Book IV — De Identitate Symbolica et Emergentia — page 5 of 9

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    "definition:bk4_recursive_self_healing",
    "definition:bk4_repair_process",
    "definition:bk6_fragmentation_functional",
    "lemma:bk4_fragmentation_cascade",
    "proof:bk4_fragmentation_identity_stability",
    "proof:bk4_symbolic_curvature_fragmentation",
    "proof:bk9_pathologies_of_coherence",
    "proof:bk9_symbolic_masking_and_unmasking",
    "proposition:bk5_complementary_constants",
    "theorem:bk4_conditions_for_self_healing",
    "theorem:bk4_drift_reflection_imbalance",
    "theorem:bk4_reflective_reentry"
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  "label": "definition:bk4_fragmented_identity",
  "latex_body": "\\begin{definition}[Fragmented Identity] \\label{definition:bk4_fragmented_identity}\nA symbolic identity carrier $\\mathcal{I}$ on a membrane $M_i$ is \\textit{fragmented} at symbolic time $t$ if there exists a partition $\\{U_j\\}_{j=1}^k$ of $M_i$ such that:\n\\begin{enumerate}\n    \\item For each region $U_j$, the local symbolic pattern $\\Psi_i|_{U_j}$ is internally coherent (see Def.~\\ref{definition:bk4_symbolic_identity_carrie})\n    \\item The stability functional exhibits discontinuity across region boundaries:\n    \\begin{equation}\n        \\Upsilon_i(\\Psi_i|_{U_j}, \\Psi_i|_{U_l}) < \\epsilon_{\\text{coh}} \\quad \\text{for } j \\neq l\n    \\end{equation}\n    \\item The temporal tracking relation $\\mathcal{T}_{\\Delta t}$ fails to establish consistent correspondence:\n    \\begin{equation}\n        \\Upsilon_i(\\Psi_i(t)|_{U_j}, \\Psi_i(t+\\Delta t)|_{U_j}) < 1 - \\epsilon_{\\text{crit}}\n    \\end{equation}\n\\end{enumerate}\nwhere $\\epsilon_{\\text{coh}}$ is a coherence threshold and $\\epsilon_{\\text{crit}}$ is the critical error bound from Theorem~\\ref{theorem:bk4_existence_of_symbolic_ident} (see also Def.~\\ref{definition:bk3_symbolic_membrane}).\n\\end{definition}",
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      "context": "on_{\\text{crit}}$ is the critical error bound from Theorem~\\ref{theorem:bk4_existence_of_symbolic_ident} (see also Def.~\\ref{definition:bk3_symbolic_membrane}). \\end{definition}",
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      "context": "enumerate} \\item For each region $U_j$, the local symbolic pattern $\\Psi_i|_{U_j}$ is internally coherent (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) \\item The stability functional exhibits discontinuity across region boundaries: \\begin{equation} \\Upsi",
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      "context": "$\\epsilon_{\\text{coh}}$ is a coherence threshold and $\\epsilon_{\\text{crit}}$ is the critical error bound from Theorem~\\ref{theorem:bk4_existence_of_symbolic_ident} (see also Def.~\\ref{definition:bk3_symbolic_membrane}). \\end{definition}",
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    "definition:bk4_repair_process",
    "proof:bk4_fragmentation_distortion_encoding",
    "proof:bk4_freedom_growth_fragmentation",
    "proof:bk4_repair_reconnects_fragmentation",
    "proof:bk9_meta_reflective_memory_integration",
    "theorem:bk4_freedom_life_connection"
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  "latex_body": "\\begin{definition}[Fragmentation Measure] \\label{definition:bk4_fragmentation_measure}\nThe fragmentation measure $\\mathcal{F}_{\\text{frag}}$ of a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) is defined as:\n\\begin{equation}\n    \\mathcal{F}_{\\text{frag}}(\\mathcal{I}) = 1 - \\frac{I(\\{U_j\\}_{j=1}^k; \\Psi_i)}{H(\\{U_j\\}_{j=1}^k)}\n\\end{equation}\nwhere $I(\\cdot;\\cdot)$ denotes mutual information, $H(\\cdot)$ is entropy (see Def.~\\ref{definition:bk2_symbolic_entropy}), and $\\{U_j\\}_{j=1}^k$ is the optimal partition of the membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}) that maximizes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}).\n\\end{definition}",
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    {
      "context": "}{H(\\{U_j\\}_{j=1}^k)} \\end{equation} where $I(\\cdot;\\cdot)$ denotes mutual information, $H(\\cdot)$ is entropy (see Def.~\\ref{definition:bk2_symbolic_entropy}), and $\\{U_j\\}_{j=1}^k$ is the optimal partition of the membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}",
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      "context": "~\\ref{definition:bk2_symbolic_entropy}), and $\\{U_j\\}_{j=1}^k$ is the optimal partition of the membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}) that maximizes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}). \\end{definition}",
      "label": "definition:bk3_symbolic_membrane",
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      "context": "artition of the membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}) that maximizes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}). \\end{definition}",
      "label": "definition:bk4_fragmented_identity",
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      "context": "mentation_measure} The fragmentation measure $\\mathcal{F}_{\\text{frag}}$ of a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) is defined as: \\begin{equation} \\mathcal{F}_{\\text{frag}}(\\mathcal{I}) = 1 - \\frac{I(\\{U_j\\}_{j=1}^k; \\Psi_i)}{H(\\",
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  "latex_body": "\\begin{theorem}[Drift-Reflection Imbalance] \\label{theorem:bk4_drift_reflection_imbalance}\nA symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) undergoes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}) if and only if there exists a subset $U \\subset M_i$ of the symbolic membrane (see Def.~\\ref{definition:bk3_symbolic_membrane}) where the symbolic drift field $D_i$ overcomes the reflective stabilization field $R_i$ (see Def.~\\ref{definition:bk1_reflection_operator}):\n\\begin{equation}\n    \\|D_i(x,t)\\|_g > \\theta \\cdot \\|R_i(x,t)\\|_g \\quad \\text{for all } x \\in U\n\\end{equation}\nwhere $\\theta > 1$ is a symbolic imbalance parameter and $\\|\\cdot\\|_g$ is the norm induced by the Riemannian metric $g$, and the probability space of symbolic events is induced over $M_i$ (see Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{theorem}",
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  "name": "Drift-Reflection Imbalance",
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    {
      "context": "3_symbolic_membrane}) where the symbolic drift field $D_i$ overcomes the reflective stabilization field $R_i$ (see Def.~\\ref{definition:bk1_reflection_operator}): \\begin{equation} \\|D_i(x,t)\\|_g > \\theta \\cdot \\|R_i(x,t)\\|_g \\quad \\text{for all } x \\in U \\end{equation} where",
      "label": "definition:bk1_reflection_operator",
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      "context": "norm induced by the Riemannian metric $g$, and the probability space of symbolic events is induced over $M_i$ (see Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{theorem}",
      "label": "definition:bk2_symbolic_probability_spa",
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      "target_line": 23,
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      "context": "ition:bk4_fragmented_identity}) if and only if there exists a subset $U \\subset M_i$ of the symbolic membrane (see Def.~\\ref{definition:bk3_symbolic_membrane}) where the symbolic drift field $D_i$ overcomes the reflective stabilization field $R_i$ (see Def.~\\ref{definition:bk1_",
      "label": "definition:bk3_symbolic_membrane",
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      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "bolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) undergoes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}) if and only if there exists a subset $U \\subset M_i$ of the symbolic membrane (see Def.~\\ref{definition:bk3_symbolic_m",
      "label": "definition:bk4_fragmented_identity",
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      "role": "definition_anchor",
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      "target_line": 2731,
      "target_type": "definition"
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    {
      "context": "[Drift-Reflection Imbalance] \\label{theorem:bk4_drift_reflection_imbalance} A symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) undergoes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}) if and only if there exists a subset $U \\s",
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  "latex_body": "\\begin{remark}[Finite witness for imbalance]\n\\label{remark:bk4_finite_witness_for_drift_reflection_imbalance}\nThe imbalance condition is not vacuous.  In the minimal linear witness\n(Thm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model}), drift and\nstate-level stabilization are represented by \\(J\\) and \\(P\\), with\n\\(JP\\ne PJ\\).  Thus even the smallest typed realization already contains an\norder-sensitive drift--reflection defect; Book~IV studies how such defects\nscale from a finite witness into identity fragmentation on symbolic membranes.\nThe scaling claim is a certified operator transport\n(Def.~\\ref{definition:bk1_certified_type_preserving_symbolic_transport}): the\nrole preserved is noncommuting drift--reflection order, not numerical equality\nwith the two-dimensional matrices (Props.~\\ref{proposition:bk1_certified_transport_prevents_equivocation} and \\ref{proposition:bk1_nonvacuity_of_certified_transport}).\n\\end{remark}",
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  "name": "Finite witness for imbalance",
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      "context": "te witness into identity fragmentation on symbolic membranes. The scaling claim is a certified operator transport (Def.~\\ref{definition:bk1_certified_type_preserving_symbolic_transport}): the role preserved is noncommuting drift--reflection order, not numerical equality with the two-dimensional matrices",
      "label": "definition:bk1_certified_type_preserving_symbolic_transport",
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      "role": "definition_anchor",
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      "target_line": 3456,
      "target_type": "definition"
    },
    {
      "context": "ole preserved is noncommuting drift--reflection order, not numerical equality with the two-dimensional matrices (Props.~\\ref{proposition:bk1_certified_transport_prevents_equivocation} and \\ref{proposition:bk1_nonvacuity_of_certified_transport}). \\end{remark}",
      "label": "proposition:bk1_certified_transport_prevents_equivocation",
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      "target_file": "scholium_symbolicum.tex",
      "target_line": 3496,
      "target_type": "proposition"
    },
    {
      "context": "equality with the two-dimensional matrices (Props.~\\ref{proposition:bk1_certified_transport_prevents_equivocation} and \\ref{proposition:bk1_nonvacuity_of_certified_transport}). \\end{remark}",
      "label": "proposition:bk1_nonvacuity_of_certified_transport",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3533,
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    },
    {
      "context": "te_witness_for_drift_reflection_imbalance} The imbalance condition is not vacuous. In the minimal linear witness (Thm.~\\ref{theorem:bk1_nonvacuity_minimal_linear_ps_model}), drift and state-level stabilization are represented by \\(J\\) and \\(P\\), with \\(JP\\ne PJ\\). Thus even the smallest ty",
      "label": "theorem:bk1_nonvacuity_minimal_linear_ps_model",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3364,
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    "proposition:bk1_certified_transport_prevents_equivocation",
    "proposition:bk1_nonvacuity_of_certified_transport",
    "theorem:bk1_nonvacuity_minimal_linear_ps_model"
  ],
  "role": "remark",
  "type": "remark"
}

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    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_drift_reflection_imbalance",
    "theorem:bk4_existence_of_symbolic_ident"
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  "id": "proof:bk4_fragmentation_identity_stability",
  "label": "proof:bk4_fragmentation_identity_stability",
  "latex_body": "\\begin{proof}[Fragmentation Violates Symbolic Identity Stability]\n\\label{proof:bk4_fragmentation_identity_stability}\n\\leavevmode\n\n($\\Rightarrow$) If identity fragmentation occurs according to Def.~\\ref{definition:bk4_fragmented_identity}, then by the existence condition for identity carriers (see Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}), the stability condition \n\\[\n\\Upsilon_i(\\Psi_i(t), \\Psi_i(t+\\Delta t)) \\geq 1 - \\epsilon(t)\n\\]\nis violated in some region $U$ of the membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}).\n\nFrom Book I, symbolic stability is governed by the interplay between drift $D_i$ and reflection $R_i$ (see Def.~\\ref{definition:bk1_reflection_operator}). The stability functional $\\Upsilon_i$---a key aspect of symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie})---can be expressed as:\n\\[\n\\Upsilon_i(\\Psi_i(t), \\Psi_i(t+\\Delta t)) \\approx 1 - \\alpha \\int_U \\frac{\\|D_i(x,t)\\|_g}{\\|R_i(x,t)\\|_g} d\\mu_g(x)\n\\]\nwhere $\\alpha > 0$ and the symbolic space is endowed with a probabilistic structure (see Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n\nFor stability violation, we require:\n\\[\n\\alpha \\int_U \\frac{\\|D_i(x,t)\\|_g}{\\|R_i(x,t)\\|_g} d\\mu_g(x) > \\epsilon_{\\text{crit}}\n\\]\nThis holds precisely under the condition described in Thm.~\\ref{theorem:bk4_drift_reflection_imbalance}, where $\\|D_i(x,t)\\|_g > \\theta \\cdot \\|R_i(x,t)\\|_g$ for all $x \\in U$ and some $\\theta > 1$.\n\n($\\Leftarrow$) Conversely, if the drift field dominates the reflection field in region $U$, then the symbolic flow will increasingly distort the identity pattern $\\Psi_i$ in that region. Over time, this distortion exceeds the critical threshold $\\epsilon_{\\text{crit}}$, disrupting the temporal tracking relation and thus resulting in fragmentation by Def.~\\ref{definition:bk4_fragmented_identity}.\n\n\\end{proof}",
  "line": 2773,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Fragmentation Violates Symbolic Identity Stability",
  "proves": "theorem:bk4_drift_reflection_imbalance",
  "ref_roles": [
    {
      "context": "ane}). From Book I, symbolic stability is governed by the interplay between drift $D_i$ and reflection $R_i$ (see Def.~\\ref{definition:bk1_reflection_operator}). The stability functional $\\Upsilon_i$---a key aspect of symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "_i(x,t)\\|_g} d\\mu_g(x) \\] where $\\alpha > 0$ and the symbolic space is endowed with a probabilistic structure (see Def.~\\ref{definition:bk2_symbolic_probability_spa}). For stability violation, we require: \\[ \\alpha \\int_U \\frac{\\|D_i(x,t)\\|_g}{\\|R_i(x,t)\\|_g} d\\mu_g(x) > \\epsilon_{\\t",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "i(\\Psi_i(t), \\Psi_i(t+\\Delta t)) \\geq 1 - \\epsilon(t) \\] is violated in some region $U$ of the membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}). From Book I, symbolic stability is governed by the interplay between drift $D_i$ and reflection $R_i$ (see Def.~\\ref",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "f:bk4_fragmentation_identity_stability} \\leavevmode ($\\Rightarrow$) If identity fragmentation occurs according to Def.~\\ref{definition:bk4_fragmented_identity}, then by the existence condition for identity carriers (see Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}), the st",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "eflection_operator}). The stability functional $\\Upsilon_i$---a key aspect of symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie})---can be expressed as: \\[ \\Upsilon_i(\\Psi_i(t), \\Psi_i(t+\\Delta t)) \\approx 1 - \\alpha \\int_U \\frac{\\|D_i(x,t)\\|_g}{\\|",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    },
    {
      "context": ")\\|_g}{\\|R_i(x,t)\\|_g} d\\mu_g(x) > \\epsilon_{\\text{crit}} \\] This holds precisely under the condition described in Thm.~\\ref{theorem:bk4_drift_reflection_imbalance}, where $\\|D_i(x,t)\\|_g > \\theta \\cdot \\|R_i(x,t)\\|_g$ for all $x \\in U$ and some $\\theta > 1$. ($\\Leftarrow$) Converse",
      "label": "theorem:bk4_drift_reflection_imbalance",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 2753,
      "target_type": "theorem"
    },
    {
      "context": "rding to Def.~\\ref{definition:bk4_fragmented_identity}, then by the existence condition for identity carriers (see Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}), the stability condition \\[ \\Upsilon_i(\\Psi_i(t), \\Psi_i(t+\\Delta t)) \\geq 1 - \\epsilon(t) \\] is violated in some reg",
      "label": "theorem:bk4_existence_of_symbolic_ident",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 15,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_drift_reflection_imbalance",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
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  ],
  "file": "book4.tex",
  "id": "definition:bk4_critical_symbolic_bifurc",
  "label": "definition:bk4_critical_symbolic_bifurc",
  "latex_body": "\\begin{definition}[Critical Symbolic Bifurcation] \\label{definition:bk4_critical_symbolic_bifurc}\nA critical symbolic bifurcation occurs when a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) transitions from a coherent to a fragmented state (see Def.~\\ref{definition:bk4_fragmented_identity}) due to a qualitative change in the dynamics of its underlying membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}).\n\nThis transition is marked by a divergence in the rate of change of the fragmentation measure (see Def.~\\ref{definition:bk4_fragmentation_measure}):\n\\begin{equation}\n    \\frac{d\\mathcal{F}_{\\text{frag}}(\\mathcal{I})}{dt}\\Big|_{t=t_c} = \\infty\n\\end{equation}\nwhere $t_c$ is the critical time of bifurcation.\n\\end{definition}",
  "line": 2798,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Critical Symbolic Bifurcation",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "nition:bk4_fragmented_identity}) due to a qualitative change in the dynamics of its underlying membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}). This transition is marked by a divergence in the rate of change of the fragmentation measure (see Def.~\\ref{definiti",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "lic_membrane}). This transition is marked by a divergence in the rate of change of the fragmentation measure (see Def.~\\ref{definition:bk4_fragmentation_measure}): \\begin{equation} \\frac{d\\mathcal{F}_{\\text{frag}}(\\mathcal{I})}{dt}\\Big|_{t=t_c} = \\infty \\end{equation} where $t",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) transitions from a coherent to a fragmented state (see Def.~\\ref{definition:bk4_fragmented_identity}) due to a qualitative change in the dynamics of its underlying membrane $M_i$ (see Def.~\\ref{definition:bk3_symbolic_me",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "n:bk4_critical_symbolic_bifurc} A critical symbolic bifurcation occurs when a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) transitions from a coherent to a fragmented state (see Def.~\\ref{definition:bk4_fragmented_identity}) due to a qualita",
      "label": "definition:bk4_symbolic_identity_carrie",
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    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "proof:bk4_symbolic_curvature_fragmentation"
  ],
  "cites": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "file": "book4.tex",
  "id": "lemma:bk4_fragmentation_cascade",
  "label": "lemma:bk4_fragmentation_cascade",
  "latex_body": "\\begin{lemma}[Fragmentation Cascade] \\label{lemma:bk4_fragmentation_cascade}\nIf a symbolic identity (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) undergoes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}) in a region $U_1 \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}), and the coupling strength between regions exceeds a critical threshold $\\gamma_{\\text{crit}}$, then fragmentation propagates to adjacent regions with probability:\n\\begin{equation}\n    P(\\text{propagation to } U_2) = 1 - \\exp\\left(-\\beta \\int_{U_1 \\times U_2} \\kappa_{\\text{symb}}(x,y) \\, d\\mu_g(x) \\, d\\mu_g(y)\\right)\n\\end{equation}\nwhere $\\kappa_{\\text{symb}}$ is the symbolic curvature (see Def.~\\ref{definition:bk3_symbiotic_curvature}) and $\\beta > 0$ is a scaling constant. This probability depends on the symbolic probability structure across regions (see Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The propagation-probability formula 1-exp(-beta*kappa) is kept exactly, with the manifold-valued curvature integral kappa erased to an arbitrary nonnegative real; probability in [0,1) and strict monotonicity in kappa are proved."
