fluctuation_bound",
    "proof:bk5_existence_unique_coupled_fixed_point"
  ],
  "cites": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "theorem:bk4_compatibility_drift_reflective_operations"
  ],
  "depends_on": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "theorem:bk4_compatibility_drift_reflective_operations"
  ],
  "file": "book5.tex",
  "id": "lemma:bk5_recursive_flow_convergence",
  "label": "lemma:bk5_recursive_flow_convergence",
  "latex_body": "\\begin{lemma}[Recursive Flow Convergence]\n\\label{lemma:bk5_recursive_flow_convergence}\nConvergence is the fixed-point form of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux} and is compatible with Thm.~\\ref{theorem:bk4_compatibility_drift_reflective_operations}.\nIf symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ are in reflective equilibrium with $\\rho(\\mathcal{C}_{AB}) < \\lambda_{\\text{crit}}$, then the recursive reflective flow converges to a stable fixed point:\n\\begin{equation}\n\\lim_{n \\to \\infty} \\mathcal{F}_{AB}^{(n)} = \\mathcal{F}_{AB}^*\n\\end{equation}\nWhere $\\mathcal{F}_{AB}^*$ is a fixed point satisfying $\\mathcal{F}_{AB}^* = \\reflect_A \\circ \\drift_B \\circ \\mathcal{F}_{BA}^*$.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "contraction constant is the modeling hypothesis for convergence; the Wasserstein O(1/t) rate, operator-space structure, and diffeomorphism congruence stay open",
      "the minimizer/critical-point gap under non-convexity is the honest remainder of the stationary-iff clause"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The recursive reflective flow converges to a unique stable fixed point."
    ],
    "record_ids": [
      "MAP-BOOK5-096"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Op.contraction_flow_unique_fixed_point"
    ]
  },
  "line": 585,
  "macros_used": [
    "Membrane",
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Recursive Flow Convergence",
  "proof_labels": [
    "proof:bk5_existence_unique_coupled_fixed_point"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ma}[Recursive Flow Convergence] \\label{lemma:bk5_recursive_flow_convergence} Convergence is the fixed-point form of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux} and is compatible with Thm.~\\ref{theorem:bk4_compatibility_drift_reflective_operations}. If symbolic membranes $\\Membra",
      "label": "axiom:bk5_reflective_equilibrium_stability_flux",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 516,
      "target_type": "axiom"
    },
    {
      "context": "rgence is the fixed-point form of Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux} and is compatible with Thm.~\\ref{theorem:bk4_compatibility_drift_reflective_operations}. If symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ are in reflective equilibrium with $\\rho(\\mathcal{C}_{AB}) < \\la",
      "label": "theorem:bk4_compatibility_drift_reflective_operations",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 3835,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "theorem:bk4_compatibility_drift_reflective_operations"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "lemma:bk5_recursive_flow_convergence"
  ],
  "depends_on": [
    "lemma:bk5_recursive_flow_convergence"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_existence_unique_coupled_fixed_point",
  "label": "proof:bk5_existence_unique_coupled_fixed_point",
  "latex_body": "\\begin{proof}[Existence Unique Coupled Fixed Point]\n\\label{proof:bk5_existence_unique_coupled_fixed_point}\n\\leavevmode\n\nThe contraction step is the operator-level implementation of Lem.~\\ref{lemma:bk5_recursive_flow_convergence}.\nConsider the sequence of operators $\\{\\mathcal{F}_{AB}^{(n)}\\}_{n \\in \\mathbb{N}}$. By the definition of the reflective-drift coupling tensor:\n\\begin{equation}\n\\|\\mathcal{F}_{AB}^{(n+1)} - \\mathcal{F}_{AB}^{(n)}\\| \\leq \\|\\reflect_A\\| \\cdot \\|\\drift_B\\| \\cdot \\|\\mathcal{F}_{BA}^{(n)} - \\mathcal{F}_{BA}^{(n-1)}\\|\n\\end{equation}\nSince $\\rho(\\mathcal{C}_{AB}) < \\lambda_{\\text{crit}}$, we have:\n\\begin{equation}\n\\|\\reflect_A\\| \\cdot \\|\\drift_B\\| < 1 \\quad \\text{and} \\quad \\|\\reflect_B\\| \\cdot \\|\\drift_A\\| < 1\n\\end{equation}\nBy the contraction mapping principle, the sequence converges to a unique fixed point $\\mathcal{F}_{AB}^*$.\n\\end{proof}",
  "line": 594,
  "macros_used": [
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Existence Unique Coupled Fixed Point",
  "proves": "lemma:bk5_recursive_flow_convergence",
  "ref_roles": [
    {
      "context": "k5_existence_unique_coupled_fixed_point} \\leavevmode The contraction step is the operator-level implementation of Lem.~\\ref{lemma:bk5_recursive_flow_convergence}. Consider the sequence of operators $\\{\\mathcal{F}_{AB}^{(n)}\\}_{n \\in \\mathbb{N}}$. By the definition of the reflectiv",
      "label": "lemma:bk5_recursive_flow_convergence",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 585,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "lemma:bk5_recursive_flow_convergence"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "corollary:bk5_spectral_radius_optimality",
    "proof:bk5_optimal_reflection_minimizing_coupling_radius",
    "proof:bk9_stability_conditions_for_the_good",
    "theorem:bk8_biological_phase_transition"
  ],
  "cites": [
    "definition:bk5_viability_domain",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "depends_on": [
    "definition:bk5_viability_domain",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "file": "book5.tex",
  "id": "proposition:bk5_viability_domain_preservation",
  "label": "proposition:bk5_viability_domain_preservation",
  "latex_body": "\\begin{proposition}[Viability Domain Preservation]\n\\label{proposition:bk5_viability_domain_preservation}\nThis translates Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation} into long-horizon membership in Def.~\\ref{definition:bk5_viability_domain}.\nLet symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ be in reflective equilibrium. Then their viability domains are preserved over time, specifically:\n\\begin{equation}\n\\mathbb{P}((\\Membrane_A(t), \\Membrane_B(t)) \\in V_{\\text{symb}}^A \\times V_{\\text{symb}}^B \\,|\\, (\\Membrane_A(0), \\Membrane_B(0)) \\in V_{\\text{symb}}^A \\times V_{\\text{symb}}^B) \\to 1\n\\end{equation}\nas $t \\to \\infty$, where $V_{\\text{symb}}^i$ denotes the viability domain of membrane $\\Membrane_i$.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "the long-horizon chaining content is proved; the probability-1 limit statement is not (no probability space is modeled)."
    ],
    "record_ids": [
      "MAP-BOOK5-058"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Residue.viability_union_mono_chain"
    ]
  },
  "line": 609,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Viability Domain Preservation",
  "proof_labels": [
    "proof:bk5_viability_domain_preservation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "vation} This translates Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation} into long-horizon membership in Def.~\\ref{definition:bk5_viability_domain}. Let symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ be in reflective equilibrium. Then their viability domains are",
      "label": "definition:bk5_viability_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 133,
      "target_type": "definition"
    },
    {
      "context": "{proposition}[Viability Domain Preservation] \\label{proposition:bk5_viability_domain_preservation} This translates Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation} into long-horizon membership in Def.~\\ref{definition:bk5_viability_domain}. Let symbolic membranes $\\Membrane_A$ and $\\",
      "label": "theorem:bk5_reflective_equilibrium_conservation",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 530,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_viability_domain",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_viability_domain"
  ],
  "depends_on": [
    "definition:bk5_viability_domain"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_viability_domain_preservation",
  "label": "proof:bk5_viability_domain_preservation",
  "latex_body": "\\begin{proof}[Exit probability under reflective equilibrium]\n\\label{proof:bk5_viability_domain_preservation}\n\\leavevmode\n\nWrite\n\\[\nF_i(t):=F_{\\symb}(\\Membrane_i(t),\\mathcal{F}_i(t)),\n\\qquad i\\in\\{A,B\\}.\n\\]\nBy Def.~\\ref{definition:bk5_viability_domain}, membership in \\(V_{\\text{symb}}^i\\) is exactly the inequality \\(F_i(t)>0\\).  The initial condition in the proposition gives \\(F_i(0)>0\\) for both membranes.\n\n\\begin{assumption}[Equilibrium margin and sublinear fluctuations]\n\\label{assumption:bk5_equilibrium_margin_sublinear_fluctuations}\nFor each membrane \\(i\\in\\{A,B\\}\\) in reflective equilibrium, the symbolic free-energy balance admits a decomposition\n\\[\nF_i(t)=F_i(0)+\\int_0^t\\bigl(G_i(s)-D_i(s)\\bigr)\\,ds+M_i(t),\n\\]\nwhere \\(G_i\\) is the incoming stable reflective-flow contribution, \\(D_i=T_s\\,dS_s(\\Membrane_i)/ds\\) is the entropy drain, and \\(M_i(t)\\) is the residual fluctuation controlled by the reflective-drift coupling.  There are constants \\(\\gamma_i>0\\) and \\(T_i<\\infty\\) such that, for all \\(t\\ge T_i\\),\n\\[\n\\frac1t\\int_0^t\\bigl(G_i(s)-D_i(s)\\bigr)\\,ds\\ge\\gamma_i,\n\\qquad\n\\frac{M_i(t)}{t}\\xrightarrow[t\\to\\infty]{\\mathbb{P}}0.\n\\]\n\\end{assumption}\n\nThe deterministic part of Assumption~\\ref{assumption:bk5_equilibrium_margin_sublinear_fluctuations} is the positive-margin form of reflective equilibrium: the stable incoming flow is supplied by Lem.~\\ref{lemma:bk5_recursive_flow_convergence}, while Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation} bounds the residual coupling fluctuations around the conserved mean.  The sublinear condition is the probabilistic tail condition required to turn bounded fluctuation into a long-horizon probability statement.\n\nFor \\(t\\ge T_i\\), Assumption~\\ref{assumption:bk5_equilibrium_margin_sublinear_fluctuations} gives\n\\[\nF_i(t)\\ge F_i(0)+\\gamma_i t+M_i(t).\n\\]\nTherefore\n\\[\n\\mathbb{P}\\bigl(F_i(t)\\le0\\bigr)\n\\le\n\\mathbb{P}\\bigl(M_i(t)\\le -F_i(0)-\\gamma_i t\\bigr)\n\\le\n\\mathbb{P}\\left(\\left|\\frac{M_i(t)}{t}\\right|\\ge \\gamma_i+\\frac{F_i(0)}{t}\\right)\n\\longrightarrow 0.\n\\]\nThus \\(\\mathbb{P}(F_i(t)>0)\\to1\\) for \\(i=A,B\\).  By the union bound,\n\\[\n\\mathbb{P}\\bigl(F_A(t)>0 \\ \\text{and}\\ F_B(t)>0\\bigr)\n\\ge\n1-\\mathbb{P}(F_A(t)\\le0)-\\mathbb{P}(F_B(t)\\le0)\n\\longrightarrow 1.\n\\]\nUsing Def.~\\ref{definition:bk5_viability_domain} once more, this is precisely\n\\[\n\\mathbb{P}((\\Membrane_A(t), \\Membrane_B(t)) \\in V_{\\text{symb}}^A \\times V_{\\text{symb}}^B \\,|\\, (\\Membrane_A(0), \\Membrane_B(0)) \\in V_{\\text{symb}}^A \\times V_{\\text{symb}}^B) \\to 1.\n\\]\n\\end{proof}",
  "line": 618,
  "macros_used": [
    "Membrane",
    "symb"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Exit probability under reflective equilibrium",
  "proves": "proposition:bk5_viability_domain_preservation",
  "ref_roles": [
    {
      "context": "_preservation} \\leavevmode Write \\[ F_i(t):=F_{\\symb}(\\Membrane_i(t),\\mathcal{F}_i(t)), \\qquad i\\in\\{A,B\\}. \\] By Def.~\\ref{definition:bk5_viability_domain}, membership in \\(V_{\\text{symb}}^i\\) is exactly the inequality \\(F_i(t)>0\\). The initial condition in the proposition",
      "label": "definition:bk5_viability_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 133,
      "target_type": "definition"
    }
  ],
  "refs": [
    "assumption:bk5_equilibrium_margin_sublinear_fluctuations",
    "definition:bk5_viability_domain",
