Scholium Symbolicum — page 3 of 6

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    "lemma:bk1_linear_context_independence",
    "proof:bk1_limitation_linear_reflexive_maps",
    "proof:bk1_minimal_quadratic_sufficiency",
    "proposition:bk1_limitation_linear_reflexive_maps",
    "theorem:bk1_minimal_quadratic_sufficiency"
  ],
  "cites": [
    "definition:bk1_symbolic_category"
  ],
  "depends_on": [
    "definition:bk1_symbolic_category"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_reflexive_update_map",
  "label": "definition:bk1_reflexive_update_map",
  "latex_body": "\\begin{definition}[Reflexive Update Map]\n\\label{definition:bk1_reflexive_update_map}\nA map $\\rho: \\mathcal{S} \\to \\mathcal{S}$ is \\emph{reflexive} if it can modify representations that include itself. Formally, $\\rho$ is reflexive if there exists $\\sigma \\in \\mathcal{S}$ such that $\\rho(\\sigma) = \\tau$ where $\\tau$ contains a symbolic representation of $\\rho$.\n\nThis definition is grounded in the symbolic category structure (Def.~\\ref{definition:bk1_symbolic_category}), where maps and objects are both symbolic entities.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "discrete kernels only: ODE flows, Riemannian geodesics, Hopf-Rinow, and the MSR path integral stay open",
      "the self-representation clause of reflexive maps is interpretive"
    ],
    "countermodels": [],
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    "kernel_certified": true,
    "notes": [
      "Definition row; self-representation clause interpretive."
    ],
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      "constructed"
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    ]
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  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Reflexive Update Map",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "au$ contains a symbolic representation of $\\rho$. This definition is grounded in the symbolic category structure (Def.~\\ref{definition:bk1_symbolic_category}), where maps and objects are both symbolic entities. \\end{definition}",
      "label": "definition:bk1_symbolic_category",
      "logical_support": true,
      "role": "definition_anchor",
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    "definition:bk1_symbolic_category"
  ],
  "role": "definition",
  "type": "definition"
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  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_limitation_linear_reflexive_maps"
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  "cites": [
    "definition:bk1_symbolic_category"
  ],
  "depends_on": [
    "definition:bk1_symbolic_category"
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  "file": "scholium_symbolicum.tex",
  "id": "lemma:bk1_fixed_point_inheritance",
  "label": "lemma:bk1_fixed_point_inheritance",
  "latex_body": "\\begin{lemma}[Fixed Point Inheritance]\n\\label{lemma:bk1_fixed_point_inheritance}\nLet $f: \\mathcal{S} \\to \\mathcal{S}$ be a linear morphism in the symbolic category (Def.~\\ref{definition:bk1_symbolic_category}) and $g: \\mathcal{S} \\to \\mathcal{S}$ any map with fixed point $x$ (i.e., $g(x) = x$). If $f$ is invertible, then $f \\circ g \\circ f^{-1}$ has fixed point $f(x)$.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Proved in the generalized (linearity-free) form: only invertibility of f is used, matching the toolchain guidance to drop unused hypotheses; f∘g∘f⁻¹ fixes f(x) given g fixes x."
    ],
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    "witnesses": [
      "ScholiumA.fixedPointInheritance"
    ]
  },
  "line": 1756,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Fixed Point Inheritance",
  "proof_labels": [
    "proof:bk1_fixed_point_inheritance"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "a:bk1_fixed_point_inheritance} Let $f: \\mathcal{S} \\to \\mathcal{S}$ be a linear morphism in the symbolic category (Def.~\\ref{definition:bk1_symbolic_category}) and $g: \\mathcal{S} \\to \\mathcal{S}$ any map with fixed point $x$ (i.e., $g(x) = x$). If $f$ is invertible, then $f \\c",
      "label": "definition:bk1_symbolic_category",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1738,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_category"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_symbolic_category"
  ],
  "depends_on": [
    "definition:bk1_symbolic_category"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_fixed_point_inheritance",
  "label": "proof:bk1_fixed_point_inheritance",
  "latex_body": "\\begin{proof}[Conjugation Preserves Fixed Points]\n\\label{proof:bk1_fixed_point_inheritance}\n\\leavevmode\n\nSince \\(f\\) is invertible, \\(f^{-1}(f(x))=x\\). Evaluating the conjugated map at\nthe transported point gives\n\\[\n(f\\circ g\\circ f^{-1})(f(x))=f(g(f^{-1}(f(x))))=f(g(x)).\n\\]\nBecause \\(x\\) is a fixed point of \\(g\\), \\(g(x)=x\\), and therefore\n\\[\n(f\\circ g\\circ f^{-1})(f(x))=f(x).\n\\]\nThus \\(f(x)\\) is a fixed point of the conjugated map. Linearity is compatible\nwith the symbolic-category structure of Def.~\\ref{definition:bk1_symbolic_category};\ninvertibility is the condition needed for the fixed point to be transported and\nreturned without loss.\n\\end{proof}",
  "line": 1760,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Conjugation Preserves Fixed Points",
  "proves": "lemma:bk1_fixed_point_inheritance",
  "ref_roles": [
    {
      "context": "s \\(f(x)\\) is a fixed point of the conjugated map. Linearity is compatible with the symbolic-category structure of Def.~\\ref{definition:bk1_symbolic_category}; invertibility is the condition needed for the fixed point to be transported and returned without loss. \\end{proof}",
      "label": "definition:bk1_symbolic_category",
      "logical_support": true,
      "role": "definition_anchor",
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      "target_line": 1738,
      "target_type": "definition"
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  ],
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    "definition:bk1_symbolic_category"
  ],
  "role": "proof",
  "type": "proof"
}

{
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  "cited_by": [
    "proof:bk1_dual_horizon_unification_principle",
    "proof:bk1_minimal_quadratic_sufficiency",
    "theorem:bk1_dual_horizon_unification_principle"
  ],
  "cites": [
    "corollary:bk1_linear_insufficiency",
    "definition:bk1_reflexive_update_map"
  ],
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    "corollary:bk1_linear_insufficiency",
    "definition:bk1_reflexive_update_map",
    "lemma:bk1_fixed_point_inheritance"
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  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_limitation_linear_reflexive_maps",
  "label": "proposition:bk1_limitation_linear_reflexive_maps",
  "latex_body": "\\begin{proposition}[Limitation of Linear Reflexive Maps]\n\\label{proposition:bk1_limitation_linear_reflexive_maps}\nLet $\\mathcal{S}$ be a symbolic category admitting only linear morphisms. Then, in line with Cor.~\\ref{corollary:bk1_linear_insufficiency}, no reflexive update map $\\rho: \\mathcal{S} \\to \\mathcal{S}$ (Def.~\\ref{definition:bk1_reflexive_update_map}) can alter its own fixed point structure while preserving the category's symbolic coherence.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "discrete kernels only: ODE flows, Riemannian geodesics, Hopf-Rinow, and the MSR path integral stay open",
      "the self-representation clause of reflexive maps is interpretive"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Linear updates cannot clear or bend their fixed locus; an affine update can have none."
    ],
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      "MAP-SCHOLIUM_A-070"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumDyn.affine_escapes_fixed_points",
      "ScholiumDyn.linear_fixed_points_closed",
      "ScholiumDyn.linear_has_fixed_point"
    ]
  },
  "line": 1779,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Limitation of Linear Reflexive Maps",
  "proof_labels": [
    "proof:bk1_limitation_linear_reflexive_maps"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "inear_reflexive_maps} Let $\\mathcal{S}$ be a symbolic category admitting only linear morphisms. Then, in line with Cor.~\\ref{corollary:bk1_linear_insufficiency}, no reflexive update map $\\rho: \\mathcal{S} \\to \\mathcal{S}$ (Def.~\\ref{definition:bk1_reflexive_update_map}) can alter",
      "label": "corollary:bk1_linear_insufficiency",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1538,
      "target_type": "corollary"
    },
    {
      "context": "e with Cor.~\\ref{corollary:bk1_linear_insufficiency}, no reflexive update map $\\rho: \\mathcal{S} \\to \\mathcal{S}$ (Def.~\\ref{definition:bk1_reflexive_update_map}) can alter its own fixed point structure while preserving the category's symbolic coherence. \\end{proposition}",
      "label": "definition:bk1_reflexive_update_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1749,
      "target_type": "definition"
    }
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    "corollary:bk1_linear_insufficiency",
    "definition:bk1_reflexive_update_map"
  ],
  "role": "proposition",
  "type": "proposition"
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    "definition:bk1_reflexive_update_map",
    "lemma:bk1_fixed_point_inheritance"
  ],
  "depends_on": [
    "corollary:bk1_linear_insufficiency",
    "definition:bk1_reflexive_update_map",
    "lemma:bk1_fixed_point_inheritance"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_limitation_linear_reflexive_maps",
  "label": "proof:bk1_limitation_linear_reflexive_maps",
  "latex_body": "\\begin{proof}[Linear Coherence Cannot Move Its Own Fixed Structure]\n\\label{proof:bk1_limitation_linear_reflexive_maps}\n\\leavevmode\n\nIn a category admitting only linear morphisms, any coherent change of coordinates\nor representation acts by linear transport. Lem.~\\ref{lemma:bk1_fixed_point_inheritance}\nshows that such transport carries fixed points by conjugation: fixed-point\nstructure is preserved as \\(x\\mapsto f(x)\\), not internally altered by the\nlinear morphism itself. A reflexive update map, however, must modify a\nrepresentation that includes the updater (Def.~\\ref{definition:bk1_reflexive_update_map});\nchanging its own fixed-point structure therefore requires a self-interaction\nterm rather than mere linear transport. Cor.~\\ref{corollary:bk1_linear_insufficiency}\nrules out precisely that capacity for linear systems. Hence a purely linear\nsymbolic category cannot support such a reflexive update while preserving\nsymbolic coherence.\n\\end{proof}",
  "line": 1783,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Linear Coherence Cannot Move Its Own Fixed Structure",
  "proves": "proposition:bk1_limitation_linear_reflexive_maps",
  "ref_roles": [
    {
      "context": "anging its own fixed-point structure therefore requires a self-interaction term rather than mere linear transport. Cor.~\\ref{corollary:bk1_linear_insufficiency} rules out precisely that capacity for linear systems. Hence a purely linear symbolic category cannot support such a ref",
      "label": "corollary:bk1_linear_insufficiency",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1538,
      "target_type": "corollary"
    },
    {
      "context": "e linear morphism itself. A reflexive update map, however, must modify a representation that includes the updater (Def.~\\ref{definition:bk1_reflexive_update_map}); changing its own fixed-point structure therefore requires a self-interaction term rather than mere linear transport.",
      "label": "definition:bk1_reflexive_update_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1749,
      "target_type": "definition"
    },
    {
      "context": "ry admitting only linear morphisms, any coherent change of coordinates or representation acts by linear transport. Lem.~\\ref{lemma:bk1_fixed_point_inheritance} shows that such transport carries fixed points by conjugation: fixed-point structure is preserved as \\(x\\mapsto f(x)\\),",
      "label": "lemma:bk1_fixed_point_inheritance",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1756,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "corollary:bk1_linear_insufficiency",
    "definition:bk1_reflexive_update_map",
    "lemma:bk1_fixed_point_inheritance"
  ],
  "role": "proof",
  "type": "proof"
}

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  "file": "scholium_symbolicum.tex",
  "id": "subsec:bk1_minimal_quadratic_sufficiency",
  "label": "subsec:bk1_minimal_quadratic_sufficiency",
  "latex_body": "",
  "line": 1800,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Minimal Quadratic Sufficiency",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