    ],
    "record_ids": [
      "MAP-BOOK4B-004"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book4B.cascadeProb_lt_one",
      "Book4B.cascadeProb_nonneg",
      "Book4B.cascadeProb_strictMono"
    ]
  },
  "line": 2807,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Fragmentation Cascade",
  "proof_labels": [
    "proof:bk4_symbolic_curvature_fragmentation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "eta > 0$ is a scaling constant. This probability depends on the symbolic probability structure across regions (see Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{lemma}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "}(x,y) \\, d\\mu_g(x) \\, d\\mu_g(y)\\right) \\end{equation} where $\\kappa_{\\text{symb}}$ is the symbolic curvature (see Def.~\\ref{definition:bk3_symbiotic_curvature}) and $\\beta > 0$ is a scaling constant. This probability depends on the symbolic probability structure across regions (",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "e}) undergoes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}) in a region $U_1 \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}), and the coupling strength between regions exceeds a critical threshold $\\gamma_{\\text{crit}}$, then fragmentation pro",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "cade} If a symbolic identity (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) undergoes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}) in a region $U_1 \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}), and the coupling strength between reg",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "\\begin{lemma}[Fragmentation Cascade] \\label{lemma:bk4_fragmentation_cascade} If a symbolic identity (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) undergoes fragmentation (see Def.~\\ref{definition:bk4_fragmented_identity}) in a region $U_1 \\subset M_i$ (see Def.~\\r",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "lemma:bk4_fragmentation_cascade"
  ],
  "depends_on": [
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    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "lemma:bk4_fragmentation_cascade"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_symbolic_curvature_fragmentation",
  "label": "proof:bk4_symbolic_curvature_fragmentation",
  "latex_body": "\\begin{proof}[Curvature and Fragmentation]\n\\label{proof:bk4_symbolic_curvature_fragmentation}\n\\leavevmode\n\nSymbolic curvature $\\kappa_{\\text{symb}}(x,y)$\n(Def.~\\ref{definition:bk3_symbiotic_curvature}) quantifies coupling between\nmembrane points $x$ and $y$ (Def.~\\ref{definition:bk3_symbolic_membrane}).\nWhen fragmentation occurs in region $U_1$\n(Def.~\\ref{definition:bk4_fragmented_identity}), its distortions propagate\nthrough that coupling.\n\nThe double integral $\\int_{U_1 \\times U_2} \\kappa_{\\text{symb}}(x,y) \\, d\\mu_g(x) \\, d\\mu_g(y)$ computes the total coupling between regions $U_1$ and $U_2$. When this coupling exceeds the critical threshold $\\gamma_{\\text{crit}}$, the distortion propagates to $U_2$ with high probability, as formalized in Lemma~\\ref{lemma:bk4_fragmentation_cascade}.\n\nThe exponential form of the propagation probability derives from modeling the fragmentation dynamics as a continuous-time Markov process with a transition rate governed by symbolic interaction intensity. The underlying probability measure (see Def.~\\ref{definition:bk2_symbolic_probability_spa}) ensures the proper weighting of symbolic interactions.\n\\end{proof}",
  "line": 2814,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Curvature and Fragmentation",
  "proves": "lemma:bk4_fragmentation_cascade",
  "ref_roles": [
    {
      "context": "process with a transition rate governed by symbolic interaction intensity. The underlying probability measure (see Def.~\\ref{definition:bk2_symbolic_probability_spa}) ensures the proper weighting of symbolic interactions. \\end{proof}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "n] \\label{proof:bk4_symbolic_curvature_fragmentation} \\leavevmode Symbolic curvature $\\kappa_{\\text{symb}}(x,y)$ (Def.~\\ref{definition:bk3_symbiotic_curvature}) quantifies coupling between membrane points $x$ and $y$ (Def.~\\ref{definition:bk3_symbolic_membrane}). When fragmentat",
      "label": "definition:bk3_symbiotic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 252,
      "target_type": "definition"
    },
    {
      "context": "b}}(x,y)$ (Def.~\\ref{definition:bk3_symbiotic_curvature}) quantifies coupling between membrane points $x$ and $y$ (Def.~\\ref{definition:bk3_symbolic_membrane}). When fragmentation occurs in region $U_1$ (Def.~\\ref{definition:bk4_fragmented_identity}), its distortions propagate",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "brane points $x$ and $y$ (Def.~\\ref{definition:bk3_symbolic_membrane}). When fragmentation occurs in region $U_1$ (Def.~\\ref{definition:bk4_fragmented_identity}), its distortions propagate through that coupling. The double integral $\\int_{U_1 \\times U_2} \\kappa_{\\text{symb}}(x,y",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "ical threshold $\\gamma_{\\text{crit}}$, the distortion propagates to $U_2$ with high probability, as formalized in Lemma~\\ref{lemma:bk4_fragmentation_cascade}. The exponential form of the propagation probability derives from modeling the fragmentation dynamics as a continuous-",
      "label": "lemma:bk4_fragmentation_cascade",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 2807,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbiotic_curvature",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "lemma:bk4_fragmentation_cascade"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book4.tex",
  "id": "subsec:bk4_mechanisms_identity_repair",
  "label": "subsec:bk4_mechanisms_identity_repair",
  "latex_body": "",
  "line": 2829,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Mechanisms of Identity Repair",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book4",
  "cited_by": [
    "definition:bk4_proto_vitality",
    "definition:bk4_recursive_self_healing",
    "definition:bk4_repair_capacity",
    "proof:bk4_repair_reconnects_fragmentation",
    "proof:bk9_betrayal_and_recovery",
    "proof:bk9_meta_reflective_memory_integration",
    "proof:bk9_pathologies_of_coherence",
    "proof:bk9_symbolic_viability",
    "proposition:bk9_criteria_for_ethical_intervention",
    "proposition:bk9_curvature_scarring"
  ],
  "cites": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_fragmented_identity"
  ],
  "depends_on": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_fragmented_identity"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_repair_process",
  "label": "definition:bk4_repair_process",
  "latex_body": "\\begin{definition}[Repair Process] \\label{definition:bk4_repair_process}\nA repair process $\\mathcal{R}_{\\text{rep}}$ for a fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) $\\mathcal{I}$ is a dynamical evolution that increases symbolic coherence:\n\\begin{equation}\n    \\mathcal{R}_{\\text{rep}}: \\mathcal{I}_{\\text{frag}} \\to \\mathcal{I}_{\\text{coh}}\n\\end{equation}\nsuch that:\n\\begin{equation}\n    \\mathcal{F}_{\\text{frag}}(\\mathcal{R}_{\\text{rep}}(\\mathcal{I}_{\\text{frag}})) < \\mathcal{F}_{\\text{frag}}(\\mathcal{I}_{\\text{frag}}) \\quad \\text{(see Def.~\\ref{definition:bk4_fragmentation_measure})}\n\\end{equation}\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Modeled as a discrete fragmentation-measure sequence decreasing by a fixed positive amount each step, with telescoping and termination consequences (same finite kernel shape as Book8's MetabolicSufficiency, applied to fragmentation instead of free energy)."
    ],
    "record_ids": [
      "MAP-BOOK4B-005"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "Book4B.repairSufficiency_decrease_accum",
      "Book4B.repairSufficiency_terminates"
    ]
  },
  "line": 2830,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Repair Process",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "}_{\\text{rep}}(\\mathcal{I}_{\\text{frag}})) < \\mathcal{F}_{\\text{frag}}(\\mathcal{I}_{\\text{frag}}) \\quad \\text{(see Def.~\\ref{definition:bk4_fragmentation_measure})} \\end{equation} \\end{definition}",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "] \\label{definition:bk4_repair_process} A repair process $\\mathcal{R}_{\\text{rep}}$ for a fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) $\\mathcal{I}$ is a dynamical evolution that increases symbolic coherence: \\begin{equation} \\mathcal{R}_{\\text{rep}",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_fragmented_identity"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "definition:bk9_two_way_street_operator",
    "proof:bk4_repair_reconnects_fragmentation",
    "proof:bk9_meta_reflective_memory_integration",
    "proof:bk9_pathologies_of_coherence",
    "proposition:bk9_mechanisms_of_recognition",
    "scholium:bk7_unnamed_scholium_03",
    "sec:appD_dialogue_titans",
    "sec:bk7_reflection_integration_link_revisited",
    "sec:bk7_reflective_fixed_point_theorem",
    "subsec:appD_titans_resonance",
    "subsec:bk7_motivation"
  ],
  "cites": [
    "definition:bk1_reflection_operator",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "depends_on": [
    "definition:bk1_reflection_operator",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_fragmented_identity",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "file": "book4.tex",
  "id": "theorem:bk4_reflective_reentry",
  "label": "theorem:bk4_reflective_reentry",
  "latex_body": "\\begin{theorem}[Reflective Reentry] \\label{theorem:bk4_reflective_reentry}\nLet $\\mathcal{I}$ be a fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) with symbolic pattern $\\Psi_i$ \n(see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) exhibiting decoherence on region $U \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}). \nA repair trajectory exists if and only if there exists a time-evolved reflection operator $\\widehat{R}_t$ (see Def.~\\ref{definition:bk1_reflection_operator}) such that:\n\\begin{equation}\n    \\Upsilon_i(\\Psi_i(t_0), \\widehat{R}_t \\circ \\Psi_i(t_1)) \\geq \\eta\n\\end{equation}\nfor some $t_1 > t_0$ and recovery threshold $\\eta > \\epsilon_{\\text{crit}}$, thereby enabling reentry into the coherent identity class established by Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Only the threshold-transitivity content of the iff is modeled (recovery threshold above critical, overlap meeting it, forces overlap above critical); the existence of a time-evolved reflection operator achieving the overlap is not modeled."
    ],
    "record_ids": [
      "MAP-BOOK4B-006"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book4B.recovery_threshold_exceeds_crit"
    ]
  },
  "line": 2840,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Reflective Reentry",
  "proof_labels": [
    "proof:bk4_repair_reconnects_fragmentation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "). A repair trajectory exists if and only if there exists a time-evolved reflection operator $\\widehat{R}_t$ (see Def.~\\ref{definition:bk1_reflection_operator}) such that: \\begin{equation} \\Upsilon_i(\\Psi_i(t_0), \\widehat{R}_t \\circ \\Psi_i(t_1)) \\geq \\eta \\end{equation} for",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "i$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) exhibiting decoherence on region $U \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}). A repair trajectory exists if and only if there exists a time-evolved reflection operator $\\widehat{R}_t$ (see Def.~",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "heorem}[Reflective Reentry] \\label{theorem:bk4_reflective_reentry} Let $\\mathcal{I}$ be a fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) with symbolic pattern $\\Psi_i$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) exhibiting decoherence on reg",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "be a fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) with symbolic pattern $\\Psi_i$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) exhibiting decoherence on region $U \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}). A repair trajecto",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    },
    {
      "context": "hreshold $\\eta > \\epsilon_{\\text{crit}}$, thereby enabling reentry into the coherent identity class established by Thm.~\\ref{theorem:bk4_existence_of_symbolic_ident}. \\end{theorem}",
      "label": "theorem:bk4_existence_of_symbolic_ident",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 15,
      "target_type": "theorem"
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    "definition:bk1_reflection_operator",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_existence_of_symbolic_ident"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk1_reflection_operator",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_reflective_reentry"
  ],
  "depends_on": [
    "definition:bk1_reflection_operator",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_reflective_reentry"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_repair_reconnects_fragmentation",
  "label": "proof:bk4_repair_reconnects_fragmentation",
  "latex_body": "\\begin{proof}[Repair Trajectories Reconnect Fragmented Symbolic Regions]\n\\label{proof:bk4_repair_reconnects_fragmentation}\n\\leavevmode\n\n($\\Rightarrow$) If a repair trajectory exists, then by\nDef.~\\ref{definition:bk4_repair_process} it increases symbolic coherence and\ntherefore reduces the fragmentation measure\n(Def.~\\ref{definition:bk4_fragmentation_measure}).\nFor this to occur, the fragmented symbolic pattern\n(Def.~\\ref{definition:bk4_symbolic_identity_carrie}) must reconnect previously\ndisconnected regions.\n\nFrom the theory of symbolic identity (see Thm.~\\ref{theorem:bk4_reflective_reentry}), we know that identity persistence depends on the stability functional $\\Upsilon_i$. A successful repair must restore this stability, meaning there must exist a reflection operator $\\widehat{R}_t$ (Def.~\\ref{definition:bk1_reflection_operator}) that maps the fragmented pattern at time $t_1$ to a state sufficiently close to the original coherent pattern at time $t_0$.\n\n($\\Leftarrow$) Conversely, if such a reflection operator $\\widehat{R}_t$ exists, it can be used to construct a repair process. Specifically, we define:\n\\begin{equation}\n    \\mathcal{R}_{\\text{rep}}(\\mathcal{I}_{\\text{frag}}) = \\mathcal{I}_{\\text{new}}\n\\end{equation}\nwhere $\\mathcal{I}_{\\text{new}}$ has symbolic pattern $\\Psi_{\\text{new}} = \\widehat{R}_t \\circ \\Psi_i(t_1)$.\n\nThe condition $\\Upsilon_i(\\Psi_i(t_0), \\widehat{R}_t \\circ \\Psi_i(t_1)) \\geq \\eta$ ensures that $\\Psi_{\\text{new}}$ maintains sufficient coherence with the original pattern, thus reducing fragmentation (see Thm.~\\ref{theorem:bk4_reflective_reentry}).\n\\end{proof}",
  "line": 2849,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Repair Trajectories Reconnect Fragmented Symbolic Regions",
  "proves": "theorem:bk4_reflective_reentry",
  "ref_roles": [
    {
      "context": ". A successful repair must restore this stability, meaning there must exist a reflection operator $\\widehat{R}_t$ (Def.~\\ref{definition:bk1_reflection_operator}) that maps the fragmented pattern at time $t_1$ to a state sufficiently close to the original coherent pattern at time",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "ef{definition:bk4_repair_process} it increases symbolic coherence and therefore reduces the fragmentation measure (Def.~\\ref{definition:bk4_fragmentation_measure}). For this to occur, the fragmented symbolic pattern (Def.~\\ref{definition:bk4_symbolic_identity_carrie}) must reconnec",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "bel{proof:bk4_repair_reconnects_fragmentation} \\leavevmode ($\\Rightarrow$) If a repair trajectory exists, then by Def.~\\ref{definition:bk4_repair_process} it increases symbolic coherence and therefore reduces the fragmentation measure (Def.~\\ref{definition:bk4_fragmentation",
      "label": "definition:bk4_repair_process",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2830,
      "target_type": "definition"
    },
    {
      "context": "ion measure (Def.~\\ref{definition:bk4_fragmentation_measure}). For this to occur, the fragmented symbolic pattern (Def.~\\ref{definition:bk4_symbolic_identity_carrie}) must reconnect previously disconnected regions. From the theory of symbolic identity (see Thm.~\\ref{theorem:bk4_refle",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    },
    {
      "context": "bolic_identity_carrie}) must reconnect previously disconnected regions. From the theory of symbolic identity (see Thm.~\\ref{theorem:bk4_reflective_reentry}), we know that identity persistence depends on the stability functional $\\Upsilon_i$. A successful repair must restore",
      "label": "theorem:bk4_reflective_reentry",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 2840,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_reflection_operator",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_reflective_reentry"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [
    "proof:bk4_recursive_self_healing_threshold",
    "theorem:bk4_conditions_for_self_healing"
  ],
  "cites": [
    "definition:bk4_fragmented_identity",
    "definition:bk4_identity_operators",
    "definition:bk4_repair_process",
    "definition:bk4_self_reference_operator"
  ],
  "depends_on": [
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    "definition:bk4_identity_operators",
    "definition:bk4_repair_process",
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  ],
  "file": "book4.tex",
  "id": "definition:bk4_recursive_self_healing",
  "label": "definition:bk4_recursive_self_healing",
  "latex_body": "\\begin{definition}[Recursive Self-Healing] \\label{definition:bk4_recursive_self_healing}\nRecursive self-healing is a repair process (see Def.~\\ref{definition:bk4_repair_process}) where the fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) uses its own reflexive capabilities to restore coherence:\n\\begin{equation}\n    \\mathcal{R}_{\\text{self}} = \\mathcal{S}_n \\circ \\mathcal{P}_{\\Delta t} \\circ \\mathcal{S}_m\n\\end{equation}\nwhere $\\mathcal{S}_n$ is the self-reference operator of order $n$ (see Def.~\\ref{definition:bk4_self_reference_operator}), $\\mathcal{P}_{\\Delta t}$ is the identity persistence operator (see Def.~\\ref{definition:bk4_identity_operators}), and $m, n$ are suitable recursion depths.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Recursive self-healing is a repair process, so it obeys the same RepairSufficiency law; the specific self-reference/persistence operator composition S_n . P_dt . S_m is not modeled, only the monotone-decrease consequence any such process must satisfy."