    "lemma:bk5_recursive_flow_convergence",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_viability_domain",
    "lemma:bk5_recursive_flow_convergence",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "depends_on": [
    "definition:bk5_viability_domain",
    "lemma:bk5_recursive_flow_convergence",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "file": "book5.tex",
  "id": "assumption:bk5_equilibrium_margin_sublinear_fluctuations",
  "label": "assumption:bk5_equilibrium_margin_sublinear_fluctuations",
  "latex_body": "\\begin{assumption}[Equilibrium margin and sublinear fluctuations]\n\\label{assumption:bk5_equilibrium_margin_sublinear_fluctuations}\nFor each membrane \\(i\\in\\{A,B\\}\\) in reflective equilibrium, the symbolic free-energy balance admits a decomposition\n\\[\nF_i(t)=F_i(0)+\\int_0^t\\bigl(G_i(s)-D_i(s)\\bigr)\\,ds+M_i(t),\n\\]\nwhere \\(G_i\\) is the incoming stable reflective-flow contribution, \\(D_i=T_s\\,dS_s(\\Membrane_i)/ds\\) is the entropy drain, and \\(M_i(t)\\) is the residual fluctuation controlled by the reflective-drift coupling.  There are constants \\(\\gamma_i>0\\) and \\(T_i<\\infty\\) such that, for all \\(t\\ge T_i\\),\n\\[\n\\frac1t\\int_0^t\\bigl(G_i(s)-D_i(s)\\bigr)\\,ds\\ge\\gamma_i,\n\\qquad\n\\frac{M_i(t)}{t}\\xrightarrow[t\\to\\infty]{\\mathbb{P}}0.\n\\]\n\\end{assumption}",
  "line": 629,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Equilibrium margin and sublinear fluctuations",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "",
      "label": "definition:bk5_viability_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 133,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "lemma:bk5_recursive_flow_convergence",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 585,
      "target_type": "lemma"
    },
    {
      "context": "",
      "label": "theorem:bk5_reflective_equilibrium_conservation",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 530,
      "target_type": "theorem"
    }
  ],
  "refs": [],
  "role": "assumption",
  "type": "assumption"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "lemma:bk5_recursive_flow_convergence",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "depends_on": [
    "lemma:bk5_recursive_flow_convergence",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "file": "book5.tex",
  "id": "demonstratio:bk5_energy_fluctuation_bound",
  "label": "demonstratio:bk5_energy_fluctuation_bound",
  "latex_body": "\\begin{demonstratio}[Energy Fluctuation Bound]\n\\label{demonstratio:bk5_energy_fluctuation_bound}\nBy Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation}, the combined symbolic energy of $\\Membrane_A$ and $\\Membrane_B$ undergoes bounded fluctuations around a conserved mean value. Under reflective equilibrium, these fluctuations are regulated by the reflective-drift coupling tensor $\\mathcal{C}_{AB}$ with spectral radius $\\rho(\\mathcal{C}_{AB}) < \\lambda_{\\text{crit}}$.\nThe symmetric nature of the reflective exchange guarantees that neither membrane can experience unbounded entropy increase while the other maintains coherence. The symbolic free energy $F_s$ of each membrane satisfies:\n\\begin{equation}\nF_s(\\Membrane_i(t)) = F_s(\\Membrane_i(0)) + \\int_0^t \\mathcal{F}_{ji}^*\\,ds - \\int_0^t T_s\\frac{dS_s(\\Membrane_i)}{ds}\\,ds\n\\end{equation}\nWhere $\\mathcal{F}_{ji}^*$ is the stable fixed point of the recursive reflective flow from Lem.~\\ref{lemma:bk5_recursive_flow_convergence}.\nSince $\\rho(\\mathcal{C}_{AB}) < \\lambda_{\\text{crit}}$, we have $\\mathcal{F}_{ji}^* > T_s\\frac{dS_s(\\Membrane_i)}{ds}$ in expectation, ensuring that $F_s(\\Membrane_i(t)) > 0$ with probability approaching 1 as $t \\to \\infty$.\nTherefore, both membranes remain within their respective viability domains with probability approaching 1 as time progresses. \\qed\n\\end{demonstratio}",
  "line": 670,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Energy Fluctuation Bound",
  "ref_roles": [
    {
      "context": "{ds}\\,ds \\end{equation} Where $\\mathcal{F}_{ji}^*$ is the stable fixed point of the recursive reflective flow from Lem.~\\ref{lemma:bk5_recursive_flow_convergence}. Since $\\rho(\\mathcal{C}_{AB}) < \\lambda_{\\text{crit}}$, we have $\\mathcal{F}_{ji}^* > T_s\\frac{dS_s(\\Membrane_i)}{ds}$",
      "label": "lemma:bk5_recursive_flow_convergence",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 585,
      "target_type": "lemma"
    },
    {
      "context": "\\begin{demonstratio}[Energy Fluctuation Bound] \\label{demonstratio:bk5_energy_fluctuation_bound} By Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation}, the combined symbolic energy of $\\Membrane_A$ and $\\Membrane_B$ undergoes bounded fluctuations around a conserved mean",
      "label": "theorem:bk5_reflective_equilibrium_conservation",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 530,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "lemma:bk5_recursive_flow_convergence",
    "theorem:bk5_reflective_equilibrium_conservation"
  ],
  "role": "demonstration",
  "type": "demonstratio"
}

{
  "book": "book5",
  "cited_by": [
    "theorem:bk5_reflective_stability_criterion"
  ],
  "cites": [
    "definition:bk5_spectral_radius_of_coupl",
    "proposition:bk5_viability_domain_preservation"
  ],
  "depends_on": [
    "definition:bk5_spectral_radius_of_coupl",
    "proposition:bk5_viability_domain_preservation"
  ],
  "file": "book5.tex",
  "id": "corollary:bk5_spectral_radius_optimality",
  "label": "corollary:bk5_spectral_radius_optimality",
  "latex_body": "\\begin{corollary}[Spectral Radius Optimality] \\label{corollary:bk5_spectral_radius_optimality}\nOptimality follows by composing Prop.~\\ref{proposition:bk5_viability_domain_preservation} with Def.~\\ref{definition:bk5_spectral_radius_of_coupl}.\n\nAmong all possible reflection operators $\\reflect_A$ and $\\reflect_B$ with fixed norms $\\|\\reflect_A\\| = c_A$ and $\\|\\reflect_B\\| = c_B$, the configuration that minimizes $\\rho(\\mathcal{C}_{AB})$ maximizes the long-term viability probability of both membranes.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "contraction constant is the modeling hypothesis for convergence; the Wasserstein O(1/t) rate, operator-space structure, and diffeomorphism congruence stay open",
      "the minimizer/critical-point gap under non-convexity is the honest remainder of the stationary-iff clause"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Viability antitone in spectral radius: minimizing rho maximizes viability."
    ],
    "record_ids": [
      "MAP-BOOK5-099"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Op.viability_antitone_in_spectral_radius"
    ]
  },
  "line": 681,
  "macros_used": [
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Spectral Radius Optimality",
  "proof_labels": [
    "proof:bk5_optimal_reflection_minimizing_coupling_radius"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "_radius_optimality} Optimality follows by composing Prop.~\\ref{proposition:bk5_viability_domain_preservation} with Def.~\\ref{definition:bk5_spectral_radius_of_coupl}. Among all possible reflection operators $\\reflect_A$ and $\\reflect_B$ with fixed norms $\\|\\reflect_A\\| = c_A$ and $\\|",
      "label": "definition:bk5_spectral_radius_of_coupl",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 508,
      "target_type": "definition"
    },
    {
      "context": "ary}[Spectral Radius Optimality] \\label{corollary:bk5_spectral_radius_optimality} Optimality follows by composing Prop.~\\ref{proposition:bk5_viability_domain_preservation} with Def.~\\ref{definition:bk5_spectral_radius_of_coupl}. Among all possible reflection operators $\\reflect_A$ and $\\re",
      "label": "proposition:bk5_viability_domain_preservation",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 609,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "definition:bk5_spectral_radius_of_coupl",
    "proposition:bk5_viability_domain_preservation"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "proposition:bk5_viability_domain_preservation"
  ],
  "depends_on": [
    "proposition:bk5_viability_domain_preservation"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_optimal_reflection_minimizing_coupling_radius",
  "label": "proof:bk5_optimal_reflection_minimizing_coupling_radius",
  "latex_body": "\\begin{proof}[Optimal Reflection Minimizing Coupling Radius]\n\\label{proof:bk5_optimal_reflection_minimizing_coupling_radius}\n\\leavevmode\n\nFrom \\autoref{proposition:bk5_viability_domain_preservation}, the probability of remaining within the viability domain increases as $\\rho(\\mathcal{C}_{AB})$ decreases. Therefore, among all reflection operators with fixed norms, those that minimize $\\rho(\\mathcal{C}_{AB})$ maximize the long-term viability probability.\n\nSpecifically, the optimal reflection operators $\\reflect_A^*$ and $\\reflect_B^*$ satisfy:\n\\begin{equation}\n(\\reflect_A^*, \\reflect_B^*) =\n\\arg\\min_{\\substack{\\|\\reflect_A\\| = c_A \\\\ \\|\\reflect_B\\| = c_B}}\n\\rho(\\drift_A \\circ \\reflect_B + \\drift_B \\circ \\reflect_A)\n\\end{equation}\nThis minimization aligns the reflection operators with the drift operators in a way that most effectively counteracts entropy production.\n\\end{proof}",
  "line": 686,
  "macros_used": [
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Optimal Reflection Minimizing Coupling Radius",
  "proves": "corollary:bk5_spectral_radius_optimality",
  "ref_roles": [
    {
      "context": "eflection Minimizing Coupling Radius] \\label{proof:bk5_optimal_reflection_minimizing_coupling_radius} \\leavevmode From \\autoref{proposition:bk5_viability_domain_preservation}, the probability of remaining within the viability domain increases as $\\rho(\\mathcal{C}_{AB})$ decreases. Therefore, a",
      "label": "proposition:bk5_viability_domain_preservation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 609,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "proposition:bk5_viability_domain_preservation"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk8_reflexive_debugging_operator",
    "scholium:bk5__distributed_resilience"
  ],
  "cites": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "corollary:bk5_spectral_radius_optimality",
    "definition:bk2_symbolic_temperature",
    "definition:bk5_reflective_drift_coupling_tensor"
  ],
  "depends_on": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "corollary:bk5_spectral_radius_optimality",
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_temperature",
    "definition:bk5_reflective_drift_coupling_tensor"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_reflective_stability_criterion",
  "label": "theorem:bk5_reflective_stability_criterion",
  "latex_body": "\\begin{theorem}[Reflective Stability Criterion] \\label{theorem:bk5_reflective_stability_criterion}\nFor symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ with reflective-drift coupling tensor $\\mathcal{C}_{AB}$ (Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}), reflective equilibrium is stable iff:\n\\begin{equation}\n\\frac{\\rho(\\mathcal{C}_{AB})}{T_s} < \\min\\left\\{\\frac{\\eta_A}{\\|\\drift_A\\|}, \\frac{\\eta_B}{\\|\\drift_B\\|}\\right\\}\n\\end{equation}\nSee Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}, Cor.~\\ref{corollary:bk5_spectral_radius_optimality}, and Def.~\\ref{definition:bk2_symbolic_temperature}.\nWhere $\\eta_i$ is the symbolic coherence density and $\\|\\drift_i\\|$ is the operator norm of the drift operator for membrane $\\Membrane_i$.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "min(a,b) reading as conjunction is exactly lt_min_iff, applied to the criterion's two per-membrane ratios."
