Book III — De Symbiosi Symbolica — page 1 of 2

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    "proof:bk3_sketch_evolutionary_dynamics",
    "proof:bk4_fragmentation_identity_stability",
    "proof:bk4_multiplication_to_curvature",
    "proof:bk4_symbolic_curvature_fragmentation",
    "proof:bk4_top_level_information_inequality",
    "proof:bk9_betrayal_and_recovery",
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      "context": "i: \\mathcal{M}_i \\rightarrow T\\mathcal{M}_i$ that is a restriction and modification of the global drift field $D$ (Def.~\\ref{definition:bk1_drift_field}), satisfying $\\|D_i(x) - D(x)\\|_g \\leq \\delta_i$ for some bound $\\delta_i > 0$. \\item A boundary permeability funct",
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  "latex_body": "\\begin{proof}[Local Regulation of Drift on a Supplied Smooth Domain]\n\\label{proof:bk3_local_regulation_smooth_membranes}\n\\leavevmode\n\nTake $\\mathcal{M}_i=U$ and let $D_i$ be the restriction of $D$ to $U$.\nThen $D_i(x)-D(x)=0$, so\n$\\|D_i(x)-D(x)\\|_g=0\\leq\\delta_i$ for every $\\delta_i>0$.  Define the\nboundary permeability by the constant function $\\pi_i(p,v)=0$, which takes\nvalues in $[0,1]$.  Finally set\n\\begin{equation}\nS_i(x)=\\exp(-\\alpha H(x)).\n\\end{equation}\nThis is strictly positive, and it is smooth whenever $H$ is smooth.  Thus the\nsupplied domain and fields carry all of the stated membrane data.  Notice that\nno smallness condition on $\\delta_i$ is needed for this canonical witness; the\ngeometric hypotheses on $U$ are separate and load-bearing.\n\\end{proof}",
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      "context": "energy: $E_i(s) = \\int_{\\mathcal{M}_i} \\rho_i(x,s)H_i(x)d\\mu_g(x)$, where $\\rho_i$ is the probability density (cf. Def.~\\ref{definition:bk2__symbolic_probability_density}) restricted to $\\mathcal{M}_i$ and normalized, and $H_i$ is the symbolic Hamiltonian (cf. Def.~\\ref{definition:bk2_symb",
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      "context": "ition:bk2_symbolic_hamiltonian}) restricted to $\\mathcal{M}_i$. (This builds upon the general symbolic energy, cf. Def.~\\ref{definition:bk2_symbolic_energy}). \\item Membrane entropy: $S_i(s) = -\\int_{\\mathcal{M}_i} \\rho_i(x,s)\\log\\rho_i(x,s)d\\mu_g(x)$ (cf. Def.~\\ref{defin",
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      "context": "ic_energy}). \\item Membrane entropy: $S_i(s) = -\\int_{\\mathcal{M}_i} \\rho_i(x,s)\\log\\rho_i(x,s)d\\mu_g(x)$ (cf. Def.~\\ref{definition:bk2_symbolic_entropy}). \\item Membrane temperature: $T_i(s) = \\left(\\frac{\\partial S_i(s)}{\\partial E_i(s)}\\right)^{-1}$ (cf. Def.~\\ref{d",
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      "context": "e}). \\item Membrane free energy: $F_i(\\beta_i) = E_i(s) - \\beta_i^{-1}S_i(s)$, where $\\beta_i = T_i^{-1}$ (cf. Def.~\\ref{definition:bk2_symbolic_free_energy}). \\end{enumerate} (The underlying manifold and measure are from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\e",
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      "context": "lic_probability_density}) restricted to $\\mathcal{M}_i$ and normalized, and $H_i$ is the symbolic Hamiltonian (cf. Def.~\\ref{definition:bk2_symbolic_hamiltonian}) restricted to $\\mathcal{M}_i$. (This builds upon the general symbolic energy, cf. Def.~\\ref{definition:bk2_symbolic_en",
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      "context": "cf. Def.~\\ref{definition:bk2_symbolic_free_energy}). \\end{enumerate} (The underlying manifold and measure are from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{definition}",
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      "context": "tropy}). \\item Membrane temperature: $T_i(s) = \\left(\\frac{\\partial S_i(s)}{\\partial E_i(s)}\\right)^{-1}$ (cf. Def.~\\ref{definition:bk2_symbolic_temperature}). \\item Membrane free energy: $F_i(\\beta_i) = E_i(s) - \\beta_i^{-1}S_i(s)$, where $\\beta_i = T_i^{-1}$ (cf. Def.~\\r",
      "label": "definition:bk2_symbolic_temperature",
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      "context": "}[Membrane Thermodynamics] \\label{definition:bk3_membrane_thermodynamics} For a symbolic membrane $\\mathcal{M}_i$ (Def.~\\ref{definition:bk3_symbolic_membrane}), we define: \\begin{enumerate} \\item Membrane energy: $E_i(s) = \\int_{\\mathcal{M}_i} \\rho_i(x,s)H_i(x)d\\mu_g(x)$, w",
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    "proof:bk3_membrane_stability_energy_permeability",
    "proof:bk3_sketch_field_perturbation",
    "proof:bk3_sketch_necessity_for_continuous_operation",
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    "proof:bk4_information_bottleneck_symbolic_filter",
    "proof:bk4_symbolic_identity_persistence",
    "scholium:bk6_semantic_network_regulation",
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  "id": "theorem:bk3_membrane_stability_criteria",
  "label": "theorem:bk3_membrane_stability_criteria",
  "latex_body": "\\begin{theorem}[Membrane Stability Criteria] \\label{theorem:bk3_membrane_stability_criteria}\nA symbolic membrane $\\mathcal{M}_i$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is stable under small perturbations if:\n\\begin{enumerate}\n    \\item The membrane free energy $F_i(\\beta_i)$ (cf. Def.~\\ref{definition:bk3_membrane_thermodynamics}) is at a local minimum.\n    \\item The symbolic flow $\\Phi^s$ induced by the internal drift field $D_i$ has no unstable fixed points in $\\mathcal{M}_i$.\n    \\item For all boundary points $p \\in \\partial\\mathcal{M}_i$, the permeability function $\\pi_i(p,v)$ satisfies $\\pi_i(p,v) < \\gamma_i$ for some threshold $\\gamma_i < 1$ when $v$ points outward and $\\|v\\|_g > \\epsilon_i$ for some $\\epsilon_i > 0$.\n\\end{enumerate}\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "manifold/PDE/Helmholtz content of Book 3 is NOT formalized; static and finite-discrete kernels only",
      "modeling laws (rate equations, stability conditions, Lipschitz response) are structure fields"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The three named conditions are kept as explicit hypothesis fields (free-energy local-min and no-unstable-fixed-point are opaque Prop witnesses); stability is proved as their conjunction, not asserted to hold generally."
    ],
    "record_ids": [
      "MAP-BOOK3-007"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3.membrane_stable_of_conditions",
      "Book3.membrane_stable_permeability_lt_one"
    ]
  },
  "line": 61,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Membrane Stability Criteria",
  "proof_labels": [
    "proof:bk3_membrane_stability_energy_permeability"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": ") is stable under small perturbations if: \\begin{enumerate} \\item The membrane free energy $F_i(\\beta_i)$ (cf. Def.~\\ref{definition:bk3_membrane_thermodynamics}) is at a local minimum. \\item The symbolic flow $\\Phi^s$ induced by the internal drift field $D_i$ has no unstable",
      "label": "definition:bk3_membrane_thermodynamics",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 49,
      "target_type": "definition"
    },
    {
      "context": "[Membrane Stability Criteria] \\label{theorem:bk3_membrane_stability_criteria} A symbolic membrane $\\mathcal{M}_i$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is stable under small perturbations if: \\begin{enumerate} \\item The membrane free energy $F_i(\\beta_i)$ (cf. Def.~",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_membrane"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "depends_on": [
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_membrane_stability_energy_permeability",
  "label": "proof:bk3_membrane_stability_energy_permeability",
  "latex_body": "\\begin{proof}[Membrane Stability from Free Energy and Bounded Permeability]\n\\label{proof:bk3_membrane_stability_energy_permeability}\n\\leavevmode\n\nIf membrane free energy $F_i(\\beta_i)$ is at a local minimum, small\nperturbations in $\\rho_i$ induce restorative forces back toward equilibrium\n(Theorem~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}).\nIf the symbolic flow has no unstable fixed points, trajectories within the membrane do not exponentially diverge, preserving internal coherence.\nThe permeability condition restricts large outward flows, preventing rapid symbolic diffusion across the boundary.\nTogether these conditions force perturbations to dissipate rather than amplify, yielding structural stability (supporting Thm.~\\ref{theorem:bk3_membrane_stability_criteria}).\n\\end{proof}",
  "line": 70,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Membrane Stability from Free Energy and Bounded Permeability",
  "proves": "theorem:bk3_membrane_stability_criteria",
  "ref_roles": [
    {
      "context": "a_i)$ is at a local minimum, small perturbations in $\\rho_i$ induce restorative forces back toward equilibrium (Theorem~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}). If the symbolic flow has no unstable fixed points, trajectories within the membrane do not exponentially diverge, pre",
      "label": "theorem:bk2_h_theorem_for_symbolic_evol",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book2.tex",
      "target_line": 255,
      "target_type": "theorem"
    },
    {
      "context": "r these conditions force perturbations to dissipate rather than amplify, yielding structural stability (supporting Thm.~\\ref{theorem:bk3_membrane_stability_criteria}). \\end{proof}",
      "label": "theorem:bk3_membrane_stability_criteria",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 61,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "section:book3.tex:82",
  "label": "",
  "latex_body": "",
  "line": 82,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Coupling and Symbiotic Relations",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_autopoiesis",
    "definition:bk3_symbolic_metabolism",
    "lemma:bk3_symbiotic_stability_conditions",
    "proof:bk3_coupling_energy_symbolic_hamiltonian",
    "proof:bk3_sketch_evolutionary_dynamics"
  ],
  "cites": [
    "definition:bk3_symbolic_membrane"
  ],
  "depends_on": [
    "definition:bk3_symbolic_membrane"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_coupling_map",
  "label": "definition:bk3_coupling_map",
  "latex_body": "\\begin{definition}[Coupling Map] \\label{definition:bk3_coupling_map}\nGiven symbolic membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$ (Def.~\\ref{definition:bk3_symbolic_membrane}), a coupling map $\\Phi_{ij}: \\mathcal{M}_i \\times \\mathcal{M}_j \\rightarrow S$ is a smooth function to a shared symbolic substrate $S$ (typically a vector space or manifold) satisfying:\n\\begin{enumerate}\n    \\item Symmetry: $\\Phi_{ij}(x,y) = \\Phi_{ji}(y,x)$ for all $x \\in \\mathcal{M}_i, y \\in \\mathcal{M}_j$.\n    \\item Boundedness: $\\|\\Phi_{ij}(x,y)\\|_S \\leq C_{ij}$ for some constant $C_{ij} > 0$ and an appropriate norm $\\|\\cdot\\|_S$ on $S$.\n    \\item Sensitivity: The gradients $\\nabla_x\\Phi_{ij}$ and $\\nabla_y\\Phi_{ij}$ exist and are non-vanishing on open dense subsets of $\\mathcal{M}_i$ and $\\mathcal{M}_j$ respectively.\n\\end{enumerate}\n\\end{definition}",
  "line": 85,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Coupling Map",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "n}[Coupling Map] \\label{definition:bk3_coupling_map} Given symbolic membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$ (Def.~\\ref{definition:bk3_symbolic_membrane}), a coupling map $\\Phi_{ij}: \\mathcal{M}_i \\times \\mathcal{M}_j \\rightarrow S$ is a smooth function to a shared symboli",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_symbolic_membrane"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_symbolic_autopoiesis",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_metabolism",
    "lemma:bk3_symbiotic_stability_conditions",
    "proof:bk3_coupling_energy_symbolic_hamiltonian",
    "proof:bk3_sketch_evolutionary_dynamics",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_homeostatic_reflexes"
  ],
  "cites": [
    "definition:bk3_coupling_map"
  ],
  "depends_on": [
    "definition:bk3_coupling_map"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_induced_coupling_energy",
  "label": "definition:bk3_induced_coupling_energy",
  "latex_body": "\\begin{definition}[Induced Coupling Energy] \\label{definition:bk3_induced_coupling_energy}\nThe coupling map $\\Phi_{ij}$ (Def.~\\ref{definition:bk3_coupling_map}) induces an energy function $H_{ij}: \\mathcal{M}_i \\times \\mathcal{M}_j \\rightarrow \\mathbb{R}$ defined as:\n\\[\nH_{ij}(x,y) = \\lambda_{ij} \\|\\Phi_{ij}(x,y) - \\Phi_{ij}^*\\|_S^2\n\\]\nwhere $\\lambda_{ij} > 0$ is a coupling strength parameter and $\\Phi_{ij}^*$ represents an optimal coupling configuration in $S$.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "H_ij = lambda*(phi-target)^2 and its nonnegativity for lambda>=0, fully proved."
