Book III — De Symbiosi Symbolica — page 2 of 2

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    "proof:bk3_knowledge_growth_integrated_condition",
    "proof:bk3_sketch_necessity_for_continuous_operation",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk3_emergence_of_symbolic_networks"
  ],
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    "definition:bk3_symbolic_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
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  "id": "theorem:bk3_conditions_sustained_symbolic_growth",
  "label": "theorem:bk3_conditions_sustained_symbolic_growth",
  "latex_body": "\\begin{theorem}[Conditions for Sustained Symbolic Growth] \\label{theorem:bk3_conditions_sustained_symbolic_growth}\nPersistent growth of symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) requires that the net contribution from integration recurrently exceeds that from differentiation along refinement flows (cf. Thm.~\\ref{theorem:bk3_evolution_of_symbolic_knowledge}):\n\\[\n\\int_{r_0}^{r_0+T} (I'(s) - D'(s)) ds > 0\n\\]\nfor some period $T > 0$ and all starting points $r_0 \\geq R_0$ for some threshold $R_0$, assuming $\\mathcal{R}(s)$ averages to zero or is dominated by the $I'-D'$ term.\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",
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    "notes": [
      "Discrete window version: positive net increment sum over a length-T window implies K strictly increases across that window, built on the telescoping lemma."
    ],
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      "Book3.sustained_growth_window"
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  },
  "line": 509,
  "macros_used": [],
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  "matter_role": "canonical_book",
  "name": "Conditions for Sustained Symbolic Growth",
  "proof_labels": [
    "proof:bk3_knowledge_growth_integrated_condition"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "\\label{theorem:bk3_conditions_sustained_symbolic_growth} Persistent growth of symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) requires that the net contribution from integration recurrently exceeds that from differentiation along refinement flo",
      "label": "definition:bk3_symbolic_knowledge_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 448,
      "target_type": "definition"
    },
    {
      "context": "at the net contribution from integration recurrently exceeds that from differentiation along refinement flows (cf. Thm.~\\ref{theorem:bk3_evolution_of_symbolic_knowledge}): \\[ \\int_{r_0}^{r_0+T} (I'(s) - D'(s)) ds > 0 \\] for some period $T > 0$ and all starting points $r_0 \\geq R_0$ for so",
      "label": "theorem:bk3_evolution_of_symbolic_knowledge",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 457,
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    "definition:bk3_symbolic_knowledge_structure",
    "theorem:bk3_evolution_of_symbolic_knowledge"
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  "file": "book3.tex",
  "id": "proof:bk3_knowledge_growth_integrated_condition",
  "label": "proof:bk3_knowledge_growth_integrated_condition",
  "latex_body": "\\begin{proof}[Secular Growth of Knowledge Under Integrated Conditions]\n\\label{proof:bk3_knowledge_growth_integrated_condition}\n\\leavevmode\n\nIf the integral condition in Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth} holds, then neglecting or assuming the average contribution of higher-order terms $\\mathcal{R}(s)$ is small over the period $T$, the change in knowledge structure $\\Delta K = K(r_0+T) - K(r_0)$ is positive. If this holds recurrently for all $r_0$ above some threshold $R_0$, it implies a secular growth trend in $K(r)$, even if there are local decreases within a period --- the unbounded, error-correcting growth of explanatory knowledge in the sense of \\citet{deutsch2011infinity}. If the condition fails, i.e., the integral is non-positive for sufficiently large $r_0$, then differentiation dominates or balances integration on average, leading to fragmentation, stagnation, or loss of symbolic coherence rather than sustained growth.\n\\end{proof}",
  "line": 517,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Secular Growth of Knowledge Under Integrated Conditions",
  "proves": "theorem:bk3_conditions_sustained_symbolic_growth",
  "ref_roles": [
    {
      "context": "ated Conditions] \\label{proof:bk3_knowledge_growth_integrated_condition} \\leavevmode If the integral condition in Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth} holds, then neglecting or assuming the average contribution of higher-order terms $\\mathcal{R}(s)$ is small over the pe",
      "label": "theorem:bk3_conditions_sustained_symbolic_growth",
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  "line": 524,
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  "name": "Conceptual Bridges and Symbolic Networks",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
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  "book": "book3",
  "cited_by": [
    "definition:bk3_conceptual_bridge_sequence",
    "definition:bk3_symbolic_network"
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  "cites": [
    "definition:bk3_symbolic_membrane"
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  "depends_on": [
    "definition:bk3_symbolic_membrane"
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  "file": "book3.tex",
  "id": "definition:bk3_compressed_relational_structure",
  "label": "definition:bk3_compressed_relational_structure",
  "latex_body": "\\begin{definition}[Compressed Relational Structure] \\label{definition:bk3_compressed_relational_structure}\nA compressed relational structure $\\sigma$ within a symbolic membrane $\\mathcal{M}$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is a lower-dimensional representation that preserves essential topological and dynamical features of a region $\\omega \\subset \\mathcal{M}$:\n\\[\n\\sigma = \\mathcal{C}(\\omega)\n\\]\nwhere $\\mathcal{C}: 2^{\\mathcal{M}} \\rightarrow \\Sigma$ is a compression operator mapping regions (subsets of $\\mathcal{M}$) to a space of compressed structures $\\Sigma$.\n\\end{definition}",
  "line": 527,
  "macros_used": [],
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  "matter_role": "canonical_book",
  "name": "Compressed Relational Structure",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "pressed_relational_structure} A compressed relational structure $\\sigma$ within a symbolic membrane $\\mathcal{M}$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is a lower-dimensional representation that preserves essential topological and dynamical features of a region $\\omega",
      "label": "definition:bk3_symbolic_membrane",
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    "definition:bk3_symbolic_membrane"
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  "cited_by": [
    "definition:bk3_conceptual_bridge_sequence",
    "definition:bk3_symbolic_autopoiesis",
    "proof:bk3_sketch_evolutionary_dynamics",
    "proof:bk3_sketch_symbolic_network_emergence",
    "theorem:bk3_emergence_of_symbolic_networks"
  ],
  "cites": [
    "definition:bk3_compressed_relational_structure",
    "definition:bk3_conceptual_bridge"
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    "definition:bk3_compressed_relational_structure",
    "definition:bk3_conceptual_bridge"
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  "file": "book3.tex",
  "id": "definition:bk3_symbolic_network",
  "label": "definition:bk3_symbolic_network",
  "latex_body": "\\begin{definition}[Symbolic Network] \\label{definition:bk3_symbolic_network}\nA symbolic network $\\mathcal{N}$ is a graph structure where:\n\\begin{enumerate}\n    \\item Nodes represent compressed relational structures $\\{\\sigma_i\\}$ (Def.~\\ref{definition:bk3_compressed_relational_structure}).\n    \\item Edges represent conceptual bridges (Definition~\\ref{definition:bk3_conceptual_bridge}) between these structures.\n    \\item The network possesses a global stability functional $\\mathcal{S}: \\mathcal{N} \\rightarrow \\mathbb{R}_+$ measuring its overall coherence.\n\\end{enumerate}\n\\end{definition}",
  "line": 536,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Network",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "raph structure where: \\begin{enumerate} \\item Nodes represent compressed relational structures $\\{\\sigma_i\\}$ (Def.~\\ref{definition:bk3_compressed_relational_structure}). \\item Edges represent conceptual bridges (Definition~\\ref{definition:bk3_conceptual_bridge}) between these struct",
      "label": "definition:bk3_compressed_relational_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 527,
      "target_type": "definition"
    },
    {
      "context": "$ (Def.~\\ref{definition:bk3_compressed_relational_structure}). \\item Edges represent conceptual bridges (Definition~\\ref{definition:bk3_conceptual_bridge}) between these structures. \\item The network possesses a global stability functional $\\mathcal{S}: \\mathcal{N} \\rig",
      "label": "definition:bk3_conceptual_bridge",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 219,
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    "definition:bk3_conceptual_bridge"
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{
  "book": "book3",
  "cited_by": [
    "abs:press",
    "definition:bk3_symbolic_metabolism",
    "sec:bk9_symbolic_ecosystems_and_emergent_governance"
  ],
  "cites": [
    "definition:bk3_symbolic_network",
    "theorem:bk3_conditions_sustained_symbolic_growth"
  ],
  "depends_on": [
    "definition:bk3_symbolic_network",
    "theorem:bk3_conditions_sustained_symbolic_growth"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_emergence_of_symbolic_networks",
  "label": "theorem:bk3_emergence_of_symbolic_networks",
  "latex_body": "\\begin{theorem}[Conditional Assembly of Symbolic Networks]\n\\label{theorem:bk3_emergence_of_symbolic_networks}\nAssume the sustained-growth condition of\nThm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth}.  In addition,\nlet $J$ be a nonempty finite index set and suppose the following assembly data\nare supplied:\n\\begin{enumerate}\n  \\item for every $j\\in J$, a selected high-coherence region $\\omega_j$ and a\n  total compression operator $\\mathcal{C}$ with\n  $\\sigma_j=\\mathcal{C}(\\omega_j)\\in\\Sigma$;\n  \\item a selected edge relation $E\\subseteq J\\times J$ and, for every\n  $(i,j)\\in E$, a reflexive encoding whose induced conceptual bridge connects\n  $\\sigma_i$ to $\\sigma_j$;\n  \\item a global stability value $s_{\\mathcal N}$ and a node-coherence lower\n  bound $m>0$ such that $s_{\\mathcal N}\\geq m$.\n\\end{enumerate}\nThen these data assemble into a symbolic network $\\mathcal N$ in the sense of\nDef.~\\ref{definition:bk3_symbolic_network}, with strictly positive global\nstability. Sustained symbolic growth alone does not supply the compression\ncodomain, nodes, edges, or stability certificate.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "bridge certificate for every selected edge",
      "finite indexed region family and total compression operator",
      "positive node-coherence floor bounded above by stage stability",
      "sustained-growth trace"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A time-indexed operational process retains the supplied region selection, total compression, bridge-certified edge relation, positive node-coherence floor, and global-stability lower bound. Execution jointly realizes the compressed nodes, bridge-backed edges, and strictly positive global stability. A paired countermodel shows that the same sustained-growth trace permits distinct compression outcomes, so growth does not identify or manufacture the assembly policy."