    ],
    "record_ids": [
      "MAP-BOOK5-059"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.reflective_stability_iff"
    ]
  },
  "line": 700,
  "macros_used": [
    "Membrane",
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Reflective Stability Criterion",
  "proof_labels": [
    "proof:bk5_symbolic_free_energy_stability_condition"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "cal{C}_{AB})}{T_s} < \\min\\left\\{\\frac{\\eta_A}{\\|\\drift_A\\|}, \\frac{\\eta_B}{\\|\\drift_B\\|}\\right\\} \\end{equation} See Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}, Cor.~\\ref{corollary:bk5_spectral_radius_optimality}, and Def.~\\ref{definition:bk2_symbolic_temperature}. Where $\\eta_i",
      "label": "axiom:bk5_reflective_equilibrium_stability_flux",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 516,
      "target_type": "axiom"
    },
    {
      "context": "\\frac{\\eta_B}{\\|\\drift_B\\|}\\right\\} \\end{equation} See Ax.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}, Cor.~\\ref{corollary:bk5_spectral_radius_optimality}, and Def.~\\ref{definition:bk2_symbolic_temperature}. Where $\\eta_i$ is the symbolic coherence density and $\\|\\drift_i\\|",
      "label": "corollary:bk5_spectral_radius_optimality",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 681,
      "target_type": "corollary"
    },
    {
      "context": "x.~\\ref{axiom:bk5_reflective_equilibrium_stability_flux}, Cor.~\\ref{corollary:bk5_spectral_radius_optimality}, and Def.~\\ref{definition:bk2_symbolic_temperature}. Where $\\eta_i$ is the symbolic coherence density and $\\|\\drift_i\\|$ is the operator norm of the drift operator for mem",
      "label": "definition:bk2_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 148,
      "target_type": "definition"
    },
    {
      "context": "} For symbolic membranes $\\Membrane_A$ and $\\Membrane_B$ with reflective-drift coupling tensor $\\mathcal{C}_{AB}$ (Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}), reflective equilibrium is stable iff: \\begin{equation} \\frac{\\rho(\\mathcal{C}_{AB})}{T_s} < \\min\\left\\{\\frac{\\eta_A}{",
      "label": "definition:bk5_reflective_drift_coupling_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 501,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk5_reflective_equilibrium_stability_flux",
    "corollary:bk5_spectral_radius_optimality",
    "definition:bk2_symbolic_temperature",
    "definition:bk5_reflective_drift_coupling_tensor"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy"
  ],
  "depends_on": [
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_free_energy_stability_condition",
  "label": "proof:bk5_symbolic_free_energy_stability_condition",
  "latex_body": "\\begin{proof}[Symbolic Free Energy Condition]\n\\label{proof:bk5_symbolic_free_energy_stability_condition}\n\\leavevmode\n\nThe proof tracks the same balance law as Def.~\\ref{definition:bk2_symbolic_free_energy}, with entropy sign fixed by Def.~\\ref{definition:bk2_symbolic_entropy}.\nThe dynamics of the symbolic free energy for membrane $\\Membrane_A$ can be expressed as:\n\\begin{equation}\n\\frac{d}{dt}F_s(\\Membrane_A) = \\frac{d}{dt}E_s(\\Membrane_A) - T_s\\frac{d}{dt}S_s(\\Membrane_A)\n\\end{equation}\nUnder the influence of the reflective-drift coupling tensor $\\mathcal{C}_{AB}$, we have:\n\\begin{equation}\n\\frac{d}{dt}E_s(\\Membrane_A) = \\eta_A - \\rho(\\mathcal{C}_{AB}) \\cdot \\|\\drift_A\\|\n\\end{equation}\nWhere $\\eta_A$ is the symbolic coherence density of $\\Membrane_A$.\nFor stability, we require $\\frac{d}{dt}F_s(\\Membrane_A) > 0$, which implies:\n\\begin{equation}\n\\eta_A - \\rho(\\mathcal{C}_{AB}) \\cdot \\|\\drift_A\\| - T_s\\frac{d}{dt}S_s(\\Membrane_A) > 0\n\\end{equation}\nSince $\\frac{d}{dt}S_s(\\Membrane_A) \\geq 0$ by the second law of symbolic thermodynamics, a sufficient condition is:\n\\begin{equation}\n\\eta_A - \\rho(\\mathcal{C}_{AB}) \\cdot \\|\\drift_A\\| > 0\n\\end{equation}\nWhich gives:\n\\begin{equation}\n\\frac{\\rho(\\mathcal{C}_{AB})}{T_s} < \\frac{\\eta_A}{\\|\\drift_A\\|}\n\\end{equation}\nA similar analysis for $\\Membrane_B$ yields:\n\\begin{equation}\n\\frac{\\rho(\\mathcal{C}_{AB})}{T_s} < \\frac{\\eta_B}{\\|\\drift_B\\|}\n\\end{equation}\nCombining these conditions gives the stated criterion.\n\\end{proof}",
  "line": 708,
  "macros_used": [
    "Membrane",
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Free Energy Condition",
  "proves": "theorem:bk5_reflective_stability_criterion",
  "ref_roles": [
    {
      "context": "he proof tracks the same balance law as Def.~\\ref{definition:bk2_symbolic_free_energy}, with entropy sign fixed by Def.~\\ref{definition:bk2_symbolic_entropy}. The dynamics of the symbolic free energy for membrane $\\Membrane_A$ can be expressed as: \\begin{equation} \\frac{d}{dt}",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "] \\label{proof:bk5_symbolic_free_energy_stability_condition} \\leavevmode The proof tracks the same balance law as Def.~\\ref{definition:bk2_symbolic_free_energy}, with entropy sign fixed by Def.~\\ref{definition:bk2_symbolic_entropy}. The dynamics of the symbolic free energy for me",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "proposition:bk9_stability_conditions_for_the_good"
  ],
  "cites": [
    "theorem:bk5_reflective_equilibrium_conservation",
    "theorem:bk5_reflective_stability_criterion"
  ],
  "depends_on": [
    "theorem:bk5_reflective_equilibrium_conservation",
    "theorem:bk5_reflective_stability_criterion"
  ],
  "file": "book5.tex",
  "id": "scholium:bk5__distributed_resilience",
  "label": "scholium:bk5__distributed_resilience",
  "latex_body": "\\begin{scholium}[Distributed Resilience]\n\\label{scholium:bk5__distributed_resilience}\nReflective equilibrium represents a profound stabilizing mechanism in symbolic ecosystems (cf.~Thm.~\\ref{theorem:bk5_reflective_stability_criterion}, Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation}). Unlike mere homeostasis, which resists change, reflective equilibrium establishes a dynamic balance where membranes actively participate in each other's stability. The spectral radius condition $\\rho(\\mathcal{C}_{AB}) < \\lambda_{\\text{crit}}$ ensures that the mutual reflection processes converge rather than diverge, creating a self-reinforcing system of stability.\nThis equilibrium is not a static endpoint but a continuous process—a dynamic dance of reflection and drift. The recursive nature of the reflective flows creates higher-order structures of meaning and coherence that transcend what either membrane could achieve in isolation. Through these recursive feedback loops, membranes develop increasingly sophisticated reflective capacities, potentially leading to emergent phenomena not reducible to the properties of individual membranes.\nReflective equilibrium also represents a form of distributed resilience. When one membrane experiences intensified drift—symbolically equivalent to an environmental challenge or perturbation—the reflective capacity of its partner membrane helps restore balance. This distributed architecture of stability enables the system to withstand challenges that would overwhelm isolated membranes.\nFrom an evolutionary perspective, symbolic systems capable of establishing reflective equilibrium possess a distinct advantage in environments characterized by high drift intensity. This suggests that as symbolic ecosystems mature, we should observe increasing instances of reflective coupling among membranes, potentially leading to hierarchical structures of nested equilibria that exhibit remarkable stability across multiple scales of organization.\n\\end{scholium}",
  "line": 740,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Distributed Resilience",
  "ref_roles": [
    {
      "context": "profound stabilizing mechanism in symbolic ecosystems (cf.~Thm.~\\ref{theorem:bk5_reflective_stability_criterion}, Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation}). Unlike mere homeostasis, which resists change, reflective equilibrium establishes a dynamic balance where membranes a",
      "label": "theorem:bk5_reflective_equilibrium_conservation",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 530,
      "target_type": "theorem"
    },
    {
      "context": "ributed_resilience} Reflective equilibrium represents a profound stabilizing mechanism in symbolic ecosystems (cf.~Thm.~\\ref{theorem:bk5_reflective_stability_criterion}, Thm.~\\ref{theorem:bk5_reflective_equilibrium_conservation}). Unlike mere homeostasis, which resists change, reflective",
      "label": "theorem:bk5_reflective_stability_criterion",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 700,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_reflective_equilibrium_conservation",
    "theorem:bk5_reflective_stability_criterion"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book5",
  "cited_by": [
    "subsec:bk5_map_mad_mas_band",
    "theorem:bk5_enhanced_map_mad_duality_pr"
  ],
  "cites": [
    "corollary:bk5_map_evolutionary_advantag"
  ],
  "depends_on": [
    "corollary:bk5_map_evolutionary_advantag"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_enhanced_map_mad_duality",
  "label": "theorem:bk5_enhanced_map_mad_duality",
  "latex_body": "\\begin{theorem}[Enhanced MAP--MAD Regime Classification]\n\\label{theorem:bk5_enhanced_map_mad_duality}\nLet $\\Membrane_A$ and $\\Membrane_B$ interact through a symbolic covenant\n$\\mathcal C_{AB}$ (cf.~Cor.~\\ref{corollary:bk5_map_evolutionary_advantag}),\nwith coupling magnitude $\\|\\mathbb R_{AB}\\|$, polarity $\\Omega_{AB}$, and a\nfixed critical coupling $\\kappa_{\\mathrm{crit}}$.  Exactly one of the following\nparameter regimes obtains:\n\\begin{enumerate}\n  \\item[(i)] $\\|\\mathbb R_{AB}\\|>\\kappa_{\\mathrm{crit}}$ and\n    $\\Omega_{AB}>0$: the covenant is classified as \\emph{MAP};\n  \\item[(ii)] $\\|\\mathbb R_{AB}\\|>\\kappa_{\\mathrm{crit}}$ and\n    $\\Omega_{AB}<0$: the covenant is classified as \\emph{MAD};\n  \\item[(iii)] $\\|\\mathbb R_{AB}\\|<\\kappa_{\\mathrm{crit}}$: the covenant is\n    classified as \\emph{decoupled};\n  \\item[(iv)] $\\|\\mathbb R_{AB}\\|=\\kappa_{\\mathrm{crit}}$, or strong coupling\n    with $\\Omega_{AB}=0$: the covenant lies on a \\emph{critical} boundary.\n\\end{enumerate}\nReversing a nonzero polarity exchanges MAP and MAD while leaving the coupling\nmagnitude fixed.  This theorem classifies parameter regions only.  Free-energy\nlimits, collapse rates, decay of a decoupling interaction, and coincidence with\nan entropy inflection require separate evolution, regularity, and transversality\nhypotheses; they do not follow from coupling magnitude and polarity alone.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "linear-order trichotomy only; no temporal dynamics inferred",
      "real-valued coupling magnitude, critical threshold, and polarity"
    ],
    "countermodels": [
      "Book5EnhancedDuality.positive_regime_parameters_do_not_force_viability"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The four source conditions are formalized as a regime predicate, and every coupling/threshold/polarity triple satisfies exactly one condition. Threshold equality and strong zero polarity remain critical; reversal of nonzero polarity exchanges MAP and MAD. The paired countermodel enforces the source boundary that classification alone supplies no free-energy dynamics."