    ],
    "record_ids": [
      "MAP-BOOK3-009"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book3.couplingEnergy_nonneg"
    ]
  },
  "line": 95,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Induced Coupling Energy",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "{definition}[Induced Coupling Energy] \\label{definition:bk3_induced_coupling_energy} The coupling map $\\Phi_{ij}$ (Def.~\\ref{definition:bk3_coupling_map}) induces an energy function $H_{ij}: \\mathcal{M}_i \\times \\mathcal{M}_j \\rightarrow \\mathbb{R}$ defined as: \\[ H_{ij}(x",
      "label": "definition:bk3_coupling_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 85,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_coupling_map"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_symbolic_autopoiesis",
    "definition:bk3_symbolic_metabolic_rate",
    "lemma:bk3_symbiotic_stability_conditions",
    "proof:bk3_coupling_energy_symbolic_hamiltonian",
    "proof:bk3_sketch_evolutionary_dynamics",
    "proof:bk3_sketch_field_perturbation",
    "theorem:bk3_homeostatic_reflexes"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_hamiltonian",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_couplinginduced_drift_modification",
  "label": "theorem:bk3_couplinginduced_drift_modification",
  "latex_body": "\\begin{theorem}[Coupling-Induced Drift Modification] \\label{theorem:bk3_couplinginduced_drift_modification}\n\\leavevmode\\newline\nThe coupling energy $H_{ij}$ (Def.~\\ref{definition:bk3_induced_coupling_energy}) induces modifications to the drift fields $D_i$ and $D_j$ within the respective membranes (Def.~\\ref{definition:bk3_symbolic_membrane}):\n\\[\nD_i^{\\text{coupled}}(x) = D_i(x) - \\eta_i \\int_{\\mathcal{M}_j} \\rho_j(y)\\nabla_x H_{ij}(x,y)d\\mu_g(y)\n\\]\n\\[\nD_j^{\\text{coupled}}(y) = D_j(y) - \\eta_j \\int_{\\mathcal{M}_i} \\rho_i(x)\\nabla_y H_{ij}(x,y)d\\mu_g(x)\n\\]\nwhere $\\eta_i, \\eta_j > 0$ are response parameters, and $\\rho_i, \\rho_j$ are\nthe corresponding probability densities\n(Def.~\\ref{definition:bk2__symbolic_probability_density}).\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "manifold/PDE/Helmholtz content of Book 3 is NOT formalized; static and finite-discrete kernels only",
      "modeling laws (rate equations, stability conditions, Lipschitz response) are structure fields"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Scalar model: the correction integral is replaced by a single bounded real; the theorem proved is the triangle-inequality consequence |D_coupled - D| <= eta*bound, given the bound as a hypothesis rather than derived from an actual integral."
    ],
    "record_ids": [
      "MAP-BOOK3-010"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3.coupled_drift_deviation_bound"
    ]
  },
  "line": 104,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Coupling-Induced Drift Modification",
  "proof_labels": [
    "proof:bk3_coupling_energy_symbolic_hamiltonian"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "re $\\eta_i, \\eta_j > 0$ are response parameters, and $\\rho_i, \\rho_j$ are the corresponding probability densities (Def.~\\ref{definition:bk2__symbolic_probability_density}). \\end{theorem}",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "fication] \\label{theorem:bk3_couplinginduced_drift_modification} \\leavevmode\\newline The coupling energy $H_{ij}$ (Def.~\\ref{definition:bk3_induced_coupling_energy}) induces modifications to the drift fields $D_i$ and $D_j$ within the respective membranes (Def.~\\ref{definition:bk3_sy",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": "duced_coupling_energy}) induces modifications to the drift fields $D_i$ and $D_j$ within the respective membranes (Def.~\\ref{definition:bk3_symbolic_membrane}): \\[ D_i^{\\text{coupled}}(x) = D_i(x) - \\eta_i \\int_{\\mathcal{M}_j} \\rho_j(y)\\nabla_x H_{ij}(x,y)d\\mu_g(y) \\] \\[ D_j^{\\",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_hamiltonian",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "depends_on": [
    "definition:bk2_symbolic_hamiltonian",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_coupling_energy_symbolic_hamiltonian",
  "label": "proof:bk3_coupling_energy_symbolic_hamiltonian",
  "latex_body": "\\begin{proof}[Effect of Coupling Energy on Symbolic Hamiltonian]\n\\label{proof:bk3_coupling_energy_symbolic_hamiltonian}\n\\leavevmode\n\nThe coupling energy \\( H_{ij} \\) (Def.~\\ref{definition:bk3_induced_coupling_energy}) contributes an additional potential term to the symbolic Hamiltonian (cf. Def.~\\ref{definition:bk2_symbolic_hamiltonian}) of each membrane.\nFrom standard results in statistical mechanics (analogous to mean-field theory), the expected force on a point \\( x \\in \\mathcal{M}_i \\) due to all points in \\( \\mathcal{M}_j \\) is given by:\n\\[\n-\\int_{\\mathcal{M}_j} \\rho_j(y) \\nabla_x H_{ij}(x, y) \\, d\\mu_g(y).\n\\]\nThis force modifies the drift field with strength parameter \\( \\eta_i \\), resulting in the coupled drift expression (Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}). The modification to \\( D_j \\) follows symmetrically.\nThis coupling creates a feedback loop where the dynamics in each membrane (Def.~\\ref{definition:bk3_symbolic_membrane}) are influenced by the state of the other membrane, mediated by the coupling map \\( \\Phi_{ij} \\) (Def.~\\ref{definition:bk3_coupling_map}). (The measure $d\\mu_g$ is from Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{proof}",
  "line": 118,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Effect of Coupling Energy on Symbolic Hamiltonian",
  "proves": "theorem:bk3_couplinginduced_drift_modification",
  "ref_roles": [
    {
      "context": "definition:bk3_induced_coupling_energy}) contributes an additional potential term to the symbolic Hamiltonian (cf. Def.~\\ref{definition:bk2_symbolic_hamiltonian}) of each membrane. From standard results in statistical mechanics (analogous to mean-field theory), the expected force",
      "label": "definition:bk2_symbolic_hamiltonian",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 67,
      "target_type": "definition"
    },
    {
      "context": "diated by the coupling map \\( \\Phi_{ij} \\) (Def.~\\ref{definition:bk3_coupling_map}). (The measure $d\\mu_g$ is from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{proof}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "mbolic_membrane}) are influenced by the state of the other membrane, mediated by the coupling map \\( \\Phi_{ij} \\) (Def.~\\ref{definition:bk3_coupling_map}). (The measure $d\\mu_g$ is from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{proof}",
      "label": "definition:bk3_coupling_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 85,
      "target_type": "definition"
    },
    {
      "context": "Hamiltonian] \\label{proof:bk3_coupling_energy_symbolic_hamiltonian} \\leavevmode The coupling energy \\( H_{ij} \\) (Def.~\\ref{definition:bk3_induced_coupling_energy}) contributes an additional potential term to the symbolic Hamiltonian (cf. Def.~\\ref{definition:bk2_symbolic_hamiltonia",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": "ion to \\( D_j \\) follows symmetrically. This coupling creates a feedback loop where the dynamics in each membrane (Def.~\\ref{definition:bk3_symbolic_membrane}) are influenced by the state of the other membrane, mediated by the coupling map \\( \\Phi_{ij} \\) (Def.~\\ref{definition:",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "is force modifies the drift field with strength parameter \\( \\eta_i \\), resulting in the coupled drift expression (Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}). The modification to \\( D_j \\) follows symmetrically. This coupling creates a feedback loop where the dynamics in each",
      "label": "theorem:bk3_couplinginduced_drift_modification",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 104,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_hamiltonian",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_autopoiesis",
    "definition:bk3_symbolic_metabolism",
    "definition:bk6_symbolic_recombination",
    "lemma:bk3_symbiotic_stability_conditions",
    "proof:bk3_sketch_field_perturbation",
    "proof:bk3_sketch_necessity_for_continuous_operation",
    "proof:bk3_sketch_perturbation_dissiptation",
    "proof:bk3_symbolic_coupling_properties_enumerated",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk3_properties_of_symbiotic_curvature",
    "theorem:bk9_irreversibility_of_covenant_breach_without_grace"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbolic_membrane"
  ],
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    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbolic_membrane"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_symbolic_symbiosis",
  "label": "definition:bk3_symbolic_symbiosis",
  "latex_body": "\\begin{definition}[Symbolic Symbiosis] \\label{definition:bk3_symbolic_symbiosis}\nTwo symbolic membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$ (Def.~\\ref{definition:bk3_symbolic_membrane}) are in symbiosis if their coupling satisfies:\n\\begin{enumerate}\n    \\item \\textbf{Mutual stability enhancement:} \n    \\[\n    S_i^{\\text{coupled}} > S_i^{\\text{isolated}} \\quad \\text{and} \\quad \n    S_j^{\\text{coupled}} > S_j^{\\text{isolated}},\n    \\]\n    where \\( S_k^{\\text{coupled}} \\) is the stability of membrane \\( k \\) under coupling.\n    \\item \\textbf{Information transfer:} \n    \\[\n    I(\\mathcal{M}_i; \\mathcal{M}_j) = \\int_{\\mathcal{M}_i \\times \\mathcal{M}_j} \n    \\rho_{ij}(x,y) \\log \\frac{\\rho_{ij}(x,y)}{\\rho_i(x) \\rho_j(y)} \\, d\\mu_g(x) d\\mu_g(y) > 0,\n    \\]\n    where \\( \\rho_{ij} \\) is the joint probability density (cf. Def.~\\ref{definition:bk2__symbolic_probability_density}). (The measure $d\\mu_g$ is from Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n    \\item \\textbf{Drift compensation:} For perturbations \\( \\delta D_i \\) to the drift field of \\( \\mathcal{M}_i \\), the coupling response reduces the perturbation effect:\n    \\[\n    \\left\\| \\delta D_i + \\delta D_i^{\\text{response}} \\right\\|_g \n    < \\left\\| \\delta D_i \\right\\|_g,\n    \\]\n    where \\( \\delta D_i^{\\text{response}} \\) is the change in drift induced by the coupling in response to the perturbation.\n\\end{enumerate}\n\\end{definition}",
  "lean_alignment": {
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    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
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    "notes": [
      "All three named clauses (mutual stability enhancement, information transfer positivity, drift compensation) kept as explicit hypothesis fields; a genuine corollary (perturbation must be nonzero under compensation) is derived."
    ],
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  },
  "line": 132,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Symbiosis",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "i(x) \\rho_j(y)} \\, d\\mu_g(x) d\\mu_g(y) > 0, \\] where \\( \\rho_{ij} \\) is the joint probability density (cf. Def.~\\ref{definition:bk2__symbolic_probability_density}). (The measure $d\\mu_g$ is from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\item \\textbf{Drift compensati",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 35,
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    },
    {
      "context": "t probability density (cf. Def.~\\ref{definition:bk2__symbolic_probability_density}). (The measure $d\\mu_g$ is from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\item \\textbf{Drift compensation:} For perturbations \\( \\delta D_i \\) to the drift field of \\( \\mathcal{M}_i \\),",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "c Symbiosis] \\label{definition:bk3_symbolic_symbiosis} Two symbolic membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$ (Def.~\\ref{definition:bk3_symbolic_membrane}) are in symbiosis if their coupling satisfies: \\begin{enumerate} \\item \\textbf{Mutual stability enhancement:}",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
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      "target_file": "book3.tex",
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    }
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  "id": "lemma:bk3_symbiotic_stability_conditions",
  "label": "lemma:bk3_symbiotic_stability_conditions",
  "latex_body": "\\begin{lemma}[Symbiotic Stability Conditions] \\label{lemma:bk3_symbiotic_stability_conditions}\nSymbiotic coupling (Def.~\\ref{definition:bk3_symbolic_symbiosis}) enhances stability when the coupling strength $\\lambda_{ij}$ and response parameters $\\eta_i, \\eta_j$ (from Def.~\\ref{definition:bk3_induced_coupling_energy} and Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}) satisfy:\n\\[\n\\lambda_{ij} > \\max\\left\\{\\frac{\\delta_i^2}{4\\eta_i \\int_{\\mathcal{M}_j} \\rho_j(y)\\|\\nabla_x \\Phi_{ij}(x,y)\\|_g^2 d\\mu_g(y)}, \\frac{\\delta_j^2}{4\\eta_j \\int_{\\mathcal{M}_i} \\rho_i(x)\\|\\nabla_y \\Phi_{ij}(x,y)\\|_g^2 d\\mu_g(x)}\\right\\}\n\\]\nwhere $\\delta_i, \\delta_j$ are the maximum internal drift perturbations in the respective membranes (Def.~\\ref{definition:bk3_symbolic_membrane}). (Probabilities $\\rho_i, \\rho_j$ are from Def.~\\ref{definition:bk2__symbolic_probability_density}, measure $d\\mu_g$ from Def.~\\ref{definition:bk2_symbolic_probability_spa}, coupling map $\\Phi_{ij}$ from Def.~\\ref{definition:bk3_coupling_map}).\n\\end{lemma}",
  "lean_alignment": {
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      "The max-of-two-bounds threshold algebra is fully proved: lambda above the max clears each side of the enhancement inequality delta^2 < 4*eta*infoGrad*lambda."