    ],
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      "MAP-BOOK3-001"
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    "witnesses": [
      "Book3.runNetworkEmergence_realizes_assembly",
      "Book3.runNetworkEmergence_stability_pos_from_floor",
      "Book3.same_growth_allows_distinct_nodes"
    ]
  },
  "line": 546,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conditional Assembly of Symbolic Networks",
  "proof_labels": [
    "proof:bk3_sketch_symbolic_network_emergence"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "{\\mathcal N}\\geq m$. \\end{enumerate} Then these data assemble into a symbolic network $\\mathcal N$ in the sense of Def.~\\ref{definition:bk3_symbolic_network}, with strictly positive global stability. Sustained symbolic growth alone does not supply the compression codomain, nod",
      "label": "definition:bk3_symbolic_network",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 536,
      "target_type": "definition"
    },
    {
      "context": "of Symbolic Networks] \\label{theorem:bk3_emergence_of_symbolic_networks} Assume the sustained-growth condition of Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth}. In addition, let $J$ be a nonempty finite index set and suppose the following assembly data are supplied: \\begin{enum",
      "label": "theorem:bk3_conditions_sustained_symbolic_growth",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 509,
      "target_type": "theorem"
    }
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    "definition:bk3_symbolic_network",
    "theorem:bk3_conditions_sustained_symbolic_growth"
  ],
  "role": "theorem",
  "type": "theorem"
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{
  "book": "book3",
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  "cites": [
    "definition:bk3_symbolic_network"
  ],
  "depends_on": [
    "definition:bk3_symbolic_network"
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  "file": "book3.tex",
  "id": "proof:bk3_sketch_symbolic_network_emergence",
  "label": "proof:bk3_sketch_symbolic_network_emergence",
  "latex_body": "\\begin{proof}[Assembly from Compression, Bridge, and Stability Witnesses]\n\\label{proof:bk3_sketch_symbolic_network_emergence}\n\\leavevmode\n\nUse the compressed structures $\\{\\sigma_j\\}_{j\\in J}$ as the node family and\nthe supplied relation $E$ as the edge relation.  By hypothesis, each selected\nedge is witnessed by a reflexive encoding and its induced conceptual bridge,\nso the edge interpretation required by\nDef.~\\ref{definition:bk3_symbolic_network} is satisfied.  Assign\n$s_{\\mathcal N}$ as the global stability value.  Since\n$s_{\\mathcal N}\\geq m>0$, it lies in $\\mathbb{R}_+$ and is strictly positive.\nThe node, edge, and stability fields therefore form the required symbolic\nnetwork.  The accompanying Lean realization retains the selected regions, total\ncompression, bridge witness for every selected edge, positive node-coherence\nfloor, and lower-bound inequality in one process certificate; its execution\nproves the node, edge, and strict-stability clauses jointly.\n\nThe sustained-growth premise identifies the intended dynamical setting but is\nnot used to manufacture any assembly datum.  In particular, positive growth is\ncompatible with an empty compression codomain, in which case even one network\nnode cannot be constructed.  This shows why the additional witnesses are\nload-bearing rather than consequences of growth alone.\n\\end{proof}",
  "line": 568,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Assembly from Compression, Bridge, and Stability Witnesses",
  "proves": "theorem:bk3_emergence_of_symbolic_networks",
  "ref_roles": [
    {
      "context": "dge is witnessed by a reflexive encoding and its induced conceptual bridge, so the edge interpretation required by Def.~\\ref{definition:bk3_symbolic_network} is satisfied. Assign $s_{\\mathcal N}$ as the global stability value. Since $s_{\\mathcal N}\\geq m>0$, it lies in $\\mat",
      "label": "definition:bk3_symbolic_network",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 536,
      "target_type": "definition"
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  "refs": [
    "definition:bk3_symbolic_network"
  ],
  "role": "proof",
  "type": "proof"
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  "book": "book3",
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    "proof:bk3_sketch_evolutionary_dynamics",
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  "cites": [
    "definition:bk3_compressed_relational_structure",
    "definition:bk3_conceptual_bridge",
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    "definition:bk3_symbolic_network"
  ],
  "depends_on": [
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  "file": "book3.tex",
  "id": "definition:bk3_conceptual_bridge_sequence",
  "label": "definition:bk3_conceptual_bridge_sequence",
  "latex_body": "\\begin{definition}[Conceptual Bridge Sequence] \\label{definition:bk3_conceptual_bridge_sequence}\nThe conceptual bridge sequence represents the progressive transformation and abstraction of symbolic structures:\n\\[\n\\Sigma_{\\mathcal{M} \\rightarrow \\sigma}, \\Sigma_{\\sigma \\rightarrow \\Sigma}, \\Sigma_{\\Sigma \\rightarrow \\mathcal{N}}, \\Sigma_{\\mathcal{N} \\rightarrow \\mathcal{M}_{\\text{meta}}}\n\\]\nwhere each $\\Sigma_{X \\rightarrow Y}$ represents a conceptual bridge (Def.~\\ref{definition:bk3_conceptual_bridge}) mapping structures of type $X$ to structures of type $Y$. This sequence maps membrane regions ($\\mathcal{M}$, Def.~\\ref{definition:bk3_symbolic_membrane}) to compressed structures ($\\sigma$, Def.~\\ref{definition:bk3_compressed_relational_structure}), relates compressed structures to the space of such structures ($\\Sigma$), organizes these into networks ($\\mathcal{N}$, Def.~\\ref{definition:bk3_symbolic_network}), and potentially leads to the emergence of an encompassing meta-level symbolic membrane ($\\mathcal{M}_{\\text{meta}}$).\n\\end{definition}",
  "line": 593,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conceptual Bridge Sequence",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "membrane regions ($\\mathcal{M}$, Def.~\\ref{definition:bk3_symbolic_membrane}) to compressed structures ($\\sigma$, Def.~\\ref{definition:bk3_compressed_relational_structure}), relates compressed structures to the space of such structures ($\\Sigma$), organizes these into networks ($\\mathcal{N}",
      "label": "definition:bk3_compressed_relational_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 527,
      "target_type": "definition"
    },
    {
      "context": "N} \\rightarrow \\mathcal{M}_{\\text{meta}}} \\] where each $\\Sigma_{X \\rightarrow Y}$ represents a conceptual bridge (Def.~\\ref{definition:bk3_conceptual_bridge}) mapping structures of type $X$ to structures of type $Y$. This sequence maps membrane regions ($\\mathcal{M}$, Def.~\\re",
      "label": "definition:bk3_conceptual_bridge",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 219,
      "target_type": "definition"
    },
    {
      "context": "ge}) mapping structures of type $X$ to structures of type $Y$. This sequence maps membrane regions ($\\mathcal{M}$, Def.~\\ref{definition:bk3_symbolic_membrane}) to compressed structures ($\\sigma$, Def.~\\ref{definition:bk3_compressed_relational_structure}), relates compressed str",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "es compressed structures to the space of such structures ($\\Sigma$), organizes these into networks ($\\mathcal{N}$, Def.~\\ref{definition:bk3_symbolic_network}), and potentially leads to the emergence of an encompassing meta-level symbolic membrane ($\\mathcal{M}_{\\text{meta}}$).",
      "label": "definition:bk3_symbolic_network",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 536,
      "target_type": "definition"
    }
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  "refs": [
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    "definition:bk3_conceptual_bridge",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network"
  ],
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{
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    "definition:bk3_conceptual_bridge_sequence",
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  "depends_on": [
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    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_closure_conceptual_bridge_sequence",
  "label": "theorem:bk3_closure_conceptual_bridge_sequence",
  "latex_body": "\\begin{theorem}[Closure of Conceptual Bridge Sequence] \\label{theorem:bk3_closure_conceptual_bridge_sequence}\n\\leavevmode\\newline\nThe conceptual bridge sequence (Def.~\\ref{definition:bk3_conceptual_bridge_sequence}) can form a closed loop.\nIn that loop, meta-level membrane $\\mathcal{M}_{\\text{meta}}$ can host symbolic processes that feed back into the original membranes $\\{\\mathcal{M}_i\\}$ (Def.~\\ref{definition:bk3_symbolic_membrane}).\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Given the loop literally closes (composite = identity on M, the honest reading of 'can form a closed loop feeding back into the originals'), the first map is injective and the return map is surjective."