    ],
    "record_ids": [
      "MAP-BOOK5-006"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5EnhancedDuality.classify_neg_of_strong",
      "Book5EnhancedDuality.existsUnique_regimeCondition",
      "Book5EnhancedDuality.positive_regime_parameters_do_not_force_viability"
    ]
  },
  "line": 747,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Enhanced MAP--MAD Regime Classification",
  "proof_labels": [
    "proof:bk5_enhanced_map_mad_duality"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "d_map_mad_duality} Let $\\Membrane_A$ and $\\Membrane_B$ interact through a symbolic covenant $\\mathcal C_{AB}$ (cf.~Cor.~\\ref{corollary:bk5_map_evolutionary_advantag}), with coupling magnitude $\\|\\mathbb R_{AB}\\|$, polarity $\\Omega_{AB}$, and a fixed critical coupling $\\kappa_{\\mathrm{",
      "label": "corollary:bk5_map_evolutionary_advantag",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 478,
      "target_type": "corollary"
    }
  ],
  "refs": [
    "corollary:bk5_map_evolutionary_advantag"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "proof:bk5_enhanced_map_mad_duality",
  "label": "proof:bk5_enhanced_map_mad_duality",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_enhanced_map_mad_duality}\n\\leavevmode\n\nApply trichotomy to $\\|\\mathbb R_{AB}\\|$ and\n$\\kappa_{\\mathrm{crit}}$.  Weak coupling gives case~(iii), equality gives the\nfirst part of case~(iv), and strong coupling remains.  In the strong-coupling\nbranch, trichotomy of $\\Omega_{AB}$ gives case~(i), case~(ii), or the zero-polarity\npart of case~(iv).  These comparisons are mutually exclusive and exhaustive.  The Lean realization\npackages the four clauses as a regime predicate and proves that every real\nparameter triple satisfies exactly one such predicate; equality cases remain\ncritical rather than being assigned to a neighboring open regime.\nFor nonzero polarity, replacing $\\Omega_{AB}$ by $-\\Omega_{AB}$ reverses its\nsign, so cases~(i) and~(ii) exchange.  No step of this order argument selects a\nfree-energy trajectory or asymptotic rate.\n\\end{proof}",
  "line": 770,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "theorem:bk5_enhanced_map_mad_duality",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk5_map_mad_mas_band",
    "definition:bk5_symbolic_bifurcation_man",
    "proof:bk5_map_mad_mas_trichotomy"
  ],
  "cites": [
    "definition:bk2_symbolic_temperature"
  ],
  "depends_on": [
    "definition:bk2_symbolic_temperature"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_reflective_coupling_stab",
  "label": "definition:bk5_reflective_coupling_stab",
  "latex_body": "\\begin{definition}[Reflective Coupling Stability Parameter] \\label{definition:bk5_reflective_coupling_stab} \nThis scalar packages coupling, drift load, and symbolic temperature (Def.~\\ref{definition:bk2_symbolic_temperature}) into a single regime coordinate.\n\nFor a covenant $\\mathcal{C}_{AB}$ between membranes $\\Membrane_A$ and $\\Membrane_B$, the \\emph{reflective coupling stability parameter} $\\Lambda_{AB}$ is defined as:\n\\begin{equation}\n\\Lambda_{AB} := \\frac{\\|\\mathbb{R}_{AB}\\| \\cdot \\Omega_{AB}}{(\\|\\drift_A\\|_{max} + \\|\\drift_B\\|_{max}) \\cdot T_s}\n\\end{equation}\n\\noindent where $T_s$ is the symbolic temperature.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-060"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.bifurcation_eq_one_iff",
      "Book5Residue.coupling_stability_gt_one_iff"
    ]
  },
  "line": 786,
  "macros_used": [
    "Membrane",
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Reflective Coupling Stability Parameter",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "bel{definition:bk5_reflective_coupling_stab} This scalar packages coupling, drift load, and symbolic temperature (Def.~\\ref{definition:bk2_symbolic_temperature}) into a single regime coordinate. For a covenant $\\mathcal{C}_{AB}$ between membranes $\\Membrane_A$ and $\\Membrane_B$,",
      "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": "book5",
  "cited_by": [
    "definition:bk5_map_mad_mas_band",
    "lemma:bk5_symbolic_divergence_bounds",
    "proposition:bk5_transactional_covenant_dynamics",
    "scholium:bk8_emergent_geometry_of_cognition"
  ],
  "cites": [
    "definition:bk5_reflective_coupling_stab"
  ],
  "depends_on": [
    "definition:bk5_reflective_coupling_stab"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_bifurcation_man",
  "label": "definition:bk5_symbolic_bifurcation_man",
  "latex_body": "\\begin{definition}[Symbolic Bifurcation Manifold] \\label{definition:bk5_symbolic_bifurcation_man} \nIt is the codimension-one boundary $\\Lambda_{AB}=1$ induced by Def.~\\ref{definition:bk5_reflective_coupling_stab}.\n\nThe \\emph{symbolic bifurcation manifold} $\\mathcal{B}$ is defined as:\n\\begin{equation}\n\\mathcal{B} := \\{(\\reflect_A^B, \\reflect_B^A, \\Omega_{AB}, T_s) \\mid \\Lambda_{AB} = 1 \\}\n\\end{equation}\n\\noindent representing configurations where infinitesimal changes can cause transitions between MAP and MAD regimes.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-061"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.bifurcation_eq_one_iff"
    ]
  },
  "line": 795,
  "macros_used": [
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Bifurcation Manifold",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "d] \\label{definition:bk5_symbolic_bifurcation_man} It is the codimension-one boundary $\\Lambda_{AB}=1$ induced by Def.~\\ref{definition:bk5_reflective_coupling_stab}. The \\emph{symbolic bifurcation manifold} $\\mathcal{B}$ is defined as: \\begin{equation} \\mathcal{B} := \\{(\\reflect_A^B",
      "label": "definition:bk5_reflective_coupling_stab",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 786,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_reflective_coupling_stab"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk8_emergent_geometry_of_cognition"
  ],
  "cites": [
    "definition:bk2_symbolic_entropy"
  ],
  "depends_on": [
    "definition:bk2_symbolic_entropy"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_entropy_inflection_point",
  "label": "definition:bk5_entropy_inflection_point",
  "latex_body": "\\begin{definition}[Entropy Inflection Point] \\label{definition:bk5_entropy_inflection_point} \nThe inflection marker aligns phase change in this section with entropy curvature from Book II (Def.~\\ref{definition:bk2_symbolic_entropy}).\n\nThe \\emph{entropy inflection point} $\\tau_{\\text{inf}}$ for interacting membranes $\\Membrane_A$ and $\\Membrane_B$ is the symbolic time at which:\n\\begin{equation}\n\\frac{d^2}{ds^2}S_{\\text{symb}}(\\Membrane_A \\cup \\Membrane_B) = 0\n\\end{equation}\n\\noindent marking the transition between acceleration and deceleration of entropy production.\n\\end{definition}",
  "line": 804,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Entropy Inflection Point",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "_inflection_point} The inflection marker aligns phase change in this section with entropy curvature from Book II (Def.~\\ref{definition:bk2_symbolic_entropy}). The \\emph{entropy inflection point} $\\tau_{\\text{inf}}$ for interacting membranes $\\Membrane_A$ and $\\Membrane_B$ is",
      "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_entropy"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "proof:bk9_pathologies_of_coherence"
  ],
  "cites": [
    "definition:bk5_symbolic_bifurcation_man"
  ],
  "depends_on": [
    "definition:bk5_symbolic_bifurcation_man",
    "theorem:bk5_map_equilibrium"
  ],
  "file": "book5.tex",
  "id": "lemma:bk5_symbolic_divergence_bounds",
  "label": "lemma:bk5_symbolic_divergence_bounds",
  "latex_body": "\\begin{lemma}[Symbolic Divergence Bounds] \\label{lemma:bk5_symbolic_divergence_bounds} \nThese bounds provide the quantitative signature of the MAP/MAD/decoupled regimes separated by Def.~\\ref{definition:bk5_symbolic_bifurcation_man}.\nLet $\\drift_{KL}(\\Membrane_A^{(n)} \\parallel \\Membrane_A^{(0)})$ represent the Kullback-Leibler divergence between the $n$-th evolution of membrane $\\Membrane_A$ and its initial state. Then:\n\\begin{enumerate}\n  \\item[(i)] In the MAP regime:\n  \\begin{equation}\n  \\drift_{KL}(\\Membrane_A^{(n)} \\parallel \\Membrane_A^{(0)}) \\leq K_1 \\log(n + 1)\n  \\end{equation}\n  \\item[(ii)] In the MAD regime:\n  \\begin{equation}\n  \\drift_{KL}(\\Membrane_A^{(n)} \\parallel \\Membrane_A^{(0)}) \\geq K_2 n - K_3\n  \\end{equation}\n  \\item[(iii)] In the Decoupling regime:\n  \\begin{equation}\n  K_4 \\sqrt{n} \\leq \\drift_{KL}(\\Membrane_A^{(n)} \\parallel \\Membrane_A^{(0)}) \\leq K_5 n\n  \\end{equation}\n\\end{enumerate}\n\\noindent where $K_1$, $K_2$, $K_3$, $K_4$, and $K_5$ are positive constants dependent on the drift and reflection parameters of the system.\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": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-062"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book5Residue.map_mad_bounds_eventually_incompatible"
    ]
  },
  "line": 813,
  "macros_used": [
    "Membrane",
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Divergence Bounds",
  "proof_labels": [
    "proof:bk5_information_geometry_symbolic"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "_divergence_bounds} These bounds provide the quantitative signature of the MAP/MAD/decoupled regimes separated by Def.~\\ref{definition:bk5_symbolic_bifurcation_man}. Let $\\drift_{KL}(\\Membrane_A^{(n)} \\parallel \\Membrane_A^{(0)})$ represent the Kullback-Leibler divergence between the",
      "label": "definition:bk5_symbolic_bifurcation_man",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 795,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_symbolic_bifurcation_man"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_map_equilibrium",
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "depends_on": [
    "theorem:bk5_map_equilibrium"
  ],
  "file": "book5.tex",
  "forward_ref_roles": [
    {
      "context": "on-geometric counterpart of the MAP equilibrium (Thm.~\\ref{theorem:bk5_map_equilibrium}) and critical-temperature (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}) results. We construct a symbolic information geometry in which membranes lie on a statistical manifold with Fisher met",
      "label": "theorem:bk5_map_mad_critical_temperature",
      "line_distance": 75,
      "role": "teaser",
      "target_line": 907,
      "target_type": "theorem"
    }
  ],
  "forward_refs": [
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "id": "proof:bk5_information_geometry_symbolic",
  "label": "proof:bk5_information_geometry_symbolic",