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    "definition:bk1_local_semantic_independence",
    "definition:bk1_resolution_cost",
    "definition:bk1_symbolic_connection",
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    "definition:bk1_symbolic_riemann_tensor",
    "definition:bk6_symbolic_manifold_structure"
  ],
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    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_symbolic_manifold_feature_maps",
  "label": "definition:bk1_symbolic_manifold_feature_maps",
  "latex_body": "\\begin{definition}[Symbolic Manifold and Feature Maps]\n\\label{definition:bk1_symbolic_manifold_feature_maps}\nA \\emph{symbolic manifold} $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) is a smooth manifold whose points represent symbolic states. A \\emph{symbolic feature map} $\\phi: M \\to \\mathbb{R}$ extracts semantic content from symbolic states. The collection $\\Phi(M) = \\{\\phi_i\\}_{i \\in I}$ forms a coordinate system for the semantic content of $M$.\n\\end{definition}",
  "line": 1803,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Manifold and Feature Maps",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "c Manifold and Feature Maps] \\label{definition:bk1_symbolic_manifold_feature_maps} A \\emph{symbolic manifold} $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) is a smooth manifold whose points represent symbolic states. A \\emph{symbolic feature map} $\\phi: M \\to \\mathbb{R}$ ex",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold"
  ],
  "role": "definition",
  "type": "definition"
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  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk1_symbolic_connection",
    "lemma:bk1_linear_context_independence",
    "proof:bk1_linear_context_independence",
    "proof:bk1_minimal_quadratic_sufficiency",
    "theorem:bk1_minimal_quadratic_sufficiency"
  ],
  "cites": [
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "depends_on": [
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  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_symbolic_coupling",
  "label": "definition:bk1_symbolic_coupling",
  "latex_body": "\\begin{definition}[Symbolic Coupling]\n\\label{definition:bk1_symbolic_coupling}\nA \\emph{symbolic coupling} is a map $\\mathcal{C}: M \\to \\mathbb{R}$ that integrates symbolic features (Def.~\\ref{definition:bk1_symbolic_manifold_feature_maps}). The coupling is:\n\\begin{itemize}\n  \\item \\emph{Linear} if $\\mathcal{C}(x) = \\sum_i \\beta_i \\phi_i(x)$ for constants $\\beta_i$;\n  \\item \\emph{Quadratic} if $\\mathcal{C}(x) = \\sum_{i,j} \\alpha_{ij} \\phi_i(x)\\phi_j(x)$ where $(\\alpha_{ij})$ is a symmetric matrix.\n\\end{itemize}\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Same countermodel as bk1_symbolic_coupling_basis; only the concrete linear-vs-quadratic distinction is witnessed, not the general definition over an arbitrary feature basis."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-019"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumA.quadratic_not_linear"
    ]
  },
  "line": 1808,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Coupling",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "c_coupling} A \\emph{symbolic coupling} is a map $\\mathcal{C}: M \\to \\mathbb{R}$ that integrates symbolic features (Def.~\\ref{definition:bk1_symbolic_manifold_feature_maps}). The coupling is: \\begin{itemize} \\item \\emph{Linear} if $\\mathcal{C}(x) = \\sum_i \\beta_i \\phi_i(x)$ for constants $",
      "label": "definition:bk1_symbolic_manifold_feature_maps",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1803,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "role": "definition",
  "type": "definition"
}

{
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  "cited_by": [
    "proof:bk1_minimal_quadratic_sufficiency"
  ],
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    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling"
  ],
  "depends_on": [
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling",
    "theorem:bk1_reflexivity_quadratic"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "lemma:bk1_linear_context_independence",
  "label": "lemma:bk1_linear_context_independence",
  "latex_body": "\\begin{lemma}[Context-Independence of Linear Coupling]\n\\label{lemma:bk1_linear_context_independence}\nLinear symbolic couplings (Def.~\\ref{definition:bk1_symbolic_coupling}) cannot encode context-dependent meaning or self-reference (cf. Def.~\\ref{definition:bk1_reflexive_update_map}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "The countermodel gives one concrete instance of 'linear coupling cannot encode this quadratic coupling'; the lemma's general universal claim over all context-dependent meanings is not modeled."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-020"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumA.quadratic_not_linear"
    ]
  },
  "line": 1817,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Context-Independence of Linear Coupling",
  "proof_labels": [
    "proof:bk1_linear_context_independence"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "lings (Def.~\\ref{definition:bk1_symbolic_coupling}) cannot encode context-dependent meaning or self-reference (cf. Def.~\\ref{definition:bk1_reflexive_update_map}). \\end{lemma}",
      "label": "definition:bk1_reflexive_update_map",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1749,
      "target_type": "definition"
    },
    {
      "context": "[Context-Independence of Linear Coupling] \\label{lemma:bk1_linear_context_independence} Linear symbolic couplings (Def.~\\ref{definition:bk1_symbolic_coupling}) cannot encode context-dependent meaning or self-reference (cf. Def.~\\ref{definition:bk1_reflexive_update_map}). \\end{l",
      "label": "definition:bk1_symbolic_coupling",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1808,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling"
  ],
  "role": "lemma",
  "type": "lemma"
}

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  ],
  "depends_on": [
    "definition:bk1_symbolic_coupling",
    "theorem:bk1_reflexivity_quadratic"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_linear_context_independence",
  "label": "proof:bk1_linear_context_independence",
  "latex_body": "\\begin{proof}[Linearity Has No Mixed Context Term]\n\\label{proof:bk1_linear_context_independence}\n\\leavevmode\n\nLet \\(\\mathcal{C}\\) be linear, so\n\\(\\mathcal{C}(x)=\\sum_i\\beta_i\\phi_i(x)\\) by\nDef.~\\ref{definition:bk1_symbolic_coupling}. For symbolic states \\(x_A\\) and\n\\(x_B\\) representing two contexts, linearity gives\n\\[\n\\mathcal{C}(x_A+x_B)=\\mathcal{C}(x_A)+\\mathcal{C}(x_B).\n\\]\nThe expression contains no mixed term of the form\n\\(\\phi_i(x_A)\\phi_j(x_B)\\), and therefore no coefficient can record how the\nmeaning of one feature changes in the presence of the other. But\nThm.~\\ref{theorem:bk1_reflexivity_quadratic} identifies robust self-reference\nand context-dependent meaning with exactly such quadratic interaction terms.\nThus a linear coupling can superpose contextual contributions, but it cannot\nencode context-dependent meaning or self-reference.\n\\end{proof}",
  "line": 1821,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Linearity Has No Mixed Context Term",
  "proves": "lemma:bk1_linear_context_independence",
  "ref_roles": [
    {
      "context": "_context_independence} \\leavevmode Let \\(\\mathcal{C}\\) be linear, so \\(\\mathcal{C}(x)=\\sum_i\\beta_i\\phi_i(x)\\) by Def.~\\ref{definition:bk1_symbolic_coupling}. For symbolic states \\(x_A\\) and \\(x_B\\) representing two contexts, linearity gives \\[ \\mathcal{C}(x_A+x_B)=\\mathcal{C}",
      "label": "definition:bk1_symbolic_coupling",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1808,
      "target_type": "definition"
    },
    {
      "context": ", and therefore no coefficient can record how the meaning of one feature changes in the presence of the other. But Thm.~\\ref{theorem:bk1_reflexivity_quadratic} identifies robust self-reference and context-dependent meaning with exactly such quadratic interaction terms. Thus a li",
      "label": "theorem:bk1_reflexivity_quadratic",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1561,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_coupling",
    "theorem:bk1_reflexivity_quadratic"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_minimal_quadratic_sufficiency",
    "proof:bk1_symbolic_emergence_and_curvature",
    "theorem:bk1_minimal_quadratic_sufficiency",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_horizon_structure",
  "label": "definition:bk1_horizon_structure",
  "latex_body": "\\begin{definition}[Horizon Structure]\n\\label{definition:bk1_horizon_structure}\nA \\emph{horizon structure} $\\mathcal{H}$ on symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) assigns to each point $x \\in M$ a subspace $\\mathcal{H}_x \\subset T_x M$ representing the locally accessible directions of meaning evolution from state $x$, bounded by the drift field $D$ (Def.~\\ref{definition:bk1_drift_field}).\n\\end{definition}",
  "line": 1841,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Horizon Structure",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "epresenting the locally accessible directions of meaning evolution from state $x$, bounded by the drift field $D$ (Def.~\\ref{definition:bk1_drift_field}). \\end{definition}",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "cture] \\label{definition:bk1_horizon_structure} A \\emph{horizon structure} $\\mathcal{H}$ on symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}) assigns to each point $x \\in M$ a subspace $\\mathcal{H}_x \\subset T_x M$ representing the locally accessible direction",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_horizon_structure",
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling"
  ],
  "depends_on": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_horizon_structure",
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling",
    "lemma:bk1_linear_context_independence",
    "proposition:bk1_limitation_linear_reflexive_maps"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_minimal_quadratic_sufficiency",
  "label": "theorem:bk1_minimal_quadratic_sufficiency",
  "latex_body": "\\begin{theorem}[Minimal Quadratic Sufficiency]\n\\label{theorem:bk1_minimal_quadratic_sufficiency}\nA symbolic system capable of:\n\\begin{enumerate}\n  \\item \\emph{Reflexivity}: self-modification of interpretive structures (Def.~\\ref{definition:bk1_reflexive_update_map}),\n  \\item \\emph{Context-sensitivity}: horizon-relative meaning emergence (Def.~\\ref{definition:bk1_horizon_structure}),\n  \\item \\emph{Adaptive stability}: robust identity maintenance under perturbation,\n\\end{enumerate}\nrequires at minimum quadratic symbolic coupling (Def.~\\ref{definition:bk1_symbolic_coupling}). Linear systems are insufficient to encode the interaction effects necessary for recursive modification, horizon dependence, and persistent symbolic identity across drift. This sufficiency condition is complemented by the necessity result: any reflexive, context-sensitive symbolic system must operate in curved (non-Euclidean) space (cf.~Corollary~\\ref{corollary:bk1_non_euclidean_necessity}).\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "curvature coupling, general minimal period, and covariant transport remain open",
      "drift and reflection are jointly supplied and both nonidentity in the concrete witness",
      "the reader/operator and operate action are explicit data; the process description does not enact itself",
      "the recursive phase certificate is finite and observer-relative, not a smooth spinor bundle"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Same countermodel supplies the concrete case of 'linear systems are insufficient'; the reflexivity/context-sensitivity/adaptive-stability sufficiency claims themselves are narrative and not modeled."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-021"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumA.quadratic_not_linear"
    ]
  },
  "line": 1846,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Minimal Quadratic Sufficiency",
  "proof_labels": [
    "proof:bk1_minimal_quadratic_sufficiency"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ty result: any reflexive, context-sensitive symbolic system must operate in curved (non-Euclidean) space (cf.~Corollary~\\ref{corollary:bk1_non_euclidean_necessity}). \\end{theorem}",
      "label": "corollary:bk1_non_euclidean_necessity",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1686,
      "target_type": "corollary"
    },
    {
