Dual Horizon – A Formal Proof by Elimination — page 3 of 3

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  "id": "proof:appC_phi_minimized_entropy_per_complexity",
  "label": "proof:appC_phi_minimized_entropy_per_complexity",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_phi_minimized_entropy_per_complexity}\nBy Theorem~\\ref{theorem:appC_phi_min_growth}, every sustainable $\\lambda$ satisfies\n$\\lambda\\ge\\varphi>1$. On $(1,\\infty)$,\n\\[\n\\mathcal{I}'(\\lambda)=1-\\frac{1}{\\lambda^2}>0,\n\\]\nso $\\mathcal{I}$ is strictly increasing throughout the feasible interval\n$[\\varphi,\\infty)$. Therefore the minimum over sustainable rates occurs at the\nleft endpoint $\\lambda=\\varphi$.\n\\end{proof}",
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      "context": "\\begin{proof} \\label{proof:appC_phi_minimized_entropy_per_complexity} By Theorem~\\ref{theorem:appC_phi_min_growth}, every sustainable $\\lambda$ satisfies $\\lambda\\ge\\varphi>1$. On $(1,\\infty)$, \\[ \\mathcal{I}'(\\lambda)=1-\\frac{1}{\\lam",
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  "id": "definition:appC_phi_stable_region",
  "label": "definition:appC_phi_stable_region",
  "latex_body": "\\begin{definition}[$\\varphi$-Stable Region]\n\\label{definition:appC_phi_stable_region}\nLet $\\Phi$ be the entropy-minimizing reflective update map on the observer's\nsymbolic manifold. A region $M_\\varphi$ is \\emph{$\\varphi$-stable} if:\n\\begin{enumerate}\n    \\item it is invariant under the update, $\\Phi(M_\\varphi)\\subseteq M_\\varphi$;\n    \\item along $M_\\varphi$ the curvature operator satisfies\n    \\[\n    \\langle K_t(v), v \\rangle = \\varphi^{-1} \\|v\\|^2;\n    \\]\n    \\item there is a neighborhood $U$ of $M_\\varphi$ and a constant $q<1$ such that\n    \\[\n    d(\\Phi(x),M_\\varphi)\\le q\\,d(x,M_\\varphi)\\qquad(x\\in U).\n    \\]\n\\end{enumerate}\n\\end{definition}",
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  "name": "$\\varphi$-Stable Region",
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  "id": "lemma:appC_geodesic_convergence",
  "label": "lemma:appC_geodesic_convergence",
  "latex_body": "\\begin{lemma}[Geodesic Convergence to $M_\\varphi$]\n\\label{lemma:appC_geodesic_convergence}\nIf an observer trajectory $x_{n+1}=\\Phi(x_n)$ remains in the neighborhood $U$ of a\n$\\varphi$-stable region $M_\\varphi$, then $x_n$ converges to $M_\\varphi$ in\nobserver-relative distance:\n\\[\n d(x_n,M_\\varphi)\\le q^n d(x_0,M_\\varphi)\\longrightarrow0 .\n\\]\n\\end{lemma}",
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  "name": "Geodesic Convergence to $M_\\varphi$",
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  "id": "proof:appC_geodesic_convergence",
  "label": "proof:appC_geodesic_convergence",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_geodesic_convergence}\nThe contraction clause in Def.~\\ref{definition:appC_phi_stable_region} gives\n$d(x_{n+1},M_\\varphi)=d(\\Phi(x_n),M_\\varphi)\\le qd(x_n,M_\\varphi)$ whenever\n$x_n\\in U$. Iterating yields\n$d(x_n,M_\\varphi)\\le q^n d(x_0,M_\\varphi)$. Since $0\\le q<1$, $q^n\\to0$, so the\ndistance from the trajectory to $M_\\varphi$ tends to zero. Invariance of\n$M_\\varphi$ ensures that once the trajectory reaches the stable region it remains\nthere.\n\\end{proof}",
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      "context": "\\begin{proof} \\label{proof:appC_geodesic_convergence} The contraction clause in Def.~\\ref{definition:appC_phi_stable_region} gives $d(x_{n+1},M_\\varphi)=d(\\Phi(x_n),M_\\varphi)\\le qd(x_n,M_\\varphi)$ whenever $x_n\\in U$. Iterating yields $d(x_n,M",
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  "latex_body": "\\begin{scholium}[Symbolic–Geometric Equivalence of $\\varphi$]\n\\label{scholium:appC_symbolic_geometric_equivalence}\nThe golden ratio appears in symbolic thermodynamics, curvature operators, and recursive observer models. It is a structural attractor unifying symbolic emergence (cf. Scholium~\\ref{scholium:appC_two_horizons_co_constitutive}) and the memory-based geometry of time (Scholium~\\ref{scholium:appC_time_as_memory}).\n\\end{scholium}",
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      "context": "ergence (cf. Scholium~\\ref{scholium:appC_two_horizons_co_constitutive}) and the memory-based geometry of time (Scholium~\\ref{scholium:appC_time_as_memory}). \\end{scholium}",