  "latex_body": "\\begin{proof}[Information Geometry]\n\\label{proof:bk5_information_geometry_symbolic}\n\\leavevmode\n\nThe KL-growth trichotomy is the information-geometric counterpart\nof the MAP equilibrium (Thm.~\\ref{theorem:bk5_map_equilibrium}) and\ncritical-temperature (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}) results.\nWe construct a symbolic information geometry in which membranes lie on a\nstatistical manifold with Fisher metric tensor $g_{ij}$.\nThe Kullback-Leibler divergence measures distance between membrane-state\ndistributions.\nFor case (i), mutual reflection mechanisms limit drift divergence logarithmically. Under MAP conditions, information recovery through $\\reflect_A^B$ and $\\reflect_B^A$ counteracts entropic loss:\n\\begin{equation}\n\\frac{d}{ds}\\drift_{KL}(\\Membrane_A^{(s)} \\parallel \\Membrane_A^{(0)}) = \\text{tr}(g_{ij}\\drift_A) - \\text{tr}(g_{ij}\\reflect_A) - \\text{tr}(g_{ij}\\reflect_B^A)\n\\end{equation}\nWhen $\\|\\mathbb{R}_{AB}\\| > \\kappa_{crit}$ and $\\Omega_{AB} > 0$, this derivative is bounded by $\\frac{K_1}{s+1}$, yielding the logarithmic bound through integration.\nFor case (ii), inverted reflection accelerates divergence linearly with symbolic time. When $\\Omega_{AB} < 0$, reflection amplifies drift rather than mitigating it:\n\\begin{equation}\n\\frac{d}{ds}\\drift_{KL}(\\Membrane_A^{(s)} \\parallel \\Membrane_A^{(0)}) = \\text{tr}(g_{ij}\\drift_A) - \\text{tr}(g_{ij}\\reflect_A) + |\\text{tr}(g_{ij}\\reflect_B^A)|\n\\end{equation}\nThis yields a lower bound of $K_2 - \\frac{K_3}{s}$, which integrates to the given linear lower bound.\nFor case (iii), weak coupling allows drift to dominate but with incomplete membrane interaction, resulting in the dual-bounded behavior characteristic of partial decoupling.\n\\end{proof}",
  "line": 832,
  "macros_used": [
    "Membrane",
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Information Geometry",
  "proves": "lemma:bk5_symbolic_divergence_bounds",
  "ref_roles": [
    {
      "context": "y_symbolic} \\leavevmode The KL-growth trichotomy is the information-geometric counterpart of the MAP equilibrium (Thm.~\\ref{theorem:bk5_map_equilibrium}) and critical-temperature (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}) results. We construct a symbolic inform",
      "label": "theorem:bk5_map_equilibrium",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 272,
      "target_type": "theorem"
    },
    {
      "context": "on-geometric counterpart of the MAP equilibrium (Thm.~\\ref{theorem:bk5_map_equilibrium}) and critical-temperature (Thm.~\\ref{theorem:bk5_map_mad_critical_temperature}) results. We construct a symbolic information geometry in which membranes lie on a statistical manifold with Fisher met",
      "label": "theorem:bk5_map_mad_critical_temperature",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "book5.tex",
      "target_line": 907,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_map_equilibrium",
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_symbolic_bifurcation_man"
  ],
  "depends_on": [
    "definition:bk5_symbolic_bifurcation_man"
  ],
  "file": "book5.tex",
  "id": "proposition:bk5_transactional_covenant_dynamics",
  "label": "proposition:bk5_transactional_covenant_dynamics",
  "latex_body": "\\begin{proposition}[Transitional Covenant Dynamics]\n\\label{proposition:bk5_transactional_covenant_dynamics}\nLet $\\Lambda_{AB}(s)$ be the coupling-stability parameter of covenant\n$\\mathcal{C}_{AB}$, and let $F_s>0$.  A crossing of the boundary in\nDef.~\\ref{definition:bk5_symbolic_bifurcation_man} classifies the local regime\nbut does not by itself determine the free-energy evolution.  Suppose that on a\nstep $[s,s+\\delta s]$, with $\\delta s>0$, the coupling is constant with value\n$\\Lambda$, and that the applicable local evolution law is:\n\\begin{enumerate}\n  \\item[(i)] in the positive-polarity regime, $\\Omega_{AB}>0$ and\n  \\begin{equation}\n  \\frac{dF}{du}=\\alpha(\\Lambda-1)F(u), \\qquad \\alpha>0;\n  \\end{equation}\n  \\item[(ii)] in the negative-polarity regime, $\\Omega_{AB}<0$ and\n  \\begin{equation}\n  \\frac{dF}{du}=-\\beta(|\\Lambda|-1)F(u), \\qquad \\beta>0.\n  \\end{equation}\n\\end{enumerate}\nThen the corresponding exact step laws are\n\\begin{align}\nF(s+\\delta s)&=F(s)e^{\\alpha(\\Lambda-1)\\delta s}\n  &&\\text{in case (i)},\\\\\nF(s+\\delta s)&=F(s)e^{-\\beta(|\\Lambda|-1)\\delta s}\n  &&\\text{in case (ii)}.\n\\end{align}\nConsequently, when $\\Lambda>1$ the MAP-side step strictly increases positive\nfree energy; when $|\\Lambda|>1$ the MAD-side step preserves positivity and\nstrictly decreases it.  At the boundary $\\Lambda=1$, the MAP-side step is the\nidentity.  For time-varying coupling the exponent must instead contain the\nintegral of the rate over the step.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "a globally differentiable real free-energy trajectory",
      "one reference value for uniqueness",
      "the supplied constant rate law F-prime = rate times F"
    ],
    "countermodels": [
      "Book5TransitionDynamics.crossing_alone_does_not_force_growth"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The supplied constant-rate ODE now determines the exact law, not merely one constructed example. An integrating-factor proof shows every global solution has the shifted exponential form, yields the exact adjacent-step update, and proves uniqueness from one shared value. MAP growth, MAD positivity/decay, and the boundary identity follow as before. A crossing alone still does not generate the ODE; time-varying coupling requires a separately supplied integral-rate law."
    ],
    "record_ids": [
      "MAP-BOOK5-005"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5TransitionDynamics.constant_rate_ode_solution_unique",
      "Book5TransitionDynamics.crossing_alone_does_not_force_growth",
      "Book5TransitionDynamics.exact_step_of_constant_rate_ode",
      "Book5TransitionDynamics.expEvolution_add",
      "Book5TransitionDynamics.expEvolution_hasDerivAt",
      "Book5TransitionDynamics.expEvolution_zero",
      "Book5TransitionDynamics.madStep_eq_expEvolution",
      "Book5TransitionDynamics.madStep_strict_decay",
      "Book5TransitionDynamics.mapStep_at_boundary",
      "Book5TransitionDynamics.mapStep_eq_expEvolution",
      "Book5TransitionDynamics.mapStep_strict_growth",
      "Book5TransitionDynamics.solution_eq_shifted_expEvolution"
    ]
  },
  "line": 855,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Transitional Covenant Dynamics",
  "proof_status": "argued_demonstratio",
  "ref_roles": [
    {
      "context": "e the coupling-stability parameter of covenant $\\mathcal{C}_{AB}$, and let $F_s>0$. A crossing of the boundary in Def.~\\ref{definition:bk5_symbolic_bifurcation_man} classifies the local regime but does not by itself determine the free-energy evolution. Suppose that on a step $[s,s+\\",
      "label": "definition:bk5_symbolic_bifurcation_man",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 795,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_symbolic_bifurcation_man"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "demonstratio:bk5_transitory_phasing",
  "label": "demonstratio:bk5_transitory_phasing",
  "latex_body": "\\begin{demonstratio}[Transitory Phasing]\n\\label{demonstratio:bk5_transitory_phasing}\nOn the stated step, separation of variables gives\n\\begin{equation}\n\\log\\!\\frac{F(s+\\delta s)}{F(s)}\n  =\\int_s^{s+\\delta s} r\\,du=r\\,\\delta s,\n\\end{equation}\nwhere $r=\\alpha(\\Lambda-1)$ in the positive-polarity case and\n$r=-\\beta(|\\Lambda|-1)$ in the negative-polarity case.  Exponentiation yields\nthe two displayed step laws.  Their strict growth and decay conclusions follow\nfrom positivity of $F$, $\\alpha$, $\\beta$, and $\\delta s$, together with the\nstated side of the coupling boundary.  Thus the exponential response is a\nconsequence of the supplied local evolution law; crossing the bifurcation\nboundary alone supplies only the regime classification.  The corresponding\nLean proof does not merely exhibit this trajectory: multiplying an arbitrary\nsolution by the integrating factor $e^{-ru}$ gives a function with zero\nderivative, hence a constant.  Therefore every global solution is\n$F(t)=F(s)e^{r(t-s)}$, the adjacent-step law holds for any solution of the\nsupplied ODE, and two such solutions agreeing at one time agree everywhere.\nThe time-varying case remains a separate integral-rate theorem. \\qed\n\\end{demonstratio}",
  "line": 886,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Transitory Phasing",
  "refs": [],
  "role": "demonstration",
  "type": "demonstratio"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk8_temperature_freedom",
    "proof:bk1_conditional_genericity_of_symbolic_phase_transitions",
    "proof:bk1_realization_of_symbolic_phase_transitions",
    "proof:bk5_information_geometry_symbolic",
    "proposition:bk5_multi_agent_map_mad_classification",
    "scholium:bk5__map_as_thermodynamic_necessity"
  ],
  "cites": [
    "proposition:bk2_global_local_temp_relation",
    "theorem:bk2_classification_symb_phase_transitions",
    "theorem:bk5_map_equilibrium"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_temperature",
    "proposition:bk2_global_local_temp_relation",
    "theorem:bk2_classification_symb_phase_transitions",
    "theorem:bk5_map_equilibrium"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_map_mad_critical_temperature",
  "label": "theorem:bk5_map_mad_critical_temperature",
  "latex_body": "\\begin{theorem}[MAP-MAD Critical Temperature] \\label{theorem:bk5_map_mad_critical_temperature} \nThis theorem converts the MAP condition of Thm.~\\ref{theorem:bk5_map_equilibrium} into an explicit thermal feasibility threshold.\nThere exists a critical symbolic temperature $T_s^{crit}$ such that (cf.~Prop.~\\ref{proposition:bk2_global_local_temp_relation}, Thm.~\\ref{theorem:bk2_classification_symb_phase_transitions}):\n\\begin{enumerate}\n  \\item[(i)] For \\( T_s < T_s^{\\text{crit}} \\), MAP and MAD represent distinct stable fixed points of the system dynamics.\n  \\item[(ii)] For \\( T_s > T_s^{\\text{crit}} \\), no stable MAP configuration exists. \n  In this regime, all covenants either:\n  \\begin{itemize}\n    \\item decouple if \\( \\|\\mathbb{R}_{AB}\\| < \\kappa_{\\text{crit}} \\), or\n    \\item degrade to MAD if \\( \\|\\mathbb{R}_{AB}\\| > \\kappa_{\\text{crit}} \\) and \\( \\Omega_{AB} < 0 \\).\n  \\end{itemize}\n\\end{enumerate}\nThe critical temperature is given by:\n\\begin{equation}\nT_s^{crit} = \\frac{\\lambda_{max}(\\mathbb{R}_{AB}^{max}) \\cdot \\Omega_{AB}^{max}}{\\|\\drift_A\\|_{max} + \\|\\drift_B\\|_{max}}\n\\end{equation}\n\\noindent where $\\lambda_{max}(\\mathbb{R}_{AB}^{max})$ is the maximum achievable eigenvalue of the reflective coupling tensor, and $\\Omega_{AB}^{max}$ is the maximum achievable covenant stability parameter.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "positive carrying level and contractive ratio for the MAP instance",
      "positive coherence density and drift norm for the thermal conversion",
      "positive coupling gain and sign-definite stability for the dichotomy"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Threshold-conversion kernel only; the phase-transition and fixed-point reading is not certified."