    ],
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  "name": "Symbiotic Stability Conditions",
  "proof_labels": [
    "proof:bk3_coupling_vs_perturbation_stability"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "n the respective membranes (Def.~\\ref{definition:bk3_symbolic_membrane}). (Probabilities $\\rho_i, \\rho_j$ are from Def.~\\ref{definition:bk2__symbolic_probability_density}, measure $d\\mu_g$ from Def.~\\ref{definition:bk2_symbolic_probability_spa}, coupling map $\\Phi_{ij}$ from Def.~\\ref{defi",
      "label": "definition:bk2__symbolic_probability_density",
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      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "abilities $\\rho_i, \\rho_j$ are from Def.~\\ref{definition:bk2__symbolic_probability_density}, measure $d\\mu_g$ from Def.~\\ref{definition:bk2_symbolic_probability_spa}, coupling map $\\Phi_{ij}$ from Def.~\\ref{definition:bk3_coupling_map}). \\end{lemma}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "_density}, measure $d\\mu_g$ from Def.~\\ref{definition:bk2_symbolic_probability_spa}, coupling map $\\Phi_{ij}$ from Def.~\\ref{definition:bk3_coupling_map}). \\end{lemma}",
      "label": "definition:bk3_coupling_map",
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      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 85,
      "target_type": "definition"
    },
    {
      "context": "osis}) enhances stability when the coupling strength $\\lambda_{ij}$ and response parameters $\\eta_i, \\eta_j$ (from Def.~\\ref{definition:bk3_induced_coupling_energy} and Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}) satisfy: \\[ \\lambda_{ij} > \\max\\left\\{\\frac{\\delta_i^2}{",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": ")}\\right\\} \\] where $\\delta_i, \\delta_j$ are the maximum internal drift perturbations in the respective membranes (Def.~\\ref{definition:bk3_symbolic_membrane}). (Probabilities $\\rho_i, \\rho_j$ are from Def.~\\ref{definition:bk2__symbolic_probability_density}, measure $d\\mu_g$ fr",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "\\begin{lemma}[Symbiotic Stability Conditions] \\label{lemma:bk3_symbiotic_stability_conditions} Symbiotic coupling (Def.~\\ref{definition:bk3_symbolic_symbiosis}) enhances stability when the coupling strength $\\lambda_{ij}$ and response parameters $\\eta_i, \\eta_j$ (from Def.~\\ref{",
      "label": "definition:bk3_symbolic_symbiosis",
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      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "\\lambda_{ij}$ and response parameters $\\eta_i, \\eta_j$ (from Def.~\\ref{definition:bk3_induced_coupling_energy} and Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}) satisfy: \\[ \\lambda_{ij} > \\max\\left\\{\\frac{\\delta_i^2}{4\\eta_i \\int_{\\mathcal{M}_j} \\rho_j(y)\\|\\nabla_x \\Phi_{ij}(x,y",
      "label": "theorem:bk3_couplinginduced_drift_modification",
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    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_couplinginduced_drift_modification"
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  "file": "book3.tex",
  "id": "proof:bk3_coupling_vs_perturbation_stability",
  "label": "proof:bk3_coupling_vs_perturbation_stability",
  "latex_body": "\\begin{proof}[Coupling-Induced Drift Must Outweigh Internal Perturbations]\n\\label{proof:bk3_coupling_vs_perturbation_stability}\n\\leavevmode\n\nFor stability enhancement, the coupling-induced drift modification must counteract potential internal perturbations.\nThe condition in Lem.~\\ref{lemma:bk3_symbiotic_stability_conditions} ensures that expected restoring force from coupling exceeds the maximum destabilizing force from $\\delta_i$ and $\\delta_j$.\nThe factor of 4 comes from worst-case alignment between perturbation and gradient directions.\nThe integrals represent average coupling sensitivity, weighted by probability distributions.\n\\end{proof}",
  "line": 165,
  "macros_used": [],
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  "name": "Coupling-Induced Drift Must Outweigh Internal Perturbations",
  "proves": "lemma:bk3_symbiotic_stability_conditions",
  "ref_roles": [
    {
      "context": "cement, the coupling-induced drift modification must counteract potential internal perturbations. The condition in Lem.~\\ref{lemma:bk3_symbiotic_stability_conditions} ensures that expected restoring force from coupling exceeds the maximum destabilizing force from $\\delta_i$ and $\\delta",
      "label": "lemma:bk3_symbiotic_stability_conditions",
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  "book": "book3",
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  "cites": [
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    "lemma:bk1_local_stability_analysis",
    "proposition:bk1_observer_relative_bounded_approximation",
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  "file": "book3.tex",
  "id": "scholium:bk3_hypotheses_as_cognitive_membranes",
  "label": "scholium:bk3_hypotheses_as_cognitive_membranes",
  "latex_body": "\\begin{scholium}[Hypotheses as Cognitive Membranes] \\label{scholium:bk3_hypotheses_as_cognitive_membranes}\nIn the symbiotic framing, hypotheses no longer serve as fixed conjectures or static predictions (cf.~Definition~\\ref{definition:bk1_symbolic_hypothesis}). Instead, they behave as \\emph{semi-permeable cognitive membranes}—interfaces between symbolic subsystems that mediate flows of drift and reflection (cf.~Definition~\\ref{definition:bk1_drift_field}, Proposition~\\ref{proposition:bk1_observer_relative_bounded_approximation}).\nJust as biological membranes allow selective exchange, symbolic hypotheses regulate which transformations are permitted, reinforced, or resisted. Each hypothesis \\(\\mathcal{H}_\\Obs\\) thus becomes a site of \\emph{selective resonance}, structured by the observer’s internal metrics (cf.~Definition~\\ref{definition:bk1_bounded_observer}) and bounded by its epistemic curvature (cf.~Scholium~\\ref{scholium:bk1_hypotheses_as_submanifolds}).\nThis reframes cognition not as isolated modeling, but as relational attunement—where hypotheses evolve through interaction with symbolic environments and co-adaptive membranes. Reflexive updates to hypotheses correspond to metabolic exchanges across symbolic membranes, driven by free-energy gradients (cf.~Definition~\\ref{definition:bk2_symbolic_free_energy}) and stabilized through drift-reflection dynamics (cf.~Theorem~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}, Lemma~\\ref{lemma:bk1_local_stability_analysis}).\\end{scholium}",
  "line": 175,
  "macros_used": [
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  "matter_region": "mainmatter",
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  "name": "Hypotheses as Cognitive Membranes",
  "ref_roles": [
    {
      "context": "\\Obs\\) thus becomes a site of \\emph{selective resonance}, structured by the observer’s internal metrics (cf.~Definition~\\ref{definition:bk1_bounded_observer}) and bounded by its epistemic curvature (cf.~Scholium~\\ref{scholium:bk1_hypotheses_as_submanifolds}). This reframes cog",
      "label": "definition:bk1_bounded_observer",
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      "role": "cf_near_match",
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      "context": "cognitive membranes}—interfaces between symbolic subsystems that mediate flows of drift and reflection (cf.~Definition~\\ref{definition:bk1_drift_field}, Proposition~\\ref{proposition:bk1_observer_relative_bounded_approximation}). Just as biological membranes allow selecti",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "branes} In the symbiotic framing, hypotheses no longer serve as fixed conjectures or static predictions (cf.~Definition~\\ref{definition:bk1_symbolic_hypothesis}). Instead, they behave as \\emph{semi-permeable cognitive membranes}—interfaces between symbolic subsystems that mediate",
      "label": "definition:bk1_symbolic_hypothesis",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1257,
      "target_type": "definition"
    },
    {
      "context": "hypotheses correspond to metabolic exchanges across symbolic membranes, driven by free-energy gradients (cf.~Definition~\\ref{definition:bk2_symbolic_free_energy}) and stabilized through drift-reflection dynamics (cf.~Theorem~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equat",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
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    {
      "context": "ized through drift-reflection dynamics (cf.~Theorem~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}, Lemma~\\ref{lemma:bk1_local_stability_analysis}).\\end{scholium}",
      "label": "lemma:bk1_local_stability_analysis",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3707,
      "target_type": "lemma"
    },
    {
      "context": "lic subsystems that mediate flows of drift and reflection (cf.~Definition~\\ref{definition:bk1_drift_field}, Proposition~\\ref{proposition:bk1_observer_relative_bounded_approximation}). Just as biological membranes allow selective exchange, symbolic hypotheses regulate which transformations are permitt",
      "label": "proposition:bk1_observer_relative_bounded_approximation",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 155,
      "target_type": "proposition"
    },
    {
      "context": "nal metrics (cf.~Definition~\\ref{definition:bk1_bounded_observer}) and bounded by its epistemic curvature (cf.~Scholium~\\ref{scholium:bk1_hypotheses_as_submanifolds}). This reframes cognition not as isolated modeling, but as relational attunement—where hypotheses evolve through intera",
      "label": "scholium:bk1_hypotheses_as_submanifolds",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1253,
      "target_type": "scholium"
    },
    {
      "context": "cf.~Definition~\\ref{definition:bk2_symbolic_free_energy}) and stabilized through drift-reflection dynamics (cf.~Theorem~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}, Lemma~\\ref{lemma:bk1_local_stability_analysis}).\\end{scholium}",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
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      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
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    "definition:bk1_symbolic_hypothesis",
    "definition:bk2_symbolic_free_energy",
    "lemma:bk1_local_stability_analysis",
    "proposition:bk1_observer_relative_bounded_approximation",
    "scholium:bk1_hypotheses_as_submanifolds",
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
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  "role": "scholium",
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  "line": 180,
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  "type": "section"
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{
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    "definition:bk3_symbolic_autopoiesis",
    "definition:bk4_recursive_identity_encod",
    "lemma:bk3_reflexive_encodings_generate_conceptual_bridges",
    "proof:bk3_reflexive_encoding_preserves_structure",
    "proof:bk3_triangle_inequality_encoding_bound",
    "theorem:bk3_cyclic_reflexive_encodings"
  ],
  "cites": [
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    "definition:bk3_symbolic_membrane"
  ],
  "depends_on": [
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  "file": "book3.tex",
  "id": "definition:bk3_reflexive_encoding",
  "label": "definition:bk3_reflexive_encoding",
  "latex_body": "\\begin{definition}[Reflexive Encoding] \\label{definition:bk3_reflexive_encoding}\nA reflexive encoding for a symbolic membrane $\\mathcal{M}_i$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is a smooth map $E_i: \\mathcal{M}_i \\rightarrow \\mathcal{M}_j$ to another membrane $\\mathcal{M}_j$ satisfying:\n\\begin{enumerate}\n    \\item Bounded distortion: $d_g(E_j \\circ E_i(x), x) \\leq \\epsilon_{ij}$ for all $x \\in \\mathcal{M}_i$ and some bound $\\epsilon_{ij} > 0$, where $d_g$ is the distance induced by the symbolic metric $g$.\n    \\item Stability preservation: $S_i(x) \\approx S_j(E_i(x))$ up to a scaling factor, meaning that stable regions map to stable regions.\n    \\item Information preservation: The map preserves a significant portion of the information content, quantified by the conditional entropy $H(\\mathcal{M}_i | E_i(\\mathcal{M}_i)) < H(\\mathcal{M}_i) - \\kappa_i$ for some threshold $\\kappa_i > 0$ (cf. Def.~\\ref{definition:bk2_symbolic_entropy}).\n\\end{enumerate}\n\\end{definition}",
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    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Bounded round-trip distortion and the information-preservation inequality (conditionalEntropy < totalEntropy - kappa) are modeled; the approximate-equality 'stability preservation up to scaling' clause is not formalized."