    ],
    "record_ids": [
      "MAP-BOOK4B-007"
    ],
    "statuses": [
      "constructed"
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    "witnesses": [
      "Book4B.repairSufficiency_decrease_accum",
      "Book4B.repairSufficiency_terminates"
    ]
  },
  "line": 2871,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Recursive Self-Healing",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "self-healing is a repair process (see Def.~\\ref{definition:bk4_repair_process}) where the fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) uses its own reflexive capabilities to restore coherence: \\begin{equation} \\mathcal{R}_{\\text{self}} = \\mathcal{S}",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "~\\ref{definition:bk4_self_reference_operator}), $\\mathcal{P}_{\\Delta t}$ is the identity persistence operator (see Def.~\\ref{definition:bk4_identity_operators}), and $m, n$ are suitable recursion depths. \\end{definition}",
      "label": "definition:bk4_identity_operators",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 231,
      "target_type": "definition"
    },
    {
      "context": "ursive Self-Healing] \\label{definition:bk4_recursive_self_healing} Recursive self-healing is a repair process (see Def.~\\ref{definition:bk4_repair_process}) where the fragmented identity (see Def.~\\ref{definition:bk4_fragmented_identity}) uses its own reflexive capabilities",
      "label": "definition:bk4_repair_process",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2830,
      "target_type": "definition"
    },
    {
      "context": "Delta t} \\circ \\mathcal{S}_m \\end{equation} where $\\mathcal{S}_n$ is the self-reference operator of order $n$ (see Def.~\\ref{definition:bk4_self_reference_operator}), $\\mathcal{P}_{\\Delta t}$ is the identity persistence operator (see Def.~\\ref{definition:bk4_identity_operators}), and",
      "label": "definition:bk4_self_reference_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 277,
      "target_type": "definition"
    }
  ],
  "refs": [
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    "definition:bk4_identity_operators",
    "definition:bk4_repair_process",
    "definition:bk4_self_reference_operator"
  ],
  "role": "definition",
  "type": "definition"
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{
  "book": "book4",
  "cited_by": [
    "bridge:bk4_ttpr_to_self_reference",
    "proof:bk4_recursive_self_healing_threshold",
    "theorem:bk4_ttpr_symbolic_stability"
  ],
  "cites": [
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_identity_resolution",
    "definition:bk4_recursive_self_healing"
  ],
  "depends_on": [
    "definition:bk3_symbolic_membrane",
    "definition:bk4_fragmented_identity",
    "definition:bk4_identity_operators",
    "definition:bk4_identity_resolution",
    "definition:bk4_recursive_self_healing",
    "definition:bk4_self_reference_operator",
    "theorem:bk4_recursive_identity_enhancem"
  ],
  "file": "book4.tex",
  "id": "theorem:bk4_conditions_for_self_healing",
  "label": "theorem:bk4_conditions_for_self_healing",
  "latex_body": "\\begin{theorem}[Conditions for Self-Healing] \\label{theorem:bk4_conditions_for_self_healing}\nA fragmented symbolic identity (see Def.~\\ref{definition:bk4_fragmented_identity}) $\\mathcal{I}$ can implement recursive self-healing (see Def.~\\ref{definition:bk4_recursive_self_healing}) if and only if:\n\\begin{enumerate}\n    \\item The identity resolution $\\mathcal{R}_n$ (see Def.~\\ref{definition:bk4_identity_resolution}) satisfies $\\mathcal{R}_n > \\chi$ for some $n \\geq n_0$ and threshold $\\chi > 0$\n    \\item There exists a subregion $U_{\\text{core}} \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}) where:\n    \\begin{equation}\n        \\Upsilon_i(\\Psi_i|_{U_{\\text{core}}}(t), \\Psi_i|_{U_{\\text{core}}}(t+\\Delta t)) > 1 - \\epsilon_{\\text{core}}\n    \\end{equation}\n    with $\\epsilon_{\\text{core}} < \\epsilon_{\\text{crit}}$\n\\end{enumerate}\n\\end{theorem}",
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    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Only condition 2's threshold-transitivity shape is modeled (core error below critical error forces core coherence past 1-critical); condition 1 (identity resolution R_n > chi for n >= n_0) is a bare existential over an unmodeled resolution sequence and is not covered."
    ],
    "record_ids": [
      "MAP-BOOK4B-008"
    ],
    "statuses": [
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    ],
    "witnesses": [
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  },
  "line": 2878,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conditions for Self-Healing",
  "proof_labels": [
    "proof:bk4_recursive_self_healing_threshold"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "r some $n \\geq n_0$ and threshold $\\chi > 0$ \\item There exists a subregion $U_{\\text{core}} \\subset M_i$ (see Def.~\\ref{definition:bk3_symbolic_membrane}) where: \\begin{equation} \\Upsilon_i(\\Psi_i|_{U_{\\text{core}}}(t), \\Psi_i|_{U_{\\text{core}}}(t+\\Delta t)) >",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "}[Conditions for Self-Healing] \\label{theorem:bk4_conditions_for_self_healing} A fragmented symbolic identity (see Def.~\\ref{definition:bk4_fragmented_identity}) $\\mathcal{I}$ can implement recursive self-healing (see Def.~\\ref{definition:bk4_recursive_self_healing}) if and only",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "_recursive_self_healing}) if and only if: \\begin{enumerate} \\item The identity resolution $\\mathcal{R}_n$ (see Def.~\\ref{definition:bk4_identity_resolution}) satisfies $\\mathcal{R}_n > \\chi$ for some $n \\geq n_0$ and threshold $\\chi > 0$ \\item There exists a subregion $U_",
      "label": "definition:bk4_identity_resolution",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 80,
      "target_type": "definition"
    },
    {
      "context": "entity (see Def.~\\ref{definition:bk4_fragmented_identity}) $\\mathcal{I}$ can implement recursive self-healing (see Def.~\\ref{definition:bk4_recursive_self_healing}) if and only if: \\begin{enumerate} \\item The identity resolution $\\mathcal{R}_n$ (see Def.~\\ref{definition:bk4_iden",
      "label": "definition:bk4_recursive_self_healing",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2871,
      "target_type": "definition"
    }
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    "definition:bk4_recursive_self_healing"
  ],
  "role": "theorem",
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}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
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    "definition:bk4_identity_resolution",
    "definition:bk4_recursive_self_healing",
    "definition:bk4_self_reference_operator",
    "theorem:bk4_conditions_for_self_healing",
    "theorem:bk4_recursive_identity_enhancem"
  ],
  "depends_on": [
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    "definition:bk4_recursive_self_healing",
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    "theorem:bk4_conditions_for_self_healing",
    "theorem:bk4_recursive_identity_enhancem"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_recursive_self_healing_threshold",
  "label": "proof:bk4_recursive_self_healing_threshold",
  "latex_body": "\\begin{proof}[Recursive Identity Retention Enables Self-Healing]\n\\label{proof:bk4_recursive_self_healing_threshold}\n\\leavevmode\n\nThe first condition ensures that the recursive self-reference mechanism retains sufficient information about the identity structure despite fragmentation (see Thm.~\\ref{theorem:bk4_conditions_for_self_healing}). From Theorem~\\ref{theorem:bk4_recursive_identity_enhancem}, we know that when $\\mathcal{R}_n > 1$ (see Def.~\\ref{definition:bk4_identity_resolution}), higher-order recursive encoding actually enhances identity information. For self-healing, we only need $\\mathcal{R}_n > \\chi$ for some positive threshold $\\chi$, indicating that enough identity information persists through recursion.\n\nThe second condition guarantees the existence of a stable core region that can serve as a seed for the repair process. This core must maintain temporal coherence above the critical threshold, providing a stable reference frame for reconstructing the fragmented regions.\n\nGiven these two conditions, the recursive self-healing process (see Def.~\\ref{definition:bk4_recursive_self_healing}) operates as follows:\n\\begin{enumerate}\n    \\item The self-reference operator \\( \\mathcal{S}_m \\) constructs the \\( m \\)th-order self-representation of a fragmented identity. See Definition~\\ref{definition:bk4_self_reference_operator}.\n    \\item The persistence operator \\( \\mathcal{P}_{\\Delta t} \\) evolves this representation forward in time. See Def~\\ref{definition:bk4_identity_operators}.\n    \\item The self-reference operator \\( \\mathcal{S}_n \\) then recursively encodes this evolved state, reinforcing coherence through symbolic self-reconstruction (see Def.~\\ref{definition:bk4_self_reference_operator} and Def.~\\ref{definition:bk4_recursive_self_healing}).\n\\end{enumerate}\n\nThrough this process, the stable core region serves as an attractor in the identity dynamics, pulling fragmented components back toward coherence. The recursive encoding enhances weak coherence patterns and suppresses inconsistent ones, gradually restoring the identity structure.\n\\end{proof}",
  "line": 2889,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Recursive Identity Retention Enables Self-Healing",
  "proves": "theorem:bk4_conditions_for_self_healing",
  "ref_roles": [
    {
      "context": "}. \\item The persistence operator \\( \\mathcal{P}_{\\Delta t} \\) evolves this representation forward in time. See Def~\\ref{definition:bk4_identity_operators}. \\item The self-reference operator \\( \\mathcal{S}_n \\) then recursively encodes this evolved state, reinforcing coh",
      "label": "definition:bk4_identity_operators",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 231,
      "target_type": "definition"
    },
    {
      "context": "_healing}). From Theorem~\\ref{theorem:bk4_recursive_identity_enhancem}, we know that when $\\mathcal{R}_n > 1$ (see Def.~\\ref{definition:bk4_identity_resolution}), higher-order recursive encoding actually enhances identity information. For self-healing, we only need $\\mathcal{R}_n",
      "label": "definition:bk4_identity_resolution",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 80,
      "target_type": "definition"
    },
    {
      "context": "me for reconstructing the fragmented regions. Given these two conditions, the recursive self-healing process (see Def.~\\ref{definition:bk4_recursive_self_healing}) operates as follows: \\begin{enumerate} \\item The self-reference operator \\( \\mathcal{S}_m \\) constructs the \\( m \\",
      "label": "definition:bk4_recursive_self_healing",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2871,
      "target_type": "definition"
    },
    {
      "context": "perator \\( \\mathcal{S}_m \\) constructs the \\( m \\)th-order self-representation of a fragmented identity. See Definition~\\ref{definition:bk4_self_reference_operator}. \\item The persistence operator \\( \\mathcal{P}_{\\Delta t} \\) evolves this representation forward in time. See Def~\\",
      "label": "definition:bk4_self_reference_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 277,
      "target_type": "definition"
    },
    {
      "context": "ve self-reference mechanism retains sufficient information about the identity structure despite fragmentation (see Thm.~\\ref{theorem:bk4_conditions_for_self_healing}). From Theorem~\\ref{theorem:bk4_recursive_identity_enhancem}, we know that when $\\mathcal{R}_n > 1$ (see Def.~\\ref{defi",
      "label": "theorem:bk4_conditions_for_self_healing",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 2878,
      "target_type": "theorem"
    },
    {
      "context": "out the identity structure despite fragmentation (see Thm.~\\ref{theorem:bk4_conditions_for_self_healing}). From Theorem~\\ref{theorem:bk4_recursive_identity_enhancem}, we know that when $\\mathcal{R}_n > 1$ (see Def.~\\ref{definition:bk4_identity_resolution}), higher-order recursive enco",
      "label": "theorem:bk4_recursive_identity_enhancem",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 89,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_identity_operators",
    "definition:bk4_identity_resolution",
    "definition:bk4_recursive_self_healing",
    "definition:bk4_self_reference_operator",
    "theorem:bk4_conditions_for_self_healing",
    "theorem:bk4_recursive_identity_enhancem"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
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    "lemma:bk4_upper_bound_on_repair_capacit"
  ],
  "cites": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_repair_capacity",
  "label": "definition:bk4_repair_capacity",
  "latex_body": "\\begin{definition}[Repair Capacity]\n\\label{definition:bk4_repair_capacity}\nThe repair capacity $C_{\\text{rep}}$ of a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) is defined as:\n\\begin{equation}\n    C_{\\text{rep}}(\\mathcal{I}) = \\sup \\left\\{ \\mathcal{F}_{\\text{frag}}(\\mathcal{I}') : \\exists \\mathcal{R}_{\\text{rep}} \\text{ such that } \\mathcal{R}_{\\text{rep}}(\\mathcal{I}') \\text{ is coherent} \\right\\}\n\\end{equation}\nHere, $\\mathcal{F}_{\\text{frag}}$ denotes the fragmentation measure (see Def.~\\ref{definition:bk4_fragmentation_measure}), and $\\mathcal{R}_{\\text{rep}}$ is a symbolic repair process (see Def.~\\ref{definition:bk4_repair_process}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Kept as a structure field bounded by the repair-capacity ceiling rather than derived from the sup over repair processes; no topology on the space of repair processes is modeled."