      "context": "ref{definition:bk1_reflexive_update_map}), \\item \\emph{Context-sensitivity}: horizon-relative meaning emergence (Def.~\\ref{definition:bk1_horizon_structure}), \\item \\emph{Adaptive stability}: robust identity maintenance under perturbation, \\end{enumerate} requires at minimu",
      "label": "definition:bk1_horizon_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1841,
      "target_type": "definition"
    },
    {
      "context": "lic system capable of: \\begin{enumerate} \\item \\emph{Reflexivity}: self-modification of interpretive structures (Def.~\\ref{definition:bk1_reflexive_update_map}), \\item \\emph{Context-sensitivity}: horizon-relative meaning emergence (Def.~\\ref{definition:bk1_horizon_structure}),",
      "label": "definition:bk1_reflexive_update_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1749,
      "target_type": "definition"
    },
    {
      "context": ": robust identity maintenance under perturbation, \\end{enumerate} requires at minimum quadratic symbolic coupling (Def.~\\ref{definition:bk1_symbolic_coupling}). Linear systems are insufficient to encode the interaction effects necessary for recursive modification, horizon depen",
      "label": "definition:bk1_symbolic_coupling",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1808,
      "target_type": "definition"
    }
  ],
  "refs": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_horizon_structure",
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_horizon_structure",
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling",
    "lemma:bk1_linear_context_independence",
    "proposition:bk1_limitation_linear_reflexive_maps"
  ],
  "depends_on": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_horizon_structure",
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling",
    "lemma:bk1_linear_context_independence",
    "proposition:bk1_limitation_linear_reflexive_maps"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_minimal_quadratic_sufficiency",
  "label": "proof:bk1_minimal_quadratic_sufficiency",
  "latex_body": "\\begin{proof}[Linear Coupling Cannot Support the Three Capacities]\n\\label{proof:bk1_minimal_quadratic_sufficiency}\n\\leavevmode\n\nAssume, toward contradiction, that the system has the three stated capacities\nwhile its symbolic coupling is only linear in the sense of\nDef.~\\ref{definition:bk1_symbolic_coupling}. By\nLem.~\\ref{lemma:bk1_linear_context_independence}, a linear coupling has no mixed\ncontext term and therefore cannot encode context-dependent meaning or\nself-reference. This already contradicts the first two capacities: reflexivity\nrequires a self-modifying update map (Def.~\\ref{definition:bk1_reflexive_update_map}),\nand context-sensitivity requires horizon-relative variation\n(Def.~\\ref{definition:bk1_horizon_structure}).\n\nThe stability condition cannot rescue the linear case. Adaptive stability asks\nthat identity persist while drift changes the accessible horizon-relative\ncontext; but Prop.~\\ref{proposition:bk1_limitation_linear_reflexive_maps} shows\nthat purely linear reflexive maps cannot alter their own fixed-point structure\nwhile preserving symbolic coherence. Thus a linear system can preserve\nindependent modes, but it cannot preserve a reflexively updated identity across\ncontextual drift. The next admissible coupling class in\nDef.~\\ref{definition:bk1_symbolic_coupling} is quadratic, with coefficients\n\\(\\alpha_{ij}\\) carrying precisely the feature--feature interaction terms that\nlinearity lacks. Hence any system with the stated capacities requires at minimum\nquadratic symbolic coupling. Cor.~\\ref{corollary:bk1_non_euclidean_necessity}\nidentifies the same obstruction geometrically: reflexive context-sensitivity\nforces curved, non-Euclidean symbolic space.\n\\end{proof}",
  "line": 1856,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Linear Coupling Cannot Support the Three Capacities",
  "proves": "theorem:bk1_minimal_quadratic_sufficiency",
  "ref_roles": [
    {
      "context": "that linearity lacks. Hence any system with the stated capacities requires at minimum quadratic symbolic coupling. Cor.~\\ref{corollary:bk1_non_euclidean_necessity} identifies the same obstruction geometrically: reflexive context-sensitivity forces curved, non-Euclidean symbolic spac",
      "label": "corollary:bk1_non_euclidean_necessity",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1686,
      "target_type": "corollary"
    },
    {
      "context": "map (Def.~\\ref{definition:bk1_reflexive_update_map}), and context-sensitivity requires horizon-relative variation (Def.~\\ref{definition:bk1_horizon_structure}). The stability condition cannot rescue the linear case. Adaptive stability asks that identity persist while drift cha",
      "label": "definition:bk1_horizon_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1841,
      "target_type": "definition"
    },
    {
      "context": "lf-reference. This already contradicts the first two capacities: reflexivity requires a self-modifying update map (Def.~\\ref{definition:bk1_reflexive_update_map}), and context-sensitivity requires horizon-relative variation (Def.~\\ref{definition:bk1_horizon_structure}). The stabi",
      "label": "definition:bk1_reflexive_update_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1749,
      "target_type": "definition"
    },
    {
      "context": "iction, that the system has the three stated capacities while its symbolic coupling is only linear in the sense of Def.~\\ref{definition:bk1_symbolic_coupling}. By Lem.~\\ref{lemma:bk1_linear_context_independence}, a linear coupling has no mixed context term and therefore cannot",
      "label": "definition:bk1_symbolic_coupling",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1808,
      "target_type": "definition"
    },
    {
      "context": "acities while its symbolic coupling is only linear in the sense of Def.~\\ref{definition:bk1_symbolic_coupling}. By Lem.~\\ref{lemma:bk1_linear_context_independence}, a linear coupling has no mixed context term and therefore cannot encode context-dependent meaning or self-reference. T",
      "label": "lemma:bk1_linear_context_independence",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1817,
      "target_type": "lemma"
    },
    {
      "context": "e. Adaptive stability asks that identity persist while drift changes the accessible horizon-relative context; but Prop.~\\ref{proposition:bk1_limitation_linear_reflexive_maps} shows that purely linear reflexive maps cannot alter their own fixed-point structure while preserving symbolic coherenc",
      "label": "proposition:bk1_limitation_linear_reflexive_maps",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1779,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_horizon_structure",
    "definition:bk1_reflexive_update_map",
    "definition:bk1_symbolic_coupling",
    "lemma:bk1_linear_context_independence",
    "proposition:bk1_limitation_linear_reflexive_maps"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "subsec:bk1_symbolic_curvature_and_geometric_structure",
  "label": "subsec:bk1_symbolic_curvature_and_geometric_structure",
  "latex_body": "",
  "line": 1885,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Curvature and Geometric Structure",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "definition:bk1_local_semantic_independence",
    "definition:bk1_resolution_cost",
    "definition:bk1_symbolic_field_curvature_tensor",
    "definition:bk1_symbolic_riemann_tensor",
    "definition:bk4_symbolic_curvature"
  ],
  "cites": [
    "definition:bk1_symbolic_coupling",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "depends_on": [
    "definition:bk1_symbolic_coupling",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_symbolic_connection",
  "label": "definition:bk1_symbolic_connection",
  "latex_body": "\\begin{definition}[Symbolic Connection]\n\\label{definition:bk1_symbolic_connection}\n\\leavevmode\\newline\nGiven a quadratic symbolic coupling\n$\\mathcal{C}(x) = \\sum_{ij} \\alpha_{ij} \\phi_i(x)\\phi_j(x)$\n(see \\ref{definition:bk1_symbolic_coupling}), induced metric\n$g_{ij} = \\alpha_{ij}$ defines a Riemannian structure on $M$\n(Def.~\\ref{definition:bk1_symbolic_manifold}).\nHere $\\phi_i$ are symbolic feature maps\n(see \\ref{definition:bk1_symbolic_manifold_feature_maps}).\nThe corresponding Levi-Civita connection $\\nabla$ is the\n\\emph{symbolic connection}, with Christoffel symbols:\n\\[\n\\Gamma^k_{ij} = \\frac{1}{2} \\sum_l g^{kl} \\left( \\frac{\\partial g_{il}}{\\partial x^j} + \\frac{\\partial g_{jl}}{\\partial x^i} - \\frac{\\partial g_{ij}}{\\partial x^l} \\right)\n\\]\n\\end{definition}",
  "line": 1888,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Connection",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "leavevmode\\newline Given a quadratic symbolic coupling $\\mathcal{C}(x) = \\sum_{ij} \\alpha_{ij} \\phi_i(x)\\phi_j(x)$ (see \\ref{definition:bk1_symbolic_coupling}), induced metric $g_{ij} = \\alpha_{ij}$ defines a Riemannian structure on $M$ (Def.~\\ref{definition:bk1_symbolic_manifo",
      "label": "definition:bk1_symbolic_coupling",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1808,
      "target_type": "definition"
    },
    {
      "context": "f{definition:bk1_symbolic_coupling}), induced metric $g_{ij} = \\alpha_{ij}$ defines a Riemannian structure on $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}). Here $\\phi_i$ are symbolic feature maps (see \\ref{definition:bk1_symbolic_manifold_feature_maps}). The corresponding",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "Riemannian structure on $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}). Here $\\phi_i$ are symbolic feature maps (see \\ref{definition:bk1_symbolic_manifold_feature_maps}). The corresponding Levi-Civita connection $\\nabla$ is the \\emph{symbolic connection}, with Christoffel symbols: \\[ \\Ga",
      "label": "definition:bk1_symbolic_manifold_feature_maps",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1803,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_coupling",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "definition:bk1_effective_horizon_signature",
    "definition:bk1_emergence_event",
    "definition:bk1_observer_horizon_structure",
    "lemma:bk2_thermodynamic_consistency_hypothesis_manifolds",
    "proof:bk1_curvature_semantic_holonomy",
    "proof:bk1_dual_horizon_unification_principle",
    "proof:bk1_sketch_observed_consequences",
    "proof:bk2_thermodynamic_consistency_hypothesis_manifolds",
    "proof:bk9_symbolic_masking_and_unmasking",
    "proposition:bk1_curvature_semantic_entanglement",
    "theorem:bk1_dual_horizon_cosmogenesis",
    "theorem:bk1_dual_horizon_unification_principle",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "cites": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "depends_on": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_symbolic_riemann_tensor",
  "label": "definition:bk1_symbolic_riemann_tensor",
  "latex_body": "\\begin{definition}[Symbolic Riemann Curvature Tensor]\n\\label{definition:bk1_symbolic_riemann_tensor}\nThe \\emph{symbolic curvature tensor} is the Riemann curvature tensor of the symbolic connection (see \\ref{definition:bk1_symbolic_connection}):\n\\[\n\\kappa(X,Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X,Y]} Z\n\\]\nfor vector fields $X, Y, Z$ on $M$, where $\\nabla$ acts on the symbolic feature manifold (see \\ref{definition:bk1_symbolic_manifold_feature_maps}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Discrete curvature as the commutator loop defect with a concrete nonzero witness; the Riemannian tensor stays open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-055"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Atlas.curvature_witness",
      "Atlas.holonomy_eps_squared"
    ]
  },
  "line": 1905,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Riemann Curvature Tensor",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ic_riemann_tensor} The \\emph{symbolic curvature tensor} is the Riemann curvature tensor of the symbolic connection (see \\ref{definition:bk1_symbolic_connection}): \\[ \\kappa(X,Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X,Y]} Z \\] for vector fields $X, Y, Z$ on $M$,",
      "label": "definition:bk1_symbolic_connection",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1888,
      "target_type": "definition"
    },
    {
      "context": "X Z - \\nabla_{[X,Y]} Z \\] for vector fields $X, Y, Z$ on $M$, where $\\nabla$ acts on the symbolic feature manifold (see \\ref{definition:bk1_symbolic_manifold_feature_maps}). \\end{definition}",
      "label": "definition:bk1_symbolic_manifold_feature_maps",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1803,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk1_semantic_non_integrability",
    "corollary:bk1_non_euclidean_necessity",
    "proof:bk1_symbolic_emergence_and_curvature",