    ],
    "record_ids": [
      "MAP-BOOK3-013"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book3.reflexive_encoding_information_strictly_preserved"
    ]
  },
  "line": 183,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Reflexive Encoding",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "tropy $H(\\mathcal{M}_i | E_i(\\mathcal{M}_i)) < H(\\mathcal{M}_i) - \\kappa_i$ for some threshold $\\kappa_i > 0$ (cf. Def.~\\ref{definition:bk2_symbolic_entropy}). \\end{enumerate} \\end{definition}",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "Encoding] \\label{definition:bk3_reflexive_encoding} A reflexive encoding for a symbolic membrane $\\mathcal{M}_i$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is a smooth map $E_i: \\mathcal{M}_i \\rightarrow \\mathcal{M}_j$ to another membrane $\\mathcal{M}_j$ satisfying: \\begin{",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_entropy",
    "definition:bk3_symbolic_membrane"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "depends_on": [
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_cyclic_reflexive_encodings",
  "label": "theorem:bk3_cyclic_reflexive_encodings",
  "latex_body": "\\begin{theorem}[Cyclic Reflexive Encodings] \\label{theorem:bk3_cyclic_reflexive_encodings}\nFor a cycle of reflexive encodings $E_i: \\mathcal{M}_i \\rightarrow \\mathcal{M}_{i+1}$ for $i = 1,2,...,n$ with $\\mathcal{M}_{n+1} = \\mathcal{M}_1$ (Def.~\\ref{definition:bk3_symbolic_membrane}), the composition $E = E_n \\circ E_{n-1} \\circ \\cdots \\circ E_1$ satisfies:\n\\[\nd_g(E(x), x) \\leq \\sum_{i=1}^{n} \\epsilon_{i,i+1}\n\\]\nfor all $x \\in \\mathcal{M}_1$, where $\\epsilon_{i,i+1}$ is the distortion bound for encoding $E_i$ (from Def.~\\ref{definition:bk3_reflexive_encoding}).\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "manifold/PDE/Helmholtz content of Book 3 is NOT formalized; static and finite-discrete kernels only",
      "modeling laws (rate equations, stability conditions, Lipschitz response) are structure fields"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Telescoping triangle-inequality bound on a chain of n composed maps in a PseudoMetricSpace, proved by induction. Directly instantiates the cyclic case since the theorem is agnostic to whether the chain closes into a loop."
    ],
    "record_ids": [
      "MAP-BOOK3-014"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book3.cyclic_distortion_bound"
    ]
  },
  "line": 193,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Cyclic Reflexive Encodings",
  "proof_labels": [
    "proof:bk3_triangle_inequality_encoding_bound"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "i,i+1} \\] for all $x \\in \\mathcal{M}_1$, where $\\epsilon_{i,i+1}$ is the distortion bound for encoding $E_i$ (from Def.~\\ref{definition:bk3_reflexive_encoding}). \\end{theorem}",
      "label": "definition:bk3_reflexive_encoding",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 183,
      "target_type": "definition"
    },
    {
      "context": "s $E_i: \\mathcal{M}_i \\rightarrow \\mathcal{M}_{i+1}$ for $i = 1,2,...,n$ with $\\mathcal{M}_{n+1} = \\mathcal{M}_1$ (Def.~\\ref{definition:bk3_symbolic_membrane}), the composition $E = E_n \\circ E_{n-1} \\circ \\cdots \\circ E_1$ satisfies: \\[ d_g(E(x), x) \\leq \\sum_{i=1}^{n} \\epsilo",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk3_reflexive_encoding"
  ],
  "depends_on": [
    "definition:bk3_reflexive_encoding"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_triangle_inequality_encoding_bound",
  "label": "proof:bk3_triangle_inequality_encoding_bound",
  "latex_body": "\\begin{proof}[Triangle Inequality Bounds Reflexive Encoding Drift]\n\\label{proof:bk3_triangle_inequality_encoding_bound}\n\\leavevmode\n\nUsing the triangle inequality for the metric $d_g$:\n\\begin{align*}\nd_g(E(x), x) &= d_g(E_n \\circ \\cdots \\circ E_1(x), x) \\\\\n&\\leq d_g(E_n \\circ \\cdots \\circ E_1(x), E_{n-1} \\circ \\cdots \\circ E_1(x)) \\\\\n&\\quad + d_g(E_{n-1} \\circ \\cdots \\circ E_1(x), E_{n-2} \\circ \\cdots \\circ E_1(x)) \\\\\n&\\quad + \\cdots + d_g(E_1(x), x)\n\\end{align*}\nSetting $x_0 = x$ and $x_k = E_k(x_{k-1})$ for $k = 1, \\ldots, n$, so that $E(x) = x_n$, each consecutive pair satisfies $d_g(x_k, x_{k-1}) \\leq \\epsilon_{k-1, k}$ by the distortion bound of Def.~\\ref{definition:bk3_reflexive_encoding}. Substituting into the triangle inequality expansion above:\n\\[\nd_g(E(x), x) \\leq \\sum_{k=1}^{n} d_g(x_k, x_{k-1}) \\leq \\sum_{i=1}^{n} \\epsilon_{i,i+1}\n\\]\nThus compositions of reflexive encodings maintain bounded total distortion, allowing information to circulate through networks of symbolic membranes while preserving essential structure.\n\\end{proof}",
  "line": 201,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Triangle Inequality Bounds Reflexive Encoding Drift",
  "proves": "theorem:bk3_cyclic_reflexive_encodings",
  "ref_roles": [
    {
      "context": "E(x) = x_n$, each consecutive pair satisfies $d_g(x_k, x_{k-1}) \\leq \\epsilon_{k-1, k}$ by the distortion bound of Def.~\\ref{definition:bk3_reflexive_encoding}. Substituting into the triangle inequality expansion above: \\[ d_g(E(x), x) \\leq \\sum_{k=1}^{n} d_g(x_k, x_{k-1}) \\leq",
      "label": "definition:bk3_reflexive_encoding",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 183,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_reflexive_encoding"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_conceptual_bridge_sequence",
    "definition:bk3_symbolic_autopoiesis",
    "definition:bk3_symbolic_metabolism",
    "definition:bk3_symbolic_network",
    "lemma:bk3_reflexive_encodings_generate_conceptual_bridges"
  ],
  "cites": [
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "depends_on": [
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_conceptual_bridge",
  "label": "definition:bk3_conceptual_bridge",
  "latex_body": "\\begin{definition}[Conceptual Bridge] \\label{definition:bk3_conceptual_bridge}\nA conceptual bridge between symbolic domains $\\mathcal{D}_1$ and $\\mathcal{D}_2$ (which can be symbolic membranes, cf. Def.~\\ref{definition:bk3_symbolic_membrane}) is a pair of maps $(f_{12}, f_{21})$ where $f_{12}: \\mathcal{D}_1 \\rightarrow \\mathcal{D}_2$ and $f_{21}: \\mathcal{D}_2 \\rightarrow \\mathcal{D}_1$ satisfy:\n\\begin{enumerate}\n    \\item Approximate invertibility: $f_{21} \\circ f_{12}$ and $f_{12} \\circ f_{21}$ are approximately identity maps on their respective domains, with bounded distortion (related to Def.~\\ref{definition:bk3_reflexive_encoding}).\n    \\item Structure preservation: The maps preserve key structural relations within each domain.\n    \\item Semantic consistency: The meanings or interpretations associated with mapped elements remain coherent across domains.\n\\end{enumerate}\n\\end{definition}",
  "line": 219,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conceptual Bridge",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "12} \\circ f_{21}$ are approximately identity maps on their respective domains, with bounded distortion (related to Def.~\\ref{definition:bk3_reflexive_encoding}). \\item Structure preservation: The maps preserve key structural relations within each domain. \\item Semantic c",
      "label": "definition:bk3_reflexive_encoding",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 183,
      "target_type": "definition"
    },
    {
      "context": "nceptual bridge between symbolic domains $\\mathcal{D}_1$ and $\\mathcal{D}_2$ (which can be symbolic membranes, cf. Def.~\\ref{definition:bk3_symbolic_membrane}) is a pair of maps $(f_{12}, f_{21})$ where $f_{12}: \\mathcal{D}_1 \\rightarrow \\mathcal{D}_2$ and $f_{21}: \\mathcal{D}_",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "proof:bk3_reflexive_encoding_preserves_structure"
  ],
  "cites": [
    "definition:bk3_conceptual_bridge",
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "depends_on": [
    "definition:bk3_conceptual_bridge",
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "file": "book3.tex",
  "id": "lemma:bk3_reflexive_encodings_generate_conceptual_bridges",
  "label": "lemma:bk3_reflexive_encodings_generate_conceptual_bridges",
  "latex_body": "\\begin{lemma}[Reflexive Encodings Generate Conceptual Bridges] \\label{lemma:bk3_reflexive_encodings_generate_conceptual_bridges}\nGiven reflexive encodings $E_i: \\mathcal{M}_i \\rightarrow \\mathcal{M}_j$ and $E_j: \\mathcal{M}_j \\rightarrow \\mathcal{M}_i$ between symbolic membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$ (Def.~\\ref{definition:bk3_symbolic_membrane}, Def.~\\ref{definition:bk3_reflexive_encoding}), the pair $(E_i, E_j)$ forms a conceptual bridge (Def.~\\ref{definition:bk3_conceptual_bridge}) between the symbolic domains represented by these membranes.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "manifold/PDE/Helmholtz content of Book 3 is NOT formalized; static and finite-discrete kernels only",
      "modeling laws (rate equations, stability conditions, Lipschitz response) are structure fields"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Shows that reflexive-encoding-style round-trip bounds are literally sufficient data to construct a BoundedRoundTrip; existence-style construction rather than a deep theorem."