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    {
      "context": "vature operators, and recursive observer models. It is a structural attractor unifying symbolic emergence (cf. Scholium~\\ref{scholium:appC_two_horizons_co_constitutive}) and the memory-based geometry of time (Scholium~\\ref{scholium:appC_time_as_memory}). \\end{scholium}",
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  "latex_body": "\\begin{definition}[Symbolic Operator Assumptions]\n\\label{definition:appC_symbolic_operator_assumptions}\nAssume symbolic emergence is represented by a positive two-step complexity\nsequence $(s_n)$ whose state vector is\n\\[\n\\mathbf{s}_n=(s_n,s_{n-1})^T.\n\\]\nThe minimal drift--reflection closure preserves current symbolic content and one\nmemory trace:\n\\[\ns_{n+1}=s_n+s_{n-1}.\n\\]\nThus Drift contributes the current term $s_n$, Reflection contributes the retained\nmemory term $s_{n-1}$, and recursive emergence is their balanced composition.\n\\end{definition}",
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  "latex_body": "\\begin{lemma}[Matrix Representation of Symbolic Operators]\n\\label{lemma:appC_matrix_representation_symbolic_operators}\nUnder the two-step closure of\nDef.~\\ref{definition:appC_symbolic_operator_assumptions}, symbolic evolution is\nrepresented by\n\\[\nM=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix},\n\\qquad\n\\mathbf{s}_{n+1}=M\\mathbf{s}_n.\n\\]\n\\end{lemma}",
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  "name": "Matrix Representation of Symbolic Operators",
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  "ref_roles": [
    {
      "context": "n of Symbolic Operators] \\label{lemma:appC_matrix_representation_symbolic_operators} Under the two-step closure of Def.~\\ref{definition:appC_symbolic_operator_assumptions}, symbolic evolution is represented by \\[ M=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}, \\qquad \\mathbf{s}_{n+1}=M\\mathbf{s}_n.",
      "label": "definition:appC_symbolic_operator_assumptions",
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  "id": "proof:appC_matrix_rep_symbolic_operators",
  "label": "proof:appC_matrix_rep_symbolic_operators",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_matrix_rep_symbolic_operators}\nBy definition,\n$s_{n+1}=s_n+s_{n-1}$ and the memory coordinate updates by\n$s_n\\mapsto s_n$. Therefore\n\\[\n\\begin{pmatrix}s_{n+1}\\\\s_n\\end{pmatrix}\n=\\begin{pmatrix}s_n+s_{n-1}\\\\s_n\\end{pmatrix}\n=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}\n\\begin{pmatrix}s_n\\\\s_{n-1}\\end{pmatrix}.\n\\]\nThis proves the claimed matrix representation.\n\\end{proof}",
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  "id": "theorem:appC_phi_eigenvalue_recursive_emergence",
  "label": "theorem:appC_phi_eigenvalue_recursive_emergence",
  "latex_body": "\\begin{theorem}[Golden Ratio as Eigenvalue of Recursive Emergence]\n\\label{theorem:appC_phi_eigenvalue_recursive_emergence}\nThe golden ratio $\\varphi$ is the Perron--Frobenius eigenvalue, hence the dominant\nasymptotic growth factor, of the minimal recursive-emergence matrix\n$M=\\bigl(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\bigr)$.\n\\end{theorem}",
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  "name": "Golden Ratio as Eigenvalue of Recursive Emergence",
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  "id": "proof:appC_phi_eigenvalue_recursive_emergence",
  "label": "proof:appC_phi_eigenvalue_recursive_emergence",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_phi_eigenvalue_recursive_emergence}\nThe characteristic polynomial is\n\\[\n\\det(M-\\mu I)=\\det\\begin{pmatrix}1-\\mu&1\\\\1&-\\mu\\end{pmatrix}\n=\\mu^2-\\mu-1.\n\\]\nIts roots are\n\\[\n\\mu_+=\\frac{1+\\sqrt5}{2}=\\varphi,\n\\qquad\n\\mu_- =\\frac{1-\\sqrt5}{2}=-\\varphi^{-1}.\n\\]\nSince $|\\mu_-|<\\mu_+$, the spectral radius is $\\varphi$. The matrix has strictly\npositive powers after finitely many steps, so the Perron--Frobenius eigenvalue is\nreal, positive, simple, and equal to this spectral radius. Hence generic positive\nstate vectors grow asymptotically at rate $\\varphi$.\n\\end{proof}",
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  "id": "lemma:appC_fibonacci_structure_matrix_powers",
  "label": "lemma:appC_fibonacci_structure_matrix_powers",