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "cites": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "depends_on": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_local_semantic_independence",
  "label": "definition:bk1_local_semantic_independence",
  "latex_body": "\\begin{definition}[Local Semantic Independence]\n\\label{definition:bk1_local_semantic_independence}\nLet $U \\subset M$ be a contractible coordinate neighborhood in the symbolic\nmanifold (Def.~\\ref{definition:bk1_symbolic_manifold_feature_maps}), equipped\nwith the symbolic connection $\\nabla$ of Def.~\\ref{definition:bk1_symbolic_connection}.\nFor a piecewise smooth path $\\gamma:x\\to y$ in $U$, let\n$P^\\nabla_\\gamma:T_xM\\to T_yM$ denote parallel transport by $\\nabla$.\nSymbolic meanings are \\emph{locally independent on $U$} when, for every\n$x,y\\in U$ and every pair of paths $\\gamma_0,\\gamma_1:x\\to y$ in $U$,\n\\[\nP^\\nabla_{\\gamma_0} = P^\\nabla_{\\gamma_1}.\n\\]\nEquivalently, first-order semantic variations represented in $T_xM$ can be\ntransported to $T_yM$ without acquiring path-dependent contextual residue.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Independence = commuting contextual updates."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-053"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "Atlas.path_dependent_iff_noncommuting"
    ]
  },
  "line": 1914,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Local Semantic Independence",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "fold (Def.~\\ref{definition:bk1_symbolic_manifold_feature_maps}), equipped with the symbolic connection $\\nabla$ of Def.~\\ref{definition:bk1_symbolic_connection}. For a piecewise smooth path $\\gamma:x\\to y$ in $U$, let $P^\\nabla_\\gamma:T_xM\\to T_yM$ denote parallel transport by $\\",
      "label": "definition:bk1_symbolic_connection",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1888,
      "target_type": "definition"
    },
    {
      "context": "local_semantic_independence} Let $U \\subset M$ be a contractible coordinate neighborhood in the symbolic manifold (Def.~\\ref{definition:bk1_symbolic_manifold_feature_maps}), equipped with the symbolic connection $\\nabla$ of Def.~\\ref{definition:bk1_symbolic_connection}. For a piecewise smoo",
      "label": "definition:bk1_symbolic_manifold_feature_maps",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1803,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_certified_type_preserving_symbolic_transport",
    "proof:bk1_curvature_semantic_entanglement",
    "proof:bk1_dimensional_bounds_emergence",
    "proof:bk1_operational_irony_requires_imagination",
    "proof:bk1_operational_irony_requires_reflexive_curvature",
    "proof:bk1_symbolic_emergence_and_curvature",
    "proof:bk1_symbolic_irony_requires_curvature"
  ],
  "cites": [],
  "depends_on": [
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "lemma:bk1_curvature_semantic_holonomy",
  "label": "lemma:bk1_curvature_semantic_holonomy",
  "latex_body": "\\begin{lemma}[Curvature as Infinitesimal Semantic Holonomy]\n\\label{lemma:bk1_curvature_semantic_holonomy}\nLet $X,Y$ be vector fields on $U$ and let $\\square_{\\epsilon}(X,Y)$ be the\ninfinitesimal rectangle obtained by flowing a distance $\\epsilon$ first along\n$X$ and then along $Y$, then back along $-X$ and $-Y$. The parallel transport\naround this loop satisfies\n\\[\nP^\\nabla_{\\partial\\square_{\\epsilon}(X,Y)}Z\n= Z + \\epsilon^2 \\kappa(X,Y)Z + O(\\epsilon^3).\n\\]\nThus $\\kappa(X,Y)Z$ is precisely the second-order semantic residue obtained by\ntransporting the same local meaning around two different infinitesimal routes.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Exact eps-squared route residue in the linear-transport model; parallel transport on genuine manifolds stays open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-054"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Atlas.holonomy_eps_squared",
      "Atlas.holonomy_zero_iff_commute"
    ]
  },
  "line": 1930,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Curvature as Infinitesimal Semantic Holonomy",
  "proof_labels": [
    "proof:bk1_curvature_semantic_holonomy"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "depends_on": [
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_curvature_semantic_holonomy",
  "label": "proof:bk1_curvature_semantic_holonomy",
  "latex_body": "\\begin{proof}[Curvature is the second-order transport defect]\n\\label{proof:bk1_curvature_semantic_holonomy}\n\\leavevmode\n\nParallel transport around the rectangle compares the two ordered covariant\nupdates $\\nabla_X\\nabla_YZ$ and $\\nabla_Y\\nabla_XZ$. Because the closing edge of\nthe infinitesimal parallelogram contributes the Lie-bracket correction\n$\\nabla_{[X,Y]}Z$, the second-order failure of the two routes to agree is\n\\[\n\\nabla_X\\nabla_YZ-\\nabla_Y\\nabla_XZ-\\nabla_{[X,Y]}Z\n=\\kappa(X,Y)Z\n\\]\nby Def.~\\ref{definition:bk1_symbolic_riemann_tensor}. Taylor expansion of the\ntransport map around the loop gives the displayed\n$\\epsilon^2$ term, with all remaining terms of order $O(\\epsilon^3)$.\n\\end{proof}",
  "line": 1944,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Curvature is the second-order transport defect",
  "proves": "lemma:bk1_curvature_semantic_holonomy",
  "ref_roles": [
    {
      "context": "r failure of the two routes to agree is \\[ \\nabla_X\\nabla_YZ-\\nabla_Y\\nabla_XZ-\\nabla_{[X,Y]}Z =\\kappa(X,Y)Z \\] by Def.~\\ref{definition:bk1_symbolic_riemann_tensor}. Taylor expansion of the transport map around the loop gives the displayed $\\epsilon^2$ term, with all remaining terms",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1905,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_non_euclidean_necessity",
    "proof:bk1_curvature_projection_residue",
    "proof:bk1_symbolic_emergence_and_curvature",
    "proof:bk8_curvature_entanglement_equivalence",
    "proof:bk8_entanglement_as_frame_artifact",
    "proof:bk8_flattening_decoherence_equivalence",
    "proof:bk8_symbolic_curvature_and_separability"
  ],
  "cites": [
    "definition:bk1_local_semantic_independence",
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "depends_on": [
    "definition:bk1_local_semantic_independence",
    "definition:bk1_symbolic_riemann_tensor",
    "lemma:bk1_curvature_semantic_holonomy"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_curvature_semantic_entanglement",
  "label": "proposition:bk1_curvature_semantic_entanglement",
  "latex_body": "\\begin{proposition}[Curvature and Semantic Entanglement]\n\\label{proposition:bk1_curvature_semantic_entanglement}\nOn any contractible coordinate neighborhood $U \\subset M$, the symbolic curvature\n$\\kappa$ of Def.~\\ref{definition:bk1_symbolic_riemann_tensor} vanishes on $U$ if\nand only if symbolic meanings are locally independent on $U$ in the sense of\nDef.~\\ref{definition:bk1_local_semantic_independence}.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Curvature vanishes iff meanings are locally independent - exactly the flatness-iff-commuting theorem."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-089"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Atlas.holonomy_zero_iff_commute"
    ]
  },
  "line": 1961,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Curvature and Semantic Entanglement",
  "proof_labels": [
    "proof:bk1_curvature_semantic_entanglement"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ic_riemann_tensor} vanishes on $U$ if and only if symbolic meanings are locally independent on $U$ in the sense of Def.~\\ref{definition:bk1_local_semantic_independence}. \\end{proposition}",
      "label": "definition:bk1_local_semantic_independence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1914,
      "target_type": "definition"
    },
    {
      "context": "mantic_entanglement} On any contractible coordinate neighborhood $U \\subset M$, the symbolic curvature $\\kappa$ of Def.~\\ref{definition:bk1_symbolic_riemann_tensor} vanishes on $U$ if and only if symbolic meanings are locally independent on $U$ in the sense of Def.~\\ref{definition:bk",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1905,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_local_semantic_independence",
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "lemma:bk1_curvature_semantic_holonomy"
  ],
  "depends_on": [
    "lemma:bk1_curvature_semantic_holonomy"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_curvature_semantic_entanglement",
  "label": "proof:bk1_curvature_semantic_entanglement",
  "latex_body": "\\begin{proof}[Flatness iff local semantic independence]\n\\label{proof:bk1_curvature_semantic_entanglement}\n\\leavevmode\n\n($\\Rightarrow$) Suppose $\\kappa=0$ on $U$. By\nLemma~\\ref{lemma:bk1_curvature_semantic_holonomy}, infinitesimal transport\naround every coordinate rectangle has zero second-order semantic residue. Since\n$U$ is contractible, any loop in $U$ can be decomposed into such infinitesimal\nrectangles. The holonomy around the whole loop is therefore trivial, so parallel\ntransport from $x$ to $y$ depends only on the endpoints and not on the chosen\npath. Hence symbolic meanings are locally independent.\n\n($\\Leftarrow$) Conversely, suppose symbolic meanings are locally independent on\n$U$. Then the transport around every sufficiently small coordinate rectangle is\nthe identity. Applying Lemma~\\ref{lemma:bk1_curvature_semantic_holonomy}, the\ncoefficient of the $\\epsilon^2$ term must vanish for all vector fields $X,Y,Z$:\n\\[\n\\kappa(X,Y)Z=0.\n\\]\nThus $\\kappa=0$ on $U$.\n\\end{proof}",
  "line": 1969,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Flatness iff local semantic independence",
  "proves": "proposition:bk1_curvature_semantic_entanglement",
  "ref_roles": [
    {
      "context": "nce] \\label{proof:bk1_curvature_semantic_entanglement} \\leavevmode ($\\Rightarrow$) Suppose $\\kappa=0$ on $U$. By Lemma~\\ref{lemma:bk1_curvature_semantic_holonomy}, infinitesimal transport around every coordinate rectangle has zero second-order semantic residue. Since $U$ is contrac",
      "label": "lemma:bk1_curvature_semantic_holonomy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1930,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "lemma:bk1_curvature_semantic_holonomy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk8_curvature_entanglement_equivalence",
    "proof:bk8_entanglement_as_frame_artifact"
  ],
  "cites": [],
  "depends_on": [
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "corollary:bk1_curvature_projection_residue",
  "label": "corollary:bk1_curvature_projection_residue",
  "latex_body": "\\begin{corollary}[Curvature Residue under Non-Expressive Projection]\n\\label{corollary:bk1_curvature_projection_residue}\nLet $\\Pi_F:M\\to F$ be an observer-frame projection whose target frame $F$ treats\nsemantic transport as path-independent on $\\Pi_F(U)$. If $\\kappa\\neq 0$ on $U$,\nthen there exist paths $\\gamma_0,\\gamma_1:x\\to y$ in $U$ and a semantic\nvariation $Z\\in T_xM$ such that\n\\[\n\\Pi_F(P^\\nabla_{\\gamma_0}Z) \\neq \\Pi_F(P^\\nabla_{\\gamma_1}Z)\n\\]\nor else the projection discards the curvature residue. In either case, the\nprojection represents curved semantic coupling as a frame artifact: either a\nvisible non-factorizable residual or an information loss.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Nonzero curvature forces a frame-artifact residue - the non-Euclidean necessity kernel."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-090"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Atlas.non_euclidean_necessity"
    ]
  },
  "line": 1991,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Curvature Residue under Non-Expressive Projection",
  "proof_labels": [
    "proof:bk1_curvature_projection_residue"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "depends_on": [
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_curvature_projection_residue",
  "label": "proof:bk1_curvature_projection_residue",
  "latex_body": "\\begin{proof}[Projection residue from failed path independence]\n\\label{proof:bk1_curvature_projection_residue}\n\\leavevmode\n\nIf $\\kappa\\neq 0$ on $U$, Prop.~\\ref{proposition:bk1_curvature_semantic_entanglement}\nimplies that local semantic independence fails. Hence some pair of paths\n$\\gamma_0,\\gamma_1:x\\to y$ and some $Z\\in T_xM$ satisfy\n$P^\\nabla_{\\gamma_0}Z\\neq P^\\nabla_{\\gamma_1}Z$. A frame $F$ that assumes\npath-independent semantic transport has no intrinsic curvature coordinate in\nwhich to store this difference. Therefore the projected images either remain\ndistinct as an observable residual, or they are identified by $\\Pi_F$, in which\ncase the curvature information has been discarded. This is the projection\nmechanism later read as frame artifact in Book VIII.\n\\end{proof}",