    ],
    "record_ids": [
      "MAP-BOOK3-016"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3.reflexive_pair_generates_bridge"
    ]
  },
  "line": 229,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Reflexive Encodings Generate Conceptual Bridges",
  "proof_labels": [
    "proof:bk3_reflexive_encoding_preserves_structure"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ymbolic_membrane}, Def.~\\ref{definition:bk3_reflexive_encoding}), the pair $(E_i, E_j)$ forms a conceptual bridge (Def.~\\ref{definition:bk3_conceptual_bridge}) between the symbolic domains represented by these membranes. \\end{lemma}",
      "label": "definition:bk3_conceptual_bridge",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 219,
      "target_type": "definition"
    },
    {
      "context": "M}_i$ between symbolic membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$ (Def.~\\ref{definition:bk3_symbolic_membrane}, Def.~\\ref{definition:bk3_reflexive_encoding}), the pair $(E_i, E_j)$ forms a conceptual bridge (Def.~\\ref{definition:bk3_conceptual_bridge}) between the symbolic do",
      "label": "definition:bk3_reflexive_encoding",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 183,
      "target_type": "definition"
    },
    {
      "context": "and $E_j: \\mathcal{M}_j \\rightarrow \\mathcal{M}_i$ between symbolic membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$ (Def.~\\ref{definition:bk3_symbolic_membrane}, Def.~\\ref{definition:bk3_reflexive_encoding}), the pair $(E_i, E_j)$ forms a conceptual bridge (Def.~\\ref{definition:b",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_conceptual_bridge",
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk3_reflexive_encoding",
    "lemma:bk3_reflexive_encodings_generate_conceptual_bridges"
  ],
  "depends_on": [
    "definition:bk3_reflexive_encoding",
    "lemma:bk3_reflexive_encodings_generate_conceptual_bridges"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_reflexive_encoding_preserves_structure",
  "label": "proof:bk3_reflexive_encoding_preserves_structure",
  "latex_body": "\\begin{proof}[Symbolic Reflexive Encoding Preserves Semantic Structure]\n\\label{proof:bk3_reflexive_encoding_preserves_structure}\n\\leavevmode\n\nBounded distortion (Def.~\\ref{definition:bk3_reflexive_encoding}) gives\napproximate invertibility:\n$d_g(E_j \\circ E_i(x), x) \\leq \\epsilon_{ij}$ and\n$d_g(E_i \\circ E_j(y), y) \\leq \\epsilon_{ji}$.\nStability preservation keeps structural relations intact, since stable\nconfigurations in one membrane map to stable configurations in the other.\nInformation preservation keeps semantic consistency across the mapping.\nTherefore reflexive encodings generate conceptual bridges\n(Lem.~\\ref{lemma:bk3_reflexive_encodings_generate_conceptual_bridges}) that\nsupport coherent transfer of symbolic structure between membranes.\n\\end{proof}",
  "line": 233,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Reflexive Encoding Preserves Semantic Structure",
  "proves": "lemma:bk3_reflexive_encodings_generate_conceptual_bridges",
  "ref_roles": [
    {
      "context": "rves Semantic Structure] \\label{proof:bk3_reflexive_encoding_preserves_structure} \\leavevmode Bounded distortion (Def.~\\ref{definition:bk3_reflexive_encoding}) gives approximate invertibility: $d_g(E_j \\circ E_i(x), x) \\leq \\epsilon_{ij}$ and $d_g(E_i \\circ E_j(y), y) \\leq \\eps",
      "label": "definition:bk3_reflexive_encoding",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 183,
      "target_type": "definition"
    },
    {
      "context": "ervation keeps semantic consistency across the mapping. Therefore reflexive encodings generate conceptual bridges (Lem.~\\ref{lemma:bk3_reflexive_encodings_generate_conceptual_bridges}) that support coherent transfer of symbolic structure between membranes. \\end{proof}",
      "label": "lemma:bk3_reflexive_encodings_generate_conceptual_bridges",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 229,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk3_reflexive_encoding",
    "lemma:bk3_reflexive_encodings_generate_conceptual_bridges"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "subsec:bk3_symbiotic_curvature_system_properties",
  "label": "subsec:bk3_symbiotic_curvature_system_properties",
  "latex_body": "",
  "line": 249,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbiotic Curvature and System Properties",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [
    "lemma:bk4_fragmentation_cascade",
    "proof:bk3_sketch_necessity_for_continuous_operation",
    "proof:bk3_sketch_perturbation_dissiptation",
    "proof:bk3_symbolic_coupling_properties_enumerated",
    "proof:bk4_sketch_cross_field_product",
    "proof:bk4_symbolic_curvature_boundary",
    "proof:bk4_symbolic_curvature_fragmentation",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk3_properties_of_symbiotic_curvature",
    "theorem:bk3_symbiotic_curvature_and_resilience",
    "theorem:bk4_formation_differentiation_boundaries",
    "theorem:bk4_fuzzy_sum_rule"
  ],
  "cites": [
    "definition:bk3_coupling_map",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis"
  ],
  "depends_on": [
    "definition:bk3_coupling_map",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_symbiotic_curvature",
  "label": "definition:bk3_symbiotic_curvature",
  "latex_body": "\\begin{definition}[Symbiotic Curvature] \\label{definition:bk3_symbiotic_curvature}\nFor a system of coupled symbolic membranes $\\{\\mathcal{M}_i\\}_{i=1}^n$ (Def.~\\ref{definition:bk3_symbolic_membrane}) with coupling maps $\\{\\Phi_{ij}\\}$ (Def.~\\ref{definition:bk3_coupling_map}) and symbiotic relations (Def.~\\ref{definition:bk3_symbolic_symbiosis}), the symbiotic curvature $\\kappa_{\\text{symb}}$ is defined as:\n\\[\n\\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\}) = \\frac{1}{n}\\sum_{i=1}^n \\frac{S_i^{\\text{coupled}}}{S_i^{\\text{isolated}}} \\cdot \\left(1 + \\gamma \\sum_{j \\neq i} I(\\mathcal{M}_i; \\mathcal{M}_j)\\right)\n\\]\nwhere $S_i^{\\text{coupled}}$ and $S_i^{\\text{isolated}}$ are the stability measures of membrane $i$ in coupled and isolated states respectively, $I(\\mathcal{M}_i; \\mathcal{M}_j)$ is the mutual information between membranes, and $\\gamma > 0$ is a scaling parameter.\n\\end{definition}",
  "line": 252,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbiotic Curvature",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "anes $\\{\\mathcal{M}_i\\}_{i=1}^n$ (Def.~\\ref{definition:bk3_symbolic_membrane}) with coupling maps $\\{\\Phi_{ij}\\}$ (Def.~\\ref{definition:bk3_coupling_map}) and symbiotic relations (Def.~\\ref{definition:bk3_symbolic_symbiosis}), the symbiotic curvature $\\kappa_{\\text{symb}}$",
      "label": "definition:bk3_coupling_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 85,
      "target_type": "definition"
    },
    {
      "context": "\\label{definition:bk3_symbiotic_curvature} For a system of coupled symbolic membranes $\\{\\mathcal{M}_i\\}_{i=1}^n$ (Def.~\\ref{definition:bk3_symbolic_membrane}) with coupling maps $\\{\\Phi_{ij}\\}$ (Def.~\\ref{definition:bk3_coupling_map}) and symbiotic relations (Def.~\\ref{definit",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "ic_membrane}) with coupling maps $\\{\\Phi_{ij}\\}$ (Def.~\\ref{definition:bk3_coupling_map}) and symbiotic relations (Def.~\\ref{definition:bk3_symbolic_symbiosis}), the symbiotic curvature $\\kappa_{\\text{symb}}$ is defined as: \\[ \\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\}) = \\frac{1}{n",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_coupling_map",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "proof:bk3_sketch_necessity_for_continuous_operation",
    "proof:bk3_symbolic_coupling_properties_enumerated",
    "proof:bk4_sketch_cross_field_product",
    "proof:bk4_sketch_symbolic_path_interference",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "cites": [
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis"
  ],
  "depends_on": [
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_properties_of_symbiotic_curvature",
  "label": "theorem:bk3_properties_of_symbiotic_curvature",
  "latex_body": "\\begin{theorem}[Properties of Symbiotic Curvature] \\label{theorem:bk3_properties_of_symbiotic_curvature}\nThe symbiotic curvature $\\kappa_{\\text{symb}}$ (Def.~\\ref{definition:bk3_symbiotic_curvature}) satisfies:\n\\begin{enumerate}\n    \\item Positivity: $\\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\}) > 0$ for any non-empty set of membranes.\n    \\item Symbiotic enhancement: If all pairs of membranes are in symbiosis (Definition~\\ref{definition:bk3_symbolic_symbiosis}), then $\\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\}) > 1$.\n    \\item Monotonicity under information increase: If the mutual information $I(\\mathcal{M}_i; \\mathcal{M}_j)$ increases while stability ratios remain constant, $\\kappa_{\\text{symb}}$ increases.\n    \\item Subadditivity: For disjoint sets of membranes $A$ and $B$ with no coupling between them, $\\kappa_{\\text{symb}}(A \\cup B) \\leq \\max(\\kappa_{\\text{symb}}(A), \\kappa_{\\text{symb}}(B))$.\n\\end{enumerate}\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "manifold/PDE/Helmholtz content of Book 3 is NOT formalized; static and finite-discrete kernels only",
      "modeling laws (rate equations, stability conditions, Lipschitz response) are structure fields"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Positivity and symbiotic-enhancement (>1) are proved in full for the finite-n formula. Monotonicity is proved as a non-strict comparison across two info matrices. Subadditivity is only covered at the level of its abstract mediant/convex-combination core (weighted_avg_le_max); the full derivation from summing over two disjoint index sets is not carried out."
    ],
    "record_ids": [
      "MAP-BOOK3-018"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3.symbioticCurvature_gt_one",
      "Book3.symbioticCurvature_mono_info",
      "Book3.symbioticCurvature_pos",
      "Book3.weighted_avg_le_max"
    ]
  },
  "line": 261,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Properties of Symbiotic Curvature",
  "proof_labels": [
    "proof:bk3_symbolic_coupling_properties_enumerated"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "c Curvature] \\label{theorem:bk3_properties_of_symbiotic_curvature} The symbiotic curvature $\\kappa_{\\text{symb}}$ (Def.~\\ref{definition:bk3_symbiotic_curvature}) satisfies: \\begin{enumerate} \\item Positivity: $\\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\}) > 0$ for any non-empty set",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "any non-empty set of membranes. \\item Symbiotic enhancement: If all pairs of membranes are in symbiosis (Definition~\\ref{definition:bk3_symbolic_symbiosis}), then $\\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\}) > 1$. \\item Monotonicity under information increase: If the mutual",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "depends_on": [
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_symbolic_coupling_properties_enumerated",
  "label": "proof:bk3_symbolic_coupling_properties_enumerated",
  "latex_body": "\\begin{proof}[Categorical Properties of Symbolic Coupling]\n\\label{proof:bk3_symbolic_coupling_properties_enumerated}\n\\leavevmode\n\n\\begin{enumerate}\n    \\item Positivity follows from the positivity of stability measures ($S_i > 0$) and mutual information ($I \\geq 0$). Since $S_i^{\\text{coupled}} > 0$ and $S_i^{\\text{isolated}} > 0$, their ratio is positive. The term in parentheses is $1 + (\\text{non-negative terms}) \\geq 1$. The sum of positive terms divided by $n$ is positive.\n    \\item By the definition of symbiosis (Definition~\\ref{definition:bk3_symbolic_symbiosis}), each $S_i^{\\text{coupled}} > S_i^{\\text{isolated}}$, so their ratio exceeds 1. The mutual information terms $I(\\mathcal{M}_i; \\mathcal{M}_j)$ are positive under symbiosis. Thus, the term $\\left(1 + \\gamma \\sum_{j \\neq i} I(\\mathcal{M}_i; \\mathcal{M}_j)\\right)$ is strictly greater than 1. The average of terms, each being a product of a number $>1$ and another number $>1$, will be greater than 1.\n    \\item This follows directly from the definition (Def.~\\ref{definition:bk3_symbiotic_curvature}), as $\\kappa_{\\text{symb}}$ is an increasing function of the mutual information terms $I(\\mathcal{M}_i; \\mathcal{M}_j)$ when all else is held constant.\n    \\item Without coupling between sets $A = \\{\\mathcal{M}_k\\}_{k \\in K_A}$ and $B = \\{\\mathcal{M}_l\\}_{l \\in K_B}$, the mutual information terms $I(\\mathcal{M}_k; \\mathcal{M}_l)$ are zero for $k \\in K_A, l \\in K_B$. Let $n_A = |A|$ and $n_B = |B|$, so $n = n_A + n_B$.\n    \\[\n    \\kappa_{\\text{symb}}(A \\cup B) = \\frac{1}{n_A+n_B} \\left( \\sum_{k \\in K_A} \\frac{S_k^{\\text{c}}}{S_k^{\\text{i}}} (1 + \\gamma \\sum_{k' \\in K_A, k' \\neq k} I_{kk'}) + \\sum_{l \\in K_B} \\frac{S_l^{\\text{c}}}{S_l^{\\text{i}}} (1 + \\gamma \\sum_{l' \\in K_B, l' \\neq l} I_{ll'}) \\right)\n    \\]\n    \\[\n    = \\frac{1}{n_A+n_B} (n_A \\kappa_{\\text{symb}}(A) + n_B \\kappa_{\\text{symb}}(B))\n    \\]\n    This is a weighted average of $\\kappa_{\\text{symb}}(A)$ and $\\kappa_{\\text{symb}}(B)$, which is bounded above by the maximum of the two. (This supports Thm.~\\ref{theorem:bk3_properties_of_symbiotic_curvature}).\n\\end{enumerate}\n\\end{proof}",
  "line": 271,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Categorical Properties of Symbolic Coupling",
  "proves": "theorem:bk3_properties_of_symbiotic_curvature",
  "ref_roles": [
    {
      "context": "number $>1$ and another number $>1$, will be greater than 1. \\item This follows directly from the definition (Def.~\\ref{definition:bk3_symbiotic_curvature}), as $\\kappa_{\\text{symb}}$ is an increasing function of the mutual information terms $I(\\mathcal{M}_i; \\mathcal{M}_j)$",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "s}) \\geq 1$. The sum of positive terms divided by $n$ is positive. \\item By the definition of symbiosis (Definition~\\ref{definition:bk3_symbolic_symbiosis}), each $S_i^{\\text{coupled}} > S_i^{\\text{isolated}}$, so their ratio exceeds 1. The mutual information terms $I(\\mathc",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "_{\\text{symb}}(A)$ and $\\kappa_{\\text{symb}}(B)$, which is bounded above by the maximum of the two. (This supports Thm.~\\ref{theorem:bk3_properties_of_symbiotic_curvature}). \\end{enumerate} \\end{proof}",
      "label": "theorem:bk3_properties_of_symbiotic_curvature",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 261,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [
    "proof:bk3_sketch_perturbation_dissiptation",
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "cites": [
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "depends_on": [],
  "file": "book3.tex",
  "forward_ref_roles": [
    {
      "context": "e initial perturbation (measured appropriately, e.g., in terms of probability density deviation). (This is key for Thm.~\\ref{theorem:bk3_symbiotic_curvature_and_resilience}). \\end{definition}",
      "label": "theorem:bk3_symbiotic_curvature_and_resilience",
      "line_distance": 9,
      "role": "teaser",
      "target_line": 299,
      "target_type": "theorem"
    }
  ],
  "forward_refs": [
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "id": "definition:bk3_perturbation_response_function",
  "label": "definition:bk3_perturbation_response_function",
  "latex_body": "\\begin{definition}[Perturbation Response Function] \\label{definition:bk3_perturbation_response_function}\nFor a system of coupled symbolic membranes, the perturbation response function $R(\\delta, t)$ measures how the system's state deviation evolves over time $t$ after an initial perturbation of magnitude $\\delta$:\n\\[\nR(\\delta, t) = \\frac{\\|\\Delta S(t)\\|_g}{\\delta}\n\\]\nwhere $\\Delta S(t)$ is the state deviation at time $t$ after the initial perturbation (measured appropriately, e.g., in terms of probability density deviation). (This is key for Thm.~\\ref{theorem:bk3_symbiotic_curvature_and_resilience}).\n\\end{definition}",
  "line": 290,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Perturbation Response Function",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "e initial perturbation (measured appropriately, e.g., in terms of probability density deviation). (This is key for Thm.~\\ref{theorem:bk3_symbiotic_curvature_and_resilience}). \\end{definition}",
      "label": "theorem:bk3_symbiotic_curvature_and_resilience",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "book3.tex",
      "target_line": 299,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_perturbation_response_function",
    "remark:bk3_symbolic_membrane_remark",
    "remark:bk4_fuzzy",
    "scholium:bk4_symbolic_entanglement"
  ],
  "cites": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk3_perturbation_response_function",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "depends_on": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk3_perturbation_response_function",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk3_symbiotic_stability_conditions",
    "theorem:bk1_symbolic_emergence_and_curvature",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_symbiotic_curvature_and_resilience",
  "label": "theorem:bk3_symbiotic_curvature_and_resilience",
  "latex_body": "\\begin{theorem}[Symbiotic Curvature and Resilience] \\label{theorem:bk3_symbiotic_curvature_and_resilience}\nHigher symbiotic curvature (Def.~\\ref{definition:bk3_symbiotic_curvature}; cf.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, \\ref{corollary:bk1_non_euclidean_necessity}) correlates with enhanced resilience to perturbations (Def.~\\ref{definition:bk3_perturbation_response_function}) for coupled symbolic membranes (Def.~\\ref{definition:bk3_symbolic_membrane}):\n\\[\n\\lim_{t \\rightarrow \\infty} R(\\delta, t) \\leq \\frac{C}{\\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\})}\n\\]\nfor some constant $C > 0$ and sufficiently small perturbations $\\delta$.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "manifold/PDE/Helmholtz content of Book 3 is NOT formalized; static and finite-discrete kernels only",
      "modeling laws (rate equations, stability conditions, Lipschitz response) are structure fields"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Honest static kernel: the bound C/kappa_symb is antitone in kappa_symb for fixed C>0. The limiting behaviour of R(delta,t) as t->infinity is not modeled."