  "latex_body": "\\begin{lemma}[Fibonacci Structure via Matrix Powers]\n\\label{lemma:appC_fibonacci_structure_matrix_powers}\nLet $F_0=0$, $F_1=1$, and $F_{n+1}=F_n+F_{n-1}$. For\n$M=\\bigl(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\bigr)$,\n\\[\nM^n=\\begin{pmatrix}F_{n+1}&F_n\\\\F_n&F_{n-1}\\end{pmatrix}\\qquad(n\\ge1).\n\\]\n\\end{lemma}",
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      "Exact matrix-power identity for n>=1, reindexed n|->n+1 to avoid Nat subtraction at the excluded n=0 case."
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  "name": "Fibonacci Structure via Matrix Powers",
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  "id": "proof:appC_fibonacci_matrix_powers",
  "label": "proof:appC_fibonacci_matrix_powers",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_fibonacci_matrix_powers}\nFor $n=1$ the formula gives\n$\\bigl(\\begin{smallmatrix}F_2&F_1\\\\F_1&F_0\\end{smallmatrix}\\bigr)\n=\\bigl(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\bigr)=M$. Assume the formula\nholds for $n$. Then\n\\[\nM^{n+1}=M^nM\n=\\begin{pmatrix}F_{n+1}&F_n\\\\F_n&F_{n-1}\\end{pmatrix}\n \\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}\n=\\begin{pmatrix}F_{n+1}+F_n&F_{n+1}\\\\F_n+F_{n-1}&F_n\\end{pmatrix}\n=\\begin{pmatrix}F_{n+2}&F_{n+1}\\\\F_{n+1}&F_n\\end{pmatrix}.\n\\]\nThis is the formula with $n$ replaced by $n+1$.\n\\end{proof}",
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  "id": "proposition:appC_conditional_minimality_2x2",
  "label": "proposition:appC_conditional_minimality_2x2",
  "latex_body": "\\begin{proposition}[Conditional Minimality of 2×2 Form]\n\\label{proposition:appC_conditional_minimality_2x2}\nUnder the assumption that symbolic emergence requires encoding both current state\nand one memory state, the 2×2 matrix form is minimal for representing the\ndrift-reflection composition (see Axiom~\\ref{axiom:appC_axiom_of_memory}).\n\\end{proposition}",
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    {
      "context": "state and one memory state, the 2×2 matrix form is minimal for representing the drift-reflection composition (see Axiom~\\ref{axiom:appC_axiom_of_memory}). \\end{proposition}",
      "label": "axiom:appC_axiom_of_memory",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 846,
      "target_type": "axiom"
    }
  ],
  "refs": [
    "axiom:appC_axiom_of_memory"
  ],
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "lemma:appC_matrix_representation_symbolic_operators"
  ],
  "depends_on": [
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  ],
  "file": "appendix_dual_horizon.tex",
  "id": "proof:appC_conditional_minimality_2x2",
  "label": "proof:appC_conditional_minimality_2x2",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_conditional_minimality_2x2}\nA one-dimensional linear state stores only one scalar degree of freedom at step\n$n$. It can represent a Markov update $s_{n+1}=a s_n$, but it cannot distinguish\ntwo histories with the same current value $s_n$ and different previous values\n$s_{n-1}$, even though the required recursion\n$s_{n+1}=f(s_n,s_{n-1})$ depends on both. Therefore dimension one is insufficient\nfor memory-dependent emergence. Dimension two is sufficient, because the state\nvector $(s_n,s_{n-1})^T$ and the matrix in\nLemma~\\ref{lemma:appC_matrix_representation_symbolic_operators} exactly encode the\ncurrent value and one retained memory trace. Hence $2\\times2$ is minimal under the\nstated one-step-memory assumption.\n\\end{proof}",
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  "matter_role": "appendix_expansion",
  "name": "",
  "proves": "proposition:appC_conditional_minimality_2x2",
  "ref_roles": [
    {
      "context": "ry-dependent emergence. Dimension two is sufficient, because the state vector $(s_n,s_{n-1})^T$ and the matrix in Lemma~\\ref{lemma:appC_matrix_representation_symbolic_operators} exactly encode the current value and one retained memory trace. Hence $2\\times2$ is minimal under the stated one-step-m",
      "label": "lemma:appC_matrix_representation_symbolic_operators",
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  "refs": [
    "lemma:appC_matrix_representation_symbolic_operators"
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{
  "book": "appendix_dual_horizon",
  "cited_by": [
    "definition:appC_symbolic_curvature_function",
    "proof:appC_geometric_interpretation_curvature"
  ],
  "cites": [
    "definition:bk4_bounded_observer"
  ],
  "depends_on": [