  "line": 2005,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Projection residue from failed path independence",
  "proves": "corollary:bk1_curvature_projection_residue",
  "ref_roles": [
    {
      "context": "m failed path independence] \\label{proof:bk1_curvature_projection_residue} \\leavevmode If $\\kappa\\neq 0$ on $U$, Prop.~\\ref{proposition:bk1_curvature_semantic_entanglement} implies that local semantic independence fails. Hence some pair of paths $\\gamma_0,\\gamma_1:x\\to y$ and some $Z\\in T_xM",
      "label": "proposition:bk1_curvature_semantic_entanglement",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1961,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk4_refinement_contraction"
  ],
  "cites": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "depends_on": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_resolution_cost",
  "label": "definition:bk1_resolution_cost",
  "latex_body": "\\begin{definition}[Resolution Cost]\n\\label{definition:bk1_resolution_cost}\nFor symbolic states $p, q \\in M$, the \\emph{resolution cost} is:\n\\[\n\\reflect(p,q) = \\inf_{\\gamma: p \\to q} \\int_\\gamma \\sqrt{g(\\dot{\\gamma}, \\dot{\\gamma})} \\, dt\n\\]\nwhere the infimum is taken over all smooth paths $\\gamma$ connecting $p$ and $q$, and $g$ is the metric induced via the symbolic connection (see \\ref{definition:bk1_symbolic_connection}) and feature map structure (see \\ref{definition:bk1_symbolic_manifold_feature_maps}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "discrete kernels only: ODE flows, Riemannian geodesics, Hopf-Rinow, and the MSR path integral stay open",
      "the self-representation clause of reflexive maps is interpretive"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The path-infimum is a pseudometric: every law proved of the infimum."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-064"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumDyn.resCost_self",
      "ScholiumDyn.resCost_symm",
      "ScholiumDyn.resCost_triangle"
    ]
  },
  "line": 2020,
  "macros_used": [
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Resolution Cost",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "n over all smooth paths $\\gamma$ connecting $p$ and $q$, and $g$ is the metric induced via the symbolic connection (see \\ref{definition:bk1_symbolic_connection}) and feature map structure (see \\ref{definition:bk1_symbolic_manifold_feature_maps}). \\end{definition}",
      "label": "definition:bk1_symbolic_connection",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1888,
      "target_type": "definition"
    },
    {
      "context": "etric induced via the symbolic connection (see \\ref{definition:bk1_symbolic_connection}) and feature map structure (see \\ref{definition:bk1_symbolic_manifold_feature_maps}). \\end{definition}",
      "label": "definition:bk1_symbolic_manifold_feature_maps",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1803,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_connection",
    "definition:bk1_symbolic_manifold_feature_maps"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk1_symbolic_primacy",
    "corollary:bk1_dimensional_bounds_emergence",
    "definition:bk6_symbolic_laplace_beltrami_operator_complete",
    "demonstratio:bk7_convergence_within_reflective_basin",
    "proof:bk1_dimensional_bounds_emergence",
    "proof:bk8_flattening_decoherence_equivalence",
    "subsec:appD_info_geometry_contribution_differentiation",
    "theorem:bk3_symbiotic_curvature_and_resilience"
  ],
  "cites": [
    "definition:bk1_emergence_event",
    "definition:bk1_horizon_structure",
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "depends_on": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_emergence_event",
    "definition:bk1_horizon_structure",
    "definition:bk1_local_semantic_independence",
    "definition:bk1_symbolic_riemann_tensor",
    "lemma:bk1_curvature_semantic_holonomy",
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_symbolic_emergence_and_curvature",
  "label": "theorem:bk1_symbolic_emergence_and_curvature",
  "latex_body": "\\begin{theorem}[Symbolic Emergence and Curvature]\n\\label{theorem:bk1_symbolic_emergence_and_curvature}\nA symbolic system exhibits emergent behavior—characterized by horizon-relative novelty (see \\ref{definition:bk1_horizon_structure}), reflexive identity, and contextual meaning (see \\ref{definition:bk1_emergence_event})—if and only if its symbolic manifold has non-zero curvature $\\kappa \\neq 0$ (see \\ref{definition:bk1_symbolic_riemann_tensor}).\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The iff at transport level: route-dependent meaning iff nonzero commutator; Riemannian form open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-084"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Atlas.non_euclidean_necessity",
      "Atlas.path_dependent_iff_noncommuting"
    ]
  },
  "line": 2029,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Emergence and Curvature",
  "proof_labels": [
    "proof:bk1_symbolic_emergence_and_curvature"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "horizon-relative novelty (see \\ref{definition:bk1_horizon_structure}), reflexive identity, and contextual meaning (see \\ref{definition:bk1_emergence_event})—if and only if its symbolic manifold has non-zero curvature $\\kappa \\neq 0$ (see \\ref{definition:bk1_symbolic_riemann_",
      "label": "definition:bk1_emergence_event",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1325,
      "target_type": "definition"
    },
    {
      "context": "ic_emergence_and_curvature} A symbolic system exhibits emergent behavior—characterized by horizon-relative novelty (see \\ref{definition:bk1_horizon_structure}), reflexive identity, and contextual meaning (see \\ref{definition:bk1_emergence_event})—if and only if its symbolic man",
      "label": "definition:bk1_horizon_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1841,
      "target_type": "definition"
    },
    {
      "context": "\\ref{definition:bk1_emergence_event})—if and only if its symbolic manifold has non-zero curvature $\\kappa \\neq 0$ (see \\ref{definition:bk1_symbolic_riemann_tensor}). \\end{theorem}",
      "label": "definition:bk1_symbolic_riemann_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1905,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_emergence_event",
    "definition:bk1_horizon_structure",
    "definition:bk1_symbolic_riemann_tensor"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_emergence_event",
    "definition:bk1_horizon_structure",
    "definition:bk1_local_semantic_independence",
    "lemma:bk1_curvature_semantic_holonomy",
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "depends_on": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_emergence_event",
    "definition:bk1_horizon_structure",
    "definition:bk1_local_semantic_independence",
    "lemma:bk1_curvature_semantic_holonomy",
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_symbolic_emergence_and_curvature",
  "label": "proof:bk1_symbolic_emergence_and_curvature",
  "latex_body": "\\begin{proof}\n\\label{proof:bk1_symbolic_emergence_and_curvature}\n\\leavevmode\n\nThe mathematical core of the equivalence is the proven curvature--semantics correspondence; the three marks of emergence are its readings. Work on a contractible neighborhood $U \\subset M$.\n\n\\emph{($\\Leftarrow$) $\\kappa \\neq 0$ on $U$ implies emergence.}\n\\emph{Contextual meaning.} By Prop.~\\ref{proposition:bk1_curvature_semantic_entanglement}, $\\kappa \\neq 0$ on $U$ is equivalent to the failure of local semantic independence (Def.~\\ref{definition:bk1_local_semantic_independence}): symbolic meanings are mutually dependent, i.e.\\ contextual.\n\\emph{Horizon-relative novelty.} By Lemma~\\ref{lemma:bk1_curvature_semantic_holonomy}, parallel transport of a symbolic feature around an infinitesimal loop returns it displaced by $\\kappa$ (semantic holonomy). Transport within the horizon structure (Def.~\\ref{definition:bk1_horizon_structure}) is therefore path-dependent, so traversal of accessible directions generates states reachable by no single direction---novelty relative to the horizon.\n\\emph{Reflexive identity.} A reflexive identity is a self-model invariant under the observer's own reflective transport loop (Def.~\\ref{definition:bk1_emergence_event}). When $\\kappa \\neq 0$ that loop acts as a nontrivial operator, whose invariant structure is a \\emph{distinguished} fixed self-model; when $\\kappa = 0$ transport is trivial and no self-model is distinguished. Thus all three marks hold.\n\n\\emph{($\\Rightarrow$) Emergence implies $\\kappa \\neq 0$.}\nContextual meaning is, by definition, the failure of local semantic independence; the converse direction of Prop.~\\ref{proposition:bk1_curvature_semantic_entanglement} then gives $\\kappa \\neq 0$ on $U$. (Equivalently, reflexivity together with context-sensitivity forces $\\kappa \\neq 0$ by Cor.~\\ref{corollary:bk1_non_euclidean_necessity}.)\n\nThe two implications give emergence $\\iff \\kappa \\neq 0$.\n\\end{proof}",
  "line": 2034,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "",
  "proves": "theorem:bk1_symbolic_emergence_and_curvature",
  "ref_roles": [
    {
      "context": "ves $\\kappa \\neq 0$ on $U$. (Equivalently, reflexivity together with context-sensitivity forces $\\kappa \\neq 0$ by Cor.~\\ref{corollary:bk1_non_euclidean_necessity}.) The two implications give emergence $\\iff \\kappa \\neq 0$. \\end{proof}",
      "label": "corollary:bk1_non_euclidean_necessity",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1686,
      "target_type": "corollary"
    },
    {
      "context": "xive identity.} A reflexive identity is a self-model invariant under the observer's own reflective transport loop (Def.~\\ref{definition:bk1_emergence_event}). When $\\kappa \\neq 0$ that loop acts as a nontrivial operator, whose invariant structure is a \\emph{distinguished} fix",
      "label": "definition:bk1_emergence_event",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1325,
      "target_type": "definition"
    },
    {
      "context": "n infinitesimal loop returns it displaced by $\\kappa$ (semantic holonomy). Transport within the horizon structure (Def.~\\ref{definition:bk1_horizon_structure}) is therefore path-dependent, so traversal of accessible directions generates states reachable by no single direction--",
      "label": "definition:bk1_horizon_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1841,
      "target_type": "definition"
    },
    {
      "context": "vature_semantic_entanglement}, $\\kappa \\neq 0$ on $U$ is equivalent to the failure of local semantic independence (Def.~\\ref{definition:bk1_local_semantic_independence}): symbolic meanings are mutually dependent, i.e.\\ contextual. \\emph{Horizon-relative novelty.} By Lemma~\\ref{lemma:bk1_",
      "label": "definition:bk1_local_semantic_independence",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1914,
      "target_type": "definition"
    },
    {
      "context": "c_independence}): symbolic meanings are mutually dependent, i.e.\\ contextual. \\emph{Horizon-relative novelty.} By Lemma~\\ref{lemma:bk1_curvature_semantic_holonomy}, parallel transport of a symbolic feature around an infinitesimal loop returns it displaced by $\\kappa$ (semantic holon",
      "label": "lemma:bk1_curvature_semantic_holonomy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1930,
      "target_type": "lemma"
    },
    {
      "context": "ood $U \\subset M$. \\emph{($\\Leftarrow$) $\\kappa \\neq 0$ on $U$ implies emergence.} \\emph{Contextual meaning.} By Prop.~\\ref{proposition:bk1_curvature_semantic_entanglement}, $\\kappa \\neq 0$ on $U$ is equivalent to the failure of local semantic independence (Def.~\\ref{definition:bk1_local_sem",
      "label": "proposition:bk1_curvature_semantic_entanglement",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1961,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "corollary:bk1_non_euclidean_necessity",
    "definition:bk1_emergence_event",
    "definition:bk1_horizon_structure",
    "definition:bk1_local_semantic_independence",
    "lemma:bk1_curvature_semantic_holonomy",
    "proposition:bk1_curvature_semantic_entanglement"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk4_topological_stability_via_spectral_and_curvature_constraints"
  ],
  "cites": [
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "depends_on": [
    "lemma:bk1_curvature_semantic_holonomy",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "corollary:bk1_dimensional_bounds_emergence",
  "label": "corollary:bk1_dimensional_bounds_emergence",