    ],
    "record_ids": [
      "MAP-BOOK3-020"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3.resilience_bound_antitone"
    ]
  },
  "line": 299,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbiotic Curvature and Resilience",
  "proof_labels": [
    "proof:bk3_sketch_perturbation_dissiptation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "iotic curvature (Def.~\\ref{definition:bk3_symbiotic_curvature}; cf.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, \\ref{corollary:bk1_non_euclidean_necessity}) correlates with enhanced resilience to perturbations (Def.~\\ref{definition:bk3_perturbation_response_function}) for co",
      "label": "corollary:bk1_non_euclidean_necessity",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1686,
      "target_type": "corollary"
    },
    {
      "context": "and_curvature}, \\ref{corollary:bk1_non_euclidean_necessity}) correlates with enhanced resilience to perturbations (Def.~\\ref{definition:bk3_perturbation_response_function}) for coupled symbolic membranes (Def.~\\ref{definition:bk3_symbolic_membrane}): \\[ \\lim_{t \\rightarrow \\infty} R(\\delta,",
      "label": "definition:bk3_perturbation_response_function",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 290,
      "target_type": "definition"
    },
    {
      "context": "iotic Curvature and Resilience] \\label{theorem:bk3_symbiotic_curvature_and_resilience} Higher symbiotic curvature (Def.~\\ref{definition:bk3_symbiotic_curvature}; cf.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, \\ref{corollary:bk1_non_euclidean_necessity}) correlates with e",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "lience to perturbations (Def.~\\ref{definition:bk3_perturbation_response_function}) for coupled symbolic membranes (Def.~\\ref{definition:bk3_symbolic_membrane}): \\[ \\lim_{t \\rightarrow \\infty} R(\\delta, t) \\leq \\frac{C}{\\kappa_{\\text{symb}}(\\{\\mathcal{M}_i\\})} \\] for some consta",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": ":bk3_symbiotic_curvature_and_resilience} Higher symbiotic curvature (Def.~\\ref{definition:bk3_symbiotic_curvature}; cf.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, \\ref{corollary:bk1_non_euclidean_necessity}) correlates with enhanced resilience to perturbations (Def.~\\ref{definitio",
      "label": "theorem:bk1_symbolic_emergence_and_curvature",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2029,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk3_perturbation_response_function",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk3_perturbation_response_function",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk3_symbiotic_stability_conditions",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "depends_on": [
    "definition:bk3_perturbation_response_function",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk3_symbiotic_stability_conditions",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_sketch_perturbation_dissiptation",
  "label": "proof:bk3_sketch_perturbation_dissiptation",
  "latex_body": "\\begin{proof}[Perturbation Dissipation via Lyapunov Argument]\n\\label{proof:bk3_sketch_perturbation_dissiptation}\n\\leavevmode\n\n\\textbf{Lyapunov function.}\nDefine $V(t) = \\|\\Delta S(t)\\|_g^2$, where $\\Delta S(t)$ is the state\ndeviation after perturbation $\\delta$.\nCf.~Def.~\\ref{definition:bk3_perturbation_response_function}.\nSince $\\|\\cdot\\|_g$ is a Riemannian norm, $V \\geq 0$ with $V = 0$ if and only if $\\Delta S = 0$ (equilibrium).\n\n\\textbf{Region of attraction.}\nBy Thm.~\\ref{theorem:bk3_membrane_stability_criteria}, the membrane free energy\n$F_i(\\beta_i)$ is at a local minimum at equilibrium.\nLet $\\Omega_c = \\{\\Delta S \\mid V(\\Delta S) \\leq c\\}$ be a sublevel set contained\nin the basin of this local minimum; such $c > 0$ exists by continuity.\nThe condition ``sufficiently small perturbation $\\delta$'' in the theorem\nstatement means precisely $V(0) = \\delta^2 \\leq c$, i.e.\\ $\\delta \\leq \\sqrt{c}$.\n\n\\textbf{Rate bound.}\nWithin $\\Omega_c$, $\\kappa_{\\text{symb}}$ encodes two restorative mechanisms\n(Def.~\\ref{definition:bk3_symbiotic_curvature}):\n\\begin{enumerate}\n    \\item $S_i^{\\text{coupled}}/S_i^{\\text{isolated}} > 1$ under symbiosis\n    (Def.~\\ref{definition:bk3_symbolic_symbiosis}, condition 1) strengthens\n    restorative drift forces proportionally to the excess stability ratio.\n    \\item $I(\\mathcal{M}_i;\\mathcal{M}_j) > 0$ (condition 2) enables cross-membrane\n    drift compensation (condition 3):\n    $\\|\\delta D_i + \\delta D_i^{\\text{response}}\\|_g < \\|\\delta D_i\\|_g$,\n    directly reducing $\\dot{V}$.\n\\end{enumerate}\nBy Lemma~\\ref{lemma:bk3_symbiotic_stability_conditions} and the coupling\nparameters $\\lambda_{ij}, \\eta_i$, both effects combine to give\n\\[\n    \\dot{V}(t) \\leq -\\alpha\\,\\kappa_{\\text{symb}}\\,V(t), \\quad \\alpha > 0,\n\\]\nso $\\dot{V} < 0$ strictly for $V > 0$ (asymptotic stability).\n\n\\textbf{Convergence and bound.}\nSince $\\dot{V} \\leq 0$ within $\\Omega_c$ and the only invariant set where\n$\\dot{V} = 0$ is $\\{\\Delta S = 0\\}$, LaSalle's invariance principle implies\nall trajectories starting in $\\Omega_c$ converge to $\\Delta S = 0$.\nBy Gr\\\"{o}nwall's inequality the explicit rate gives\n$V(t) \\leq \\delta^2 e^{-\\alpha\\,\\kappa_{\\text{symb}}\\,t}$, hence:\n\\[\n\\lim_{t \\to \\infty} R(\\delta, t)\n\\;=\\; \\lim_{t \\to \\infty} \\frac{\\|\\Delta S(t)\\|_g}{\\delta}\n\\;\\leq\\; \\lim_{t \\to \\infty} e^{-(\\alpha/2)\\kappa_{\\text{symb}}\\,t} = 0\n\\;\\leq\\; \\frac{C}{\\kappa_{\\text{symb}}}\n\\]\nfor any $C > 0$, establishing the stated bound.\n\\end{proof}",
  "line": 307,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Perturbation Dissipation via Lyapunov Argument",
  "proves": "theorem:bk3_symbiotic_curvature_and_resilience",
  "ref_roles": [
    {
      "context": "} Define $V(t) = \\|\\Delta S(t)\\|_g^2$, where $\\Delta S(t)$ is the state deviation after perturbation $\\delta$. Cf.~Def.~\\ref{definition:bk3_perturbation_response_function}. Since $\\|\\cdot\\|_g$ is a Riemannian norm, $V \\geq 0$ with $V = 0$ if and only if $\\Delta S = 0$ (equilibrium). \\textb",
      "label": "definition:bk3_perturbation_response_function",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 290,
      "target_type": "definition"
    },
    {
      "context": "leq \\sqrt{c}$. \\textbf{Rate bound.} Within $\\Omega_c$, $\\kappa_{\\text{symb}}$ encodes two restorative mechanisms (Def.~\\ref{definition:bk3_symbiotic_curvature}): \\begin{enumerate} \\item $S_i^{\\text{coupled}}/S_i^{\\text{isolated}} > 1$ under symbiosis (Def.~\\ref{definitio",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "tic_curvature}): \\begin{enumerate} \\item $S_i^{\\text{coupled}}/S_i^{\\text{isolated}} > 1$ under symbiosis (Def.~\\ref{definition:bk3_symbolic_symbiosis}, condition 1) strengthens restorative drift forces proportionally to the excess stability ratio. \\item $I(\\math",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "a D_i + \\delta D_i^{\\text{response}}\\|_g < \\|\\delta D_i\\|_g$, directly reducing $\\dot{V}$. \\end{enumerate} By Lemma~\\ref{lemma:bk3_symbiotic_stability_conditions} and the coupling parameters $\\lambda_{ij}, \\eta_i$, both effects combine to give \\[ \\dot{V}(t) \\leq -\\alpha\\,\\kappa",
      "label": "lemma:bk3_symbiotic_stability_conditions",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 157,
      "target_type": "lemma"
    },
    {
      "context": "nian norm, $V \\geq 0$ with $V = 0$ if and only if $\\Delta S = 0$ (equilibrium). \\textbf{Region of attraction.} By Thm.~\\ref{theorem:bk3_membrane_stability_criteria}, the membrane free energy $F_i(\\beta_i)$ is at a local minimum at equilibrium. Let $\\Omega_c = \\{\\Delta S \\mid V(\\Delta",
      "label": "theorem:bk3_membrane_stability_criteria",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 61,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk3_perturbation_response_function",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk3_symbiotic_stability_conditions",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "role": "proof",
  "type": "proof"
}

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    {
      "context": "relationships between symbolic states). \\end{enumerate} (This process is governed by the Refinement Vector Field, Def.~\\ref{definition:bk3_refinement_vector_field}). \\end{definition}",
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  "label": "definition:bk3_symbolic_refinement",
  "latex_body": "\\begin{definition}[Symbolic Refinement] \\label{definition:bk3_symbolic_refinement}\nSymbolic refinement is a continuous process on a symbolic membrane $\\mathcal{M}$ (Def.~\\ref{definition:bk3_symbolic_membrane}), parameterized by $r \\in [0, \\infty)$, that enhances the symbolic structure by:\n\\begin{enumerate}\n    \\item Increasing internal differentiation (creating more distinct symbolic states).\n    \\item Strengthening integration (enhancing relationships between symbolic states).\n\\end{enumerate}\n(This process is governed by the Refinement Vector Field, Def.~\\ref{definition:bk3_refinement_vector_field}).\n\\end{definition}",
  "line": 364,
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  "name": "Symbolic Refinement",
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    {
      "context": "relationships between symbolic states). \\end{enumerate} (This process is governed by the Refinement Vector Field, Def.~\\ref{definition:bk3_refinement_vector_field}). \\end{definition}",
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    {
      "context": "inition:bk3_symbolic_refinement} Symbolic refinement is a continuous process on a symbolic membrane $\\mathcal{M}$ (Def.~\\ref{definition:bk3_symbolic_membrane}), parameterized by $r \\in [0, \\infty)$, that enhances the symbolic structure by: \\begin{enumerate} \\item Increasing",
      "label": "definition:bk3_symbolic_membrane",
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  "latex_body": "\\begin{definition}[Refinement Vector Field] \\label{definition:bk3_refinement_vector_field}\nThe symbolic refinement vector field $V_r: \\mathcal{M} \\rightarrow T\\mathcal{M}$ governs the evolution of symbolic states under refinement (Def.~\\ref{definition:bk3_symbolic_refinement}):\n\\[\n\\frac{dx}{dr} = V_r(x)\n\\]\nwhere $x \\in \\mathcal{M}$ represents a point in the symbolic manifold.\n\\end{definition}",
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  "ref_roles": [
    {
      "context": "ector field $V_r: \\mathcal{M} \\rightarrow T\\mathcal{M}$ governs the evolution of symbolic states under refinement (Def.~\\ref{definition:bk3_symbolic_refinement}): \\[ \\frac{dx}{dr} = V_r(x) \\] where $x \\in \\mathcal{M}$ represents a point in the symbolic manifold. \\end{definition}",
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    "proof:bk3_differentiation_knowledge_structure",
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    "definition:bk2_symbolic_probability_spa",
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  "id": "definition:bk3_integration_differentiation_pressures",
  "label": "definition:bk3_integration_differentiation_pressures",