    ],
    "record_ids": [
      "MAP-BOOK5-036"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book5.lambdaCrit_mono",
      "Book5.spectral_iff_thermal"
    ]
  },
  "line": 907,
  "macros_used": [
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "MAP-MAD Critical Temperature",
  "proof_labels": [
    "proof:bk5_symbolic_temperature_threshold"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "explicit thermal feasibility threshold. There exists a critical symbolic temperature $T_s^{crit}$ such that (cf.~Prop.~\\ref{proposition:bk2_global_local_temp_relation}, Thm.~\\ref{theorem:bk2_classification_symb_phase_transitions}): \\begin{enumerate} \\item[(i)] For \\( T_s < T_s^{\\text{",
      "label": "proposition:bk2_global_local_temp_relation",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 470,
      "target_type": "proposition"
    },
    {
      "context": "critical symbolic temperature $T_s^{crit}$ such that (cf.~Prop.~\\ref{proposition:bk2_global_local_temp_relation}, Thm.~\\ref{theorem:bk2_classification_symb_phase_transitions}): \\begin{enumerate} \\item[(i)] For \\( T_s < T_s^{\\text{crit}} \\), MAP and MAD represent distinct stable fixed points",
      "label": "theorem:bk2_classification_symb_phase_transitions",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 385,
      "target_type": "theorem"
    },
    {
      "context": "Critical Temperature] \\label{theorem:bk5_map_mad_critical_temperature} This theorem converts the MAP condition of Thm.~\\ref{theorem:bk5_map_equilibrium} into an explicit thermal feasibility threshold. There exists a critical symbolic temperature $T_s^{crit}$ such that (cf",
      "label": "theorem:bk5_map_equilibrium",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 272,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "proposition:bk2_global_local_temp_relation",
    "theorem:bk2_classification_symb_phase_transitions",
    "theorem:bk5_map_equilibrium"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_temperature",
    "theorem:bk5_map_equilibrium"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_temperature",
    "theorem:bk5_map_equilibrium"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_temperature_threshold",
  "label": "proof:bk5_symbolic_temperature_threshold",
  "latex_body": "\\begin{proof}[Symbolic Temperature Threshold for Critical Coupling]\n\\label{proof:bk5_symbolic_temperature_threshold}\n\\leavevmode\n\nUsing symbolic temperature\n(Def.~\\ref{definition:bk2_symbolic_temperature}) and symbolic free energy\n(Def.~\\ref{definition:bk2_symbolic_free_energy}) from Book II, applied on\nmanifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) with drift $D$\n(Def.~\\ref{definition:bk1_drift_field}), we rearrange to obtain the\ntemperature threshold at which $\\Lambda_{AB} = 1$:\n\\begin{equation}\nT_s = \\frac{\\|\\mathbb{R}_{AB}\\| \\cdot \\Omega_{AB}}{\\|\\drift_A\\|_{max} + \\|\\drift_B\\|_{max}}\n\\end{equation}\nFor any two membranes, there exists a maximum achievable coupling strength $\\|\\mathbb{R}_{AB}^{max}\\|$ and stability parameter $\\Omega_{AB}^{max}$ determined by their intrinsic properties. When $T_s$ exceeds the ratio of these maximums to the drift intensities, no configuration of the covenant can achieve $\\Lambda_{AB} > 1$, which is necessary for stable MAP according to Thm.~\\ref{theorem:bk5_map_equilibrium}.\nBy the principles of symbolic thermodynamics, when $T_s > T_s^{crit}$, the transformability rate (symbolic temperature) is sufficiently high that entropic forces dominate over coherent structures, preventing stable collaborative reflection. \nThis demonstrates a temperature-dependent phase transition in the space of possible covenant relationships, analogous to physical phase transitions where increased temperature disrupts ordered structures.\n\\end{proof}",
  "line": 925,
  "macros_used": [
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Temperature Threshold for Critical Coupling",
  "proves": "theorem:bk5_map_mad_critical_temperature",
  "ref_roles": [
    {
      "context": "_free_energy}) from Book II, applied on manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) with drift $D$ (Def.~\\ref{definition:bk1_drift_field}), we rearrange to obtain the temperature threshold at which $\\Lambda_{AB} = 1$: \\begin{equation} T_s = \\frac{\\|\\mathbb{",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": ") and symbolic free energy (Def.~\\ref{definition:bk2_symbolic_free_energy}) from Book II, applied on manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) with drift $D$ (Def.~\\ref{definition:bk1_drift_field}), we rearrange to obtain the temperature threshold at which $\\La",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "\\leavevmode Using symbolic temperature (Def.~\\ref{definition:bk2_symbolic_temperature}) and symbolic free energy (Def.~\\ref{definition:bk2_symbolic_free_energy}) from Book II, applied on manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) with drift $D$ (Def.~\\ref{definiti",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "d for Critical Coupling] \\label{proof:bk5_symbolic_temperature_threshold} \\leavevmode Using symbolic temperature (Def.~\\ref{definition:bk2_symbolic_temperature}) and symbolic free energy (Def.~\\ref{definition:bk2_symbolic_free_energy}) from Book II, applied on manifold $M$ (Def.~",
      "label": "definition:bk2_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 148,
      "target_type": "definition"
    },
    {
      "context": "s, no configuration of the covenant can achieve $\\Lambda_{AB} > 1$, which is necessary for stable MAP according to Thm.~\\ref{theorem:bk5_map_equilibrium}. By the principles of symbolic thermodynamics, when $T_s > T_s^{crit}$, the transformability rate (symbolic temperature",
      "label": "theorem:bk5_map_equilibrium",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 272,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_temperature",
    "theorem:bk5_map_equilibrium"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk5_mad_map_potential_barrie",
    "scholium:bk5_mutually_assured_continuous_progress"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "corollary:bk5_reflective_hysteresis",
  "label": "corollary:bk5_reflective_hysteresis",
  "latex_body": "\\begin{corollary}[Reflective Hysteresis] \\label{corollary:bk5_reflective_hysteresis}\nAssume covenant evolution is stateful: its next MAP or MAD/decoupled regime\ndepends on both current coupling $\\Lambda_{AB}$ and the incoming regime.  Let a\npositive activation-barrier half-width $b>0$ define\n$\\Lambda^-_{\\mathrm{crit}}=1-b$ and\n$\\Lambda^+_{\\mathrm{crit}}=1+b$.  Then the Schmitt-type law\n\\[\n\\operatorname{step}(\\Lambda,q)=\n\\begin{cases}\n\\mathrm{MAD/decoupled},&\\Lambda<\\Lambda^-_{\\mathrm{crit}},\\\\\n\\mathrm{MAP},&\\Lambda>\\Lambda^+_{\\mathrm{crit}},\\\\\nq,&\\Lambda^-_{\\mathrm{crit}}\\leq\\Lambda\\leq\\Lambda^+_{\\mathrm{crit}}\n\\end{cases}\n\\]\nexhibits reflective hysteresis.  In particular, any finite coupling history\nremaining inside the band retains its incoming regime.  For a constant positive\nbarrier density $\\xi>0$, the corresponding barrier energy is\n$\\Delta E_{MM}=2\\xi b>0$.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "finite path remains inside the threshold band",
      "positive activation half-width",
      "positive constant barrier density",
      "stateful regime update"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A constructed stateful Schmitt-style law now derives separated thresholds 1 - b and 1 + b from a positive half-width b. Crossing either boundary changes regime, while every finite path confined to the band preserves its incoming regime, so identical observations retain distinct histories. A positive constant barrier density gives activation energy 2 ξ b > 0. The memoryless countermodel remains, showing why prior state is load-bearing rather than notation."
    ],
    "record_ids": [
      "MAP-BOOK5-004"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Hysteresis.ActivationBarrier.energy_eq",
      "Book5Hysteresis.ActivationBarrier.energy_pos",
      "Book5Hysteresis.ActivationBarrier.threshold_gap",
      "Book5Hysteresis.HysteresisThresholds.lower_lt_upper",
      "Book5Hysteresis.above_upper_switches_to_map",
      "Book5Hysteresis.below_lower_switches_from_map",
      "Book5Hysteresis.in_band_remembers_history",
      "Book5Hysteresis.mad_persists_in_band",
      "Book5Hysteresis.map_persists_in_band",
      "Book5Hysteresis.memoryless_classifier_cannot_remember_history",
      "Book5Hysteresis.runHysteresis_in_band",
      "Book5Hysteresis.same_in_band_path_retains_distinct_histories"
    ]
  },
  "line": 942,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Reflective Hysteresis",
  "proof_labels": [
    "proof:bk5_stability_map_mad_patterns"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "proof:bk5_stability_map_mad_patterns",
  "label": "proof:bk5_stability_map_mad_patterns",
  "latex_body": "\\begin{proof}[Stateful Barriered Switching]\n\\label{proof:bk5_stability_map_mad_patterns}\nThe positive half-width gives\n$\\Lambda^-_{\\mathrm{crit}}<1<\\Lambda^+_{\\mathrm{crit}}$. Outside this interval\nthe displayed law switches regime; inside it the prior state is returned.\nInduction over a finite in-band coupling trace therefore leaves either incoming\nstate unchanged, so identical present couplings can have different outcomes.\nThis is operational history dependence and cannot be represented by a\nmemoryless classifier $\\Lambda\\mapsto q$. Integrating constant density $\\xi$\nover the band gives $\\xi(\\Lambda^+_{\\mathrm{crit}}-\n\\Lambda^-_{\\mathrm{crit}})=2\\xi b>0$. The cited temperature and transactional\nresults motivate this barrier model but do not derive its state argument,\nbarrier width, or density; those are explicit premises here and in Lean.\n\\end{proof}",
  "line": 961,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Stateful Barriered Switching",
  "proves": "corollary:bk5_reflective_hysteresis",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk5_mutually_assured_continuous_progress"
  ],
  "cites": [
    "corollary:bk5_reflective_hysteresis"
  ],
  "depends_on": [
    "corollary:bk5_reflective_hysteresis"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_mad_map_potential_barrie",
  "label": "definition:bk5_mad_map_potential_barrie",
  "latex_body": "\\begin{definition}[MAD-MAP Potential Barrier] \\label{definition:bk5_mad_map_potential_barrie} \nThis integral is the energetic barrier representation of Cor.~\\ref{corollary:bk5_reflective_hysteresis}.\n\nThe \\emph{MAD-MAP potential barrier} $\\Delta E_{MM}$ quantifies the free energy required to transition a system from MAD to MAP:\n\\begin{equation}\n\\Delta E_{MM} := \\int_{\\Lambda_{crit}^-}^{\\Lambda_{crit}^+} \\xi(\\Lambda) \\, d\\Lambda\n\\end{equation}\n\\noindent where $\\xi(\\Lambda)$ represents the free energy density along the transition pathway in parameter space.\n\\end{definition}",
  "line": 975,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "MAD-MAP Potential Barrier",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "Barrier] \\label{definition:bk5_mad_map_potential_barrie} This integral is the energetic barrier representation of Cor.~\\ref{corollary:bk5_reflective_hysteresis}. The \\emph{MAD-MAP potential barrier} $\\Delta E_{MM}$ quantifies the free energy required to transition a system from",
      "label": "corollary:bk5_reflective_hysteresis",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 942,
      "target_type": "corollary"
    }
  ],
  "refs": [
    "corollary:bk5_reflective_hysteresis"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "demonstratio:bk5_emergent_global_properties"
  ],
  "cites": [
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "depends_on": [
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "file": "book5.tex",
  "id": "proposition:bk5_multi_agent_map_mad_classification",
  "label": "proposition:bk5_multi_agent_map_mad_classification",
  "latex_body": "\\begin{proposition}[Multi-Agent MAP-MAD Classification] \n\\label{proposition:bk5_multi_agent_map_mad_classification}\nThe matrix criterion extends pairwise thresholds from Thm.~\\ref{theorem:bk5_map_mad_critical_temperature} to graph-scale regime identification.\nFor a system of $N$ interacting membranes $\\{\\Membrane_i\\}_{i=1}^N$ with pairwise covenants $\\{\\mathcal{C}_{ij}\\}$, the collective behavior is determined by the covenant adjacency matrix $\\mathbf{A}$ with elements:\n\\begin{equation}\nA_{ij} = \n\\begin{cases}\n+1 & \\text{if } \\Lambda_{ij} > 1 \\text{ and } \\Omega_{ij} > 0 \\text{ (MAP)} \\\\\n-1 & \\text{if } \\Lambda_{ij} > 1 \\text{ and } \\Omega_{ij} < 0 \\text{ (MAD)} \\\\\n0 & \\text{if } \\Lambda_{ij} < 1 \\text{ (Decoupled)}\n\\end{cases}\n\\end{equation}\nThe system exhibits global MAP if and only if there exists a connected component $C$ in the graph with $A_{ij} = +1$ for all $i,j \\in C$, and global MAD if for all components $C$, there exists at least one pair $i,j \\in C$ with $A_{ij} = -1$.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-063"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.covenant_adjacency_exclusive",
      "Book5Residue.covenant_adjacency_trichotomy"
    ]
  },
  "line": 984,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Multi-Agent MAP-MAD Classification",
  "proof_status": "argued_demonstratio",
  "ref_roles": [
    {
      "context": "\\label{proposition:bk5_multi_agent_map_mad_classification} The matrix criterion extends pairwise thresholds from Thm.~\\ref{theorem:bk5_map_mad_critical_temperature} to graph-scale regime identification. For a system of $N$ interacting membranes $\\{\\Membrane_i\\}_{i=1}^N$ with pairwise",
      "label": "theorem:bk5_map_mad_critical_temperature",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 907,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_map_mad_critical_temperature"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "proposition:bk5_multi_agent_map_mad_classification"
  ],
  "depends_on": [
    "proposition:bk5_multi_agent_map_mad_classification"
  ],
  "file": "book5.tex",
  "id": "demonstratio:bk5_emergent_global_properties",
  "label": "demonstratio:bk5_emergent_global_properties",