    ],
    "record_ids": [
      "MAP-BOOK3-031"
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    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book7B.conceptualBridgeLoop_toM_surjective",
      "Book7B.conceptualBridgeLoop_toSigma1_injective"
    ]
  },
  "line": 602,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Closure of Conceptual Bridge Sequence",
  "proof_labels": [
    "proof:bk3_sketch_evolutionary_dynamics"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "quence] \\label{theorem:bk3_closure_conceptual_bridge_sequence} \\leavevmode\\newline The conceptual bridge sequence (Def.~\\ref{definition:bk3_conceptual_bridge_sequence}) can form a closed loop. In that loop, meta-level membrane $\\mathcal{M}_{\\text{meta}}$ can host symbolic processes that",
      "label": "definition:bk3_conceptual_bridge_sequence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 593,
      "target_type": "definition"
    },
    {
      "context": "hcal{M}_{\\text{meta}}$ can host symbolic processes that feed back into the original membranes $\\{\\mathcal{M}_i\\}$ (Def.~\\ref{definition:bk3_symbolic_membrane}). \\end{theorem}",
      "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_sequence",
    "definition:bk3_symbolic_membrane"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk1_symbolic_manifold",
    "definition:bk3_conceptual_bridge_sequence",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold",
    "definition:bk3_conceptual_bridge_sequence",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_sketch_evolutionary_dynamics",
  "label": "proof:bk3_sketch_evolutionary_dynamics",
  "latex_body": "\\begin{proof}[Closure of Conceptual Bridge Sequence]\n\\label{proof:bk3_sketch_evolutionary_dynamics}\n\\leavevmode\n\nThe conceptual bridge sequence (Def.~\\ref{definition:bk3_conceptual_bridge_sequence})\nmaps $\\mathcal{M} \\to \\sigma \\to \\Sigma \\to \\mathcal{N} \\to \\mathcal{M}_{\\text{meta}}$.\nWe show the last step closes the loop.\n\n\\textbf{Existence of $\\mathcal{M}_{\\text{meta}}$ within $M$.}\nThe symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) is the\nspace of all symbolic structures on the observer's domain. The network\n$\\mathcal{N}$ (Def.~\\ref{definition:bk3_symbolic_network}), being a finite graph\nof compressed relational structures with a stability functional\n$\\mathcal{S}(\\mathcal{N}) \\in \\mathbb{R}_+$, is itself a symbolic structure and\ntherefore an element of $M$. By Def.~\\ref{definition:bk3_symbolic_membrane},\nany sufficiently coherent sub-region of $M$ with a well-defined boundary and\ndrift field qualifies as a symbolic membrane; $\\mathcal{N}$ and its dynamics\nsatisfy these conditions, constituting $\\mathcal{M}_{\\text{meta}} \\subset M$.\n\n\\textbf{Feedback into $\\{\\mathcal{M}_i\\}$.}\nSince $\\mathcal{M}_{\\text{meta}} \\subset M$ and the $\\mathcal{M}_i \\subset M$,\nthe coupling map construction (Def.~\\ref{definition:bk3_coupling_map}) applies\nbetween $\\mathcal{M}_{\\text{meta}}$ and each $\\mathcal{M}_i$. The state of\n$\\mathcal{M}_{\\text{meta}}$ can therefore modulate the coupling strengths\n$\\lambda_{ij}$ (Def.~\\ref{definition:bk3_induced_coupling_energy}) and response\nparameters $\\eta_i$ (Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification})\nof the lower-level membranes, closing the loop\n$\\mathcal{M}_{\\text{meta}} \\to \\{\\mathcal{M}_i\\}$.\n\nThe composition of this feedback with the forward sequence\n$\\{\\mathcal{M}_i\\} \\to \\mathcal{M}_{\\text{meta}}$ is therefore a well-defined\nendomorphism of the symbolic manifold $M$, establishing the closed loop.\n\\end{proof}",
  "line": 608,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Closure of Conceptual Bridge Sequence",
  "proves": "theorem:bk3_closure_conceptual_bridge_sequence",
  "ref_roles": [
    {
      "context": "st step closes the loop. \\textbf{Existence of $\\mathcal{M}_{\\text{meta}}$ within $M$.} The symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) is the space of all symbolic structures on the observer's domain. The network $\\mathcal{N}$ (Def.~\\ref{definition:bk3_",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "ptual Bridge Sequence] \\label{proof:bk3_sketch_evolutionary_dynamics} \\leavevmode The conceptual bridge sequence (Def.~\\ref{definition:bk3_conceptual_bridge_sequence}) maps $\\mathcal{M} \\to \\sigma \\to \\Sigma \\to \\mathcal{N} \\to \\mathcal{M}_{\\text{meta}}$. We show the last step closes t",
      "label": "definition:bk3_conceptual_bridge_sequence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 593,
      "target_type": "definition"
    },
    {
      "context": "}$.} Since $\\mathcal{M}_{\\text{meta}} \\subset M$ and the $\\mathcal{M}_i \\subset M$, the coupling map construction (Def.~\\ref{definition:bk3_coupling_map}) applies between $\\mathcal{M}_{\\text{meta}}$ and each $\\mathcal{M}_i$. The state of $\\mathcal{M}_{\\text{meta}}$ can the",
      "label": "definition:bk3_coupling_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 85,
      "target_type": "definition"
    },
    {
      "context": "hcal{M}_i$. The state of $\\mathcal{M}_{\\text{meta}}$ can therefore modulate the coupling strengths $\\lambda_{ij}$ (Def.~\\ref{definition:bk3_induced_coupling_energy}) and response parameters $\\eta_i$ (Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}) of the lower-level membra",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": "al $\\mathcal{S}(\\mathcal{N}) \\in \\mathbb{R}_+$, is itself a symbolic structure and therefore an element of $M$. By Def.~\\ref{definition:bk3_symbolic_membrane}, any sufficiently coherent sub-region of $M$ with a well-defined boundary and drift field qualifies as a symbolic membr",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "1_symbolic_manifold}) is the space of all symbolic structures on the observer's domain. The network $\\mathcal{N}$ (Def.~\\ref{definition:bk3_symbolic_network}), being a finite graph of compressed relational structures with a stability functional $\\mathcal{S}(\\mathcal{N}) \\in \\m",
      "label": "definition:bk3_symbolic_network",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 536,
      "target_type": "definition"
    },
    {
      "context": "ing strengths $\\lambda_{ij}$ (Def.~\\ref{definition:bk3_induced_coupling_energy}) and response parameters $\\eta_i$ (Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}) of the lower-level membranes, closing the loop $\\mathcal{M}_{\\text{meta}} \\to \\{\\mathcal{M}_i\\}$. The composition of",
      "label": "theorem:bk3_couplinginduced_drift_modification",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 104,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold",
    "definition:bk3_conceptual_bridge_sequence",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "sec:bk3_symbolic_metabolism_persistent_life",
  "label": "sec:bk3_symbolic_metabolism_persistent_life",
  "latex_body": "",
  "line": 642,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Metabolism and Persistent Life",
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "section:book3.tex:643",
  "label": "",
  "latex_body": "",
  "line": 643,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Metabolism",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_autophagic_drift",
    "proof:bk3_sketch_necessity_for_continuous_operation",
    "subsec:bk9_executio_final"
  ],
  "cites": [
    "definition:bk1_reflection_operator",
    "definition:bk3_conceptual_bridge",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_closure_conceptual_bridge_sequence",
    "theorem:bk3_emergence_of_symbolic_networks"
  ],
  "depends_on": [
    "definition:bk1_reflection_operator",
    "definition:bk3_conceptual_bridge",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_closure_conceptual_bridge_sequence",
    "theorem:bk3_emergence_of_symbolic_networks"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_symbolic_metabolism",
  "label": "definition:bk3_symbolic_metabolism",
  "latex_body": "\\begin{definition}[Symbolic Metabolism] \\label{definition:bk3_symbolic_metabolism}\n\\leavevmode\\newline\nSymbolic metabolism is the regulated transformation and flow of symbolic\nstructures across membranes (Def.~\\ref{definition:bk3_symbolic_membrane}) and\nconceptual bridges (Def.~\\ref{definition:bk3_conceptual_bridge}) in a system.\nIt is characterized by (cf.~Thm.~\\ref{theorem:bk3_emergence_of_symbolic_networks},\nThm.~\\ref{theorem:bk3_closure_conceptual_bridge_sequence},\nDef.~\\ref{definition:bk1_reflection_operator}):\n\\begin{enumerate}\n    \\item Energy utilization: transformation of symbolic potential energy\n    (e.g., $H_{ij}$; Def.~\\ref{definition:bk3_induced_coupling_energy}) into\n    structured information (e.g., maintained $\\rho_{ij}$ and stable $\\sigma_i$).\n    \\item Homeostasis: maintenance of essential symbolic parameters\n    (e.g., stability $S_i$ and mutual information $I_{ij}$ from\n    Def.~\\ref{definition:bk3_symbolic_symbiosis}) within viable ranges under\n    perturbation.\n    \\item Adaptive response: modification of internal processes\n    (e.g., drift fields $D_i$ and coupling $\\Phi_{ij}$ from\n    Def.~\\ref{definition:bk3_coupling_map}) in response to external or internal\n    symbolic perturbations.\n\\end{enumerate}\n\\end{definition}",
  "line": 646,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Metabolism",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "~Thm.~\\ref{theorem:bk3_emergence_of_symbolic_networks}, Thm.~\\ref{theorem:bk3_closure_conceptual_bridge_sequence}, Def.~\\ref{definition:bk1_reflection_operator}): \\begin{enumerate} \\item Energy utilization: transformation of symbolic potential energy (e.g., $H_{ij}$; Def.",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "flow of symbolic structures across membranes (Def.~\\ref{definition:bk3_symbolic_membrane}) and conceptual bridges (Def.~\\ref{definition:bk3_conceptual_bridge}) in a system. It is characterized by (cf.~Thm.~\\ref{theorem:bk3_emergence_of_symbolic_networks}, Thm.~\\ref{theorem:bk3_",
      "label": "definition:bk3_conceptual_bridge",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 219,
      "target_type": "definition"
    },
    {
      "context": "ptive response: modification of internal processes (e.g., drift fields $D_i$ and coupling $\\Phi_{ij}$ from Def.~\\ref{definition:bk3_coupling_map}) in response to external or internal symbolic perturbations. \\end{enumerate} \\end{definition}",
      "label": "definition:bk3_coupling_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 85,
      "target_type": "definition"
    },
    {
      "context": ": \\begin{enumerate} \\item Energy utilization: transformation of symbolic potential energy (e.g., $H_{ij}$; Def.~\\ref{definition:bk3_induced_coupling_energy}) into structured information (e.g., maintained $\\rho_{ij}$ and stable $\\sigma_i$). \\item Homeostasis: maintenan",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": "mode\\newline Symbolic metabolism is the regulated transformation and flow of symbolic structures across membranes (Def.~\\ref{definition:bk3_symbolic_membrane}) and conceptual bridges (Def.~\\ref{definition:bk3_conceptual_bridge}) in a system. It is characterized by (cf.~Thm.~\\re",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": ": maintenance of essential symbolic parameters (e.g., stability $S_i$ and mutual information $I_{ij}$ from Def.~\\ref{definition:bk3_symbolic_symbiosis}) within viable ranges under perturbation. \\item Adaptive response: modification of internal processes (e.g.",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "onceptual_bridge}) in a system. It is characterized by (cf.~Thm.~\\ref{theorem:bk3_emergence_of_symbolic_networks}, Thm.~\\ref{theorem:bk3_closure_conceptual_bridge_sequence}, Def.~\\ref{definition:bk1_reflection_operator}): \\begin{enumerate} \\item Energy utilization: transformation of symb",
      "label": "theorem:bk3_closure_conceptual_bridge_sequence",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 602,
      "target_type": "theorem"
    },
    {
      "context": "ne}) and conceptual bridges (Def.~\\ref{definition:bk3_conceptual_bridge}) in a system. It is characterized by (cf.~Thm.~\\ref{theorem:bk3_emergence_of_symbolic_networks}, Thm.~\\ref{theorem:bk3_closure_conceptual_bridge_sequence}, Def.~\\ref{definition:bk1_reflection_operator}): \\begin{enum",
      "label": "theorem:bk3_emergence_of_symbolic_networks",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 546,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_reflection_operator",
    "definition:bk3_conceptual_bridge",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_closure_conceptual_bridge_sequence",
    "theorem:bk3_emergence_of_symbolic_networks"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk3_symbolic_homeostasis",
    "proof:bk3_sketch_field_perturbation",
    "remark:bk3_symbolic_membrane_remark",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk3_homeostatic_reflexes"
  ],
  "cites": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "depends_on": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_symbolic_metabolic_rate",
  "label": "definition:bk3_symbolic_metabolic_rate",
  "latex_body": "\\begin{definition}[Symbolic Metabolic Rate] \\label{definition:bk3_symbolic_metabolic_rate}\nThe symbolic metabolic rate $R_{\\text{meta}}$ of a system of coupled symbolic membranes $\\{\\mathcal{M}_i\\}$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is defined as (cf.~Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk2_symbolic_free_energy}):\n\\[\nR_{\\text{meta}} = \\sum_{i,j} \\int_{\\mathcal{M}_i \\times \\mathcal{M}_j} \\rho_{ij}(x,y) \\|\\nabla_g H_{ij}(x,y)\\|_g \\, d\\mu_g(x) \\, d\\mu_g(y)\n\\]\nwhere:\n\\begin{itemize}\n    \\item $\\rho_{ij}$ is the joint symbolic probability density (Def.~\\ref{definition:bk2__symbolic_probability_density}) over the coupled membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$,\n    \\item $H_{ij}$ is the coupling Hamiltonian (energy) between membranes (Definition~\\ref{definition:bk3_induced_coupling_energy}),\n    \\item $\\nabla_g$ is the gradient with respect to the symbolic metric $g$ (from Def.~\\ref{definition:bk2_symbolic_probability_spa}) (acting on both $x$ and $y$ components, norm taken in the product tangent space),\n    \\item and the integral quantifies the total symbolic flux or activity driven by coupling-induced forces, weighted by the probability density.\n\\end{itemize}\n\\end{definition}",
  "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": false,
    "notes": [
      "Finite double-sum analogue of the continuous double integral over paired membranes; only nonnegativity of the analogue is proved."