  "latex_body": "\\begin{corollary}[Dimensional Bounds on Emergence]\n\\label{corollary:bk1_dimensional_bounds_emergence}\nBy Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, the complexity of symbolic emergence is bounded below by the rank of the curvature tensor $\\kappa$. Systems with richer curvature structure support more complex emergent phenomena.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "manifold metric, exact rank bound, and the interpretive unification/primacy claims stay open per row notes"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Nonzero curvature has an active mode (complexity >= 1); the exact rank bound stays open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-088"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumBridge.nonzero_curvature_has_active_mode"
    ]
  },
  "line": 2051,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Dimensional Bounds on Emergence",
  "proof_labels": [
    "proof:bk1_dimensional_bounds_emergence"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "\\begin{corollary}[Dimensional Bounds on Emergence] \\label{corollary:bk1_dimensional_bounds_emergence} By Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, the complexity of symbolic emergence is bounded below by the rank of the curvature tensor $\\kappa$. Systems with riche",
      "label": "theorem:bk1_symbolic_emergence_and_curvature",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2029,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "lemma:bk1_curvature_semantic_holonomy",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "depends_on": [
    "lemma:bk1_curvature_semantic_holonomy",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_dimensional_bounds_emergence",
  "label": "proof:bk1_dimensional_bounds_emergence",
  "latex_body": "\\begin{proof}\n\\label{proof:bk1_dimensional_bounds_emergence}\n\\leavevmode\nBy Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, emergence occurs exactly where $\\kappa \\neq 0$. Each independent emergent mode is an independent direction of semantic holonomy (Lemma~\\ref{lemma:bk1_curvature_semantic_holonomy}): a feature whose parallel transport around an infinitesimal loop returns displaced by a nonzero amount. Two emergent modes are distinguishable as separate phenomena only when their holonomy displacements are linearly independent vectors on the feature manifold; by the holonomy identity, those displacements are the images of the loop's tangent bivector under $\\kappa$. Hence the count of linearly independent emergent modes equals the dimension of the image of $\\kappa$ as an operator---that is, $\\operatorname{rank}\\kappa$---which therefore bounds the complexity of emergence from below. A curvature tensor of higher rank opens a strictly larger space of independent emergent directions, so richer curvature supports richer emergence.\n\\end{proof}",
  "line": 2056,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "",
  "proves": "corollary:bk1_dimensional_bounds_emergence",
  "ref_roles": [
    {
      "context": "s exactly where $\\kappa \\neq 0$. Each independent emergent mode is an independent direction of semantic holonomy (Lemma~\\ref{lemma:bk1_curvature_semantic_holonomy}): a feature whose parallel transport around an infinitesimal loop returns displaced by a nonzero amount. Two emergent m",
      "label": "lemma:bk1_curvature_semantic_holonomy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1930,
      "target_type": "lemma"
    },
    {
      "context": "\\begin{proof} \\label{proof:bk1_dimensional_bounds_emergence} \\leavevmode By Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, emergence occurs exactly where $\\kappa \\neq 0$. Each independent emergent mode is an independent direction of semantic",
      "label": "theorem:bk1_symbolic_emergence_and_curvature",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2029,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "lemma:bk1_curvature_semantic_holonomy",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "theorem:bk4_paradoxical_arrow_of_time"
  ],
  "cites": [
    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "sec:bk1_category_errors_in_classical_models",
  "label": "sec:bk1_category_errors_in_classical_models",
  "latex_body": "",
  "line": 2066,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Category Errors in Classical Models",
  "ref_roles": [
    {
      "context": "",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": false,
      "role": "navigation",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "subsec:bk1_limits_of_classical_frameworks",
  "label": "subsec:bk1_limits_of_classical_frameworks",
  "latex_body": "",
  "line": 2071,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Limits of Classical Frameworks",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_newtonian_category_error",
  "label": "definition:bk1_newtonian_category_error",
  "latex_body": "\\begin{definition}[Newtonian Category Error]\n\\label{definition:bk1_newtonian_category_error}\nA modeling framework exhibits the Newtonian Category Error when it presupposes manifold smoothness and continuity \\emph{a priori}, thereby violating bounded observer logic (see \\ref{definition:bk1_bounded_observer}). Specifically, if $\\mathcal{O}$ denotes a bounded observer with access function $\\alpha: \\mathcal{O} \\to \\mathcal{O}$ where $\\alpha(\\mathcal{O}) \\subsetneq \\mathcal{O}$, then any framework assuming global differentiability disconnects form from relation, rendering the drift operator $D$ (see \\ref{definition:bk1_drift_field}) non-constructible within the observer's horizon on the symbolic manifold $M$ (see \\ref{definition:bk1_symbolic_manifold}).\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "application order in the composite is not interpreted as ontological origin order",
      "catS, observer detection, stage continuity, and geometric realization remain distinct supplied interfaces",
      "drift and reflection are fields of one OperationalStage witness; neither is derived from the other"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Only the access-function clause (alpha(O) properly contained in O, read as non-surjectivity) is modeled, yielding an inaccessible state; the manifold-smoothness/drift-non-constructibility conclusion is not."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-038"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "ScholiumC.exists_inaccessible_of_not_surjective"
    ]
  },
  "line": 2074,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Newtonian Category Error",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "r when it presupposes manifold smoothness and continuity \\emph{a priori}, thereby violating bounded observer logic (see \\ref{definition:bk1_bounded_observer}). Specifically, if $\\mathcal{O}$ denotes a bounded observer with access function $\\alpha: \\mathcal{O} \\to \\mathcal{O}$",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "n any framework assuming global differentiability disconnects form from relation, rendering the drift operator $D$ (see \\ref{definition:bk1_drift_field}) non-constructible within the observer's horizon on the symbolic manifold $M$ (see \\ref{definition:bk1_symbolic_manifol",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "see \\ref{definition:bk1_drift_field}) non-constructible within the observer's horizon on the symbolic manifold $M$ (see \\ref{definition:bk1_symbolic_manifold}). \\end{definition}",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_drift_field",
    "definition:bk1_symbolic_manifold"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "proof:bk1_dual_horizon_unification_principle",
    "theorem:bk1_dual_horizon_unification_principle"
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  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proposition:bk1_newtonian_incompleteness",
  "label": "proposition:bk1_newtonian_incompleteness",
  "latex_body": "\\begin{proposition}[Newtonian Incompleteness]\n\\label{proposition:bk1_newtonian_incompleteness}\nLet \\(V\\) be a real normed vector space and let the Newtonian force law be\n\\(F_m(a)=m a\\) for \\(m\\in\\mathbb{R}\\) and \\(a\\in V\\).  The map \\(F_m\\) is\nequivariant under every continuous linear change of frame \\(L:V\\to V\\):\n\\[\n  L(F_m(a))=F_m(L(a)).\n\\]\nHowever, if an accelerated frame contributes a nonzero acceleration\n\\(w\\in V\\), its transformed acceleration \\(a+2w\\) is not \\(a\\).  Hence the\nlinear-frame covariance of the Newtonian law does not by itself extend to\naccelerated observer frames: such an extension requires an explicit\nframe-correction term.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "continuous linear change of frame",
      "nonzero frame acceleration for strict defect",
      "nonzero uniform frame acceleration",
      "real normed vector space",
      "twice differentiable trajectory at the stated point"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Unflattened covariance-boundary kernel: on every real normed vector space, scalar Newtonian force commutes with every continuous linear frame map, while each nonzero uniform frame acceleration produces a nonzero 2*w defect. The NVec derivative construction remains the concrete dynamical witness. Extending covariance to accelerated observers therefore requires an explicit correction; relativistic gravity and a general spacetime theory are not claimed."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-020"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumD.newtonian_incompleteness_kernel",
      "ScholiumD.newtonian_incompleteness_normedSpace",
      "accelerated_frame_defect",
      "accelerated_frame_defect_ne",
      "newtonForce_equivariant"
    ]
  },
  "line": 2079,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Newtonian Incompleteness",
  "proof_labels": [
    "proof:bk1_newtonian_incompleteness"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_newtonian_incompleteness",
  "label": "proof:bk1_newtonian_incompleteness",
  "latex_body": "\\begin{proof}[Covariance Boundary]\n\\label{proof:bk1_newtonian_incompleteness}\n\\leavevmode\n\nFor every continuous linear \\(L\\),\n\\[\n  L(F_m(a))=L(ma)=mL(a)=F_m(L(a)),\n\\]\nso \\(F_m\\) is equivariant under the stated covariance class.  If \\(w\\ne0\\),\nthen \\(2w\\ne0\\), and cancellation in the additive group of \\(V\\) gives\n\\(a+2w\\ne a\\).  The accelerated-frame defect therefore cannot be obtained by\nordinary linear-frame equivariance alone and must be represented explicitly.\n\\end{proof}",
  "line": 2093,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Covariance Boundary",
  "proves": "proposition:bk1_newtonian_incompleteness",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "definition:bk1_quantum_category_error",
  "label": "definition:bk1_quantum_category_error",
  "latex_body": "\\begin{definition}[Quantum Tensor-Closure Category Error]\n\\label{definition:bk1_quantum_category_error}\nA tensor-closure category error occurs when an update valued in one state\ncarrier \\(\\mathcal H\\) is identified directly with a tensor\n\\(\\psi\\otimes\\varphi\\in\\mathcal H\\otimes\\mathcal H\\) without specifying a\nmap from the tensor product back to \\(\\mathcal H\\).  A \\emph{lossless linear\nclosure} is a linear isomorphism\n\\[\n C:\\mathcal H\\otimes_{\\mathbb K}\\mathcal H\n   \\overset{\\sim}{\\longrightarrow}\\mathcal H.\n\\]\nWhether a Hamiltonian is externally controlled, dynamically updated, or\nrepresented inside a larger quantum system is a separate modeling question;\nstrict linearity alone does not prohibit such constructions.\n\\end{definition}",
  "line": 2107,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Quantum Tensor-Closure Category Error",
  "proof_status": "definitional",
  "refs": [],
  "role": "definition",
  "type": "definition"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "lemma:bk1_symbolic_quantum_incompatibility",
  "label": "lemma:bk1_symbolic_quantum_incompatibility",
  "latex_body": "\\begin{lemma}[Finite-Dimensional Symbolic--Quantum Tensor Obstruction]\n\\label{lemma:bk1_symbolic_quantum_incompatibility}\nLet \\(\\mathcal H\\) be a finite-dimensional vector space over a field\n\\(\\mathbb K\\), with \\(1<\\dim_{\\mathbb K}\\mathcal H<\\infty\\).  Then there is no\nlossless linear closure\n\\[\n \\mathcal H\\otimes_{\\mathbb K}\\mathcal H\n   \\overset{\\sim}{\\longrightarrow}\\mathcal H.\n\\]\nConsequently, a symbolic reflexive update whose preservation requires the\npair tensor \\(\\phi(s)\\otimes\\phi(s')\\) to be represented losslessly in the\nsame finite-dimensional state carrier cannot be preserved by such a closure.\nUnitary reflection evolution does not remove this dimension obstruction.\nThe conclusion is sharp: in dimension one the tensor square is linearly\nisomorphic to the original carrier.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "See the receipted theorem statement and coverage note for explicit premises."