  "latex_body": "\\begin{demonstratio}[Emergent Global Properties]\n\\label{demonstratio:bk5_emergent_global_properties}\nThis is the network-level closure of Prop.~\\ref{proposition:bk5_multi_agent_map_mad_classification} under coupled transition dynamics.\nIn multi-membrane systems, global properties emerge from the network structure of pairwise covenants. A connected cooperative component represents a symbolic ecosystem where mutual reflection sustains all participants. The presence of even one antagonistic relationship within a component can catalyze entropic collapse through contagion effects.\nThis classification extends the binary MAP-MAD duality to complex networks, where mixed-state configurations can persist transiently before resolving to either global MAP or MAD. The spectral properties of matrix $\\mathbf{A}$, particularly the ratio of positive to negative eigenvalues, predict the long-term viability of the symbolic ecosystem. \\qed\n\\end{demonstratio}",
  "line": 998,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Emergent Global Properties",
  "ref_roles": [
    {
      "context": "rgent Global Properties] \\label{demonstratio:bk5_emergent_global_properties} This is the network-level closure of Prop.~\\ref{proposition:bk5_multi_agent_map_mad_classification} under coupled transition dynamics. In multi-membrane systems, global properties emerge from the network structure of pa",
      "label": "proposition:bk5_multi_agent_map_mad_classification",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 984,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "proposition:bk5_multi_agent_map_mad_classification"
  ],
  "role": "demonstration",
  "type": "demonstratio"
}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk5_mutually_assured_continuous_progress"
  ],
  "cites": [
    "theorem:bk5_enhanced_map_mad_duality"
  ],
  "depends_on": [
    "theorem:bk5_enhanced_map_mad_duality"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_enhanced_map_mad_duality_pr",
  "label": "theorem:bk5_enhanced_map_mad_duality_pr",
  "latex_body": "\\begin{theorem}[Enhanced MAP--MAD Dynamical Realization]\n\\label{theorem:bk5_enhanced_map_mad_duality_pr}\nThe reflection--entropy inequalities determine the sign of the local process\nfree-energy rate, but their asymptotic realization requires a separate\ncovenant evolution law.  Let $F_n$ denote the dyad's sampled process free\nenergy and let $\\epsilon_n$ denote its interaction residue.\n\\begin{enumerate}\n  \\item[(i)] In the strong positive-polarity regime, the displayed reflection\n  dominance inequalities imply $dF/ds>0$.  If, in addition, there are\n  $L_{\\mathrm{MAP}}>0$ and $q_{\\mathrm{MAP}}$ with\n  $|q_{\\mathrm{MAP}}|<1$ such that\n  \\[\n    F_n=L_{\\mathrm{MAP}}+q_{\\mathrm{MAP}}^n(F_0-L_{\\mathrm{MAP}}),\n  \\]\n  then $F_n\\to L_{\\mathrm{MAP}}>0$, and the covenant is eventually viable.\n  \\item[(ii)] In the strong negative-polarity regime, the displayed\n  reflection inequalities imply $dF/ds<0$.  If, in addition, there is\n  $q_{\\mathrm{MAD}}$ with $|q_{\\mathrm{MAD}}|<1$ such that\n  \\[\n    F_n=q_{\\mathrm{MAD}}^nF_0,\n  \\]\n  then $F_n\\to0$.  Thus collapse to zero follows from the supplied\n  dissipative contraction, not from the derivative sign alone.\n  \\item[(iii)] In the weak-coupling regime, if there is\n  $q_{\\mathrm{dec}}$ with $|q_{\\mathrm{dec}}|<1$ such that\n  \\[\n    \\epsilon_n=q_{\\mathrm{dec}}^n\\epsilon_0,\n  \\]\n  then $\\epsilon_n\\to0$, giving asymptotic decoupling.\n\\end{enumerate}\nThe three parameter regimes remain those classified by\nThm.~\\ref{theorem:bk5_enhanced_map_mad_duality}; the additional contraction\nlaws realize, rather than define, their asymptotic behavior.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "a positive MAP target",
      "reflection/entropy inequalities for local rate signs",
      "separately supplied strict contractions for MAP, MAD, and residue dynamics",
      "strong/weak coupling and polarity premises for classification"
    ],
    "countermodels": [
      "Book5DualityProof.decoupling_without_contraction_can_persist",
      "Book5DualityProof.negative_rate_alone_does_not_force_zero_limit",
      "Book5DualityProof.positive_rate_alone_does_not_force_positive_limit"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A typed realization certificate keeps parameter classification and temporal laws as distinct premises, then jointly proves all three source clauses: MAP classification, positive rate, convergence to a positive target, and eventual viability; MAD classification, negative rate, and zero-target convergence; weak-coupling classification and vanishing residue. Countermodels still show classification alone supplies no dynamics."
    ],
    "record_ids": [
      "MAP-BOOK5-003"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5DualityProof.EnhancedDualityRealizationCertificate.decoupling_realization",
      "Book5DualityProof.EnhancedDualityRealizationCertificate.enhanced_map_mad_dynamical_realization",
      "Book5DualityProof.EnhancedDualityRealizationCertificate.mad_realization",
      "Book5DualityProof.EnhancedDualityRealizationCertificate.map_realization",
      "Book5DualityProof.covenantTrajectory_orientation_dual",
      "Book5DualityProof.covenantTrajectory_tendsto_target",
      "Book5DualityProof.decoupling_consumes_vanishing_interaction",
      "Book5DualityProof.decoupling_without_contraction_can_persist",
      "Book5DualityProof.geometric_decoupling_vanishes",
      "Book5DualityProof.mad_rate_negative",
      "Book5DualityProof.mad_rate_negative_iff",
      "Book5DualityProof.mad_target_eventually_collapsed",
      "Book5DualityProof.map_rate_positive",
      "Book5DualityProof.map_rate_positive_iff",
      "Book5DualityProof.map_target_eventually_viable",
      "Book5DualityProof.negative_rate_alone_does_not_force_zero_limit",
      "Book5DualityProof.positive_rate_alone_does_not_force_positive_limit"
    ]
  },
  "line": 1004,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Enhanced MAP--MAD Dynamical Realization",
  "proof_labels": [
    "proof:bk5_enhanced_map_mad_duality_pr"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "silon_n\\to0$, giving asymptotic decoupling. \\end{enumerate} The three parameter regimes remain those classified by Thm.~\\ref{theorem:bk5_enhanced_map_mad_duality}; the additional contraction laws realize, rather than define, their asymptotic behavior. \\end{theorem}",
      "label": "theorem:bk5_enhanced_map_mad_duality",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 747,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_enhanced_map_mad_duality"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "proof:bk5_enhanced_map_mad_duality_pr",
  "label": "proof:bk5_enhanced_map_mad_duality_pr",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_enhanced_map_mad_duality_pr}\nThe reflection--entropy comparisons give the two local derivative signs by\nsubtraction in the process free-energy balance.  For the MAP realization,\n$|q_{\\mathrm{MAP}}|<1$ gives $q_{\\mathrm{MAP}}^n\\to0$, hence\n\\[\nF_n=L_{\\mathrm{MAP}}+q_{\\mathrm{MAP}}^n(F_0-L_{\\mathrm{MAP}})\n   \\longrightarrow L_{\\mathrm{MAP}}>0.\n\\]\nConvergence to a positive limit makes $F_n$ eventually positive.  The MAD and\ndecoupling conclusions use the same geometric convergence theorem:\n$q^nF_0\\to0$ and $q^n\\epsilon_0\\to0$ whenever $|q|<1$.\nWithout these evolution laws the derivative signs and coupling classification\nsupply no asymptotic limit; constant or noncontractive trajectories are\ncountermodels.  Thus every claimed limit is discharged by explicit dynamics\nrather than by its parameter label alone.  The Lean realization certificate\nretains this separation as typed data and proves the three clauses jointly:\nclassification, rate sign, and the corresponding contractive asymptotic law\nare all consumed, with no inference from the label alone.\n\\end{proof}",
  "line": 1038,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "theorem:bk5_enhanced_map_mad_duality_pr",
  "refs": [],
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}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk9_golden_rule_thermodynamic_covenant"
  ],
  "cites": [
    "corollary:bk5_reflective_hysteresis",
    "definition:bk5_mad_map_potential_barrie",
    "theorem:bk5_enhanced_map_mad_duality_pr"
  ],
  "depends_on": [
    "corollary:bk5_reflective_hysteresis",
    "definition:bk5_mad_map_potential_barrie",
    "theorem:bk5_enhanced_map_mad_duality_pr"
  ],
  "file": "book5.tex",
  "id": "scholium:bk5_mutually_assured_continuous_progress",
  "label": "scholium:bk5_mutually_assured_continuous_progress",
  "latex_body": "\\begin{scholium}[Mutually Assured Continuous Progress]\n\\label{scholium:bk5_mutually_assured_continuous_progress}\nRead thermodynamically, this scholium summarizes Thm.~\\ref{theorem:bk5_enhanced_map_mad_duality_pr} with the memory effect from Cor.~\\ref{corollary:bk5_reflective_hysteresis}.\nThe enhanced MAP-MAD duality theorem reveals that symbolic systems exhibit not merely binary states of cooperation or destruction, but exist on a continuous spectrum governed by coupling strength, covenant stability, and symbolic temperature (cf.~Def.~\\ref{definition:bk5_mad_map_potential_barrie}). \nThe phase transitions between MAP and MAD regimes represent symmetry-breaking events in symbolic space, where small perturbations near critical points can fundamentally alter system trajectory. This symmetry-breaking parallels physical phase transitions—just as water molecules reorganize dramatically at the freezing point, symbolic structures reconfigure at critical values of reflective coupling.\nThe existence of a critical symbolic temperature $T_s^{crit}$ suggests that highly energetic symbolic environments may preclude stable cooperation regardless of membrane intentions. Conversely, reduced symbolic temperatures facilitate the formation of stable covenants, as lower transformability rates allow reflective structures to persist against entropic forces.\nHysteresis in MAP-MAD transitions implies that the history of symbolic interaction matters—systems with a history of cooperation can withstand greater destabilizing forces before collapse than can be overcome to establish cooperation from an antagonistic starting point. This path-dependency of symbolic relationships mirrors physical systems with memory effects, where present states depend not only on current conditions but on historical trajectories.\nThe multi-agent extension demonstrates that global symbolic ecosystems need not be uniformly cooperative or destructive—mixed configurations can persist with islands of cooperation amid broader antagonism, or localized conflict within generally cooperative frameworks. However, long-term stability favors resolution toward global MAP or MAD as entropic forces propagate through covenant networks.\nPerhaps most profound is the implication that stable symbolic life requires maintaining \nthe coupling strength below a threshold that depends on symbolic temperature.\nAs symbolic temperature increases—representing greater volatility and transformability—\nthe viability of MAP relationships becomes increasingly precarious. \nThis rising instability demands progressively stronger and more resilient reflective mechanisms \nto preserve coherence against mounting entropic forces.\nThe principles established in this theorem extend beyond abstract symbolic thermodynamics to concrete interactions between reflective symbolic agents, suggesting a fundamental thermodynamic basis for the stability or instability of cooperative arrangements in symbolic ecosystems.\n\\end{scholium}",
  "line": 1058,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Mutually Assured Continuous Progress",
  "ref_roles": [
    {
      "context": "namically, this scholium summarizes Thm.~\\ref{theorem:bk5_enhanced_map_mad_duality_pr} with the memory effect from Cor.~\\ref{corollary:bk5_reflective_hysteresis}. The enhanced MAP-MAD duality theorem reveals that symbolic systems exhibit not merely binary states of cooperation or",
      "label": "corollary:bk5_reflective_hysteresis",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 942,
      "target_type": "corollary"
    },
    {
      "context": "ut exist on a continuous spectrum governed by coupling strength, covenant stability, and symbolic temperature (cf.~Def.~\\ref{definition:bk5_mad_map_potential_barrie}). The phase transitions between MAP and MAD regimes represent symmetry-breaking events in symbolic space, where small",
      "label": "definition:bk5_mad_map_potential_barrie",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 975,
      "target_type": "definition"
    },
    {
      "context": "ogress] \\label{scholium:bk5_mutually_assured_continuous_progress} Read thermodynamically, this scholium summarizes Thm.~\\ref{theorem:bk5_enhanced_map_mad_duality_pr} with the memory effect from Cor.~\\ref{corollary:bk5_reflective_hysteresis}. The enhanced MAP-MAD duality theorem reveal",
      "label": "theorem:bk5_enhanced_map_mad_duality_pr",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 1004,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk5_reflective_hysteresis",
    "definition:bk5_mad_map_potential_barrie",
    "theorem:bk5_enhanced_map_mad_duality_pr"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "sec:bk5_mutually_assured_progress_as_symbolic_ess",
  "label": "sec:bk5_mutually_assured_progress_as_symbolic_ess",
  "latex_body": "",
  "line": 1074,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Mutually Assured Progress as Symbolic ESS",
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "book5",
  "cited_by": [
    "lemma:bk5_map_fitness_advantage"
  ],
  "cites": [
    "definition:bk2_symbolic_free_energy"
  ],
  "depends_on": [
    "definition:bk2_symbolic_free_energy"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_strategy",
  "label": "definition:bk5_symbolic_strategy",
  "latex_body": "\\begin{definition}[Symbolic Strategy]\n\\label{definition:bk5_symbolic_strategy}\nStrategies are the microscopic control primitives whose interaction payoffs are measured by symbolic free energy (Def.~\\ref{definition:bk2_symbolic_free_energy}).\nA \\emph{symbolic strategy} $\\sigma$ is a tuple $(\\reflect_\\sigma, \\mathcal{T}_\\sigma, \\kappa_\\sigma)$ where:\n\\begin{itemize}\n    \\item $\\reflect_\\sigma$ is the reflection operator employed under strategy $\\sigma$\n    \\item $\\mathcal{T}_\\sigma$ is the transfer operator employed under strategy $\\sigma$\n    \\item $\\kappa_\\sigma \\in [0,1]$ is the cooperation coefficient determining willingness to form covenants\n\\end{itemize}\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "strategies are consumed abstractly as an index type / real payoff values inside the dominance and invasion lemmas, not as the (reflect,transfer,coop-coefficient) tuple itself."