    ],
    "record_ids": [
      "MAP-BOOK3-026"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book3.metabolicRate_nonneg"
    ]
  },
  "line": 670,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Metabolic Rate",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "d\\mu_g(x) \\, d\\mu_g(y) \\] where: \\begin{itemize} \\item $\\rho_{ij}$ is the joint symbolic probability density (Def.~\\ref{definition:bk2__symbolic_probability_density}) over the coupled membranes $\\mathcal{M}_i$ and $\\mathcal{M}_j$, \\item $H_{ij}$ is the coupling Hamiltonian (energy",
      "label": "definition:bk2__symbolic_probability_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 35,
      "target_type": "definition"
    },
    {
      "context": "f{definition:bk3_symbolic_membrane}) is defined as (cf.~Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk2_symbolic_free_energy}): \\[ R_{\\text{meta}} = \\sum_{i,j} \\int_{\\mathcal{M}_i \\times \\mathcal{M}_j} \\rho_{ij}(x,y) \\|\\nabla_g H_{ij}(x,y)\\|_g \\",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": ":bk3_induced_coupling_energy}), \\item $\\nabla_g$ is the gradient with respect to the symbolic metric $g$ (from Def.~\\ref{definition:bk2_symbolic_probability_spa}) (acting on both $x$ and $y$ components, norm taken in the product tangent space), \\item and the integral quantifie",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "athcal{M}_i$ and $\\mathcal{M}_j$, \\item $H_{ij}$ is the coupling Hamiltonian (energy) between membranes (Definition~\\ref{definition:bk3_induced_coupling_energy}), \\item $\\nabla_g$ is the gradient with respect to the symbolic metric $g$ (from Def.~\\ref{definition:bk2_symbolic_",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": "rate} The symbolic metabolic rate $R_{\\text{meta}}$ of a system of coupled symbolic membranes $\\{\\mathcal{M}_i\\}$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is defined as (cf.~Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk2_symbolic_free_e",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "of coupled symbolic membranes $\\{\\mathcal{M}_i\\}$ (Def.~\\ref{definition:bk3_symbolic_membrane}) is defined as (cf.~Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk2_symbolic_free_energy}): \\[ R_{\\text{meta}} = \\sum_{i,j} \\int_{\\mathcal{M}_i \\times \\mathcal{M",
      "label": "theorem:bk3_couplinginduced_drift_modification",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 104,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2__symbolic_probability_density",
    "definition:bk2_symbolic_free_energy",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_membrane",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "depends_on": [
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "file": "book3.tex",
  "id": "remark:bk3_symbolic_membrane_remark",
  "label": "remark:bk3_symbolic_membrane_remark",
  "latex_body": "\\begin{remark} \\label{remark:bk3_symbolic_membrane_remark}\nThe symbolic metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) measures the system's internal symbolic \"activity\" — the intensity of regulated information and energy flows that sustain structural coherence and dynamics across the coupled membranes (cf.~Def.~\\ref{definition:bk3_membrane_thermodynamics}, Thm.~\\ref{theorem:bk3_symbiotic_curvature_and_resilience}). It reflects the magnitude of the forces mediating the interactions.\n\\end{remark}",
  "line": 685,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "ref_roles": [
    {
      "context": "ated information and energy flows that sustain structural coherence and dynamics across the coupled membranes (cf.~Def.~\\ref{definition:bk3_membrane_thermodynamics}, Thm.~\\ref{theorem:bk3_symbiotic_curvature_and_resilience}). It reflects the magnitude of the forces mediating the inte",
      "label": "definition:bk3_membrane_thermodynamics",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 49,
      "target_type": "definition"
    },
    {
      "context": "\\begin{remark} \\label{remark:bk3_symbolic_membrane_remark} The symbolic metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) measures the system's internal symbolic \"activity\" — the intensity of regulated information and energy flows that sust",
      "label": "definition:bk3_symbolic_metabolic_rate",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 670,
      "target_type": "definition"
    },
    {
      "context": "ctural coherence and dynamics across the coupled membranes (cf.~Def.~\\ref{definition:bk3_membrane_thermodynamics}, Thm.~\\ref{theorem:bk3_symbiotic_curvature_and_resilience}). It reflects the magnitude of the forces mediating the interactions. \\end{remark}",
      "label": "theorem:bk3_symbiotic_curvature_and_resilience",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 299,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk3_membrane_thermodynamics",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "book3",
  "cited_by": [
    "proof:bk3_sketch_necessity_for_continuous_operation",
    "proof:bk9_isolation_dissociation_theorem",
    "theorem:bk9_isolation_dissociation_theorem"
  ],
  "cites": [
    "corollary:bk9_emergence_of_moral_agency",
    "definition:bk1_observer_relative_interpretability",
    "definition:bk1_proto_drift_field",
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_metabolism"
  ],
  "depends_on": [
    "corollary:bk9_emergence_of_moral_agency",
    "definition:bk1_observer_relative_interpretability",
    "definition:bk1_proto_drift_field",
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_metabolism"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_autophagic_drift",
  "label": "definition:bk3_autophagic_drift",
  "latex_body": "\\begin{definition}[Autophagic Drift] \\label{definition:bk3_autophagic_drift}\nAutophagic drift is a symbolic phase in which agency $\\mathcal{A}$ is suspended (cf.~\\ref{corollary:bk9_emergence_of_moral_agency}) \nand symbolic drift $\\mathcal{D}$ proceeds without immediate constraint (cf.~\\ref{definition:bk1_proto_drift_field}). \nThis phase allows symbolic membranes (cf.~\\ref{definition:bk3_symbolic_membrane}) \nto perform selective self-digestion, pruning unstable or incoherent forms and redistributing symbolic free energy (cf.~\\ref{definition:bk2_symbolic_free_energy}).\n\nIt is metabolically essential: a regenerative drift cycle that supports long-term coherence (cf.~\\ref{definition:bk3_symbolic_metabolism}) \nby enabling spontaneous symbolic recomposition beneath the horizon of active regulation (cf.~\\ref{definition:bk1_observer_relative_interpretability}).\n\\end{definition}",
  "line": 690,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Autophagic Drift",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "l{definition:bk3_autophagic_drift} Autophagic drift is a symbolic phase in which agency $\\mathcal{A}$ is suspended (cf.~\\ref{corollary:bk9_emergence_of_moral_agency}) and symbolic drift $\\mathcal{D}$ proceeds without immediate constraint (cf.~\\ref{definition:bk1_proto_drift_field}).",
      "label": "corollary:bk9_emergence_of_moral_agency",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book9.tex",
      "target_line": 127,
      "target_type": "corollary"
    },
    {
      "context": "bk3_symbolic_metabolism}) by enabling spontaneous symbolic recomposition beneath the horizon of active regulation (cf.~\\ref{definition:bk1_observer_relative_interpretability}). \\end{definition}",
      "label": "definition:bk1_observer_relative_interpretability",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 190,
      "target_type": "definition"
    },
    {
      "context": "{corollary:bk9_emergence_of_moral_agency}) and symbolic drift $\\mathcal{D}$ proceeds without immediate constraint (cf.~\\ref{definition:bk1_proto_drift_field}). This phase allows symbolic membranes (cf.~\\ref{definition:bk3_symbolic_membrane}) to perform selective self-digesti",
      "label": "definition:bk1_proto_drift_field",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2797,
      "target_type": "definition"
    },
    {
      "context": "to perform selective self-digestion, pruning unstable or incoherent forms and redistributing symbolic free energy (cf.~\\ref{definition:bk2_symbolic_free_energy}). It is metabolically essential: a regenerative drift cycle that supports long-term coherence (cf.~\\ref{definition:bk3",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "s without immediate constraint (cf.~\\ref{definition:bk1_proto_drift_field}). This phase allows symbolic membranes (cf.~\\ref{definition:bk3_symbolic_membrane}) to perform selective self-digestion, pruning unstable or incoherent forms and redistributing symbolic free energy (cf",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "mbolic_free_energy}). It is metabolically essential: a regenerative drift cycle that supports long-term coherence (cf.~\\ref{definition:bk3_symbolic_metabolism}) by enabling spontaneous symbolic recomposition beneath the horizon of active regulation (cf.~\\ref{definition:bk1_obse",
      "label": "definition:bk3_symbolic_metabolism",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 646,
      "target_type": "definition"
    }
  ],
  "refs": [
    "corollary:bk9_emergence_of_moral_agency",
    "definition:bk1_observer_relative_interpretability",
    "definition:bk1_proto_drift_field",
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_metabolism"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "section:book3.tex:700",
  "label": "",
  "latex_body": "",
  "line": 700,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Metabolic Stability and Regulation",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [
    "axiom:bk9_emergent_autonomy",
    "definition:bk6_symbolic_confidence_field",
    "definition:bk6_symbolic_mutation",
    "lemma:bk6_conservation_of_symbolic_information",
    "proof:bk3_sketch_field_perturbation",
    "remark:bk8_symbolic_repair_loop",
    "subsec:bk6_from_map_to_operator_formalism",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk3_homeostatic_reflexes"
  ],
  "cites": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "depends_on": [
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_symbolic_homeostasis",
  "label": "definition:bk3_symbolic_homeostasis",
  "latex_body": "\\begin{definition}[Symbolic Homeostasis] \\label{definition:bk3_symbolic_homeostasis}\nA symbolic system maintains homeostasis if, for a bounded range of perturbations $\\delta$ (affecting, e.g., drift fields or external potentials), the symbolic metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) remains within a stable operating band (cf.~Thm.~\\ref{theorem:bk3_membrane_stability_criteria}, Def.~\\ref{definition:bk2_symbolic_free_energy}):\n\\[\nR_{\\text{min}} \\leq R_{\\text{meta}}(\\delta) \\leq R_{\\text{max}}\n\\]\nwhere $R_{\\text{min}}, R_{\\text{max}}$ are threshold bounds set by system\nstructure (e.g., membranes, Def.~\\ref{definition:bk3_symbolic_membrane}) and\nviability requirements.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Operating-band definition and the trivial-but-real consequence that a homeostatic state forces rmin<=rmax."