    "definition:bk4_bounded_observer"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "definition:appC_bounded_symbolic_observer_dynamics",
  "label": "definition:appC_bounded_symbolic_observer_dynamics",
  "latex_body": "\\begin{definition}[Bounded Symbolic Observer Dynamics]\n\\label{definition:appC_bounded_symbolic_observer_dynamics}\nConsider a symbolic observer with bounded attention radius $\\delta$ (cf.~\\ref{definition:bk4_bounded_observer}) navigating meaning space. The observer experiences:\n\\begin{itemize}\n\\item Forward drift: tendency to explore new symbolic territory at rate $\\theta$\n\\item Reflective curvature: memory-based constraint pulling back with strength $1/\\theta$\n\\item Bounded exploration: total symbolic displacement must remain finite\n\\end{itemize}\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
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    ],
    "countermodels": [],
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    "kernel_certified": true,
    "notes": [
      "theta as forward-drift rate, 1/theta as curvature penalty, kept as plain reals rather than a manifold dynamical system."
    ],
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  "line": 1332,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Bounded Symbolic Observer Dynamics",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ition:appC_bounded_symbolic_observer_dynamics} Consider a symbolic observer with bounded attention radius $\\delta$ (cf.~\\ref{definition:bk4_bounded_observer}) navigating meaning space. The observer experiences: \\begin{itemize} \\item Forward drift: tendency to explore new symbo",
      "label": "definition:bk4_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book4.tex",
      "target_line": 427,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk4_bounded_observer"
  ],
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  "type": "definition"
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "definition:appC_bounded_symbolic_observer_dynamics"
  ],
  "depends_on": [
    "definition:appC_bounded_symbolic_observer_dynamics"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "definition:appC_symbolic_curvature_function",
  "label": "definition:appC_symbolic_curvature_function",
  "latex_body": "\\begin{definition}[Symbolic Curvature Function]\n\\label{definition:appC_symbolic_curvature_function}\nThe total symbolic curvature experienced by the observer (Def.~\\ref{definition:appC_bounded_symbolic_observer_dynamics}) is:\n\\[\n\\kappa(\\theta) = \\theta + \\frac{1}{\\theta}\n\\]\nwhere $\\theta > 0$ represents the ratio of forward drift to reflective strength.\n\\end{definition}",
  "line": 1342,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Symbolic Curvature Function",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ion] \\label{definition:appC_symbolic_curvature_function} The total symbolic curvature experienced by the observer (Def.~\\ref{definition:appC_bounded_symbolic_observer_dynamics}) is: \\[ \\kappa(\\theta) = \\theta + \\frac{1}{\\theta} \\] where $\\theta > 0$ represents the ratio of forward drift to refle",
      "label": "definition:appC_bounded_symbolic_observer_dynamics",
      "logical_support": true,
      "role": "definition_anchor",
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  ],
  "refs": [
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  "role": "definition",
  "type": "definition"
}

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  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "definition:bk6_symbolic_curvature_tensor"
  ],
  "depends_on": [
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    "definition:bk6_symbolic_curvature_tensor"
  ],
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  "id": "lemma:appC_geometric_interpretation_curvature",
  "label": "lemma:appC_geometric_interpretation_curvature",
  "latex_body": "\\begin{lemma}[Geometric Interpretation of Curvature Terms]\n\\label{lemma:appC_geometric_interpretation_curvature}\nThe term $\\theta$ represents symbolic drift velocity, while $1/\\theta$ represents\nthe curvature penalty imposed by bounded memory. The sum $\\kappa(\\theta)$ measures\ntotal symbolic effort required to maintain coherent exploration (cf.\nDef.~\\ref{definition:bk6_symbolic_curvature_tensor}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "AM-GM lower bound on kappa(theta)=theta+1/theta plus its equality trichotomy at theta=1."