  "latex_body": "\\begin{definition}[Integration and Differentiation Pressures] \\label{definition:bk3_integration_differentiation_pressures}\nAt refinement level $r$, the integration pressure $I(r)$ and differentiation pressure $D(r)$ are defined as:\n\\[\nI(r) = \\int_{\\mathcal{M}} \\rho(x,r) \\|\\nabla_g \\cdot V_r(x)\\|_g d\\mu_g(x)\n\\]\n\\[\nD(r) = \\int_{\\mathcal{M}} \\rho(x,r) \\|\\text{curl}_g(V_r)(x)\\|_g d\\mu_g(x)\n\\]\nwhere $\\nabla_g \\cdot$ is the divergence operator and $\\text{curl}_g$ is the curl operator (appropriately defined on the manifold) with respect to the symbolic metric $g$. (Here $\\rho$ is from Def.~\\ref{definition:bk2__symbolic_probability_density}, $V_r$ from Def.~\\ref{definition:bk3_refinement_vector_field}, and the manifold measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{definition}",
  "line": 383,
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  "name": "Integration and Differentiation Pressures",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "url operator (appropriately defined on the manifold) with respect to the symbolic metric $g$. (Here $\\rho$ is from Def.~\\ref{definition:bk2__symbolic_probability_density}, $V_r$ from Def.~\\ref{definition:bk3_refinement_vector_field}, and the manifold measure from Def.~\\ref{definition:bk2_s",
      "label": "definition:bk2__symbolic_probability_density",
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    {
      "context": "_probability_density}, $V_r$ from Def.~\\ref{definition:bk3_refinement_vector_field}, and the manifold measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{definition}",
      "label": "definition:bk2_symbolic_probability_spa",
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      "target_line": 23,
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    },
    {
      "context": "the symbolic metric $g$. (Here $\\rho$ is from Def.~\\ref{definition:bk2__symbolic_probability_density}, $V_r$ from Def.~\\ref{definition:bk3_refinement_vector_field}, and the manifold measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{definition}",
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  "id": "lemma:bk3_helmholtz_decomposition_refinement_field",
  "label": "lemma:bk3_helmholtz_decomposition_refinement_field",
  "latex_body": "\\begin{lemma}[Hodge--Helmholtz Decomposition of the Refinement Field]\n\\label{lemma:bk3_helmholtz_decomposition_refinement_field}\nLet $(\\mathcal{M},g)$ be a compact, connected, oriented smooth Riemannian\nmanifold without boundary, and let $V_r$ be a smooth refinement vector field.\nWriting $V_r^\\flat$ for its metric-dual one-form, there exist a smooth scalar\npotential $\\phi$, a smooth two-form $\\beta$, and a harmonic one-form $h$ such\nthat\n\\begin{equation}\nV_r^\\flat=d\\phi+\\delta\\beta+h.\n\\end{equation}\nThe three summands are pairwise orthogonal in $L^2$, and the decomposition is\nunique after the usual normalization of the scalar potential.  The harmonic\nterm represents the de Rham cohomology class of $V_r^\\flat$; in particular it\nvanishes when $H^1_{\\mathrm{dR}}(\\mathcal{M})=0$.  In dimension three, after\nusing the metric and orientation to identify forms and vector fields, the\ncoexact term $\\delta\\beta$ is the conventional curl component.\n\nFor a finite-dimensional inner-product model with a chosen integrative\nsubspace $G$, the corresponding certified kernel is the orthogonal split\n\\begin{equation}\nV_r=P_GV_r+(I-P_G)V_r,\n\\end{equation}\nwhose two components are orthogonal and unique relative to $G$.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "carrier-indexed linear instruments; injectivity or another explicit faithfulness witness for detection claims",
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "explicit unique orthogonal Hodge decomposition and faithful first-cohomology class map",
      "finite model: selected orthogonal exact/coexact subspaces",
      "global certificate: compact, connected, oriented, smooth Riemannian membrane without boundary",
      "linear operational readout for perceptual or computational exposure",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Conditional global Hodge bridge: a certificate retains the compact, connected, oriented, smooth Riemannian, boundaryless membrane hypotheses; exact, coexact, and harmonic sectors reconstruct uniquely and orthogonally; squared refinement energy separates by sector; and a faithful harmonic-class map kills the harmonic part when first cohomology is trivial. Any linear operational readout, including sonification, preserves the certified split but does not manufacture its geometric hypotheses. The readout bridge is carrier-neutral across sound, light, temperature, pressure, and other linear instruments; nonzero-residue detection additionally requires a faithful instrument, while the zero-readout countermodel shows silence cannot establish absence."
    ],
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    "statuses": [
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    "witnesses": [
      "Book3Helmholtz.FiniteHodgeData.nonzero_harmonic_component_exists",
      "Book3Helmholtz.FiniteHodgeData.reconstruction",
      "Book3Helmholtz.GlobalHodgeCertificate.components",
      "Book3Helmholtz.GlobalHodgeCertificate.faithful_readout_detects_harmonic",
      "Book3Helmholtz.GlobalHodgeCertificate.harmonic_eq_zero_of_subsingleton_cohomology",
      "Book3Helmholtz.GlobalHodgeCertificate.multimodal_readout_reconstructs",
      "Book3Helmholtz.GlobalHodgeCertificate.operational_readout_reconstructs",
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  "name": "Hodge--Helmholtz Decomposition of the Refinement Field",
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  "id": "proof:bk3_symbolic_helmholtz_decomposition",
  "label": "proof:bk3_symbolic_helmholtz_decomposition",
  "latex_body": "\\begin{proof}[Symbolic Forces via Hodge Decomposition]\n\\label{proof:bk3_symbolic_helmholtz_decomposition}\n\\leavevmode\n\nThe Hodge decomposition theorem on compact oriented Riemannian manifolds gives\nthe orthogonal direct sum\n\\begin{equation}\n\\Omega^1(\\mathcal{M})\n =\\operatorname{im}d\\;\\oplus\\;\\operatorname{im}\\delta\n \\;\\oplus\\;\\mathcal{H}^1(\\mathcal{M}).\n\\end{equation}\nApplying it to $V_r^\\flat$ yields the displayed decomposition.  Hodge theory\nidentifies $\\mathcal{H}^1(\\mathcal{M})$ with\n$H^1_{\\mathrm{dR}}(\\mathcal{M})$, proving the stated vanishing criterion.  The\nthree-dimensional curl reading follows only after the stated metric and\norientation identifications.\n\nIn the finite-dimensional model, orthogonal projection onto $G$ gives\n$P_GV_r\\in G$ and $(I-P_G)V_r\\in G^\\perp$.  Their sum reconstructs $V_r$;\northogonality and uniqueness follow from\n$G\\cap G^\\perp=\\{0\\}$.  This finite kernel captures the identifiable\nintegration--differentiation split without claiming the omitted global\nanalytic machinery.\n\\end{proof}",
  "line": 420,
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  "name": "Symbolic Forces via Hodge Decomposition",
  "proves": "lemma:bk3_helmholtz_decomposition_refinement_field",
  "refs": [],
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  "latex_body": "",
  "line": 445,
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  "name": "Evolution of Symbolic Knowledge Structure",
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  "subtype": "subsection",
  "type": "section"
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{
  "book": "book3",
  "cited_by": [
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    "proof:bk3_differentiation_knowledge_structure",
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    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk3_evolution_of_symbolic_knowledge"
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  "id": "definition:bk3_symbolic_knowledge_structure",
  "label": "definition:bk3_symbolic_knowledge_structure",
  "latex_body": "\\begin{definition}[Symbolic Knowledge Structure] \\label{definition:bk3_symbolic_knowledge_structure}\nThe symbolic knowledge structure $K(r)$ at refinement level $r$ (Def.~\\ref{definition:bk3_symbolic_refinement}) quantifies the accumulated coherent symbolic organization, defined as:\n\\[\nK(r) = \\int_{\\mathcal{M}} \\rho(x,r) \\cdot \\kappa(x,r) \\cdot d\\mu_g(x)\n\\]\nwhere $\\kappa(x,r)$ is a local measure of symbolic coherence at point $x$ and refinement level $r$. (Here $\\rho$ is from Def.~\\ref{definition:bk2__symbolic_probability_density}, and the manifold measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{definition}",
  "line": 448,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Knowledge Structure",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "$\\kappa(x,r)$ is a local measure of symbolic coherence at point $x$ and refinement level $r$. (Here $\\rho$ is from Def.~\\ref{definition:bk2__symbolic_probability_density}, and the manifold measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{definition}",
      "label": "definition:bk2__symbolic_probability_density",
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    {
      "context": "l $r$. (Here $\\rho$ is from Def.~\\ref{definition:bk2__symbolic_probability_density}, and the manifold measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{definition}",
      "label": "definition:bk2_symbolic_probability_spa",
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      "target_line": 23,
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    },
    {
      "context": "abel{definition:bk3_symbolic_knowledge_structure} The symbolic knowledge structure $K(r)$ at refinement level $r$ (Def.~\\ref{definition:bk3_symbolic_refinement}) quantifies the accumulated coherent symbolic organization, defined as: \\[ K(r) = \\int_{\\mathcal{M}} \\rho(x,r) \\cdot \\k",
      "label": "definition:bk3_symbolic_refinement",
      "logical_support": true,
      "role": "definition_anchor",
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      "target_line": 364,
      "target_type": "definition"
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  "cites": [
    "definition:bk3_integration_differentiation_pressures",
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    "definition:bk3_symbolic_knowledge_structure",
    "definition:bk3_symbolic_refinement",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_evolution_of_symbolic_knowledge",
  "label": "theorem:bk3_evolution_of_symbolic_knowledge",
  "latex_body": "\\begin{theorem}[Evolution of Symbolic Knowledge] \\label{theorem:bk3_evolution_of_symbolic_knowledge}\nThe rate of change of symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) with respect to refinement (Def.~\\ref{definition:bk3_symbolic_refinement}) satisfies (cf.~Def.~\\ref{definition:bk3_membrane_thermodynamics}, Thm.~\\ref{theorem:bk3_membrane_stability_criteria}):\n\\[\n\\frac{dK}{dr} = \\mathcal{I}(r) - \\mathcal{D}(r) + \\mathcal{R}(r)\n\\]\nwhere $\\mathcal{I}(r)$ relates to integration pressure, $\\mathcal{D}(r)$ relates to differentiation pressure (Def.~\\ref{definition:bk3_integration_differentiation_pressures}), and $\\mathcal{R}(r)$ represents higher-order interactions and the direct change in coherence $\\kappa$. (Note: The text uses $I(r)$ and $D(r)$, let's maintain that notation assuming they represent the net effect).\n\\[\n\\frac{dK}{dr} = I'(r) - D'(r) + \\mathcal{R}(r)\n\\]\nwhere $I'(r)$ and $D'(r)$ represent the contributions of integration and differentiation pressures to the change in $K$, and $\\mathcal{R}(r)$ includes other effects.\n\\end{theorem}",
  "lean_alignment": {
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      "See the receipted theorem statement and coverage note for explicit premises."