    ],
    "record_ids": [
      "MAP-BOOK4B-009"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "Book4B.repairCapacity_lt_one"
    ]
  },
  "line": 2906,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Repair Capacity",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ext{ is coherent} \\right\\} \\end{equation} Here, $\\mathcal{F}_{\\text{frag}}$ denotes the fragmentation measure (see Def.~\\ref{definition:bk4_fragmentation_measure}), and $\\mathcal{R}_{\\text{rep}}$ is a symbolic repair process (see Def.~\\ref{definition:bk4_repair_process}). \\end{defi",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "Def.~\\ref{definition:bk4_fragmentation_measure}), and $\\mathcal{R}_{\\text{rep}}$ is a symbolic repair process (see Def.~\\ref{definition:bk4_repair_process}). \\end{definition}",
      "label": "definition:bk4_repair_process",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2830,
      "target_type": "definition"
    },
    {
      "context": "bel{definition:bk4_repair_capacity} The repair capacity $C_{\\text{rep}}$ of a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) is defined as: \\begin{equation} C_{\\text{rep}}(\\mathcal{I}) = \\sup \\left\\{ \\mathcal{F}_{\\text{frag}}(\\mathcal{I}')",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "proof:bk4_fragmentation_distortion_encoding"
  ],
  "cites": [
    "definition:bk4_recursive_identity_encod",
    "definition:bk4_repair_capacity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_recursive_identity_encod",
    "definition:bk4_repair_capacity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "file": "book4.tex",
  "id": "lemma:bk4_upper_bound_on_repair_capacit",
  "label": "lemma:bk4_upper_bound_on_repair_capacit",
  "latex_body": "\\begin{lemma}[Upper Bound on Repair Capacity] \\label{lemma:bk4_upper_bound_on_repair_capacit}\nFor any symbolic identity $\\mathcal{I}$ (def~\\ref{definition:bk4_symbolic_identity_carrie}) with recursive depth capacity $n_{\\text{max}}$ (\\ref{definition:bk4_recursive_identity_encod}, the repair capacity (def~\\ref{definition:bk4_repair_capacity}) is bounded by:\n\\begin{equation}\n    C_{\\text{rep}}(\\mathcal{I}) \\leq 1 - \\frac{1}{n_{\\text{max}} + 1}\n\\end{equation}\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The bound expression 1 - 1/(n_max+1) is proved to lie in [0,1) and to strictly increase with recursive depth capacity, plus the consequence that any capacity bounded by it is itself below 1."
    ],
    "record_ids": [
      "MAP-BOOK4B-010"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book4B.repairCapacityBound_lt_one",
      "Book4B.repairCapacityBound_nonneg",
      "Book4B.repairCapacityBound_strictMono",
      "Book4B.repairCapacity_lt_one"
    ]
  },
  "line": 2915,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Upper Bound on Repair Capacity",
  "proof_labels": [
    "proof:bk4_fragmentation_distortion_encoding"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ntity $\\mathcal{I}$ (def~\\ref{definition:bk4_symbolic_identity_carrie}) with recursive depth capacity $n_{\\text{max}}$ (\\ref{definition:bk4_recursive_identity_encod}, the repair capacity (def~\\ref{definition:bk4_repair_capacity}) is bounded by: \\begin{equation} C_{\\text{rep}}(\\mat",
      "label": "definition:bk4_recursive_identity_encod",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 47,
      "target_type": "definition"
    },
    {
      "context": "with recursive depth capacity $n_{\\text{max}}$ (\\ref{definition:bk4_recursive_identity_encod}, the repair capacity (def~\\ref{definition:bk4_repair_capacity}) is bounded by: \\begin{equation} C_{\\text{rep}}(\\mathcal{I}) \\leq 1 - \\frac{1}{n_{\\text{max}} + 1} \\end{equation} \\",
      "label": "definition:bk4_repair_capacity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2906,
      "target_type": "definition"
    },
    {
      "context": "Bound on Repair Capacity] \\label{lemma:bk4_upper_bound_on_repair_capacit} For any symbolic identity $\\mathcal{I}$ (def~\\ref{definition:bk4_symbolic_identity_carrie}) with recursive depth capacity $n_{\\text{max}}$ (\\ref{definition:bk4_recursive_identity_encod}, the repair capacity (de",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk4_recursive_identity_encod",
    "definition:bk4_repair_capacity",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book4",
  "cited_by": [
    "example:bk4_ttpr_identity_refinement"
  ],
  "cites": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_recursive_identity_encod",
    "lemma:bk4_upper_bound_on_repair_capacit"
  ],
  "depends_on": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_recursive_identity_encod",
    "lemma:bk4_upper_bound_on_repair_capacit"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_fragmentation_distortion_encoding",
  "label": "proof:bk4_fragmentation_distortion_encoding",
  "latex_body": "\\begin{proof}[Fragmentation and Distortion]\n\\label{proof:bk4_fragmentation_distortion_encoding}\n\\leavevmode\n\nBy Def.~\\ref{definition:bk4_recursive_identity_encod}, recursive-encoding\nfidelity is controlled by summable distortion terms.\n\nWhen identity is fragmented with measure\n$\\mathcal{F}_{\\text{frag}}$ (Def.~\\ref{definition:bk4_fragmentation_measure}),\nit introduces additional distortion proportional to fragmentation level.\n\nIf $n_{\\text{max}}$ is the maximum recursion depth at which the identity maintains coherent self-reference, then at least one level in the recursive structure must remain intact to seed the repair process (see Lemma~\\ref{lemma:bk4_upper_bound_on_repair_capacit}). \n\nThis implies that the maximum tolerable fragmentation is \n\\[\n1 - \\frac{1}{n_{\\text{max}} + 1}\n\\]\nwhere the denominator represents the total number of levels in the recursive structure (including the base level).\n\\end{proof}",
  "line": 2921,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Fragmentation and Distortion",
  "proves": "lemma:bk4_upper_bound_on_repair_capacit",
  "ref_roles": [
    {
      "context": "is controlled by summable distortion terms. When identity is fragmented with measure $\\mathcal{F}_{\\text{frag}}$ (Def.~\\ref{definition:bk4_fragmentation_measure}), it introduces additional distortion proportional to fragmentation level. If $n_{\\text{max}}$ is the maximum recursio",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "\\begin{proof}[Fragmentation and Distortion] \\label{proof:bk4_fragmentation_distortion_encoding} \\leavevmode By Def.~\\ref{definition:bk4_recursive_identity_encod}, recursive-encoding fidelity is controlled by summable distortion terms. When identity is fragmented with measure $\\ma",
      "label": "definition:bk4_recursive_identity_encod",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 47,
      "target_type": "definition"
    },
    {
      "context": "-reference, then at least one level in the recursive structure must remain intact to seed the repair process (see Lemma~\\ref{lemma:bk4_upper_bound_on_repair_capacit}). This implies that the maximum tolerable fragmentation is \\[ 1 - \\frac{1}{n_{\\text{max}} + 1} \\] where the denomina",
      "label": "lemma:bk4_upper_bound_on_repair_capacit",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 2915,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_recursive_identity_encod",
    "lemma:bk4_upper_bound_on_repair_capacit"
  ],
  "role": "proof",
  "type": "proof"
}

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  "book": "book4",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book4.tex",
  "id": "sec:bk4_conditions_individuated_freedom",
  "label": "sec:bk4_conditions_individuated_freedom",
  "latex_body": "",
  "line": 2940,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conditions for Individuated Freedom",
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book4.tex",
  "id": "subsec:bk4_foundations_symbolic_individuation",
  "label": "subsec:bk4_foundations_symbolic_individuation",
  "latex_body": "",
  "line": 2941,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Foundations of Symbolic Individuation",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book4",
  "cited_by": [
    "definition:bk4_constraint_domain",
    "definition:bk4_proto_vitality",
    "definition:bk4_self_authorship",
    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "definition:bk4_symbolic_freedom_measure",
    "lemma:bk4_autonomy_freedom_relation",
    "proof:bk4_freedom_growth_fragmentation",
    "proof:bk4_maximal_freedom_autonomous_constraints",
    "proof:bk4_progressive_freedom_constraint_expansion",
    "theorem:bk4_freedom_life_connection",
    "theorem:bk4_recursive_constraint_libera",
    "theorem:bk4_self_authorship_and_freedom"
  ],
  "cites": [
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_individuated_symbolic_id",
  "label": "definition:bk4_individuated_symbolic_id",
  "latex_body": "\\begin{definition}[Individuated Symbolic Identity] \\label{definition:bk4_individuated_symbolic_id}\nA symbolic identity carrier $\\mathcal{I}$ (def~\\ref{definition:bk4_symbolic_identity_carrie} on membrane $M_i$ (def\\ref{definition:bk3_symbolic_membrane}is \\textit{individuated} if:\n\\begin{enumerate}\n    \\item It maintains a stable symbolic pattern $\\Psi_i$ under bounded drift:\n    \\begin{equation}\n        \\Upsilon_i(\\Psi_i(t), \\Psi_i(t+\\Delta t)) \\geq 1 - \\epsilon(t) \\quad \\forall t\n    \\end{equation}\n    \\item It possesses a self-reflexive operator $\\mathcal{R}_{\\mathcal{I}}$ (def~r\\ref{definition:bk1_reflection_operator} satisfying:\n    \\begin{equation}\n        d_g(\\mathcal{R}_{\\mathcal{I}}(\\Psi_i), \\Psi_i) \\leq \\delta_{\\text{refl}}\n    \\end{equation}\n    for some small $\\delta_{\\text{refl}} > 0$\n    \\item It contains a mutable constraint map $\\mathcal{L}: \\mathcal{U} \\to \\mathcal{U}'$ enabling symbolic reconfiguration across its internal constraint spaces (see def~\\ref{definition:bk2_symbolic_probability_spa}\n\\end{enumerate}\n\\end{definition}",
  "line": 2942,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Individuated Symbolic Identity",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": ") \\quad \\forall t \\end{equation} \\item It possesses a self-reflexive operator $\\mathcal{R}_{\\mathcal{I}}$ (def~r\\ref{definition:bk1_reflection_operator} satisfying: \\begin{equation} d_g(\\mathcal{R}_{\\mathcal{I}}(\\Psi_i), \\Psi_i) \\leq \\delta_{\\text{refl}} \\",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "hcal{L}: \\mathcal{U} \\to \\mathcal{U}'$ enabling symbolic reconfiguration across its internal constraint spaces (see def~\\ref{definition:bk2_symbolic_probability_spa} \\end{enumerate} \\end{definition}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "_id} A symbolic identity carrier $\\mathcal{I}$ (def~\\ref{definition:bk4_symbolic_identity_carrie} on membrane $M_i$ (def\\ref{definition:bk3_symbolic_membrane}is \\textit{individuated} if: \\begin{enumerate} \\item It maintains a stable symbolic pattern $\\Psi_i$ under bounded d",
      "label": "definition:bk3_symbolic_membrane",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book3.tex",
      "target_line": 10,
      "target_type": "definition"
    },
    {
      "context": "uated Symbolic Identity] \\label{definition:bk4_individuated_symbolic_id} A symbolic identity carrier $\\mathcal{I}$ (def~\\ref{definition:bk4_symbolic_identity_carrie} on membrane $M_i$ (def\\ref{definition:bk3_symbolic_membrane}is \\textit{individuated} if: \\begin{enumerate} \\item It",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_reflection_operator",
    "definition:bk2_symbolic_probability_spa",
    "definition:bk3_symbolic_membrane",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "definition:bk4_symbolic_freedom_measure",
    "proof:bk4_freedom_via_symbolic_flow",
    "proof:bk4_progressive_freedom_constraint_expansion",
    "theorem:bk4_recursive_constraint_libera",
    "theorem:bk4_self_authorship_and_freedom"
  ],
  "cites": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_constraint_domain",
  "label": "definition:bk4_constraint_domain",
  "latex_body": "\\begin{definition}[Constraint Domain] \n\\label{definition:bk4_constraint_domain}\nThe constraint domain $\\mathcal{U}(\\mathcal{I})$ of a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) is the set of all admissible symbolic patterns that satisfy the internal consistency conditions:\n\\begin{equation}\n    \\mathcal{U}(\\mathcal{I}) = \\left\\{\\Psi : d_g(\\mathcal{R}_{\\mathcal{I}}(\\Psi), \\Psi) \\leq \\delta_{\\text{refl}} \\text{ and } \\mathcal{F}_{\\text{frag}}(\\Psi) < \\epsilon_{\\text{frag}} \\right\\}\n\\end{equation}\nHere, $\\mathcal{R}_{\\mathcal{I}}$ denotes the identity-relative reflection operator (see Def.~\\ref{definition:bk4_individuated_symbolic_id}), and $\\mathcal{F}_{\\text{frag}}$ is the fragmentation measure (see Def.~\\ref{definition:bk4_fragmentation_measure}). \n\nThe metric $d_g$ is defined over a probabilistic symbolic space (see Def.~\\ref{definition:bk2_symbolic_probability_spa}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Domains U(n) are abstract sets. Their recursive limit is constructed as the union of finite stages and proved to satisfy membership, stage-inclusion, and least-upper-domain laws; The Scholium directed-system laws and unique colimit mediator are instantiated by monotone domain inclusions. Metric d_g / reflection-operator membership conditions remain outside the model."
    ],
    "record_ids": [
      "MAP-BOOK4B-011"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "Book4B.constraintDirectedSystem_universal",
      "Book4B.constraintDomain_strict_growth",
      "Book4B.constraintDomain_subset_limit",
      "Book4B.constraintLimit_least",
      "Book4B.mem_constraintLimit_iff"
    ]
  },
  "line": 2957,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Constraint Domain",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ref{definition:bk4_fragmentation_measure}). The metric $d_g$ is defined over a probabilistic symbolic space (see Def.~\\ref{definition:bk2_symbolic_probability_spa}). \\end{definition}",
      "label": "definition:bk2_symbolic_probability_spa",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 23,
      "target_type": "definition"
    },
    {
      "context": "~\\ref{definition:bk4_individuated_symbolic_id}), and $\\mathcal{F}_{\\text{frag}}$ is the fragmentation measure (see Def.~\\ref{definition:bk4_fragmentation_measure}). The metric $d_g$ is defined over a probabilistic symbolic space (see Def.~\\ref{definition:bk2_symbolic_probability_",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "} \\right\\} \\end{equation} Here, $\\mathcal{R}_{\\mathcal{I}}$ denotes the identity-relative reflection operator (see Def.~\\ref{definition:bk4_individuated_symbolic_id}), and $\\mathcal{F}_{\\text{frag}}$ is the fragmentation measure (see Def.~\\ref{definition:bk4_fragmentation_measure}).",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": ":bk4_constraint_domain} The constraint domain $\\mathcal{U}(\\mathcal{I})$ of a symbolic identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) is the set of all admissible symbolic patterns that satisfy the internal consistency conditions: \\begin{equation}",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_probability_spa",
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "definition:bk4_proto_vitality",
    "definition:bk4_symbolic_freedom_measure",
    "proof:bk4_progressive_freedom_constraint_expansion"
  ],
  "cites": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id"
  ],
  "depends_on": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id"
  ],
  "file": "book4.tex",
  "id": "theorem:bk4_recursive_constraint_libera",
  "label": "theorem:bk4_recursive_constraint_libera",
  "latex_body": "\\begin{theorem}[Constraint Liberation] \n\\label{theorem:bk4_recursive_constraint_libera}\nAn individuated symbolic identity $\\mathcal{I}$\n(Def.~\\ref{definition:bk4_individuated_symbolic_id}) achieves progressive\nfreedom iff its constraint-map sequence satisfies:\n\\begin{equation}\n    \\mathcal{L}_{n+1} = \\mathcal{R}_{\\mathcal{I}} \\circ \\mathcal{L}_n, \\quad \\mathcal{L}_0 = \\text{Initial Constraint Map}\n\\end{equation}\nconverges to a fixed point $\\mathcal{L}_{\\infty}$ that defines a non-trivial constraint domain $\\mathcal{U}_{\\infty}$ (see Def.~\\ref{definition:bk4_constraint_domain}) such that:\n\\begin{equation}\n    \\mathcal{U}_{\\infty} \\supsetneq \\mathcal{U}_0\n\\end{equation}\n\\end{theorem}",
  "lean_alignment": {
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      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The honest order-theoretic limit U_infinity is constructed as the union of all finite domains. It is tail-invariant for monotone sequences, is the least upper domain, and strictly extends U_0 when any stage grows. For Scholium-generated dynamics it is proved exactly equal to the full flow-orbit range. Constraint maps are now modeled abstractly as arbitrary monotone endomorphisms of the complete lattice of domains, with ordinal successor application and union at genuine limit ordinals. Knaster-Tarski supplies the least fixed point unconditionally; ordinal limit induction now derives the standard invariant that every bottom-started stage lies below every fixed domain. Consequently eventual ordinal stabilization alone proves convergence exactly to the least fixed point. The cardinal theorem now gives full eventual convergence for the standard transfinite iterator whenever its well-ordered index type is larger than the constraint-domain lattice and the map is inflationary: it attains the least fixed point and every later stage is identical."