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Faithful logical kernel with exact preservation predicates: no map can preserve both reflection and binary reflexive update when joint preservation entails a Hamiltonian-level meta-update and the target quantum model forbids that update. These two category-error premises remain explicit because unitary linear evolution and tensor structure alone do not establish them; a concrete Hilbert-space no-go theorem remains open."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-021"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "ScholiumD.symbolic_quantum_incompatibility_kernel"
    ]
  },
  "line": 2123,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Finite-Dimensional Symbolic--Quantum Tensor Obstruction",
  "proof_labels": [
    "proof:bk1_symbolic_quantum_incompatibility"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_symbolic_quantum_incompatibility",
  "label": "proof:bk1_symbolic_quantum_incompatibility",
  "latex_body": "\\begin{proof}[Tensor Dimension Obstruction]\n\\label{proof:bk1_symbolic_quantum_incompatibility}\n\\leavevmode\n\nWrite \\(n=\\dim_{\\mathbb K}\\mathcal H\\).  Finite-dimensional tensor products\nsatisfy\n\\[\n \\dim_{\\mathbb K}(\\mathcal H\\otimes_{\\mathbb K}\\mathcal H)=n^2.\n\\]\nA linear isomorphism to \\(\\mathcal H\\) would therefore imply \\(n^2=n\\).\nFor positive finite \\(n\\), this forces \\(n=1\\), contradicting \\(n>1\\).\nThus no lossless linear self-tensor closure exists in the stated regime.\nWhen \\(n=1\\), both sides have dimension one and a linear isomorphism does\nexist, proving sharpness.  The argument concerns the typed tensor closure; it\ndoes not assert a universal prohibition on Hamiltonian control or quantum\nmodels of self-reference.\n\\end{proof}",
  "line": 2139,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Tensor Dimension Obstruction",
  "proves": "lemma:bk1_symbolic_quantum_incompatibility",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "subsec:bk1_reflexivity_requires_quadratic_framing",
  "label": "subsec:bk1_reflexivity_requires_quadratic_framing",
  "latex_body": "",
  "line": 2157,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Conclusion: Reflexivity Requires Quadratic Framing",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
    "corollary:bk7_stability_innovation_equilibrium",
    "proof:bk7_stability_innovation_equilibrium"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "theorem:bk1_symbolic_emergence_theorem_thermodynamics",
  "label": "theorem:bk1_symbolic_emergence_theorem_thermodynamics",
  "latex_body": "\\begin{theorem}[Symbolic Emergence Theorem---Contextual Cross-Error]\n\\label{theorem:bk1_symbolic_emergence_theorem_thermodynamics}\nLet $\\mathcal{S}$ be a symbolic system over a manifold $M$, and let\n$\\mathcal{U}:\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}$ be a local residual\nupdate in state and context coordinates. Suppose $\\mathcal{S}$ supports:\n\\begin{itemize}\n  \\item horizon-relative novelty: $\\exists s\\in\\mathcal{S}$ such that $D(s)\\notin\\alpha(\\mathcal{S})$ (Def.~\\ref{definition:bk1_drift_field});\n  \\item reflexive symbolic identity: $R(\\mathcal{S})\\cap\\mathcal{S}\\neq\\emptyset$ (Def.~\\ref{definition:bk1_reflection_operator});\n  \\item contextual structural growth: $\\mathcal{U}$ is not additively separable as $A(\\xi)+B(\\chi)$.\n\\end{itemize}\nThen there exist state and context displacements $\\xi,\\chi$ for which the\nmixed cross-error\n\\[\n\\Delta\\mathcal{U}(\\xi,\\chi)\n =\\mathcal{U}(\\xi,\\chi)-\\mathcal{U}(\\xi,0)\n  -\\mathcal{U}(0,\\chi)+\\mathcal{U}(0,0)\n\\]\nis nonzero.  This cross-error is the Scholium-level certificate supplied to\nBook~IV.  The later transport construction may geometrize it as noncommuting\nstate--context transport; that geometric realization is not used as a premise\nof this theorem.  Likewise, identifying the leading mixed term as bilinear\nrequires the smooth local-expansion hypotheses stated with the later\nquadratic construction.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "a real-valued state-context update",
      "failure of additive separation into independent state and context contributions"
    ],
    "countermodels": [
      "ScholiumD.emergence_premises_do_not_force_curvature"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Layered repair: contextual nonseparability locally forces a nonzero mixed cross-error by an explicit additive-decomposition contradiction. The Scholium stops at that certificate; Book IV consumes it to construct noncommuting transport, and Book VII consumes the Book IV geometry. The zero-curvature Bool model remains the negative control showing novelty, reflective identity, and abstract dimension growth alone do not supply the certificate or curvature."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_B-022"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "ScholiumD.contextualGrowth_exposes_crossError",
      "ScholiumD.emergence_premises_do_not_force_curvature"
    ]
  },
  "line": 2160,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Emergence Theorem---Contextual Cross-Error",
  "proof_labels": [
    "proof:bk1_symbolic_emergence_theorem_thermodynamics"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "in{itemize} \\item horizon-relative novelty: $\\exists s\\in\\mathcal{S}$ such that $D(s)\\notin\\alpha(\\mathcal{S})$ (Def.~\\ref{definition:bk1_drift_field}); \\item reflexive symbolic identity: $R(\\mathcal{S})\\cap\\mathcal{S}\\neq\\emptyset$ (Def.~\\ref{definition:bk1_reflectio",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "f{definition:bk1_drift_field}); \\item reflexive symbolic identity: $R(\\mathcal{S})\\cap\\mathcal{S}\\neq\\emptyset$ (Def.~\\ref{definition:bk1_reflection_operator}); \\item contextual structural growth: $\\mathcal{U}$ is not additively separable as $A(\\xi)+B(\\chi)$. \\end{itemize} Th",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_symbolic_emergence_theorem_thermodynamics",
  "label": "proof:bk1_symbolic_emergence_theorem_thermodynamics",
  "latex_body": "\\begin{proof}[Nonseparability Forces a Cross-Error]\n\\label{proof:bk1_symbolic_emergence_theorem_thermodynamics}\n\\leavevmode\n\nAssume for contradiction that every mixed cross-error vanishes.  Define\n\\[\n A(\\xi)=\\mathcal U(\\xi,0),\n \\qquad\n B(\\chi)=\\mathcal U(0,\\chi)-\\mathcal U(0,0).\n\\]\nThe vanishing cross-difference identity rearranges pointwise to\n$\\mathcal U(\\xi,\\chi)=A(\\xi)+B(\\chi)$, contradicting contextual structural\ngrowth.  Hence some $\\Delta\\mathcal U(\\xi,\\chi)$ is nonzero.  Novelty and\nreflexive identity retain their emergence roles, while contextual\nnonseparability is the load-bearing premise for this certificate.  Book~IV\nconsumes the certificate downstream; it does not discharge the Scholium proof\nbackward.\n\\end{proof}",
  "line": 2184,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Nonseparability Forces a Cross-Error",
  "proves": "theorem:bk1_symbolic_emergence_theorem_thermodynamics",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk1_symbolic_primacy"
  ],
  "cites": [
    "theorem:bk1_symbolic_emergence_theorem_thermodynamics"
  ],
  "depends_on": [
    "theorem:bk1_symbolic_emergence_theorem_thermodynamics"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
  "label": "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
  "latex_body": "\\begin{corollary}[Necessity of Non-Euclidean Symbolic Transport]\n\\label{corollary:bk1_necessity_of_non_euclidean_symbolic_space}\nUnder the hypotheses of\nThm.~\\ref{theorem:bk1_symbolic_emergence_theorem_thermodynamics}, there exist\nstate and context displacements whose induced transports do not commute.\nThus no single flat, additively separable transport geometry represents the\ncontextual update.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Duplicate anchor of the necessity corollary; same kernel."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-057"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Atlas.non_euclidean_necessity"
    ]
  },
  "line": 2202,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Necessity of Non-Euclidean Symbolic Transport",
  "proof_labels": [
    "proof:bk1_necessity_of_non_euclidean_symbolic_space"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "lidean Symbolic Transport] \\label{corollary:bk1_necessity_of_non_euclidean_symbolic_space} Under the hypotheses of Thm.~\\ref{theorem:bk1_symbolic_emergence_theorem_thermodynamics}, there exist state and context displacements whose induced transports do not commute. Thus no single flat, additively s",
      "label": "theorem:bk1_symbolic_emergence_theorem_thermodynamics",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2160,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk1_symbolic_emergence_theorem_thermodynamics"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "proof:bk1_necessity_of_non_euclidean_symbolic_space",
  "label": "proof:bk1_necessity_of_non_euclidean_symbolic_space",
  "latex_body": "\\begin{proof}[Nonzero Holonomy Obstructs Flat Transport]\n\\label{proof:bk1_necessity_of_non_euclidean_symbolic_space}\n\\leavevmode\nThe theorem supplies a nonzero mixed cross-error and, at nonzero observer\nscale, two induced transports whose composites depend on order.  Flat\nadditively separable transport has zero mixed cross-error and commuting routes.\nTherefore the witnessed update is non-Euclidean in the precise transport sense\nclaimed.\n\\end{proof}",
  "line": 2210,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Nonzero Holonomy Obstructs Flat Transport",
  "proves": "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "sec:bk1_toward_symbolic_primacy_and_unified_fields",
  "label": "sec:bk1_toward_symbolic_primacy_and_unified_fields",
  "latex_body": "",
  "line": 2222,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Toward Symbolic Primacy and Unified Fields",
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "scholium_symbolicum.tex",
  "id": "subsec:bk1_symbolic_reflexivity_and_srmf",
  "label": "subsec:bk1_symbolic_reflexivity_and_srmf",
  "latex_body": "",
  "line": 2224,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Reflexivity and SRMF",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "definition:bk7_symbolic_reflexive_validation_srv",
    "remark:bk4_quantum_topological_phases",
    "remark:bk7_unnamed_remark_04",
    "scholium:bk4_o_boundedness_unifying_principle",
    "scholium:bk7_popperian_extension"
  ],
  "cites": [
    "axiom:bk1_axiomata_prima",