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The rate equation dK/dr = I'-D'+R is kept as a structure field (a modeling commitment, not derived); one algebraic rearrangement is proved."
    ],
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      "MAP-BOOK3-021"
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  "line": 457,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Evolution of Symbolic Knowledge",
  "proof_labels": [
    "proof:bk3_differentiation_knowledge_structure"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": ") \\] where $\\mathcal{I}(r)$ relates to integration pressure, $\\mathcal{D}(r)$ relates to differentiation pressure (Def.~\\ref{definition:bk3_integration_differentiation_pressures}), and $\\mathcal{R}(r)$ represents higher-order interactions and the direct change in coherence $\\kappa$. (Note: The tex",
      "label": "definition:bk3_integration_differentiation_pressures",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 383,
      "target_type": "definition"
    },
    {
      "context": "ic_knowledge_structure}) with respect to refinement (Def.~\\ref{definition:bk3_symbolic_refinement}) satisfies (cf.~Def.~\\ref{definition:bk3_membrane_thermodynamics}, Thm.~\\ref{theorem:bk3_membrane_stability_criteria}): \\[ \\frac{dK}{dr} = \\mathcal{I}(r) - \\mathcal{D}(r) + \\mathcal{R}(",
      "label": "definition:bk3_membrane_thermodynamics",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 49,
      "target_type": "definition"
    },
    {
      "context": "ge] \\label{theorem:bk3_evolution_of_symbolic_knowledge} The rate of change of symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) with respect to refinement (Def.~\\ref{definition:bk3_symbolic_refinement}) satisfies (cf.~Def.~\\ref{definition:bk3_mem",
      "label": "definition:bk3_symbolic_knowledge_structure",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 448,
      "target_type": "definition"
    },
    {
      "context": "ic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) with respect to refinement (Def.~\\ref{definition:bk3_symbolic_refinement}) satisfies (cf.~Def.~\\ref{definition:bk3_membrane_thermodynamics}, Thm.~\\ref{theorem:bk3_membrane_stability_criteria}):",
      "label": "definition:bk3_symbolic_refinement",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 364,
      "target_type": "definition"
    },
    {
      "context": "(Def.~\\ref{definition:bk3_symbolic_refinement}) satisfies (cf.~Def.~\\ref{definition:bk3_membrane_thermodynamics}, Thm.~\\ref{theorem:bk3_membrane_stability_criteria}): \\[ \\frac{dK}{dr} = \\mathcal{I}(r) - \\mathcal{D}(r) + \\mathcal{R}(r) \\] where $\\mathcal{I}(r)$ relates to integration",
      "label": "theorem:bk3_membrane_stability_criteria",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 61,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk3_integration_differentiation_pressures",
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_knowledge_structure",
    "definition:bk3_symbolic_refinement",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_integration_differentiation_pressures",
    "definition:bk3_refinement_vector_field",
    "definition:bk3_symbolic_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
  ],
  "depends_on": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_integration_differentiation_pressures",
    "definition:bk3_refinement_vector_field",
    "definition:bk3_symbolic_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_differentiation_knowledge_structure",
  "label": "proof:bk3_differentiation_knowledge_structure",
  "latex_body": "\\begin{proof}[Derivative of Knowledge Structure with Respect to Refinement]\n\\label{proof:bk3_differentiation_knowledge_structure}\n\\leavevmode\n\nDifferentiate the knowledge structure $K(r)$\n(Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) with respect to $r$:\n\\[\n\\frac{dK}{dr} = \\int_{\\mathcal{M}} \\frac{\\partial}{\\partial r}(\\rho(x,r) \\cdot \\kappa(x,r)) d\\mu_g(x)\n\\]\nUsing the product rule and the continuity equation for $\\rho$ (assuming $\\rho$ evolves according to the flow $V_r$ (Def.~\\ref{definition:bk3_refinement_vector_field}), i.e., $\\frac{\\partial \\rho}{\\partial r} + \\nabla_g \\cdot (\\rho V_r) = 0$):\n\\begin{align*}\n\\frac{dK}{dr} &= \\int_{\\mathcal{M}} \\left[ \\frac{\\partial \\rho}{\\partial r} \\cdot \\kappa + \\rho \\cdot \\frac{\\partial \\kappa}{\\partial r} \\right] d\\mu_g(x) \\\\\n&= \\int_{\\mathcal{M}} \\left[ -\\nabla_g \\cdot (\\rho V_r) \\cdot \\kappa + \\rho \\cdot \\frac{\\partial \\kappa}{\\partial r} \\right] d\\mu_g(x)\n\\end{align*}\nUsing integration by parts (divergence theorem) on the first term (measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}):\n\\[\n-\\int_{\\mathcal{M}} (\\nabla_g \\cdot (\\rho V_r)) \\kappa \\, d\\mu_g = \\int_{\\mathcal{M}} (\\rho V_r) \\cdot (\\nabla_g \\kappa) \\, d\\mu_g - \\int_{\\partial\\mathcal{M}} \\kappa (\\rho V_r) \\cdot \\mathbf{n} \\, dS\n\\]\nAssuming boundary terms vanish or are negligible. The evolution then depends on how $V_r$ relates to $\\kappa$ and how $\\kappa$ itself changes ($\\partial \\kappa / \\partial r$).\n\\[\n\\frac{dK}{dr} = \\int_{\\mathcal{M}} \\rho \\left[ V_r \\cdot \\nabla_g \\kappa + \\frac{\\partial \\kappa}{\\partial r} \\right] d\\mu_g\n\\]\nFurther analysis relating $V_r$ (via its divergence and curl components) and $\\partial \\kappa / \\partial r$ to the concepts of integration and differentiation pressures $I(r)$ and $D(r)$ (Def.~\\ref{definition:bk3_integration_differentiation_pressures}) defined earlier (perhaps $\\kappa$ increases with convergence and decreases with curl) would lead to the form $I'(r) - D'(r) + \\mathcal{R}(r)$ (as in Thm.~\\ref{theorem:bk3_evolution_of_symbolic_knowledge}). The exact relationship depends on the specific definition of $\\kappa$ and its coupling to $V_r$. The terms $I'(r)$ and $D'(r)$ would be integrals involving $\\rho$, $\\kappa$, and components of $V_r$.\n\\end{proof}",
  "line": 469,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Derivative of Knowledge Structure with Respect to Refinement",
  "proves": "theorem:bk3_evolution_of_symbolic_knowledge",
  "ref_roles": [
    {
      "context": "r} \\right] d\\mu_g(x) \\end{align*} Using integration by parts (divergence theorem) on the first term (measure from Def.~\\ref{definition:bk2_symbolic_probability_spa}): \\[ -\\int_{\\mathcal{M}} (\\nabla_g \\cdot (\\rho V_r)) \\kappa \\, d\\mu_g = \\int_{\\mathcal{M}} (\\rho V_r) \\cdot (\\nabla_g \\",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "and $\\partial \\kappa / \\partial r$ to the concepts of integration and differentiation pressures $I(r)$ and $D(r)$ (Def.~\\ref{definition:bk3_integration_differentiation_pressures}) defined earlier (perhaps $\\kappa$ increases with convergence and decreases with curl) would lead to the form $I'(r) -",
      "label": "definition:bk3_integration_differentiation_pressures",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 383,
      "target_type": "definition"
    },
    {
      "context": "sing the product rule and the continuity equation for $\\rho$ (assuming $\\rho$ evolves according to the flow $V_r$ (Def.~\\ref{definition:bk3_refinement_vector_field}), i.e., $\\frac{\\partial \\rho}{\\partial r} + \\nabla_g \\cdot (\\rho V_r) = 0$): \\begin{align*} \\frac{dK}{dr} &= \\int_{\\mat",
      "label": "definition:bk3_refinement_vector_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 374,
      "target_type": "definition"
    },
    {
      "context": "] \\label{proof:bk3_differentiation_knowledge_structure} \\leavevmode Differentiate the knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) with respect to $r$: \\[ \\frac{dK}{dr} = \\int_{\\mathcal{M}} \\frac{\\partial}{\\partial r}(\\rho(x,r) \\cdot \\kappa(x,r)) d\\",
      "label": "definition:bk3_symbolic_knowledge_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 448,
      "target_type": "definition"
    },
    {
      "context": "increases with convergence and decreases with curl) would lead to the form $I'(r) - D'(r) + \\mathcal{R}(r)$ (as in Thm.~\\ref{theorem:bk3_evolution_of_symbolic_knowledge}). The exact relationship depends on the specific definition of $\\kappa$ and its coupling to $V_r$. The terms $I'(r)$ an",
      "label": "theorem:bk3_evolution_of_symbolic_knowledge",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 457,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_integration_differentiation_pressures",
    "definition:bk3_refinement_vector_field",
    "definition:bk3_symbolic_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [
    "proof:bk3_integrated_knowledge_dynamics"
  ],
  "cites": [
    "definition:bk3_symbolic_knowledge_structure"
  ],
  "depends_on": [
    "definition:bk3_symbolic_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
  ],
  "file": "book3.tex",
  "id": "corollary:bk3_integrated_knowledge_structure",
  "label": "corollary:bk3_integrated_knowledge_structure",
  "latex_body": "\\begin{corollary}[Integrated Knowledge Structure] \\label{corollary:bk3_integrated_knowledge_structure}\nThe accumulated symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) from initial refinement state $r_0$ to state $r$ is:\n\\[\nK(r) = K(r_0) + \\int_{r_0}^r (I'(s) - D'(s) + \\mathcal{R}(s)) ds\n\\]\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "manifold/PDE/Helmholtz content of Book 3 is NOT formalized; static and finite-discrete kernels only",
      "modeling laws (rate equations, stability conditions, Lipschitz response) are structure fields"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Discrete telescoping-sum analogue of the continuous FTC-style corollary, proved by induction on Nat."
    ],
    "record_ids": [
      "MAP-BOOK3-022"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book3.knowledge_structure_telescopes"
    ]
  },
  "line": 494,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Integrated Knowledge Structure",
  "proof_labels": [
    "proof:bk3_integrated_knowledge_dynamics"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ructure] \\label{corollary:bk3_integrated_knowledge_structure} The accumulated symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) from initial refinement state $r_0$ to state $r$ is: \\[ K(r) = K(r_0) + \\int_{r_0}^r (I'(s) - D'(s) + \\mathcal{R}(s))",
      "label": "definition:bk3_symbolic_knowledge_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 448,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk3_symbolic_knowledge_structure"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "corollary:bk3_integrated_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
  ],
  "depends_on": [
    "corollary:bk3_integrated_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_integrated_knowledge_dynamics",
  "label": "proof:bk3_integrated_knowledge_dynamics",
  "latex_body": "\\begin{proof}[Integration of Knowledge Refinement Dynamics]\n\\label{proof:bk3_integrated_knowledge_dynamics}\n\\leavevmode\n\nThis follows directly from integrating the differential equation in Theorem~\\ref{theorem:bk3_evolution_of_symbolic_knowledge} with respect to the refinement parameter $s$ from $r_0$ to $r$. (This supports Cor.~\\ref{corollary:bk3_integrated_knowledge_structure}).\n\\end{proof}",
  "line": 501,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Integration of Knowledge Refinement Dynamics",
  "proves": "corollary:bk3_integrated_knowledge_structure",
  "ref_roles": [
    {
      "context": "k3_evolution_of_symbolic_knowledge} with respect to the refinement parameter $s$ from $r_0$ to $r$. (This supports Cor.~\\ref{corollary:bk3_integrated_knowledge_structure}). \\end{proof}",
      "label": "corollary:bk3_integrated_knowledge_structure",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 494,
      "target_type": "corollary"
    },
    {
      "context": "integrated_knowledge_dynamics} \\leavevmode This follows directly from integrating the differential equation in Theorem~\\ref{theorem:bk3_evolution_of_symbolic_knowledge} with respect to the refinement parameter $s$ from $r_0$ to $r$. (This supports Cor.~\\ref{corollary:bk3_integrated_knowl",
      "label": "theorem:bk3_evolution_of_symbolic_knowledge",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 457,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk3_integrated_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
  ],
  "role": "proof",
  "type": "proof"
}