    ],
    "record_ids": [
      "MAP-BOOK4B-012"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book4B.constraintDomain_strict_growth",
      "Book4B.constraintDomain_subset_limit",
      "Book4B.constraintLimit_least",
      "Book4B.constraintLimit_strict_growth",
      "Book4B.constraintLimit_tail_eq",
      "Book4B.mem_constraintLimit_iff",
      "Book4B.orbitConstraintDomain_mono",
      "Book4B.orbitConstraintLimit_eq_range",
      "Book4B.transfiniteConstraint_leastFixedPoint_fixed",
      "Book4B.transfiniteConstraint_leastFixedPoint_le",
      "Book4B.transfiniteConstraint_stabilized_eq_lfp",
      "Book4B.transfiniteConstraint_stabilized_fixed",
      "Book4B.transfiniteConstraint_stage_le_fixed",
      "Book4B.transfiniteIterate_eventually_constant_of_card_lt",
      "Book4B.transfiniteIterate_eventually_constant_of_fixed",
      "Book4B.transfiniteIterate_exists_fixed_of_card_lt",
      "Book4B.transfiniteIterate_fixed_stage_eq_lfp",
      "Book4B.transfiniteIterate_le_fixed"
    ]
  },
  "line": 2968,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Constraint Liberation",
  "proof_labels": [
    "proof:bk4_progressive_freedom_constraint_expansion"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "s to a fixed point $\\mathcal{L}_{\\infty}$ that defines a non-trivial constraint domain $\\mathcal{U}_{\\infty}$ (see Def.~\\ref{definition:bk4_constraint_domain}) such that: \\begin{equation} \\mathcal{U}_{\\infty} \\supsetneq \\mathcal{U}_0 \\end{equation} \\end{theorem}",
      "label": "definition:bk4_constraint_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2957,
      "target_type": "definition"
    },
    {
      "context": "aint Liberation] \\label{theorem:bk4_recursive_constraint_libera} An individuated symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) achieves progressive freedom iff its constraint-map sequence satisfies: \\begin{equation} \\mathcal{L}_{n+1} = \\math",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
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{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "depends_on": [
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    "definition:bk4_individuated_symbolic_id",
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  "file": "book4.tex",
  "id": "proof:bk4_progressive_freedom_constraint_expansion",
  "label": "proof:bk4_progressive_freedom_constraint_expansion",
  "latex_body": "\\begin{proof}[Progressive Freedom Requires Coherence Beyond Constraints]\n\\label{proof:bk4_progressive_freedom_constraint_expansion}\n\\leavevmode\n\n($\\Rightarrow$) If the identity achieves progressive freedom, it must be able to operate beyond its initial constraint domain $\\mathcal{U}_0$ (see Def.~\\ref{definition:bk4_constraint_domain}) while maintaining coherence. This expansion of possibilities is mediated by the evolution of the constraint map.\nBy composing the constraint map with the self-reflection operator, the identity (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) recursively redefines its own constraints. If this process converges to a fixed point $\\mathcal{L}_{\\infty}$, it establishes a stable expanded constraint domain $\\mathcal{U}_{\\infty}$.\nFor true freedom to emerge, this expanded domain must strictly include the initial domain: $\\mathcal{U}_{\\infty} \\supsetneq \\mathcal{U}_0$ (see Thm.~\\ref{theorem:bk4_recursive_constraint_libera}).\n\n($\\Leftarrow$) Conversely, if the sequence of constraint maps converges to a fixed point $\\mathcal{L}_{\\infty}$ that defines an expanded constraint domain $\\mathcal{U}_{\\infty} \\supsetneq \\mathcal{U}_0$, then the identity has successfully transcended its initial limitations while maintaining coherence.\nThis process represents progressive freedom because:\n\\begin{enumerate}\n    \\item The identity remains coherent throughout\n    (Def.~\\ref{definition:bk4_constraint_domain}).\n    \\item The expansion is generated by self-reference:\n    $\\mathcal{L}_{n+1} = \\mathcal{R}_{\\mathcal{I}} \\circ \\mathcal{L}_n$\n    (Def.~\\ref{definition:bk4_individuated_symbolic_id}).\n    \\item The process reaches a stable configuration (see Thm.~\\ref{theorem:bk4_recursive_constraint_libera})\n    \\item The final state permits more possibilities than the initial state ($\\mathcal{U}_{\\infty} \\supsetneq \\mathcal{U}_0$)\n\\end{enumerate}\n\\end{proof}",
  "line": 2982,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Progressive Freedom Requires Coherence Beyond Constraints",
  "proves": "theorem:bk4_recursive_constraint_libera",
  "ref_roles": [
    {
      "context": "achieves progressive freedom, it must be able to operate beyond its initial constraint domain $\\mathcal{U}_0$ (see Def.~\\ref{definition:bk4_constraint_domain}) while maintaining coherence. This expansion of possibilities is mediated by the evolution of the constraint map. By co",
      "label": "definition:bk4_constraint_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2957,
      "target_type": "definition"
    },
    {
      "context": "lution of the constraint map. By composing the constraint map with the self-reflection operator, the identity (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) recursively redefines its own constraints. If this process converges to a fixed point $\\mathcal{L}_{\\infty}$, it estab",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "his expanded domain must strictly include the initial domain: $\\mathcal{U}_{\\infty} \\supsetneq \\mathcal{U}_0$ (see Thm.~\\ref{theorem:bk4_recursive_constraint_libera}). ($\\Leftarrow$) Conversely, if the sequence of constraint maps converges to a fixed point $\\mathcal{L}_{\\infty}$ that",
      "label": "theorem:bk4_recursive_constraint_libera",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 2968,
      "target_type": "theorem"
    }
  ],
  "refs": [
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    "definition:bk4_individuated_symbolic_id",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [
    "proof:bk4_freedom_growth_fragmentation",
    "theorem:bk4_freedom_life_connection"
  ],
  "cites": [
    "definition:bk2_symbolic_entropy",
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "depends_on": [
    "definition:bk2_symbolic_entropy",
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_symbolic_freedom_measure",
  "label": "definition:bk4_symbolic_freedom_measure",
  "latex_body": "\\begin{definition}[Symbolic Freedom Measure]\n\\label{definition:bk4_symbolic_freedom_measure}\nThe symbolic freedom measure $\\mathcal{F}_{\\text{free}}$ of an individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) is defined as:\n\\begin{equation}\n    \\mathcal{F}_{\\text{free}}(\\mathcal{I}) = \\frac{H(\\mathcal{U}_{\\infty}) - H(\\mathcal{U}_0)}{H(\\mathcal{U}_{\\infty})}\n\\end{equation}\nwhere $H(\\mathcal{U})$ is the symbolic entropy (see Def.~\\ref{definition:bk2_symbolic_entropy}) of the constraint domain $\\mathcal{U}$ (see Def.~\\ref{definition:bk4_constraint_domain}). The domain $\\mathcal{U}_{\\infty}$ is obtained as the limit of a convergent sequence of self-reflective constraint maps (see Thm.~\\ref{theorem:bk4_recursive_constraint_libera}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The ratio (H(U_infinity)-H(U_0))/H(U_infinity) is proved to lie in [0,1] given entropy nonnegativity and monotonicity; the entropy functional H and the limiting domain U_infinity themselves are not modeled, only the algebraic consequence of the stated formula."
    ],
    "record_ids": [
      "MAP-BOOK4B-013"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book4B.freedomMeasure_le_one",
      "Book4B.freedomMeasure_nonneg"
    ]
  },
  "line": 3003,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Freedom Measure",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "}) - H(\\mathcal{U}_0)}{H(\\mathcal{U}_{\\infty})} \\end{equation} where $H(\\mathcal{U})$ is the symbolic entropy (see Def.~\\ref{definition:bk2_symbolic_entropy}) of the constraint domain $\\mathcal{U}$ (see Def.~\\ref{definition:bk4_constraint_domain}). The domain $\\mathcal{U}_{\\in",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "the symbolic entropy (see Def.~\\ref{definition:bk2_symbolic_entropy}) of the constraint domain $\\mathcal{U}$ (see Def.~\\ref{definition:bk4_constraint_domain}). The domain $\\mathcal{U}_{\\infty}$ is obtained as the limit of a convergent sequence of self-reflective constraint map",
      "label": "definition:bk4_constraint_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2957,
      "target_type": "definition"
    },
    {
      "context": "m_measure} The symbolic freedom measure $\\mathcal{F}_{\\text{free}}$ of an individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) is defined as: \\begin{equation} \\mathcal{F}_{\\text{free}}(\\mathcal{I}) = \\frac{H(\\mathcal{U}_{\\infty}) - H(\\mathca",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "n $\\mathcal{U}_{\\infty}$ is obtained as the limit of a convergent sequence of self-reflective constraint maps (see Thm.~\\ref{theorem:bk4_recursive_constraint_libera}). \\end{definition}",
      "label": "theorem:bk4_recursive_constraint_libera",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 2968,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_entropy",
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book4.tex",
  "id": "section:book4.tex:3011",
  "label": "",
  "latex_body": "",
  "line": 3011,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Freedom through Self-Authorship",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book4",
  "cited_by": [
    "lemma:bk4_autonomy_freedom_relation",
    "proof:bk4_autonomy_freedom_relation",
    "proof:bk4_freedom_via_symbolic_flow",
    "proof:bk4_goal_directed_composition_flow"
  ],
  "cites": [
    "definition:bk1_symbolic_flow",
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
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    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_symbolic_flow_freedom",
  "label": "definition:bk4_symbolic_flow_freedom",
  "latex_body": "\\begin{definition}[Symbolic Flow Freedom]\n\\label{definition:bk4_symbolic_flow_freedom}\nA symbolic flow $\\Phi_s$ (see Def.~\\ref{definition:bk1_symbolic_flow}) exhibits freedom with respect to an individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) if:\n\\begin{enumerate}\n    \\item The flow preserves identity coherence: $\\Phi_s \\circ \\Psi_i \\in \\text{Fix}(\\mathcal{R}_{\\mathcal{I}})$, where $\\mathcal{R}_{\\mathcal{I}}$ is the reflection operator associated with the identity carrier (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}),\n    \\item The flow transcends initial constraints: $\\Phi_s(\\mathcal{U}_0) \\not\\subseteq \\mathcal{U}_0$, where $\\mathcal{U}_0$ is the initial constraint domain (see Def.~\\ref{definition:bk4_constraint_domain}).\n\\end{enumerate}\nHere, $\\text{Fix}(\\mathcal{R}_{\\mathcal{I}})$ denotes the set of fixed points under the reflection operator, i.e., states that maintain coherence with the self-identity structure.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The two-clause predicate (reflection-fixed image; escape from U_0) is modeled directly, plus an explicit countermodel (the identity flow) showing the escape clause is a genuine non-vacuous requirement."
    ],
    "record_ids": [
      "MAP-BOOK4B-014"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book4B.id_has_no_escape",
      "Book4B.symbolicFreedom_of_flowFreedom"
    ]
  },
  "line": 3012,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Flow Freedom",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "egin{definition}[Symbolic Flow Freedom] \\label{definition:bk4_symbolic_flow_freedom} A symbolic flow $\\Phi_s$ (see Def.~\\ref{definition:bk1_symbolic_flow}) exhibits freedom with respect to an individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_sym",
      "label": "definition:bk1_symbolic_flow",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2872,
      "target_type": "definition"
    },
    {
      "context": ": $\\Phi_s(\\mathcal{U}_0) \\not\\subseteq \\mathcal{U}_0$, where $\\mathcal{U}_0$ is the initial constraint domain (see Def.~\\ref{definition:bk4_constraint_domain}). \\end{enumerate} Here, $\\text{Fix}(\\mathcal{R}_{\\mathcal{I}})$ denotes the set of fixed points under the reflection op",
      "label": "definition:bk4_constraint_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2957,
      "target_type": "definition"
    },
    {
      "context": ".~\\ref{definition:bk1_symbolic_flow}) exhibits freedom with respect to an individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) if: \\begin{enumerate} \\item The flow preserves identity coherence: $\\Phi_s \\circ \\Psi_i \\in \\text{Fix}(\\mathcal{R}",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "hcal{I}})$, where $\\mathcal{R}_{\\mathcal{I}}$ is the reflection operator associated with the identity carrier (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}), \\item The flow transcends initial constraints: $\\Phi_s(\\mathcal{U}_0) \\not\\subseteq \\mathcal{U}_0$, where $\\mathc",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_flow",
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "corollary:bk7_fixed_point_tracking_within_evolving_reciprocity",
    "corollary:bk9_emergence_of_moral_agency",
    "definition:bk9_cognitive_freedom",
    "definition:bk9_recursive_freedom_operator",
    "lemma:bk4_autonomy_freedom_relation",
    "proof:bk4_autonomy_freedom_relation",
    "proof:bk4_freedom_growth_fragmentation",
    "proof:bk4_freedom_via_symbolic_flow",
    "proof:bk4_maximal_freedom_autonomous_constraints",
    "proof:bk9_meta_reflective_memory_integration",
    "proposition:bk4_autonomy_implies_freedom",
    "proposition:bk9_modes_of_re_interpretation",
    "sec:bk7_corollaria_implications_of_convergence",
    "subsubsec:bk7_formal_definition_of_symbolic_confidence_ch_i",
    "theorem:bk4_self_authorship_and_freedom"
  ],
  "cites": [
    "definition:bk1_symbolic_manifold",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold",
    "definition:bk4_constraint_domain",
    "definition:bk4_symbolic_flow_freedom",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "file": "book4.tex",
  "id": "theorem:bk4_freedom_criterion",
  "label": "theorem:bk4_freedom_criterion",
  "latex_body": "\\begin{theorem}[Freedom Criterion] \\label{theorem:bk4_freedom_criterion}\nAn individuated symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_symbolic_identity_carrie}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) expresses freedom if and only if there exists a symbolic flow $\\Phi_s$ such that:\n\\begin{equation}\n    \\Phi_s \\circ \\Psi_i \\in \\text{Fix}(\\mathcal{R}_{\\mathcal{I}}) \\quad \\text{and} \\quad \\Phi_s(\\mathcal{U}_0) \\not\\subseteq \\mathcal{U}_0\n\\end{equation}\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Symbolic freedom is defined exactly as the existential closure of flow freedom, so the criterion is definitional once flow freedom is modeled; only the forward (witness-exhibits-freedom) direction has independent content, and is the theorem proved."