    ],
    "record_ids": [
      "MAP-BOOK3-027"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book3.homeostatic_band_nonempty"
    ]
  },
  "line": 703,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Homeostasis",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "c_metabolic_rate}) remains within a stable operating band (cf.~Thm.~\\ref{theorem:bk3_membrane_stability_criteria}, Def.~\\ref{definition:bk2_symbolic_free_energy}): \\[ R_{\\text{min}} \\leq R_{\\text{meta}}(\\delta) \\leq R_{\\text{max}} \\] where $R_{\\text{min}}, R_{\\text{max}}$ are thre",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "ext{max}} \\] where $R_{\\text{min}}, R_{\\text{max}}$ are threshold bounds set by system structure (e.g., membranes, Def.~\\ref{definition:bk3_symbolic_membrane}) and viability requirements. \\end{definition}",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "ns $\\delta$ (affecting, e.g., drift fields or external potentials), the symbolic metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) remains within a stable operating band (cf.~Thm.~\\ref{theorem:bk3_membrane_stability_criteria}, Def.~\\ref{definition:b",
      "label": "definition:bk3_symbolic_metabolic_rate",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 670,
      "target_type": "definition"
    },
    {
      "context": "$R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) remains within a stable operating band (cf.~Thm.~\\ref{theorem:bk3_membrane_stability_criteria}, Def.~\\ref{definition:bk2_symbolic_free_energy}): \\[ R_{\\text{min}} \\leq R_{\\text{meta}}(\\delta) \\leq R_{\\text{max}} \\]",
      "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:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "proof:bk3_sketch_field_perturbation"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk1_existence_of_metric",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_homeostatic_reflexes",
  "label": "theorem:bk3_homeostatic_reflexes",
  "latex_body": "\\begin{theorem}[Homeostatic Reflexes] \\label{theorem:bk3_homeostatic_reflexes}\nA symbolic system exhibits homeostatic reflexes if perturbations $\\delta$ trigger compensatory adjustments $\\Delta D_i$ in the drift fields (or other regulatory parameters like $\\eta_i, \\lambda_{ij}$ from Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification} and Def.~\\ref{definition:bk3_induced_coupling_energy}) such that the sensitivity of the metabolic rate (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) to the perturbation is bounded (cf.~Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk1_drift_field}):\n\\[\n\\left| \\frac{d R_{\\text{meta}}}{d \\delta} \\right| \\leq C\n\\]\nfor some bounded constant $C > 0$, across a specified operating regime. This implies that the system actively counteracts disturbances to maintain its metabolic rate (supporting Def.~\\ref{definition:bk3_symbolic_homeostasis}).\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 derivative bound |dR/dDelta|<=C is modeled as a Lipschitz condition on the response function (a modeling commitment); a genuine two-sided deviation-from-baseline bound is derived from it."
    ],
    "record_ids": [
      "MAP-BOOK3-028"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3.metabolic_response_deviation_bound"
    ]
  },
  "line": 714,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Homeostatic Reflexes",
  "proof_labels": [
    "proof:bk3_sketch_field_perturbation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "n:bk3_symbolic_metabolic_rate}) to the perturbation is bounded (cf.~Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk1_drift_field}): \\[ \\left| \\frac{d R_{\\text{meta}}}{d \\delta} \\right| \\leq C \\] for some bounded constant $C > 0$, across a specified",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "vity of the metabolic rate (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) to the perturbation is bounded (cf.~Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk1_drift_field}): \\[ \\left| \\frac{d R_{\\text{meta}}}{d \\delta} \\right| \\leq C \\] for some bounde",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "gulatory parameters like $\\eta_i, \\lambda_{ij}$ from Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification} and Def.~\\ref{definition:bk3_induced_coupling_energy}) such that the sensitivity of the metabolic rate (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) to the perturbatio",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": "regime. This implies that the system actively counteracts disturbances to maintain its metabolic rate (supporting Def.~\\ref{definition:bk3_symbolic_homeostasis}). \\end{theorem}",
      "label": "definition:bk3_symbolic_homeostasis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 703,
      "target_type": "definition"
    },
    {
      "context": "ification} and Def.~\\ref{definition:bk3_induced_coupling_energy}) such that the sensitivity of the metabolic rate (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) to the perturbation is bounded (cf.~Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk1_drift_fiel",
      "label": "definition:bk3_symbolic_metabolic_rate",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 670,
      "target_type": "definition"
    },
    {
      "context": "tory adjustments $\\Delta D_i$ in the drift fields (or other regulatory parameters like $\\eta_i, \\lambda_{ij}$ from Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification} and Def.~\\ref{definition:bk3_induced_coupling_energy}) such that the sensitivity of the metabolic rate (Def.~\\ref{defin",
      "label": "theorem:bk3_couplinginduced_drift_modification",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 104,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_metabolic_rate",
    "theorem:bk3_couplinginduced_drift_modification"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk1_existence_of_metric",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_homeostatic_reflexes",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "depends_on": [
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk1_existence_of_metric",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_homeostatic_reflexes",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_sketch_field_perturbation",
  "label": "proof:bk3_sketch_field_perturbation",
  "latex_body": "\\begin{proof}[Bounded Sensitivity via Drift Compensation]\n\\label{proof:bk3_sketch_field_perturbation}\n\\leavevmode\n\nLet $\\delta$ be a perturbation to the drift fields: $D_i \\mapsto D_i + \\delta D_i$,\nwith $\\|\\delta D_i\\| \\leq \\delta$ for small $\\delta > 0$.\n\n\\textbf{Compensatory response.}\nBy Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, the coupling mechanism\nproduces compensatory drift adjustments $\\Delta D_i$ that oppose deviations from the\nsymbiotic equilibrium (Def.~\\ref{definition:bk3_symbolic_symbiosis}, condition 3):\n\\[\n\\Delta D_i = -\\kappa_{\\text{symb}} \\cdot \\delta D_i + O(\\delta^2),\n\\]\nfor a coupling constant $\\kappa_{\\text{symb}} > 0$ derived from the membrane stability\nanalysis (Thm.~\\ref{theorem:bk3_membrane_stability_criteria}).\n\n\\textbf{Sensitivity bound via Grönwall.}\nThe metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate})\ndepends on $D_i$ through the coupling energies $H_{ij}$ and probability flows $\\rho_{ij}$.\nLet $r(t) = |R_{\\text{meta}}(t) - R_{\\text{meta}}^0|$ be the deviation from unperturbed\nrate. The compensated dynamics give:\n\\[\n\\dot{r}(t) \\leq (1 - \\kappa_{\\text{symb}})\\|\\delta D_i\\| + L_H\\cdot r(t),\n\\]\nwhere $L_H$ is the Lipschitz constant of $\\nabla_g H_{ij}$ (finite by smoothness of $M$,\nLemma~\\ref{lemma:bk1_existence_of_metric}). By Grönwall's inequality:\n\\[\nr(t) \\leq \\frac{(1-\\kappa_{\\text{symb}})\\delta}{L_H}(e^{L_H t} - 1).\n\\]\nOn bounded observation horizons $t \\in [0,T]$, the sensitivity is bounded by\n$C = (1-\\kappa_{\\text{symb}})(e^{L_H T}-1)$, giving\n$|dR_{\\text{meta}}/d\\delta| \\leq C < \\infty$ as required.\n\n\\textbf{Homeostasis.}\nSince $C$ is finite and the operating band $[R_{\\text{min}}, R_{\\text{max}}]$\n(Def.~\\ref{definition:bk3_symbolic_homeostasis}) has positive width $\\geq 2C\\delta$\nfor sufficiently small $\\delta$, the perturbed metabolic rate remains within bounds.\nHence the system exhibits homeostatic reflexes (Thm.~\\ref{theorem:bk3_homeostatic_reflexes}).\n\\end{proof}",
  "line": 722,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Bounded Sensitivity via Drift Compensation",
  "proves": "theorem:bk3_homeostatic_reflexes",
  "ref_roles": [
    {
      "context": "as required. \\textbf{Homeostasis.} Since $C$ is finite and the operating band $[R_{\\text{min}}, R_{\\text{max}}]$ (Def.~\\ref{definition:bk3_symbolic_homeostasis}) has positive width $\\geq 2C\\delta$ for sufficiently small $\\delta$, the perturbed metabolic rate remains within bounds",
      "label": "definition:bk3_symbolic_homeostasis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 703,
      "target_type": "definition"
    },
    {
      "context": "bk3_membrane_stability_criteria}). \\textbf{Sensitivity bound via Grönwall.} The metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) depends on $D_i$ through the coupling energies $H_{ij}$ and probability flows $\\rho_{ij}$. Let $r(t) = |R_{\\text{meta}",
      "label": "definition:bk3_symbolic_metabolic_rate",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 670,
      "target_type": "definition"
    },
    {
      "context": "hanism produces compensatory drift adjustments $\\Delta D_i$ that oppose deviations from the symbiotic equilibrium (Def.~\\ref{definition:bk3_symbolic_symbiosis}, condition 3): \\[ \\Delta D_i = -\\kappa_{\\text{symb}} \\cdot \\delta D_i + O(\\delta^2), \\] for a coupling constant $\\kappa",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "_i\\| + L_H\\cdot r(t), \\] where $L_H$ is the Lipschitz constant of $\\nabla_g H_{ij}$ (finite by smoothness of $M$, Lemma~\\ref{lemma:bk1_existence_of_metric}). By Grönwall's inequality: \\[ r(t) \\leq \\frac{(1-\\kappa_{\\text{symb}})\\delta}{L_H}(e^{L_H t} - 1). \\] On bounded obser",
      "label": "lemma:bk1_existence_of_metric",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2886,
      "target_type": "lemma"
    },
    {
      "context": "o D_i + \\delta D_i$, with $\\|\\delta D_i\\| \\leq \\delta$ for small $\\delta > 0$. \\textbf{Compensatory response.} By Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, the coupling mechanism produces compensatory drift adjustments $\\Delta D_i$ that oppose deviations from the symbiotic",