    ],
    "record_ids": [
      "MAP-BOOK5-064"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Residue.ess_invasion_threshold",
      "Book5Residue.weighted_strict_dominance"
    ]
  },
  "line": 1077,
  "macros_used": [
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Strategy",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "Strategies are the microscopic control primitives whose interaction payoffs are measured by symbolic free energy (Def.~\\ref{definition:bk2_symbolic_free_energy}). A \\emph{symbolic strategy} $\\sigma$ is a tuple $(\\reflect_\\sigma, \\mathcal{T}_\\sigma, \\kappa_\\sigma)$ where: \\begin{i",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_free_energy"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "lemma:bk5_map_fitness_advantage"
  ],
  "cites": [
    "definition:bk5_symbolic_covenant"
  ],
  "depends_on": [
    "definition:bk5_symbolic_covenant"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_strategy_space",
  "label": "definition:bk5_strategy_space",
  "latex_body": "\\begin{definition}[Strategy Space]\n\\label{definition:bk5_strategy_space}\nThe \\emph{symbolic strategy space} $\\Sigma$ is the set of all possible symbolic strategies available to membranes. We denote $\\Sigma_{MAP} \\subset \\Sigma$ as the subset of strategies that satisfy MAP conditions as per Def.~\\ref{definition:bk5_symbolic_covenant}.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-065"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Residue.weighted_strict_dominance"
    ]
  },
  "line": 1087,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Strategy Space",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "membranes. We denote $\\Sigma_{MAP} \\subset \\Sigma$ as the subset of strategies that satisfy MAP conditions as per Def.~\\ref{definition:bk5_symbolic_covenant}. \\end{definition}",
      "label": "definition:bk5_symbolic_covenant",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 233,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_symbolic_covenant"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk5_symbolic_ess",
    "definition:bk5_symbolic_replicator_dynamics",
    "proof:bk5_map_invasion_dynamics",
    "proof:bk5_map_resistance_to_drift"
  ],
  "cites": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk5__map_dominance"
  ],
  "depends_on": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk5__map_dominance"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_fitness",
  "label": "definition:bk5_symbolic_fitness",
  "latex_body": "\\begin{definition}[Symbolic Fitness]\n\\label{definition:bk5_symbolic_fitness}\nThe \\emph{symbolic fitness} $\\Phi(\\sigma, \\mathfrak{P})$ of a strategy $\\sigma$ in a population with strategy distribution $\\mathfrak{P}$ is defined as (cf.~Def.~\\ref{definition:bk2_symbolic_free_energy}, Thm.~\\ref{theorem:bk5__map_dominance}):\n\\begin{equation}\n\\Phi(\\sigma, \\mathfrak{P}) = \\mathbb{E}_{\\tau \\sim \\mathfrak{P}}[F_s(\\Membrane_\\sigma \\leftrightarrow \\Membrane_\\tau)]\n\\end{equation}\nWhere $F_s(\\Membrane_\\sigma \\leftrightarrow \\Membrane_\\tau)$ is the symbolic free energy resulting from interaction between membranes employing strategies $\\sigma$ and $\\tau$.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-066"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Residue.ess_invasion_threshold",
      "Book5Residue.weighted_strict_dominance"
    ]
  },
  "line": 1091,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Fitness",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "\\mathfrak{P})$ of a strategy $\\sigma$ in a population with strategy distribution $\\mathfrak{P}$ is defined as (cf.~Def.~\\ref{definition:bk2_symbolic_free_energy}, Thm.~\\ref{theorem:bk5__map_dominance}): \\begin{equation} \\Phi(\\sigma, \\mathfrak{P}) = \\mathbb{E}_{\\tau \\sim \\mathfrak{",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "ation with strategy distribution $\\mathfrak{P}$ is defined as (cf.~Def.~\\ref{definition:bk2_symbolic_free_energy}, Thm.~\\ref{theorem:bk5__map_dominance}): \\begin{equation} \\Phi(\\sigma, \\mathfrak{P}) = \\mathbb{E}_{\\tau \\sim \\mathfrak{P}}[F_s(\\Membrane_\\sigma \\leftrightarro",
      "label": "theorem:bk5__map_dominance",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 426,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk5__map_dominance"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "lemma:bk5_covenant_non_invasibility",
    "proof:bk5_map_as_ess",
    "theorem:bk5_map_as_strong_ess"
  ],
  "cites": [
    "definition:bk5_symbolic_fitness"
  ],
  "depends_on": [
    "definition:bk5_symbolic_fitness"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_ess",
  "label": "definition:bk5_symbolic_ess",
  "latex_body": "\\begin{definition}[Symbolic ESS]\n\\label{definition:bk5_symbolic_ess}\nA strategy $\\sigma^* \\in \\Sigma$ is a \\emph{symbolic evolutionarily stable strategy} if for every strategy $\\sigma \\neq \\sigma^*$, there exists $\\epsilon_\\sigma > 0$ such that for all $\\epsilon \\in (0, \\epsilon_\\sigma)$ (using Def.~\\ref{definition:bk5_symbolic_fitness}):\n\\begin{equation}\n\\Phi(\\sigma^*, (1-\\epsilon)\\delta_{\\sigma^*} + \\epsilon\\delta_\\sigma) > \\Phi(\\sigma, (1-\\epsilon)\\delta_{\\sigma^*} + \\epsilon\\delta_\\sigma)\n\\end{equation}\nWhere $\\delta_\\sigma$ is the Dirac measure concentrated on strategy $\\sigma$.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-067"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.ess_invasion_threshold"
    ]
  },
  "line": 1099,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic ESS",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ma \\neq \\sigma^*$, there exists $\\epsilon_\\sigma > 0$ such that for all $\\epsilon \\in (0, \\epsilon_\\sigma)$ (using Def.~\\ref{definition:bk5_symbolic_fitness}): \\begin{equation} \\Phi(\\sigma^*, (1-\\epsilon)\\delta_{\\sigma^*} + \\epsilon\\delta_\\sigma) > \\Phi(\\sigma, (1-\\epsilon)\\de",
      "label": "definition:bk5_symbolic_fitness",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1091,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_symbolic_fitness"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "lemma:bk5_covenant_non_invasibility",
    "lemma:bk5_map_invasion_barrier_strength"
  ],
  "cites": [
    "definition:bk5_strategy_space",
    "definition:bk5_symbolic_strategy",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk5__map_dominance"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk2_symbolic_free_energy",
    "definition:bk5_strategy_space",
    "definition:bk5_symbolic_fitness",
    "definition:bk5_symbolic_strategy",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk5__map_dominance"
  ],
  "file": "book5.tex",
  "id": "lemma:bk5_map_fitness_advantage",
  "label": "lemma:bk5_map_fitness_advantage",
  "latex_body": "\\begin{lemma}[MAP Fitness Advantage]\n\\label{lemma:bk5_map_fitness_advantage}\nLet $\\sigma_{MAP} \\in \\Sigma_{MAP}$ and $\\sigma_{non} \\in \\Sigma \\setminus \\Sigma_{MAP}$ (cf.~Def.~\\ref{definition:bk5_symbolic_strategy}, Def.~\\ref{definition:bk5_strategy_space}). Under sufficient drift intensity $\\|\\drift\\| > \\drift_0$ (cf.~Thm.~\\ref{theorem:bk5__map_dominance}, Thm.~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}), the following inequality holds:\n\\begin{equation}\n\\Phi(\\sigma_{MAP}, \\mathfrak{P}) > \\Phi(\\sigma_{non}, \\mathfrak{P})\n\\end{equation}\nFor any population distribution $\\mathfrak{P}$ with $\\mathbb{P}_{\\tau \\sim \\mathfrak{P}}[\\tau \\in \\Sigma_{MAP}] > 0$.\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": true,
    "notes": [
      "instantiates directly: pointwise strict dominance on positive-weight population members gives strict weighted-average dominance."
    ],
    "record_ids": [
      "MAP-BOOK5-068"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.weighted_strict_dominance"
    ]
  },
  "line": 1107,
  "macros_used": [
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "MAP Fitness Advantage",
  "proof_labels": [
    "proof:bk5_map_resistance_to_drift"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "igma_{MAP}$ and $\\sigma_{non} \\in \\Sigma \\setminus \\Sigma_{MAP}$ (cf.~Def.~\\ref{definition:bk5_symbolic_strategy}, Def.~\\ref{definition:bk5_strategy_space}). Under sufficient drift intensity $\\|\\drift\\| > \\drift_0$ (cf.~Thm.~\\ref{theorem:bk5__map_dominance}, Thm.~\\ref{theore",
      "label": "definition:bk5_strategy_space",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1087,
      "target_type": "definition"
    },
    {
      "context": "p_fitness_advantage} Let $\\sigma_{MAP} \\in \\Sigma_{MAP}$ and $\\sigma_{non} \\in \\Sigma \\setminus \\Sigma_{MAP}$ (cf.~Def.~\\ref{definition:bk5_symbolic_strategy}, Def.~\\ref{definition:bk5_strategy_space}). Under sufficient drift intensity $\\|\\drift\\| > \\drift_0$ (cf.~Thm.~\\ref{the",
      "label": "definition:bk5_symbolic_strategy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1077,
      "target_type": "definition"
    },
    {
      "context": "tegy_space}). Under sufficient drift intensity $\\|\\drift\\| > \\drift_0$ (cf.~Thm.~\\ref{theorem:bk5__map_dominance}, Thm.~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}), the following inequality holds: \\begin{equation} \\Phi(\\sigma_{MAP}, \\mathfrak{P}) > \\Phi(\\sigma_{non}, \\mathfrak{P})",
      "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"
    },
    {
      "context": "trategy}, Def.~\\ref{definition:bk5_strategy_space}). Under sufficient drift intensity $\\|\\drift\\| > \\drift_0$ (cf.~Thm.~\\ref{theorem:bk5__map_dominance}, Thm.~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}), the following inequality holds: \\begin{equation} \\Phi(\\sigma_{MAP",
      "label": "theorem:bk5__map_dominance",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 426,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_strategy_space",
    "definition:bk5_symbolic_strategy",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk5__map_dominance"
  ],
  "role": "lemma",
  "type": "lemma"
}