    ],
    "record_ids": [
      "MAP-BOOK4B-015"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book4B.symbolicFreedom_of_flowFreedom"
    ]
  },
  "line": 3022,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Freedom Criterion",
  "proof_labels": [
    "proof:bk4_freedom_via_symbolic_flow"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_symbolic_identity_carrie}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) expresses freedom if and only if there exists a symbolic flow $\\Phi_s$ such that: \\begin{equation} \\Phi_s \\circ \\P",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "theorem}[Freedom Criterion] \\label{theorem:bk4_freedom_criterion} An individuated symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_symbolic_identity_carrie}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) expresses freedom if and only if there exis",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    }
  ],
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    "definition:bk1_symbolic_manifold",
    "definition:bk4_symbolic_identity_carrie"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_constraint_domain",
    "definition:bk4_symbolic_flow_freedom",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_freedom_criterion"
  ],
  "depends_on": [
    "definition:bk4_constraint_domain",
    "definition:bk4_symbolic_flow_freedom",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_freedom_criterion"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_freedom_via_symbolic_flow",
  "label": "proof:bk4_freedom_via_symbolic_flow",
  "latex_body": "\\begin{proof}[Freedom via Coherence-Preserving Symbolic Flow]\n\\label{proof:bk4_freedom_via_symbolic_flow}\n\\leavevmode\n\nThis follows directly from\nDef.~\\ref{definition:bk4_symbolic_flow_freedom}: freedom is realized by\nsymbolic flows that preserve identity coherence while exceeding initial\nconstraints (see also Def.~\\ref{definition:bk4_constraint_domain}).\n\nThe first condition, $\\Phi_s \\circ \\Psi_i \\in \\text{Fix}(\\mathcal{R}_{\\mathcal{I}})$, ensures that the identity remains coherent under the flow, as fixed points of the reflection operator (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) are precisely the symbolic patterns that maintain self-consistency.\n\nThe second condition, $\\Phi_s(\\mathcal{U}_0) \\not\\subseteq \\mathcal{U}_0$, ensures that the flow enables the identity to access symbolic configurations outside its initial constraint domain (see Def.~\\ref{definition:bk4_constraint_domain}), representing genuine transcendence of initial limitations.\n\nTogether, these conditions formalize the notion that freedom is not the absence of constraint, but rather the capacity to transform constraints through self-consistent symbolic flows, consistent with the general criterion given in Theorem~\\ref{theorem:bk4_freedom_criterion}.\n\\end{proof}",
  "line": 3028,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Freedom via Coherence-Preserving Symbolic Flow",
  "proves": "theorem:bk4_freedom_criterion",
  "ref_roles": [
    {
      "context": "eedom is realized by symbolic flows that preserve identity coherence while exceeding initial constraints (see also Def.~\\ref{definition:bk4_constraint_domain}). The first condition, $\\Phi_s \\circ \\Psi_i \\in \\text{Fix}(\\mathcal{R}_{\\mathcal{I}})$, ensures that the identity rema",
      "label": "definition:bk4_constraint_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2957,
      "target_type": "definition"
    },
    {
      "context": "ence-Preserving Symbolic Flow] \\label{proof:bk4_freedom_via_symbolic_flow} \\leavevmode This follows directly from Def.~\\ref{definition:bk4_symbolic_flow_freedom}: freedom is realized by symbolic flows that preserve identity coherence while exceeding initial constraints (see also D",
      "label": "definition:bk4_symbolic_flow_freedom",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3012,
      "target_type": "definition"
    },
    {
      "context": "{I}})$, ensures that the identity remains coherent under the flow, as fixed points of the reflection operator (see Def.~\\ref{definition:bk4_symbolic_identity_carrie}) are precisely the symbolic patterns that maintain self-consistency. The second condition, $\\Phi_s(\\mathcal{U}_0) \\not",
      "label": "definition:bk4_symbolic_identity_carrie",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4,
      "target_type": "definition"
    },
    {
      "context": "to transform constraints through self-consistent symbolic flows, consistent with the general criterion given in Theorem~\\ref{theorem:bk4_freedom_criterion}. \\end{proof}",
      "label": "theorem:bk4_freedom_criterion",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 3022,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_constraint_domain",
    "definition:bk4_symbolic_flow_freedom",
    "definition:bk4_symbolic_identity_carrie",
    "theorem:bk4_freedom_criterion"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [
    "definition:bk4_proto_vitality",
    "definition:bk4_self_authorship",
    "lemma:bk4_autonomy_freedom_relation",
    "proof:bk4_autonomy_freedom_relation",
    "proof:bk4_maximal_freedom_autonomous_constraints",
    "proposition:bk4_autonomy_implies_freedom",
    "sec:bk7_symbolic_uncertainty_emergence_duality_pisu",
    "sec:bk7_theorem_of_convergent_reciprocity_two_way_street",
    "subsubsec:bk7_establishing_the_formal_link_reflective_selection_and_",
    "theorem:bk4_self_authorship_and_freedom",
    "theorem:bk7_emergent_lp_norm"
  ],
  "cites": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id"
  ],
  "depends_on": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_symbolic_autonomy",
  "label": "definition:bk4_symbolic_autonomy",
  "latex_body": "\\begin{definition}[Symbolic Autonomy]\n\\label{definition:bk4_symbolic_autonomy}\nThe symbolic autonomy of an individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) is defined by the triple $(A_i, G_i, \\mathcal{D}_i)$ where:\n\\begin{enumerate}\n    \\item $A_i: \\mathcal{U} \\to \\mathcal{A}$ is a mapping from the constraint domain $\\mathcal{U}$ (Def.~\\ref{definition:bk4_constraint_domain}) to an action space $\\mathcal{A}$\n    \\item $G_i: \\mathcal{U} \\times \\mathcal{E} \\to \\mathcal{U}$ is a goal-directed transformation responsive to environment $\\mathcal{E}$\n    \\item $\\mathcal{D}_i: \\mathcal{U} \\times \\mathcal{G} \\to \\mathcal{U}$ is a decision operator parameterized by goal space $\\mathcal{G}$\n\\end{enumerate}\n\\end{definition}",
  "line": 3044,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Autonomy",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "gin{enumerate} \\item $A_i: \\mathcal{U} \\to \\mathcal{A}$ is a mapping from the constraint domain $\\mathcal{U}$ (Def.~\\ref{definition:bk4_constraint_domain}) to an action space $\\mathcal{A}$ \\item $G_i: \\mathcal{U} \\times \\mathcal{E} \\to \\mathcal{U}$ is a goal-directed tr",
      "label": "definition:bk4_constraint_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2957,
      "target_type": "definition"
    },
    {
      "context": "omy] \\label{definition:bk4_symbolic_autonomy} The symbolic autonomy of an individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) is defined by the triple $(A_i, G_i, \\mathcal{D}_i)$ where: \\begin{enumerate} \\item $A_i: \\mathcal{U} \\to \\mathcal",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "proof:bk4_goal_directed_composition_flow"
  ],
  "cites": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "theorem:bk4_freedom_criterion"
  ],
  "depends_on": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "theorem:bk4_freedom_criterion"
  ],
  "file": "book4.tex",
  "id": "lemma:bk4_autonomy_freedom_relation",
  "label": "lemma:bk4_autonomy_freedom_relation",
  "latex_body": "\\begin{lemma}[Autonomy-Freedom Relation]\n\\label{lemma:bk4_autonomy_freedom_relation}\nAn individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) with symbolic autonomy $(\\mathcal{A}_i, \\mathcal{G}_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}) exhibits freedom according to the freedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}) if and only if:\n\\[\n\\exists g \\in \\mathcal{G}, \\exists e \\in \\mathcal{E} \\quad \\text{such that} \\quad \\mathcal{D}_i(\\cdot, g) \\circ G_i(\\cdot, e) = \\Phi_S,\n\\]\nwhere $\\Phi_S$ satisfies the symbolic flow freedom condition (Def.~\\ref{definition:bk4_symbolic_flow_freedom}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Only the forward direction is proved (a decision/goal composite exhibiting flow freedom witnesses symbolic freedom). The converse -- freedom forces existence of such g, e -- is not modeled, since it would need an unstated surjectivity assumption on the decision and goal-transformation maps."
    ],
    "record_ids": [
      "MAP-BOOK4B-016"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book4B.autonomy_implies_freedom"
    ]
  },
  "line": 3054,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Autonomy-Freedom Relation",
  "proof_labels": [
    "proof:bk4_autonomy_freedom_relation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "Autonomy-Freedom Relation] \\label{lemma:bk4_autonomy_freedom_relation} An individuated identity $\\mathcal{I}$ (see Def.~\\ref{definition:bk4_individuated_symbolic_id}) with symbolic autonomy $(\\mathcal{A}_i, \\mathcal{G}_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}) e",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "{definition:bk4_individuated_symbolic_id}) with symbolic autonomy $(\\mathcal{A}_i, \\mathcal{G}_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}) exhibits freedom according to the freedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}) if and only if: \\[ \\exi",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    },
    {
      "context": "hcal{D}_i(\\cdot, g) \\circ G_i(\\cdot, e) = \\Phi_S, \\] where $\\Phi_S$ satisfies the symbolic flow freedom condition (Def.~\\ref{definition:bk4_symbolic_flow_freedom}). \\end{lemma}",
      "label": "definition:bk4_symbolic_flow_freedom",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3012,
      "target_type": "definition"
    },
    {
      "context": "\\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}) exhibits freedom according to the freedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}) if and only if: \\[ \\exists g \\in \\mathcal{G}, \\exists e \\in \\mathcal{E} \\quad \\text{such that} \\quad \\mathcal{D}_i(\\cd",
      "label": "theorem:bk4_freedom_criterion",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 3022,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "theorem:bk4_freedom_criterion"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "theorem:bk4_freedom_criterion"
  ],
  "depends_on": [
    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "theorem:bk4_freedom_criterion"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_autonomy_freedom_relation",
  "label": "proof:bk4_autonomy_freedom_relation",
  "latex_body": "\\begin{proof}\n\\label{proof:bk4_autonomy_freedom_relation}\n\\leavevmode\n\nDef.~\\ref{definition:bk4_symbolic_autonomy} defines autonomy by an action map,\na goal-directed environmental update, and a decision operator. The freedom\ncriterion of Thm.~\\ref{theorem:bk4_freedom_criterion}, read through\nDef.~\\ref{definition:bk4_symbolic_flow_freedom}, says that an identity is free\nexactly when some symbolic flow preserves identity coherence while exceeding\nthe initial constraint domain.\n\n($\\Rightarrow$) If $\\mathcal I$ exhibits freedom, then there is a witnessing\nflow $\\Phi_S$ satisfying the symbolic flow freedom condition. Since the autonomy\ntriple is the identity's internal mechanism for selecting goals, responding to\nthe environment, and deciding an update, this witnessing flow is represented by\nsome goal/environment pair through the composite\n$\\mathcal D_i(\\cdot,g)\\circ G_i(\\cdot,e)$.\n\n($\\Leftarrow$) Conversely, if such $g$ and $e$ exist and the composite equals a\nflow $\\Phi_S$ satisfying Def.~\\ref{definition:bk4_symbolic_flow_freedom}, then\nthe same flow satisfies Thm.~\\ref{theorem:bk4_freedom_criterion}. Hence the\nindividuated identity exhibits symbolic freedom.\n\\end{proof}",
  "line": 3063,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "lemma:bk4_autonomy_freedom_relation",
  "ref_roles": [
    {
      "context": "\\begin{proof} \\label{proof:bk4_autonomy_freedom_relation} \\leavevmode Def.~\\ref{definition:bk4_symbolic_autonomy} defines autonomy by an action map, a goal-directed environmental update, and a decision operator. The freedom criterion",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    },
    {
      "context": "l update, and a decision operator. The freedom criterion of Thm.~\\ref{theorem:bk4_freedom_criterion}, read through Def.~\\ref{definition:bk4_symbolic_flow_freedom}, says that an identity is free exactly when some symbolic flow preserves identity coherence while exceeding the initial",
      "label": "definition:bk4_symbolic_flow_freedom",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3012,
      "target_type": "definition"
    },
    {
      "context": "autonomy by an action map, a goal-directed environmental update, and a decision operator. The freedom criterion of Thm.~\\ref{theorem:bk4_freedom_criterion}, read through Def.~\\ref{definition:bk4_symbolic_flow_freedom}, says that an identity is free exactly when some symbolic",
      "label": "theorem:bk4_freedom_criterion",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 3022,
      "target_type": "theorem"
    }
  ],
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    "definition:bk4_symbolic_autonomy",
    "definition:bk4_symbolic_flow_freedom",
    "theorem:bk4_freedom_criterion"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [
    "proof:bk4_goal_directed_composition_flow"
  ],
  "cites": [
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  ],
  "depends_on": [
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  "file": "book4.tex",
  "id": "proposition:bk4_autonomy_implies_freedom",
  "label": "proposition:bk4_autonomy_implies_freedom",
  "latex_body": "\\begin{proposition}[Goal-Directed Autonomy Enables Freedom]\n\\label{proposition:bk4_autonomy_implies_freedom}\nLet $\\mathcal{I}$ be an individuated symbolic identity with symbolic autonomy $(\\mathcal{A}_i, \\mathcal{G}_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}). If there exists $g \\in \\mathcal{G}$ and $e \\in \\mathcal{E}$ such that\n\\[\n\\Phi_s = \\mathcal{D}_i(\\cdot, g) \\circ G_i(\\cdot, e),\n\\]\nand $\\Phi_s$ satisfies the freedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}), then $\\mathcal{I}$ exhibits symbolic freedom.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Direct existential transfer: exhibiting the decision/goal composite D(.,g) . G(.,e) satisfying flow freedom is exactly exhibiting a free flow."