      "label": "theorem:bk3_couplinginduced_drift_modification",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 104,
      "target_type": "theorem"
    },
    {
      "context": "mall $\\delta$, the perturbed metabolic rate remains within bounds. Hence the system exhibits homeostatic reflexes (Thm.~\\ref{theorem:bk3_homeostatic_reflexes}). \\end{proof}",
      "label": "theorem:bk3_homeostatic_reflexes",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 714,
      "target_type": "theorem"
    },
    {
      "context": "+ O(\\delta^2), \\] for a coupling constant $\\kappa_{\\text{symb}} > 0$ derived from the membrane stability analysis (Thm.~\\ref{theorem:bk3_membrane_stability_criteria}). \\textbf{Sensitivity bound via Grönwall.} The metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_met",
      "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_symbolic_homeostasis",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "lemma:bk1_existence_of_metric",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_homeostatic_reflexes",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "section:book3.tex:763",
  "label": "",
  "latex_body": "",
  "line": 763,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Persistent Symbolic Life",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [
    "definition:bk6_symbolic_operator_canon",
    "remark:bk3_toward_symbolic_evolution",
    "scholium:bk8_metabolic_programming_as_proto_freedom",
    "scholium:bk9_concluding_reflection_c"
  ],
  "cites": [
    "definition:bk3_conceptual_bridge",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "depends_on": [
    "definition:bk3_conceptual_bridge",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "file": "book3.tex",
  "id": "definition:bk3_symbolic_autopoiesis",
  "label": "definition:bk3_symbolic_autopoiesis",
  "latex_body": "\\begin{definition}[Symbolic Autopoiesis] \\label{definition:bk3_symbolic_autopoiesis}\nA symbolic system exhibits autopoiesis (self-production and maintenance) if it sustains a closed loop of symbolic production, maintenance, and regulation of its own constituent components (membranes (Def.~\\ref{definition:bk3_symbolic_membrane}), coupling maps (Def.~\\ref{definition:bk3_coupling_map}), etc.), characterized by:\n\\begin{enumerate}\n    \\item Self-Maintenance: Membranes $\\{\\mathcal{M}_i\\}$ persist over time via\n    internal stability (Thm.~\\ref{theorem:bk3_membrane_stability_criteria}) and\n    symbiotic stabilization (Def.~\\ref{definition:bk3_symbolic_symbiosis}).\n    \\item Self-Modification: Reflexive encodings (Def.~\\ref{definition:bk3_reflexive_encoding}) and coupling dynamics (e.g. Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk3_induced_coupling_energy}) allow the system to modify its own drift fields, coupling configurations, and potentially membrane boundaries or permeability in response to experience or internal states.\n    \\item Self-Extension: Conceptual bridges (Def.~\\ref{definition:bk3_conceptual_bridge}) can evolve or be newly formed, allowing the system to incorporate new symbolic domains or refine its internal network structure ($\\mathcal{N}$, Def.~\\ref{definition:bk3_symbolic_network}).\n\\end{enumerate}\n\\end{definition}",
  "line": 765,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Autopoiesis",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "daries or permeability in response to experience or internal states. \\item Self-Extension: Conceptual bridges (Def.~\\ref{definition:bk3_conceptual_bridge}) can evolve or be newly formed, allowing the system to incorporate new symbolic domains or refine its internal network",
      "label": "definition:bk3_conceptual_bridge",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 219,
      "target_type": "definition"
    },
    {
      "context": "ulation of its own constituent components (membranes (Def.~\\ref{definition:bk3_symbolic_membrane}), coupling maps (Def.~\\ref{definition:bk3_coupling_map}), etc.), characterized by: \\begin{enumerate} \\item Self-Maintenance: Membranes $\\{\\mathcal{M}_i\\}$ persist over tim",
      "label": "definition:bk3_coupling_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 85,
      "target_type": "definition"
    },
    {
      "context": "on:bk3_reflexive_encoding}) and coupling dynamics (e.g. Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk3_induced_coupling_energy}) allow the system to modify its own drift fields, coupling configurations, and potentially membrane boundaries or perme",
      "label": "definition:bk3_induced_coupling_energy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 95,
      "target_type": "definition"
    },
    {
      "context": "ic stabilization (Def.~\\ref{definition:bk3_symbolic_symbiosis}). \\item Self-Modification: Reflexive encodings (Def.~\\ref{definition:bk3_reflexive_encoding}) and coupling dynamics (e.g. Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk3_induce",
      "label": "definition:bk3_reflexive_encoding",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 183,
      "target_type": "definition"
    },
    {
      "context": "ns a closed loop of symbolic production, maintenance, and regulation of its own constituent components (membranes (Def.~\\ref{definition:bk3_symbolic_membrane}), coupling maps (Def.~\\ref{definition:bk3_coupling_map}), etc.), characterized by: \\begin{enumerate} \\item Self-Mai",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": ", allowing the system to incorporate new symbolic domains or refine its internal network structure ($\\mathcal{N}$, Def.~\\ref{definition:bk3_symbolic_network}). \\end{enumerate} \\end{definition}",
      "label": "definition:bk3_symbolic_network",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 536,
      "target_type": "definition"
    },
    {
      "context": "e via internal stability (Thm.~\\ref{theorem:bk3_membrane_stability_criteria}) and symbiotic stabilization (Def.~\\ref{definition:bk3_symbolic_symbiosis}). \\item Self-Modification: Reflexive encodings (Def.~\\ref{definition:bk3_reflexive_encoding}) and coupling dynamics",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "Self-Modification: Reflexive encodings (Def.~\\ref{definition:bk3_reflexive_encoding}) and coupling dynamics (e.g. Thm.~\\ref{theorem:bk3_couplinginduced_drift_modification}, Def.~\\ref{definition:bk3_induced_coupling_energy}) allow the system to modify its own drift fields, coupling configura",
      "label": "theorem:bk3_couplinginduced_drift_modification",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 104,
      "target_type": "theorem"
    },
    {
      "context": "enumerate} \\item Self-Maintenance: Membranes $\\{\\mathcal{M}_i\\}$ persist over time via internal stability (Thm.~\\ref{theorem:bk3_membrane_stability_criteria}) and symbiotic stabilization (Def.~\\ref{definition:bk3_symbolic_symbiosis}). \\item Self-Modification: Reflexive",
      "label": "theorem:bk3_membrane_stability_criteria",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 61,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk3_conceptual_bridge",
    "definition:bk3_coupling_map",
    "definition:bk3_induced_coupling_energy",
    "definition:bk3_reflexive_encoding",
    "definition:bk3_symbolic_membrane",
    "definition:bk3_symbolic_network",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_couplinginduced_drift_modification",
    "theorem:bk3_membrane_stability_criteria"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "corollary:bk4_emergence_of_meaning",
    "proof:bk4_freedom_growth_fragmentation",
    "proof:bk9_stability_conditions_for_the_good",
    "remark:bk3_toward_symbolic_evolution",
    "scholium:bk4_symbolic_self_organization",
    "sec:bk9_emergence_ethics_and_compassion",
    "theorem:bk3_symbolic_life_satisfies_canonical_definitions",
    "theorem:bk4_freedom_life_connection",
    "theorem:bk8_biological_phase_transition"
  ],
  "cites": [
    "definition:bk1_observer_horizon_structure",
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk2_symbolic_entropy",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_knowledge_structure",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_conditions_sustained_symbolic_growth",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_observer_horizon_structure",
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk2_symbolic_entropy",
    "definition:bk3_autophagic_drift",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_knowledge_structure",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_metabolism",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk3_conditions_sustained_symbolic_growth",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_criteria_persistent_symbolic_life",
  "label": "theorem:bk3_criteria_persistent_symbolic_life",
  "latex_body": "\\begin{theorem}[Persistent Symbolic Life Criteria] \\label{theorem:bk3_criteria_persistent_symbolic_life}\nA symbolic system supports persistent symbolic life (understood as a dynamically stable, adaptive, and potentially growing symbolic organization) if (cf.~Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}, Def.~\\ref{definition:bk2_symbolic_entropy}, Def.~\\ref{definition:bk1_observer_horizon_structure}):\n\\begin{enumerate}\n    \\item Symbolic metabolic rate $R_{\\text{meta}}$\n    (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) stays within stable\n    operating bands $[R_{\\text{min}}, R_{\\text{max}}]$, indicating sustained\n    regulated activity (symbolic homeostasis,\n    Def.~\\ref{definition:bk3_symbolic_homeostasis}).\n    \\item Symbolic knowledge structure $K(r)$\n    (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) grows recurrently\n    (e.g., Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth}),\n    indicating ongoing refinement and complexification.\n    \\item Symbiotic curvature $\\kappa_{\\text{symb}}$\n    (Def.~\\ref{definition:bk3_symbiotic_curvature}) stays strictly positive and\n    bounded away from zero, ensuring persistent coupling, stability\n    enhancement, and information exchange\n    (Def.~\\ref{definition:bk3_symbolic_symbiosis},\n    Thm.~\\ref{theorem:bk3_properties_of_symbiotic_curvature}).\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": [
      "Capstone structure assembling the three named criteria (homeostatic metabolic rate, positive growth increment, curvature bounded away from zero) as fields, with two projection consequences proved."