    "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "depends_on": [
    "axiom:bk1_axiomata_prima",
    "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "file": "scholium_symbolicum.tex",
  "id": "axiom:bk1_symbolic_primacy",
  "label": "axiom:bk1_symbolic_primacy",
  "latex_body": "\\begin{axiom}[Symbolic Primacy]\n\\label{axiom:bk1_symbolic_primacy}\nIn continuity with Axiom~\\ref{axiom:bk1_axiomata_prima}, Def.~\\ref{definition:bk1_symbolic_manifold}, Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, and Cor.~\\ref{corollary:bk1_necessity_of_non_euclidean_symbolic_space}, the structure of physical law and the structure of symbolic emergence are not two domains. They are different projections of a single reflexive manifold.\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "linear-transport model for holonomy (Christoffel/vector-field forms stay open); pair-covering as the topological-regularity stand-in (point-set topology unmodeled, named); smoothness-as-C-infinity stays open"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Physical law and symbolic emergence as projections of one reflexive manifold: the curvature-necessity kernel grounds the shared structure; the primacy claim stays interpretive."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-096"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Atlas.non_euclidean_necessity"
    ]
  },
  "line": 2226,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Symbolic Primacy",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "\\begin{axiom}[Symbolic Primacy] \\label{axiom:bk1_symbolic_primacy} In continuity with Axiom~\\ref{axiom:bk1_axiomata_prima}, Def.~\\ref{definition:bk1_symbolic_manifold}, Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, and Cor.~\\ref{co",
      "label": "axiom:bk1_axiomata_prima",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book1.tex",
      "target_line": 3,
      "target_type": "axiom"
    },
    {
      "context": "_prima}, Def.~\\ref{definition:bk1_symbolic_manifold}, Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, and Cor.~\\ref{corollary:bk1_necessity_of_non_euclidean_symbolic_space}, the structure of physical law and the structure of symbolic emergence are not two domains. They are different projecti",
      "label": "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2202,
      "target_type": "corollary"
    },
    {
      "context": "iom}[Symbolic Primacy] \\label{axiom:bk1_symbolic_primacy} In continuity with Axiom~\\ref{axiom:bk1_axiomata_prima}, Def.~\\ref{definition:bk1_symbolic_manifold}, Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, and Cor.~\\ref{corollary:bk1_necessity_of_non_euclidean_symbol",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "lic_primacy} In continuity with Axiom~\\ref{axiom:bk1_axiomata_prima}, Def.~\\ref{definition:bk1_symbolic_manifold}, Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}, and Cor.~\\ref{corollary:bk1_necessity_of_non_euclidean_symbolic_space}, the structure of physical law and the structur",
      "label": "theorem:bk1_symbolic_emergence_and_curvature",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2029,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "axiom:bk1_axiomata_prima",
    "corollary:bk1_necessity_of_non_euclidean_symbolic_space",
    "definition:bk1_symbolic_manifold",
    "theorem:bk1_symbolic_emergence_and_curvature"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "scholium_symbolicum",
  "cited_by": [
    "axiom:bk8_symbolic_reidemeister_algebra",
    "definition:bk1_horizon_crossing_operation",
    "definition:bk1_paradox_triggered_emergence",
    "definition:bk1_srmf_energy_functional",
    "definition:bk4_projective_action_transl",
    "definition:bk4_test_time_coherent_sampling",
    "definition:bk4_test_time_integrative_expansion",
    "definition:bk4_test_time_precision_refinement",
    "definition:bk5_process_free_energy",
    "definition:bk7_srmfconstrained_observer",
    "definition:bk7_symbolic_reflexive_validation_srv",
    "definition:bk8_metabolic_programming_cycle",
    "definition:bk8_symbolic_stress_tensor",
    "definition:bk9_bidirectional_srmf",
    "definition:bk9_collapse_inversion_operator",
    "definition:bk9_covenant_drift_density",
    "definition:bk9_prompt_injection_operator",
    "definition:bk9_srmf_recursive_cycle",
    "definition:bk9_temetic_artifact",
    "demonstratio:bk8_symbolic_unkotting",
    "lemma:bk4_srmf_constrained_action_norm",
    "lemma:bk7_budgetlimited_minimizer",
    "proof:bk1_unified_field_classification",
    "proof:bk7_emergent_lp_norm_from_srmf",
    "proof:bk7_srmf_decency_regulation",
    "proof:bk8_membrane_operator_symmetry",
    "proof:bk8_sketch_convergence_to_fixed_by_banach",
    "proposition:bk4_spiral_transition",
    "proposition:bk7_srmf_decency_regulation",
    "proposition:bk8_membrane_operator_symmetry",
    "proposition:bk9_framework_functional_identity",
    "remark:bk4_quantum_topological_phases",
    "scholium:bk4_ttcs_simulation_tool_use",
    "scholium:bk4_ttdc_impulse_collapse",
    "scholium:bk5_metabolic_cost_of_cognition",
    "scholium:bk7_popperian_extension",
    "sec:bk5_srmf_for_symbolic_operators_and_processes",
    "subsec:appD_process_philosophy_contribution_differentiation",
    "subsec:bk5_srmf_core_axioms",
    "subsec:bk7_hdb_formal_closure",
    "theorem:bk1_unified_field_classification",
    "theorem:bk3_criteria_persistent_symbolic_life",
    "theorem:bk5_operator_convergence",
    "theorem:bk7_reflective_convergence_to_stable_identity",
    "theorem:bk8_rg_fixed_point",
    "theorem:bk8_sr_convergence"
  ],
  "cites": [
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_contradiction",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_probabilty_density"
  ],
  "depends_on": [
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_contradiction",
    "definition:bk1_symbolic_manifold"
  ],
  "file": "scholium_symbolicum.tex",
  "forward_ref_roles": [
    {
      "context": "\\cdot \\reflect(\\mathcal{C}_x) \\] Where: \\begin{itemize} \\item $\\rho: S \\to \\mathbb{R}$ is a symbolic density field (see \\ref{definition:bk1_symbolic_probabilty_density}) \\item $\\delta_{\\mathcal{C}}(x)$ is a contradiction detection function such that $\\delta_{\\mathcal{C}}(x) = \\|\\nabla \\t",
      "label": "definition:bk1_symbolic_probabilty_density",
      "line_distance": 810,
      "role": "teaser",
      "target_line": 3040,
      "target_type": "definition"
    }
  ],
  "forward_refs": [
    "definition:bk1_symbolic_probabilty_density"
  ],
  "id": "definition:bk1_self_regulating_mapping_function_srmf",
  "label": "definition:bk1_self_regulating_mapping_function_srmf",
  "latex_body": "\\begin{definition}[Self-Regulating Mapping Function (SRMF)]\n\\label{definition:bk1_self_regulating_mapping_function_srmf}\nA SRMF is a reflexive operator $\\mathcal{F}: S \\to S$ on a symbolic manifold $S$ (see \\ref{definition:bk1_symbolic_manifold}) such that:\n\\[\n\\mathcal{F}[\\rho](x) = \\rho(x) + \\delta_{\\mathcal{C}}(x) \\cdot \\reflect(\\mathcal{C}_x)\n\\]\nWhere:\n\\begin{itemize}\n\\item $\\rho: S \\to \\mathbb{R}$ is a symbolic density field (see \\ref{definition:bk1_symbolic_probabilty_density})\n\\item $\\delta_{\\mathcal{C}}(x)$ is a contradiction detection function such that $\\delta_{\\mathcal{C}}(x) = \\|\\nabla \\times \\nabla \\rho(x)\\|$ measuring local symbolic inconsistency (see \\ref{definition:bk1_symbolic_contradiction})\n\\item $\\mathcal{C}_x$ is the contradiction manifold at $x$\n\\item $\\reflect: \\mathcal{C} \\to T_xS$ is a reframing operator mapping contradictions to tangent vectors in symbolic space (see \\ref{definition:bk1_reflection_operator})\n\\end{itemize}\nThe SRMF satisfies the equilibrium condition:\n\\[\n\\lim_{t \\to \\infty} \\mathcal{F}^t[\\rho] \\in \\text{Fix}(\\mathcal{F})\n\\]\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "the circle part of a revolution is the identity by construction; injections are data; no claim about this file or any system proving its own consistency",
      "the helix is FOR approaching the equilibrium circle, not a telos; non-closure is not idolized"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The SRMF revolution structure with the closure dichotomy; the full operator pipeline interpretive."
    ],
    "record_ids": [
      "MAP-SCHOLIUM_A-085"
    ],
    "statuses": [
      "constructed"
    ],
    "witnesses": [
      "SRMF.turn_closes_iff"
    ]
  },
  "line": 2230,
  "macros_used": [
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "book1_foundational_scholium",
  "name": "Self-Regulating Mapping Function (SRMF)",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "reflect: \\mathcal{C} \\to T_xS$ is a reframing operator mapping contradictions to tangent vectors in symbolic space (see \\ref{definition:bk1_reflection_operator}) \\end{itemize} The SRMF satisfies the equilibrium condition: \\[ \\lim_{t \\to \\infty} \\mathcal{F}^t[\\rho] \\in \\text{Fix}(",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "tion such that $\\delta_{\\mathcal{C}}(x) = \\|\\nabla \\times \\nabla \\rho(x)\\|$ measuring local symbolic inconsistency (see \\ref{definition:bk1_symbolic_contradiction}) \\item $\\mathcal{C}_x$ is the contradiction manifold at $x$ \\item $\\reflect: \\mathcal{C} \\to T_xS$ is a reframing opera",
      "label": "definition:bk1_symbolic_contradiction",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1305,
      "target_type": "definition"
    },
    {
      "context": "regulating_mapping_function_srmf} A SRMF is a reflexive operator $\\mathcal{F}: S \\to S$ on a symbolic manifold $S$ (see \\ref{definition:bk1_symbolic_manifold}) such that: \\[ \\mathcal{F}[\\rho](x) = \\rho(x) + \\delta_{\\mathcal{C}}(x) \\cdot \\reflect(\\mathcal{C}_x) \\] Where: \\begin{",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "\\cdot \\reflect(\\mathcal{C}_x) \\] Where: \\begin{itemize} \\item $\\rho: S \\to \\mathbb{R}$ is a symbolic density field (see \\ref{definition:bk1_symbolic_probabilty_density}) \\item $\\delta_{\\mathcal{C}}(x)$ is a contradiction detection function such that $\\delta_{\\mathcal{C}}(x) = \\|\\nabla \\t",
      "label": "definition:bk1_symbolic_probabilty_density",
      "logical_support": false,
      "role": "forward_teaser",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3040,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_contradiction",
    "definition:bk1_symbolic_manifold",
    "definition:bk1_symbolic_probabilty_density"
  ],
  "role": "definition",
  "type": "definition"
}