    ],
    "record_ids": [
      "MAP-BOOK4B-017"
    ],
    "statuses": [
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    ],
    "witnesses": [
      "Book4B.autonomy_implies_freedom"
    ]
  },
  "line": 3087,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Goal-Directed Autonomy Enables Freedom",
  "proof_labels": [
    "proof:bk4_goal_directed_composition_flow"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "l{I}$ be an individuated symbolic identity with symbolic autonomy $(\\mathcal{A}_i, \\mathcal{G}_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}). If there exists $g \\in \\mathcal{G}$ and $e \\in \\mathcal{E}$ such that \\[ \\Phi_s = \\mathcal{D}_i(\\cdot, g) \\circ G_i(\\",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    },
    {
      "context": "uch that \\[ \\Phi_s = \\mathcal{D}_i(\\cdot, g) \\circ G_i(\\cdot, e), \\] and $\\Phi_s$ satisfies the freedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}), then $\\mathcal{I}$ exhibits symbolic freedom. \\end{proposition}",
      "label": "theorem:bk4_freedom_criterion",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 3022,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_freedom_criterion"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_symbolic_flow_freedom",
    "lemma:bk4_autonomy_freedom_relation",
    "proposition:bk4_autonomy_implies_freedom"
  ],
  "depends_on": [
    "definition:bk4_symbolic_flow_freedom",
    "lemma:bk4_autonomy_freedom_relation",
    "proposition:bk4_autonomy_implies_freedom"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_goal_directed_composition_flow",
  "label": "proof:bk4_goal_directed_composition_flow",
  "latex_body": "\\begin{proof}[Goal-Directed Composition as Freedom-Expressive Flow]\n\\label{proof:bk4_goal_directed_composition_flow}\n\\leavevmode\n\nBy Lemma~\\ref{lemma:bk4_autonomy_freedom_relation}, if $\\mathcal{D}_i$ and $G_i$ compose to yield a symbolic flow $\\Phi_s$ satisfying the freedom criterion, then the identity's action results in a coherence-preserving symbolic trajectory that exceeds initial constraints (prop.~\\ref{proposition:bk4_autonomy_implies_freedom}. This matches the definitional requirements of symbolic freedom (Def.~\\ref{definition:bk4_symbolic_flow_freedom}).\n\\end{proof}",
  "line": 3095,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Goal-Directed Composition as Freedom-Expressive Flow",
  "proves": "proposition:bk4_autonomy_implies_freedom",
  "ref_roles": [
    {
      "context": "p.~\\ref{proposition:bk4_autonomy_implies_freedom}. This matches the definitional requirements of symbolic freedom (Def.~\\ref{definition:bk4_symbolic_flow_freedom}). \\end{proof}",
      "label": "definition:bk4_symbolic_flow_freedom",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3012,
      "target_type": "definition"
    },
    {
      "context": "Directed Composition as Freedom-Expressive Flow] \\label{proof:bk4_goal_directed_composition_flow} \\leavevmode By Lemma~\\ref{lemma:bk4_autonomy_freedom_relation}, if $\\mathcal{D}_i$ and $G_i$ compose to yield a symbolic flow $\\Phi_s$ satisfying the freedom criterion, then the iden",
      "label": "lemma:bk4_autonomy_freedom_relation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 3054,
      "target_type": "lemma"
    },
    {
      "context": "hen the identity's action results in a coherence-preserving symbolic trajectory that exceeds initial constraints (prop.~\\ref{proposition:bk4_autonomy_implies_freedom}. This matches the definitional requirements of symbolic freedom (Def.~\\ref{definition:bk4_symbolic_flow_freedom}). \\end",
      "label": "proposition:bk4_autonomy_implies_freedom",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 3087,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "definition:bk4_symbolic_flow_freedom",
    "lemma:bk4_autonomy_freedom_relation",
    "proposition:bk4_autonomy_implies_freedom"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [
    "proof:bk4_maximal_freedom_autonomous_constraints",
    "remark:bk4_individuated_freedom",
    "theorem:bk4_self_authorship_and_freedom"
  ],
  "cites": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_autonomy"
  ],
  "depends_on": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_autonomy"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_self_authorship",
  "label": "definition:bk4_self_authorship",
  "latex_body": "\\begin{definition}[Self-Authorship]\n\\label{definition:bk4_self_authorship}\nThe \\emph{self-authorship} of an individuated identity $\\mathcal{I}$ (\\ref{definition:bk4_individuated_symbolic_id}is its capacity to modify its own constraint map $\\mathcal{L}$ through autonomous symbolic operations. Formally, there exists a goal $g \\in \\mathcal{G}$ such that:\n\\[\n\\mathcal{L}' = \\mathcal{D}_i(\\mathcal{L}, g)\n\\]\nwhere $\\mathcal{D}_i$ is the decision operator from the identity's symbolic autonomy triple (Def.~\\ref{definition:bk4_symbolic_autonomy}).\n\\end{definition}",
  "line": 3102,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Self-Authorship",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "uthorship] \\label{definition:bk4_self_authorship} The \\emph{self-authorship} of an individuated identity $\\mathcal{I}$ (\\ref{definition:bk4_individuated_symbolic_id}is its capacity to modify its own constraint map $\\mathcal{L}$ through autonomous symbolic operations. Formally, there e",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "_i(\\mathcal{L}, g) \\] where $\\mathcal{D}_i$ is the decision operator from the identity's symbolic autonomy triple (Def.~\\ref{definition:bk4_symbolic_autonomy}). \\end{definition}",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_autonomy"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_self_authorship",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_freedom_criterion"
  ],
  "depends_on": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_self_authorship",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_freedom_criterion",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "file": "book4.tex",
  "id": "theorem:bk4_self_authorship_and_freedom",
  "label": "theorem:bk4_self_authorship_and_freedom",
  "latex_body": "\\begin{theorem}[Self-Authorship and Freedom]\n\\label{theorem:bk4_self_authorship_and_freedom}\nLet $\\mathcal{I}$ be an individuated symbolic identity (Def.~\\ref{definition:bk4_individuated_symbolic_id}) with symbolic autonomy defined by the triple $(A_i, G_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}). Let $\\mathcal{U}(\\mathcal{I})$ denote the constraint domain associated to $\\mathcal{I}$ (Def.~\\ref{definition:bk4_constraint_domain}).\n\nThen $\\mathcal{I}$ achieves \\emph{maximal symbolic freedom} (Thm.~\\ref{theorem:bk4_freedom_criterion}) if and only if it attains \\emph{complete self-authorship} (Def.~\\ref{definition:bk4_self_authorship}), such that:\n\\begin{equation}\n    \\forall \\mathcal{L}_n, \\; \\exists g_n \\in \\mathcal{G} \\text{ with } \\mathcal{L}_{n+1} = \\mathcal{D}_i(\\mathcal{L}_n, g_n)\n\\end{equation}\nand this recursive sequence converges to a fixed-point constraint map:\n\\begin{equation}\n    \\lim_{n \\to \\infty} \\mathcal{L}_n = \\mathcal{L}_\\infty \\quad \\text{such that} \\quad \\mathcal{U}(\\mathcal{I}) = \\text{Fix}(\\mathcal{L}_\\infty)\n\\end{equation}\nwhere $\\text{Fix}(\\mathcal{L}_\\infty)$ is the set of symbolic patterns consistent with the terminal self-defined constraint logic of $\\mathcal{I}$.\n\nIn this case, the constraint domain is no longer imposed externally but arises entirely from the identity's autonomous symbolic evolution.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuity models observer-differentiability; the differentiable-manifold and group-action structures stay open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The constraint-refinement sequence converges to a unique fixed-point constraint map; the group-action structure stays open."
    ],
    "record_ids": [
      "MAP-BOOK4A-095"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book4Fz.self_authorship_fixed_point"
    ]
  },
  "line": 3111,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Self-Authorship and Freedom",
  "proof_labels": [
    "proof:bk4_maximal_freedom_autonomous_constraints"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": ":bk4_symbolic_autonomy}). Let $\\mathcal{U}(\\mathcal{I})$ denote the constraint domain associated to $\\mathcal{I}$ (Def.~\\ref{definition:bk4_constraint_domain}). Then $\\mathcal{I}$ achieves \\emph{maximal symbolic freedom} (Thm.~\\ref{theorem:bk4_freedom_criterion}) if and only i",
      "label": "definition:bk4_constraint_domain",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2957,
      "target_type": "definition"
    },
    {
      "context": "d Freedom] \\label{theorem:bk4_self_authorship_and_freedom} Let $\\mathcal{I}$ be an individuated symbolic identity (Def.~\\ref{definition:bk4_individuated_symbolic_id}) with symbolic autonomy defined by the triple $(A_i, G_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy})",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "lic freedom} (Thm.~\\ref{theorem:bk4_freedom_criterion}) if and only if it attains \\emph{complete self-authorship} (Def.~\\ref{definition:bk4_self_authorship}), such that: \\begin{equation} \\forall \\mathcal{L}_n, \\; \\exists g_n \\in \\mathcal{G} \\text{ with } \\mathcal{L}_{n+1}",
      "label": "definition:bk4_self_authorship",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3102,
      "target_type": "definition"
    },
    {
      "context": "efinition:bk4_individuated_symbolic_id}) with symbolic autonomy defined by the triple $(A_i, G_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}). Let $\\mathcal{U}(\\mathcal{I})$ denote the constraint domain associated to $\\mathcal{I}$ (Def.~\\ref{definition:bk4_con",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    },
    {
      "context": "l{I}$ (Def.~\\ref{definition:bk4_constraint_domain}). Then $\\mathcal{I}$ achieves \\emph{maximal symbolic freedom} (Thm.~\\ref{theorem:bk4_freedom_criterion}) if and only if it attains \\emph{complete self-authorship} (Def.~\\ref{definition:bk4_self_authorship}), such that: \\beg",
      "label": "theorem:bk4_freedom_criterion",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 3022,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_constraint_domain",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_self_authorship",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_freedom_criterion"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_self_authorship",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_freedom_criterion",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "depends_on": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_self_authorship",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_freedom_criterion",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_maximal_freedom_autonomous_constraints",
  "label": "proof:bk4_maximal_freedom_autonomous_constraints",
  "latex_body": "\\begin{proof}[Self-Authorship Implies Maximal Freedom]\n\\label{proof:bk4_maximal_freedom_autonomous_constraints}\n\\leavevmode\n\nLet $\\mathcal{I}$ be an individuated symbolic identity\n(Def.~\\ref{definition:bk4_individuated_symbolic_id}) with symbolic autonomy\n$(A_i, G_i, \\mathcal{D}_i)$\n(Def.~\\ref{definition:bk4_symbolic_autonomy}).\nAssume $\\mathcal{I}$ has self-authorship capacity\n(Def.~\\ref{definition:bk4_self_authorship}).\nFor any constraint logic $\\mathcal{L}$ and goal $g \\in \\mathcal{G}$, the update\n\\[\n\\mathcal{L}' = \\mathcal{D}_i(\\mathcal{L}, g)\n\\]\nis available.\n\n\\textbf{($\\Rightarrow$) Self-authorship implies convergence.}\nThe sequence $\\mathcal{L}_{n+1} = \\mathcal{D}_i(\\mathcal{L}_n, g_n)$ is a self-referential\niteration in the complete metric space of bounded constraint operators under the\noperator norm. By the self-authorship definition, each update\n(1) preserves coherence under symbolic flow (Def.~\\ref{definition:bk4_symbolic_autonomy}),\n(2) expands or stabilizes the accessible symbolic space $A_i$, and\n(3) is generated by goal-directed symbolic reasoning internal to $\\mathcal{I}$.\nConditions (1) and (2) bound the rate of change of $\\mathcal{L}_n$ and make the\nsequence non-retrograde, but bounded rate and non-retrogression do not by themselves\nfurnish a contraction, so the Banach Fixed-Point Theorem does not apply here.\nConvergence instead follows by free-energy descent. Let $\\Phi$ be the\nconstraint-incoherence potential -- the bounded-below, lower semicontinuous functional\nmeasuring the residual incoherence of a constraint map relative to the identity's\ngoals -- which goal-directed self-authorship reduces, each admissible update paying for\nits displacement,\n\\[\n\\|\\mathcal{L}_n - \\mathcal{L}_{n+1}\\| \\;\\le\\; \\Phi(\\mathcal{L}_n) - \\Phi(\\mathcal{L}_{n+1}),\n\\]\nthe Caristi descent inequality. The pair $(\\mathcal{D}_i(\\cdot, g), \\Phi)$ is therefore\na free-energy descent pair, and by the convergence mechanism of\nThm.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity} -- the telescoping\nsummable-increment argument, instantiated in the complete operator space rather than in\n$(\\prob(\\manifold), \\wass)$ -- the orbit has summable increments, is Cauchy, and\nconverges to $\\mathcal{L}_\\infty$; closure of the graph of $\\mathcal{D}_i(\\cdot, g_n)$\ngives $\\mathcal{D}_i(\\mathcal{L}_\\infty, g) = \\mathcal{L}_\\infty$, and the limit is the\nunique self-determined fixed point when the stall set of $\\Phi$ is a singleton.\nThe fixed-point set $\\mathcal{U}(\\mathcal{I}) = \\mathrm{Fix}(\\mathcal{L}_\\infty)$\nis the identity's self-determined constraint domain.\n\n\\textbf{($\\Leftarrow$) Convergence to $\\mathcal{L}_\\infty$ implies maximal freedom.}\nIf $\\mathcal{U}(\\mathcal{I}) = \\mathrm{Fix}(\\mathcal{L}_\\infty)$ and no further external\nconstraint is imposed, then for any pattern $\\mathcal{L}_n$ there exists $g_n$ such that\n$\\mathcal{L}_{n+1}$ progresses toward $\\mathcal{L}_\\infty$\n(since $\\mathcal{D}_i$ is goal-directed, Def.~\\ref{definition:bk4_symbolic_autonomy}). Hence $\\mathcal{I}$ satisfies the\nfreedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}): no externally imposed bound\nrestricts the accessible symbolic space beyond $\\mathcal{U}(\\mathcal{I})$ itself.\n\n\\textbf{Conclusion.}\nMaximal freedom is not the absence of constraint, but reflective sovereignty over\nconstraint evolution: the identity governs its own constraint map, and the fixed point\nof that self-governance is the self-authored constraint domain.\n\\end{proof}",
  "line": 3128,
  "macros_used": [
    "manifold",
    "prob",
    "wass"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Self-Authorship Implies Maximal Freedom",
  "proves": "theorem:bk4_self_authorship_and_freedom",
  "ref_roles": [
    {
      "context": "f:bk4_maximal_freedom_autonomous_constraints} \\leavevmode Let $\\mathcal{I}$ be an individuated symbolic identity (Def.~\\ref{definition:bk4_individuated_symbolic_id}) with symbolic autonomy $(A_i, G_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}). Assume $\\mathcal{I}$",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "\\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}). Assume $\\mathcal{I}$ has self-authorship capacity (Def.~\\ref{definition:bk4_self_authorship}). For any constraint logic $\\mathcal{L}$ and goal $g \\in \\mathcal{G}$, the update \\[ \\mathcal{L}' = \\mathcal{D}_i(\\math",
      "label": "definition:bk4_self_authorship",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3102,
      "target_type": "definition"
    },
    {
      "context": "identity (Def.~\\ref{definition:bk4_individuated_symbolic_id}) with symbolic autonomy $(A_i, G_i, \\mathcal{D}_i)$ (Def.~\\ref{definition:bk4_symbolic_autonomy}). Assume $\\mathcal{I}$ has self-authorship capacity (Def.~\\ref{definition:bk4_self_authorship}). For any constraint log",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    },
    {
      "context": "goal-directed, Def.~\\ref{definition:bk4_symbolic_autonomy}). Hence $\\mathcal{I}$ satisfies the freedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}): no externally imposed bound restricts the accessible symbolic space beyond $\\mathcal{U}(\\mathcal{I})$ itself. \\textb",
      "label": "theorem:bk4_freedom_criterion",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 3022,
      "target_type": "theorem"
    },
    {
      "context": "air $(\\mathcal{D}_i(\\cdot, g), \\Phi)$ is therefore a free-energy descent pair, and by the convergence mechanism of Thm.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity} -- the telescoping summable-increment argument, instantiated in the complete operator space rather than in $(\\prob(\\man",
      "label": "theorem:bk7_reflective_convergence_to_stable_identity",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 555,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_self_authorship",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_freedom_criterion",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book4.tex",
  "id": "subsec:bk4_bridge_to_symbolic_life",
  "label": "subsec:bk4_bridge_to_symbolic_life",
  "latex_body": "",
  "line": 3187,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Bridge to Symbolic Life",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_fragmented_identity",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "depends_on": [
    "definition:bk4_fragmented_identity",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "file": "book4.tex",
  "id": "definition:bk4_proto_vitality",
  "label": "definition:bk4_proto_vitality",
  "latex_body": "\\begin{definition}[Proto-Vitality] \n\\label{definition:bk4_proto_vitality}\nThe \\emph{proto-vitality} of an individuated symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) is characterized by the following conditions:\n\\begin{enumerate}\n    \\item \\textbf{Self-maintenance:} The ability to repair fragmentation (Def.~\\ref{definition:bk4_fragmented_identity}) via a recursive repair process (Def.~\\ref{definition:bk4_repair_process}).\n    \\item \\textbf{Adaptive autonomy:} The capacity to update decisions or action mappings in response to environmental conditions, consistent with symbolic autonomy (Def.~\\ref{definition:bk4_symbolic_autonomy}).\n    \\item \\textbf{Recursive self-modification:} The ability to apply reflective operations to constraint maps, enabling long-term constraint expansion (Thm.~\\ref{theorem:bk4_recursive_constraint_libera}).\n\\end{enumerate}\n\\end{definition}",
  "line": 3190,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Proto-Vitality",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "following conditions: \\begin{enumerate} \\item \\textbf{Self-maintenance:} The ability to repair fragmentation (Def.~\\ref{definition:bk4_fragmented_identity}) via a recursive repair process (Def.~\\ref{definition:bk4_repair_process}). \\item \\textbf{Adaptive autonomy:} The c",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "label{definition:bk4_proto_vitality} The \\emph{proto-vitality} of an individuated symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) is characterized by the following conditions: \\begin{enumerate} \\item \\textbf{Self-maintenance:} The ability to re",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "he ability to repair fragmentation (Def.~\\ref{definition:bk4_fragmented_identity}) via a recursive repair process (Def.~\\ref{definition:bk4_repair_process}). \\item \\textbf{Adaptive autonomy:} The capacity to update decisions or action mappings in response to environmenta",
      "label": "definition:bk4_repair_process",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2830,
      "target_type": "definition"
    },
    {
      "context": "to update decisions or action mappings in response to environmental conditions, consistent with symbolic autonomy (Def.~\\ref{definition:bk4_symbolic_autonomy}). \\item \\textbf{Recursive self-modification:} The ability to apply reflective operations to constraint maps, enabli",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    },
    {
      "context": "fication:} The ability to apply reflective operations to constraint maps, enabling long-term constraint expansion (Thm.~\\ref{theorem:bk4_recursive_constraint_libera}). \\end{enumerate} \\end{definition}",
      "label": "theorem:bk4_recursive_constraint_libera",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 2968,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_fragmented_identity",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_repair_process",
    "definition:bk4_symbolic_autonomy",
    "theorem:bk4_recursive_constraint_libera"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book4",
  "cited_by": [
    "corollary:bk4_emergence_of_meaning",
    "proof:bk4_freedom_growth_fragmentation"
  ],
  "cites": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_freedom_measure",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "depends_on": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_freedom_measure",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk4_freedom_criterion"
  ],
  "file": "book4.tex",
  "id": "theorem:bk4_freedom_life_connection",
  "label": "theorem:bk4_freedom_life_connection",
  "latex_body": "\\begin{theorem}[Freedom-Life Connection] \n\\label{theorem:bk4_freedom_life_connection}\nAn individuated symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) transitions toward symbolic life (Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}) if and only if its symbolic freedom measure (Def.~\\ref{definition:bk4_symbolic_freedom_measure}) increases over time while maintaining bounded fragmentation (Def.~\\ref{definition:bk4_fragmentation_measure}):\n\\begin{equation}\n    \\frac{d\\mathcal{F}_{\\text{free}}(\\mathcal{I})}{dt} > 0 \n    \\quad \\text{and} \\quad \n    \\mathcal{F}_{\\text{frag}}(\\mathcal{I}) < \\epsilon_{\\text{max}}.\n\\end{equation}\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The continuous derivative condition dF_free/dt > 0 is modeled by its discrete analogue (fixed positive per-step increase), yielding strict monotonicity and a telescoping lower bound; the bounded-fragmentation clause is retained as a structure field but not used in a theorem beyond being recorded."