    ],
    "record_ids": [
      "MAP-BOOK3-029"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3.persistentLife_kappa_pos",
      "Book3.persistentLife_rmin_le_rmax"
    ]
  },
  "line": 777,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Persistent Symbolic Life Criteria",
  "proof_labels": [
    "proof:bk3_sketch_necessity_for_continuous_operation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "(cf.~Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}, Def.~\\ref{definition:bk2_symbolic_entropy}, Def.~\\ref{definition:bk1_observer_horizon_structure}): \\begin{enumerate} \\item Symbolic metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metaboli",
      "label": "definition:bk1_observer_horizon_structure",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1232,
      "target_type": "definition"
    },
    {
      "context": "ymbolic life (understood as a dynamically stable, adaptive, and potentially growing symbolic organization) if (cf.~Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}, Def.~\\ref{definition:bk2_symbolic_entropy}, Def.~\\ref{definition:bk1_observer_horizon_structure}): \\begin{enumerate}",
      "label": "definition:bk1_self_regulating_mapping_function_srmf",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2230,
      "target_type": "definition"
    },
    {
      "context": "otentially growing symbolic organization) if (cf.~Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}, Def.~\\ref{definition:bk2_symbolic_entropy}, Def.~\\ref{definition:bk1_observer_horizon_structure}): \\begin{enumerate} \\item Symbolic metabolic rate $R_{\\text{m",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "indicating ongoing refinement and complexification. \\item Symbiotic curvature $\\kappa_{\\text{symb}}$ (Def.~\\ref{definition:bk3_symbiotic_curvature}) stays strictly positive and bounded away from zero, ensuring persistent coupling, stability enhancement, and i",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "g bands $[R_{\\text{min}}, R_{\\text{max}}]$, indicating sustained regulated activity (symbolic homeostasis, Def.~\\ref{definition:bk3_symbolic_homeostasis}). \\item Symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) grows recu",
      "label": "definition:bk3_symbolic_homeostasis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 703,
      "target_type": "definition"
    },
    {
      "context": "meostasis, Def.~\\ref{definition:bk3_symbolic_homeostasis}). \\item Symbolic knowledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) grows recurrently (e.g., Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth}), indicating ongoing refi",
      "label": "definition:bk3_symbolic_knowledge_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 448,
      "target_type": "definition"
    },
    {
      "context": "ition:bk1_observer_horizon_structure}): \\begin{enumerate} \\item Symbolic metabolic rate $R_{\\text{meta}}$ (Def.~\\ref{definition:bk3_symbolic_metabolic_rate}) stays within stable operating bands $[R_{\\text{min}}, R_{\\text{max}}]$, indicating sustained regulated activit",
      "label": "definition:bk3_symbolic_metabolic_rate",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 670,
      "target_type": "definition"
    },
    {
      "context": "bounded away from zero, ensuring persistent coupling, stability enhancement, and information exchange (Def.~\\ref{definition:bk3_symbolic_symbiosis}, Thm.~\\ref{theorem:bk3_properties_of_symbiotic_curvature}). \\end{enumerate} \\end{theorem}",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "owledge structure $K(r)$ (Def.~\\ref{definition:bk3_symbolic_knowledge_structure}) grows recurrently (e.g., Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth}), indicating ongoing refinement and complexification. \\item Symbiotic curvature $\\kappa_{\\text{symb}}$ (Def",
      "label": "theorem:bk3_conditions_sustained_symbolic_growth",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 509,
      "target_type": "theorem"
    },
    {
      "context": "upling, stability enhancement, and information exchange (Def.~\\ref{definition:bk3_symbolic_symbiosis}, Thm.~\\ref{theorem:bk3_properties_of_symbiotic_curvature}). \\end{enumerate} \\end{theorem}",
      "label": "theorem:bk3_properties_of_symbiotic_curvature",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 261,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_observer_horizon_structure",
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk2_symbolic_entropy",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_homeostasis",
    "definition:bk3_symbolic_knowledge_structure",
    "definition:bk3_symbolic_metabolic_rate",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk3_conditions_sustained_symbolic_growth",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk3_autophagic_drift",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_metabolism",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk3_conditions_sustained_symbolic_growth",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk3_autophagic_drift",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_metabolism",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk3_conditions_sustained_symbolic_growth",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "file": "book3.tex",
  "id": "proof:bk3_sketch_necessity_for_continuous_operation",
  "label": "proof:bk3_sketch_necessity_for_continuous_operation",
  "latex_body": "\\begin{proof}[Necessity of Each Condition for Persistent Symbolic Life]\n\\label{proof:bk3_sketch_necessity_for_continuous_operation}\n\\leavevmode\n\nWe prove each condition is necessary by contradiction.\n\n\\textbf{Necessity of Condition 1} ($R_{\\text{meta}} \\in [R_{\\text{min}}, R_{\\text{max}}]$).\nSuppose homeostasis fails: either $R_{\\text{meta}} < R_{\\text{min}}$ or\n$R_{\\text{meta}} > R_{\\text{max}}$ persistently.\nIf $R_{\\text{meta}} < R_{\\text{min}}$, symbolic metabolism\n(Def.~\\ref{definition:bk3_symbolic_metabolism}) falls below the threshold\nrequired to maintain membrane coherence; by\nThm.~\\ref{theorem:bk3_membrane_stability_criteria} the free energy $F_i(\\beta_i)$\nis no longer at a local minimum and restorative forces are lost, driving the\nsystem toward collapse.\nIf $R_{\\text{meta}} > R_{\\text{max}}$, autophagic drift\n(Def.~\\ref{definition:bk3_autophagic_drift}) accelerates unboundedly; by the\nH-theorem (Thm.~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}) entropy grows\nmonotonically and symbolic coherence is destroyed.\nIn either case persistent symbolic life is impossible.\n\n\\textbf{Necessity of Condition 2} (Recurrent growth of $K(r)$).\nSuppose $K(r)$ does not grow recurrently: there exists $R_0$ such that for all\n$r_0 \\geq R_0$, $\\int_{r_0}^{r_0+T}(I'(s)-D'(s))\\,ds \\leq 0$ for every $T>0$.\nBy Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth}, differentiation\ndominates or balances integration, so $K(r)$ stagnates or fragments.\nA stagnant $K(r)$ cannot adapt to perturbations in drift fields or coupling\nparameters; under persistent drift (Def.~\\ref{definition:bk1_drift_field}),\nstatic symbolic structures lose coherence over time, and the system eventually\nfalls below the viability threshold, contradicting persistence.\n\n\\textbf{Necessity of Condition 3} ($\\kappa_{\\text{symb}} > \\epsilon > 0$).\nSuppose $\\kappa_{\\text{symb}} \\to 0$.\nBy Def.~\\ref{definition:bk3_symbiotic_curvature}, this requires either\n$S_i^{\\text{coupled}}/S_i^{\\text{isolated}} \\to 1$ (coupling ceases to enhance\nstability) or $I(\\mathcal{M}_i;\\mathcal{M}_j) \\to 0$ (membranes become\ninformationally independent) for all pairs.\nIn either case the drift compensation condition of\nDef.~\\ref{definition:bk3_symbolic_symbiosis} (condition 3) fails:\n$\\|\\delta D_i + \\delta D_i^{\\text{response}}\\|_g \\to \\|\\delta D_i\\|_g$,\nso membranes can no longer buffer each other's perturbations.\nBy the subadditivity property (Thm.~\\ref{theorem:bk3_properties_of_symbiotic_curvature},\nclause 4), once coupling vanishes the system reduces to isolated membranes with\n$\\kappa_{\\text{symb}}(A \\cup B) \\leq \\max(\\kappa_{\\text{symb}}(A),\\kappa_{\\text{symb}}(B))$,\neach surviving independently — which is not persistent \\emph{symbolic life} in the\nsymbiotic sense required by the theorem statement.\n\nSince the failure of any single condition destroys persistence, all three are necessary.\n\\end{proof}",
  "line": 798,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Necessity of Each Condition for Persistent Symbolic Life",
  "proves": "theorem:bk3_criteria_persistent_symbolic_life",
  "ref_roles": [
    {
      "context": "s. A stagnant $K(r)$ cannot adapt to perturbations in drift fields or coupling parameters; under persistent drift (Def.~\\ref{definition:bk1_drift_field}), static symbolic structures lose coherence over time, and the system eventually falls below the viability threshold, c",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "tive forces are lost, driving the system toward collapse. If $R_{\\text{meta}} > R_{\\text{max}}$, autophagic drift (Def.~\\ref{definition:bk3_autophagic_drift}) accelerates unboundedly; by the H-theorem (Thm.~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}) entropy grows monotonic",
      "label": "definition:bk3_autophagic_drift",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 690,
      "target_type": "definition"
    },
    {
      "context": "textbf{Necessity of Condition 3} ($\\kappa_{\\text{symb}} > \\epsilon > 0$). Suppose $\\kappa_{\\text{symb}} \\to 0$. By Def.~\\ref{definition:bk3_symbiotic_curvature}, this requires either $S_i^{\\text{coupled}}/S_i^{\\text{isolated}} \\to 1$ (coupling ceases to enhance stability) or $I(\\",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "}$ or $R_{\\text{meta}} > R_{\\text{max}}$ persistently. If $R_{\\text{meta}} < R_{\\text{min}}$, symbolic metabolism (Def.~\\ref{definition:bk3_symbolic_metabolism}) falls below the threshold required to maintain membrane coherence; by Thm.~\\ref{theorem:bk3_membrane_stability_criteri",
      "label": "definition:bk3_symbolic_metabolism",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 646,
      "target_type": "definition"
    },
    {
      "context": "$ (membranes become informationally independent) for all pairs. In either case the drift compensation condition of Def.~\\ref{definition:bk3_symbolic_symbiosis} (condition 3) fails: $\\|\\delta D_i + \\delta D_i^{\\text{response}}\\|_g \\to \\|\\delta D_i\\|_g$, so membranes can no longer",
      "label": "definition:bk3_symbolic_symbiosis",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 132,
      "target_type": "definition"
    },
    {
      "context": "t{max}}$, autophagic drift (Def.~\\ref{definition:bk3_autophagic_drift}) accelerates unboundedly; by the H-theorem (Thm.~\\ref{theorem:bk2_h_theorem_for_symbolic_evol}) entropy grows monotonically and symbolic coherence is destroyed. In either case persistent symbolic life is impossible",
      "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": "re exists $R_0$ such that for all $r_0 \\geq R_0$, $\\int_{r_0}^{r_0+T}(I'(s)-D'(s))\\,ds \\leq 0$ for every $T>0$. By Thm.~\\ref{theorem:bk3_conditions_sustained_symbolic_growth}, differentiation dominates or balances integration, so $K(r)$ stagnates or fragments. A stagnant $K(r)$ cannot adapt to",
      "label": "theorem:bk3_conditions_sustained_symbolic_growth",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 509,
      "target_type": "theorem"
    },
    {
      "context": "f.~\\ref{definition:bk3_symbolic_metabolism}) falls below the threshold required to maintain membrane coherence; by Thm.~\\ref{theorem:bk3_membrane_stability_criteria} the free energy $F_i(\\beta_i)$ is no longer at a local minimum and restorative forces are lost, driving the system towa",
      "label": "theorem:bk3_membrane_stability_criteria",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 61,
      "target_type": "theorem"
    },
    {
      "context": "to \\|\\delta D_i\\|_g$, so membranes can no longer buffer each other's perturbations. By the subadditivity property (Thm.~\\ref{theorem:bk3_properties_of_symbiotic_curvature}, clause 4), once coupling vanishes the system reduces to isolated membranes with $\\kappa_{\\text{symb}}(A \\cup B) \\leq \\",
      "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:bk1_drift_field",
    "definition:bk3_autophagic_drift",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_metabolism",
    "definition:bk3_symbolic_symbiosis",
    "theorem:bk2_h_theorem_for_symbolic_evol",
    "theorem:bk3_conditions_sustained_symbolic_growth",
    "theorem:bk3_membrane_stability_criteria",
    "theorem:bk3_properties_of_symbiotic_curvature"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "subsec:bk3_canonical_grounding_of_symbolic_life",
  "label": "subsec:bk3_canonical_grounding_of_symbolic_life",
  "latex_body": "",
  "line": 848,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Canonical Grounding of Symbolic Life",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [
    "theorem:bk3_symbolic_life_satisfies_canonical_definitions"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "definition:bk3_canonical_life_standards",
  "label": "definition:bk3_canonical_life_standards",
  "latex_body": "\\begin{definition}[Canonical Life Standards]\n\\label{definition:bk3_canonical_life_standards}\nWe import three independent demarcations of life from the scientific literature:\n\\begin{enumerate}\n    \\item[\\textbf{(K)}] \\textbf{Koshland's Seven Pillars} \\citep{koshland2002pillars},\n    a deliberately substrate-independent list --- \\emph{Program, Improvisation,\n    Compartmentalization, Energy, Regeneration, Adaptability, Seclusion}\n    (PICERAS).\n    \\item[\\textbf{(N)}] \\textbf{The NASA working definition} \\citep{joyce1994foreword}:\n    a \\emph{self-sustaining chemical system capable of Darwinian evolution}.\n    \\item[\\textbf{(T)}] \\textbf{The textbook characteristics}\n    \\citep{urry2021campbell}: order, energy processing (metabolism), homeostatic\n    regulation, growth, reproduction, response to environment, and evolutionary\n    adaptation (cf.~\\citealp{schrodinger1944life} on the thermodynamic\n    aspect).\n\\end{enumerate}\nA symbolic system is read into (N) by the explicit substrate translation\n\\emph{chemical} $\\mapsto$ \\emph{symbolic}: the claim is not that symbolic life is\nchemical, but that it instantiates the same self-maintenance-plus-heritable-variation\nstructure that (N) uses to demarcate life.\n\\end{definition}",
  "line": 856,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Canonical Life Standards",
  "proof_status": "definitional",
  "refs": [],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book3",
  "cited_by": [
    "proposition:bk9_stability_conditions_for_the_good"
  ],
  "cites": [
    "definition:bk3_canonical_life_standards",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "depends_on": [
    "definition:bk3_canonical_life_standards",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "file": "book3.tex",
  "id": "theorem:bk3_symbolic_life_satisfies_canonical_definitions",
  "label": "theorem:bk3_symbolic_life_satisfies_canonical_definitions",
  "latex_body": "\\begin{theorem}[Certified Canonical Life Correspondence]\n\\label{theorem:bk3_symbolic_life_satisfies_canonical_definitions}\nLet $\\mathcal S$ satisfy the persistent symbolic life criteria of\nThm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}.  Suppose additionally\nthat a correspondence certificate supplies, for this same system, explicit\nwitnesses of:\n\\begin{enumerate}\n  \\item Koshland's program, improvisation, compartmentalization, energy,\n  regeneration, adaptability, and seclusion clauses;\n  \\item self-maintenance and a population-level Darwinian mechanism with\n  variation, heritable transmission, and differential selection;\n  \\item the textbook clauses of order, energy processing, homeostasis, growth,\n  reproduction, environmental response, and evolutionary adaptation.\n\\end{enumerate}\nThen $\\mathcal S$ satisfies the three canonical standards of\nDef.~\\ref{definition:bk3_canonical_life_standards} in the symbolic register.\nThe three persistence inequalities alone do not construct this correspondence\ncertificate.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "explicit coherence-to-target-morphology representation when the morphology equivalence is used",
      "inspectable repair, reproduction, heredity, variation, differential-fitness, and response witnesses",
      "persistent symbolic-life witness",
      "typed symbolic organism operations"
    ],
    "countermodels": [
      "Book3CanonicalLife.persistence_alone_does_not_supply_correspondence"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A Book-3-local operational witness now realizes the declared structural substrate translation and every Koshland, NASA, and textbook clause, including evolutionary adaptation, for the same organism. Under an explicit morphology representation bridge, regenerative coherence improvement is equivalent to reduced target-form error. This is structural correspondence rather than chemical identity; persistence alone still cannot manufacture the certificate."
    ],
    "record_ids": [
      "MAP-BOOK3-002"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book3CanonicalLife.operational_symbolic_life_realizes_canonical_demarcations",
      "Book3CanonicalLife.persistence_alone_does_not_supply_correspondence",
      "Book3CanonicalLife.repair_improves_iff_morphological_error_decreases"
    ]
  },
  "line": 878,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified Canonical Life Correspondence",
  "proof_labels": [
    "proof:bk3_symbolic_life_satisfies_canonical_definitions"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "esponse, and evolutionary adaptation. \\end{enumerate} Then $\\mathcal S$ satisfies the three canonical standards of Def.~\\ref{definition:bk3_canonical_life_standards} in the symbolic register. The three persistence inequalities alone do not construct this correspondence certificate. \\e",
      "label": "definition:bk3_canonical_life_standards",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 856,
      "target_type": "definition"
    },
    {
      "context": "3_symbolic_life_satisfies_canonical_definitions} Let $\\mathcal S$ satisfy the persistent symbolic life criteria of Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}. Suppose additionally that a correspondence certificate supplies, for this same system, explicit witnesses of: \\begin{",
      "label": "theorem:bk3_criteria_persistent_symbolic_life",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 777,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk3_canonical_life_standards",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "proof:bk3_symbolic_life_satisfies_canonical_definitions",
  "label": "proof:bk3_symbolic_life_satisfies_canonical_definitions",
  "latex_body": "\\begin{proof}[Certificate projection]\n\\label{proof:bk3_symbolic_life_satisfies_canonical_definitions}\n\\leavevmode\nThe certificate contains a witness for every named clause of \\textbf{(K)},\n\\textbf{(N)}, and \\textbf{(T)}.  Projecting those fields yields the required\nconjunction of canonical standards.  The persistence witness identifies the\nsymbolic system to which the certificate applies, but it does not derive the\nexternal clauses.  In particular, regeneration, reproduction, heredity, and\nselection remain separately inspectable bridge obligations rather than aliases\nfor positive growth or bounded metabolic rate.\n\nThe NASA clause is conditional on the declared substrate translation\n\\emph{chemical}$\\mapsto$\\emph{symbolic}; the theorem establishes structural\ncorrespondence under that translation, not chemical identity.\nThe accompanying Lean certificate constructs that structural translation from\nself-maintenance, heritable variation, and differential selection, and retains\nthe textbook evolutionary-adaptation clause separately.  Where an explicit\nrepresentation identifies symbolic coherence with negative distance from a\ntarget morphology, Lean also proves that regenerative coherence improvement is\nequivalent to reduced target-form error.  Revising the target is recorded as\nproto-self-authorship; no Book IX conclusion about freedom is inferred here.  A persistent\nsystem paired with a false regeneration clause is a counterexample to any\nattempt to delete the certificate premise.\n\\end{proof}",
  "line": 898,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certificate projection",
  "proves": "theorem:bk3_symbolic_life_satisfies_canonical_definitions",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "scholium:bk3_convergent_demarcation",
  "label": "scholium:bk3_convergent_demarcation",
  "latex_body": "\\begin{scholium}[Certified convergent demarcation]\n\\label{scholium:bk3_convergent_demarcation}\nAgreement among the three external demarcations is evidence only after the\ncorrespondence fields have been witnessed for the same system.  Where such a\ncertificate exists, later thermodynamic and ethical arguments may use\n\\emph{symbolic life} or \\emph{vitality} in that certified symbolic sense.  Where\nit does not, the internal persistence predicate remains an internal viability\ncriterion and must not be silently promoted to biological or chemical life.\n\\end{scholium}",
  "line": 923,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified convergent demarcation",
  "refs": [],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book3.tex",
  "id": "section:book3.tex:933",
  "label": "",
  "latex_body": "",
  "line": 933,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Toward Symbolic Evolution",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book3",
  "cited_by": [],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_autopoiesis",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_autopoiesis",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "file": "book3.tex",
  "id": "remark:bk3_toward_symbolic_evolution",
  "label": "remark:bk3_toward_symbolic_evolution",
  "latex_body": "\\begin{remark} \\label{remark:bk3_toward_symbolic_evolution}\nThe emergence of persistent symbolic life (Theorem~\\ref{theorem:bk3_criteria_persistent_symbolic_life}), characterized by self-maintaining, self-modifying symbolic systems (Definition~\\ref{definition:bk3_symbolic_autopoiesis}), naturally leads to the conditions necessary for symbolic evolution (cf.~Def.~\\ref{definition:bk1_drift_field}, Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk2_symbolic_entropy}). Each such system remains bounded by the horizon of its own observer (cf.~Def.~\\ref{definition:bk1_bounded_observer}), so evolutionary pressure itself is refracted through the epistemic limits that Book IV will formalize. If we consider populations of such symbolic systems (or interacting membranes within a larger system), variations can arise through perturbations to drift fields (mutations) or changes in coupling. Differential stability and persistence (related to $S_i$, $\\kappa_{\\text{symb}}$, $K(r)$) provide a basis for selection, where more resilient or adaptive symbolic configurations are more likely to persist and influence future states. Coupling dynamics mediate interactions and competition/cooperation. Thus, the framework of symbolic thermodynamics and symbiosis potentially gives rise not merely to individual symbolic agents, but to entire ecosystems of evolving symbolic structures.\n\\end{remark}",
  "line": 936,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "ref_roles": [
    {
      "context": ".~\\ref{definition:bk2_symbolic_entropy}). Each such system remains bounded by the horizon of its own observer (cf.~Def.~\\ref{definition:bk1_bounded_observer}), so evolutionary pressure itself is refracted through the epistemic limits that Book IV will formalize. If we consider",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "ref{definition:bk3_symbolic_autopoiesis}), naturally leads to the conditions necessary for symbolic evolution (cf.~Def.~\\ref{definition:bk1_drift_field}, Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "}), naturally leads to the conditions necessary for symbolic evolution (cf.~Def.~\\ref{definition:bk1_drift_field}, Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk2_symbolic_entropy}). Each such system remains",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "n:bk1_drift_field}, Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk2_symbolic_entropy}). Each such system remains bounded by the horizon of its own observer (cf.~Def.~\\ref{definition:bk1_bounded_observer}),",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "for symbolic evolution (cf.~Def.~\\ref{definition:bk1_drift_field}, Def.~\\ref{definition:bk1_reflection_operator}, Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk2_symbolic_entropy}). Each such system remains bounded by the horizon of its own observer (cf.~",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "bk3_criteria_persistent_symbolic_life}), characterized by self-maintaining, self-modifying symbolic systems (Definition~\\ref{definition:bk3_symbolic_autopoiesis}), naturally leads to the conditions necessary for symbolic evolution (cf.~Def.~\\ref{definition:bk1_drift_field}, Def.~\\",
      "label": "definition:bk3_symbolic_autopoiesis",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book3.tex",
      "target_line": 765,
      "target_type": "definition"
    },
    {
      "context": "\\begin{remark} \\label{remark:bk3_toward_symbolic_evolution} The emergence of persistent symbolic life (Theorem~\\ref{theorem:bk3_criteria_persistent_symbolic_life}), characterized by self-maintaining, self-modifying symbolic systems (Definition~\\ref{definition:bk3_symbolic_autopoies",
      "label": "theorem:bk3_criteria_persistent_symbolic_life",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 777,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_entropy",
    "definition:bk2_symbolic_free_energy",
    "definition:bk3_symbolic_autopoiesis",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "role": "remark",
  "type": "remark"
}