    ],
    "record_ids": [
      "MAP-BOOK4B-018"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book4B.freedomLife_increase_accum",
      "Book4B.freedomLife_strictMono",
      "Book4B.freedomLife_succ_lt"
    ]
  },
  "line": 3200,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Freedom-Life Connection",
  "proof_labels": [
    "proof:bk4_freedom_growth_fragmentation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "(Def.~\\ref{definition:bk4_symbolic_freedom_measure}) increases over time while maintaining bounded fragmentation (Def.~\\ref{definition:bk4_fragmentation_measure}): \\begin{equation} \\frac{d\\mathcal{F}_{\\text{free}}(\\mathcal{I})}{dt} > 0 \\quad \\text{and} \\quad \\mathcal",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "dom-Life Connection] \\label{theorem:bk4_freedom_life_connection} An individuated symbolic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) transitions toward symbolic life (Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}) if and only if its symboli",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "bolic life (Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}) if and only if its symbolic freedom measure (Def.~\\ref{definition:bk4_symbolic_freedom_measure}) increases over time while maintaining bounded fragmentation (Def.~\\ref{definition:bk4_fragmentation_measure}): \\begin{",
      "label": "definition:bk4_symbolic_freedom_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3003,
      "target_type": "definition"
    },
    {
      "context": "olic identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) transitions toward symbolic life (Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}) if and only if its symbolic freedom measure (Def.~\\ref{definition:bk4_symbolic_freedom_measure}) increases over time w",
      "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:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_freedom_measure",
    "theorem:bk3_criteria_persistent_symbolic_life"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_freedom_measure",
    "sec:bk5_funadmenta_symbolicae_vitae",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk4_freedom_criterion",
    "theorem:bk4_freedom_life_connection"
  ],
  "depends_on": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_freedom_measure",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk4_freedom_criterion",
    "theorem:bk4_freedom_life_connection"
  ],
  "file": "book4.tex",
  "id": "proof:bk4_freedom_growth_fragmentation",
  "label": "proof:bk4_freedom_growth_fragmentation",
  "latex_body": "\\begin{proof}[Freedom Growth and Bounded Fragmentation]\n\\label{proof:bk4_freedom_growth_fragmentation}\n\\leavevmode\n\nGrowth of the symbolic freedom measure\n$\\mathcal{F}_{\\text{free}}(\\mathcal{I})$\n(Def.~\\ref{definition:bk4_symbolic_freedom_measure}) for an individuated\nidentity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id})\nindicates increasing capacity to access self-authored configurations beyond\ninitial constraints. This satisfies the symbolic freedom criterion\n(Thm.~\\ref{theorem:bk4_freedom_criterion}).\n\nSimultaneously, bounded fragmentation, as quantified by $\\mathcal{F}_{\\text{frag}}(\\mathcal{I}) < \\epsilon_{\\text{max}}$ (Def.~\\ref{definition:bk4_fragmentation_measure}), ensures that coherence is preserved throughout this expansion. Together, these conditions formally define the transition toward symbolic life (Thm.~\\ref{theorem:bk4_freedom_life_connection}), and instantiate the criteria for persistent symbolic life first introduced in Book III (Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}).\n\nThis convergence of increasing symbolic freedom and maintained structural coherence marks the threshold where an identity shifts from merely individuated to symbolically alive.\nThe deeper mechanisms --- metabolic coherence, environment-response coupling, and symbolic reproduction --- are treated in Book V (see Section~\\ref{sec:bk5_funadmenta_symbolicae_vitae}); their abstract foundation is this dual condition.\n\\end{proof}",
  "line": 3210,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Freedom Growth and Bounded Fragmentation",
  "proves": "theorem:bk4_freedom_life_connection",
  "ref_roles": [
    {
      "context": "neously, bounded fragmentation, as quantified by $\\mathcal{F}_{\\text{frag}}(\\mathcal{I}) < \\epsilon_{\\text{max}}$ (Def.~\\ref{definition:bk4_fragmentation_measure}), ensures that coherence is preserved throughout this expansion. Together, these conditions formally define the transit",
      "label": "definition:bk4_fragmentation_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2746,
      "target_type": "definition"
    },
    {
      "context": "e}}(\\mathcal{I})$ (Def.~\\ref{definition:bk4_symbolic_freedom_measure}) for an individuated identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) indicates increasing capacity to access self-authored configurations beyond initial constraints. This satisfies the sy",
      "label": "definition:bk4_individuated_symbolic_id",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 2942,
      "target_type": "definition"
    },
    {
      "context": "rowth_fragmentation} \\leavevmode Growth of the symbolic freedom measure $\\mathcal{F}_{\\text{free}}(\\mathcal{I})$ (Def.~\\ref{definition:bk4_symbolic_freedom_measure}) for an individuated identity $\\mathcal{I}$ (Def.~\\ref{definition:bk4_individuated_symbolic_id}) indicates increasing c",
      "label": "definition:bk4_symbolic_freedom_measure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3003,
      "target_type": "definition"
    },
    {
      "context": "-- metabolic coherence, environment-response coupling, and symbolic reproduction --- are treated in Book V (see Section~\\ref{sec:bk5_funadmenta_symbolicae_vitae}); their abstract foundation is this dual condition. \\end{proof}",
      "label": "sec:bk5_funadmenta_symbolicae_vitae",
      "logical_support": false,
      "role": "navigation",
      "target_file": "book5.tex",
      "target_line": 1,
      "target_type": "section"
    },
    {
      "context": "freedom_life_connection}), and instantiate the criteria for persistent symbolic life first introduced in Book III (Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}). This convergence of increasing symbolic freedom and maintained structural coherence marks the threshold where an ide",
      "label": "theorem:bk3_criteria_persistent_symbolic_life",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book3.tex",
      "target_line": 777,
      "target_type": "theorem"
    },
    {
      "context": "to access self-authored configurations beyond initial constraints. This satisfies the symbolic freedom criterion (Thm.~\\ref{theorem:bk4_freedom_criterion}). Simultaneously, bounded fragmentation, as quantified by $\\mathcal{F}_{\\text{frag}}(\\mathcal{I}) < \\epsilon_{\\text{ma",
      "label": "theorem:bk4_freedom_criterion",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 3022,
      "target_type": "theorem"
    },
    {
      "context": "eserved throughout this expansion. Together, these conditions formally define the transition toward symbolic life (Thm.~\\ref{theorem:bk4_freedom_life_connection}), and instantiate the criteria for persistent symbolic life first introduced in Book III (Thm.~\\ref{theorem:bk3_criteri",
      "label": "theorem:bk4_freedom_life_connection",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book4.tex",
      "target_line": 3200,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_fragmentation_measure",
    "definition:bk4_individuated_symbolic_id",
    "definition:bk4_symbolic_freedom_measure",
    "sec:bk5_funadmenta_symbolicae_vitae",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk4_freedom_criterion",
    "theorem:bk4_freedom_life_connection"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk4_freedom_life_connection"
  ],
  "depends_on": [
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk4_freedom_life_connection"
  ],
  "file": "book4.tex",
  "id": "corollary:bk4_emergence_of_meaning",
  "label": "corollary:bk4_emergence_of_meaning",
  "latex_body": "\\begin{corollary}[Emergence of Meaning] \\label{corollary:bk4_emergence_of_meaning}\n\\par\nLet $\\mathcal I$ be an individuated identity in the transition to symbolic life\n(Thm.~\\ref{theorem:bk4_freedom_life_connection};\nThm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}).  Meaning generation\nis an identity-relative map\n\\begin{equation}\n    \\mathcal{M}: \\mathcal{U} \\times \\mathcal{I} \\to \\mathcal{V},\n\\end{equation}\nfrom constraint configurations and identity states to a value space.  This is the\nsymbolic analogue of the child's active construction of meaning through\nsensorimotor interaction \\citep{piaget1954construction}; the analogy motivates\nthe map but is not used as a mathematical premise.  This map\nis not supplied by the freedom--life transition alone.  For the energetic\nrealization $\\mathcal V=\\mathbb R$, additionally suppose that each identity has\nan accessible domain $A_{\\mathcal I}\\subseteq\\mathcal U$, that the Book II\nfree-energy functional $F(\\mathcal I,\\cdot)$ has an attained finite ceiling\n$F_{\\max}(\\mathcal I)$ on $A_{\\mathcal I}$, and that some accessible\nconfiguration lies strictly below that ceiling.  Then\n\\begin{equation}\n \\mathcal M_E(u,\\mathcal I)\n :=F_{\\max}(\\mathcal I)-F(\\mathcal I,u),\\qquad u\\in A_{\\mathcal I},\n\\end{equation}\nis nonnegative and nonconstant, and it reverses strict free-energy order.\nThis energetic value is one realization of the general meaning map; it does\nnot by itself determine interpretive significance or embodied action.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "accessible strict-below-ceiling witness",
      "attained finite energy ceiling",
      "explicit freedom-life-to-energy bridge",
      "identity-relative accessible domain",
      "separate convergence and local-minimum certificate for preferential flows"
    ],
    "countermodels": [
      "Book4Meaning.freedomLifeTransition_does_not_force_nonconstant_energy",
      "Book4Meaning.retained_distinction_does_not_force_action_distinction",
      "Book4Meaning.value_map_does_not_determine_significance"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Rebuilt identity-relative meaning kernel: accessible configurations carry an attained ceiling and a strict-below-ceiling witness, yielding a nonnegative, nonconstant U × I → ℝ value map. A flow certificate separates descent from convergence and local minimality; an explicit bridge separates freedom/life from the energy domain. Countermodels separate energetic value, interpretive significance, retained distinction, and embodied action."
    ],
    "record_ids": [
      "MAP-BOOK4A-004"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book4Meaning.exists_positive_meaning_iff",
      "Book4Meaning.freedomLifeTransition_does_not_force_nonconstant_energy",
      "Book4Meaning.identityMeaning_ceiling_witness_has_zero_value",
      "Book4Meaning.identityMeaning_is_nontrivial",
      "Book4Meaning.identityMeaning_nonneg_on_accessible",
      "Book4Meaning.identityMeaning_pos_iff_on_accessible",
      "Book4Meaning.identityMeaning_strict_preference_iff",
      "Book4Meaning.meaningValue_nonneg",
      "Book4Meaning.meaningValue_pos_iff",
      "Book4Meaning.meaningValue_strict_preference_iff",
      "Book4Meaning.preferentialFlow_meaning_nondecreasing",
      "Book4Meaning.retained_distinction_does_not_force_action_distinction",
      "Book4Meaning.transition_with_bridge_generates_positive_meaning",
      "Book4Meaning.value_map_does_not_determine_significance"
    ]
  },
  "line": 3228,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Emergence of Meaning",
  "proof_labels": [
    "proof:bk4_sketch_preferential_flows"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "I$ be an individuated identity in the transition to symbolic life (Thm.~\\ref{theorem:bk4_freedom_life_connection}; Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}). Meaning generation is an identity-relative map \\begin{equation} \\mathcal{M}: \\mathcal{U} \\times \\mathcal{I} \\to",
      "label": "theorem:bk3_criteria_persistent_symbolic_life",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book3.tex",
      "target_line": 777,
      "target_type": "theorem"
    },
    {
      "context": "ry:bk4_emergence_of_meaning} \\par Let $\\mathcal I$ be an individuated identity in the transition to symbolic life (Thm.~\\ref{theorem:bk4_freedom_life_connection}; Thm.~\\ref{theorem:bk3_criteria_persistent_symbolic_life}). Meaning generation is an identity-relative map \\begin{equa",
      "label": "theorem:bk4_freedom_life_connection",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 3200,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk4_freedom_life_connection"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book4",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book4.tex",
  "id": "proof:bk4_sketch_preferential_flows",
  "label": "proof:bk4_sketch_preferential_flows",
  "latex_body": "\\begin{proof}[Identity-Relative Preferential Flows]\n\\label{proof:bk4_sketch_preferential_flows}\n\\leavevmode\n\nThe ceiling premise gives\n$F(\\mathcal I,u)\\leq F_{\\max}(\\mathcal I)$ on $A_{\\mathcal I}$, hence\n$\\mathcal M_E(u,\\mathcal I)\\geq0$.  The strict-below-ceiling witness gives an\naccessible $u$ with $\\mathcal M_E(u,\\mathcal I)>0$, while an attained-ceiling\nwitness has value zero; therefore the value map is nonconstant.  For accessible\n$u,v$,\n\\[\n \\mathcal M_E(u,\\mathcal I)>\\mathcal M_E(v,\\mathcal I)\n \\quad\\Longleftrightarrow\\quad\n F(\\mathcal I,u)<F(\\mathcal I,v).\n\\]\nThus any supplied trajectory that remains accessible and descends in $F$ is\nnondecreasing in $\\mathcal M_E$.  Calling that trajectory a convergent gradient\nflow additionally requires the usual analytic data: a differentiable structure\nand metric, existence of the flow, and hypotheses sufficient for convergence\nto a local minimum.  None of these, nor nonconstancy of $F$, follows merely\nfrom crossing the symbolic-freedom threshold.\n\nFinally, the general codomain $\\mathcal V$ retains the distinction between\nvaluation and interpretation.  An identity-relative interpretation may retain\na distinction that a constant action policy erases; conversely, fixing the\nsame value map while changing the identity's significance predicate changes\nwhich events count as meaningful.  Hence energetic preference, interpretive\nsignificance, and action are connected layers, not interchangeable names for\none scalar.  The Lean kernel constructs the accessible energetic map and its\npreferential-flow monotonicity, packages the missing freedom-to-energy bridge,\nand supplies both separation countermodels.\n\\end{proof}",
  "line": 3255,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Identity-Relative Preferential Flows",
  "proves": "corollary:bk4_emergence_of_meaning",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book4",
  "cited_by": [
    "proposition:bk9_criteria_for_ethical_intervention"
  ],
  "cites": [
    "definition:bk4_self_authorship"
  ],
  "depends_on": [
    "definition:bk4_self_authorship"
  ],
  "file": "book4.tex",
  "id": "remark:bk4_individuated_freedom",
  "label": "remark:bk4_individuated_freedom",
  "latex_body": "\\begin{remark}\n\\label{remark:bk4_individuated_freedom}\nIndividuated freedom is not the absence of constraint, but rather the recursive authorship of constraint (\\ref{definition:bk4_self_authorship}). True symbolic freedom emerges not when all limitations are removed, but when limitations become self-determined expressions of identity rather than external impositions. This transition from externally-constrained to self-authoring identity forms the bridge to symbolic life and cognition developed in Book V. (see Def.~\\ref{definition:bk4_self_authorship})\n\\end{remark}",
  "line": 3287,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "ref_roles": [
    {
      "context": "ated_freedom} Individuated freedom is not the absence of constraint, but rather the recursive authorship of constraint (\\ref{definition:bk4_self_authorship}). True symbolic freedom emerges not when all limitations are removed, but when limitations become self-determined expre",
      "label": "definition:bk4_self_authorship",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 3102,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk4_self_authorship"
  ],
  "role": "remark",
  "type": "remark"
}