    ],
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      "MAP-APPENDIX_DUAL_HORIZON-039"
    ],
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      "exact"
    ],
    "witnesses": [
      "Book7B.symbolicEffort_eq_two_iff",
      "Book7B.symbolicEffort_ge_two"
    ]
  },
  "line": 1351,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Geometric Interpretation of Curvature Terms",
  "proof_labels": [
    "proof:appC_geometric_interpretation_curvature"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ded memory. The sum $\\kappa(\\theta)$ measures total symbolic effort required to maintain coherent exploration (cf. Def.~\\ref{definition:bk6_symbolic_curvature_tensor}). \\end{lemma}",
      "label": "definition:bk6_symbolic_curvature_tensor",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book6.tex",
      "target_line": 16,
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    }
  ],
  "refs": [
    "definition:bk6_symbolic_curvature_tensor"
  ],
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "definition:appC_bounded_symbolic_observer_dynamics"
  ],
  "depends_on": [
    "definition:appC_bounded_symbolic_observer_dynamics"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "proof:appC_geometric_interpretation_curvature",
  "label": "proof:appC_geometric_interpretation_curvature",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_geometric_interpretation_curvature}\nBy Def.~\\ref{definition:appC_bounded_symbolic_observer_dynamics}, $\\theta$ is the\nforward exploration rate, so its contribution to one-step effort is linear in\n$\\theta$ after normalization of units. The reflective term must decrease as drift\nincreases and increase as drift slows, because slower forward motion forces a\nlarger fraction of the step to be spent maintaining memory coherence. The\nscale-free reciprocal $1/\\theta$ is the unique reciprocal penalty normalized to\nbe $1$ at the balanced point $\\theta=1$. Since the two costs are paid in the same\nstep and in the same normalized units, finite symbolic effort is their additive\nsum $\\kappa(\\theta)=\\theta+1/\\theta$.\n\\end{proof}",
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  "macros_used": [],
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  "matter_role": "appendix_expansion",
  "name": "",
  "proves": "lemma:appC_geometric_interpretation_curvature",
  "ref_roles": [
    {
      "context": "\\begin{proof} \\label{proof:appC_geometric_interpretation_curvature} By Def.~\\ref{definition:appC_bounded_symbolic_observer_dynamics}, $\\theta$ is the forward exploration rate, so its contribution to one-step effort is linear in $\\theta$ after normaliza",
      "label": "definition:appC_bounded_symbolic_observer_dynamics",
      "logical_support": true,
      "role": "definition_anchor",
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    }
  ],
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}

{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "appendix_dual_horizon.tex",
  "id": "theorem:appC_phi_minimal_curvature_parameter",
  "label": "theorem:appC_phi_minimal_curvature_parameter",
  "latex_body": "\\begin{theorem}[Golden Ratio as Minimal Curvature Parameter]\n\\label{theorem:appC_phi_minimal_curvature_parameter}\nThe parameter $\\theta = \\varphi$ minimizes the symbolic curvature function\n$\\kappa(\\theta)$ among nondegenerate recursively sustainable exploration\nparameters, i.e. among $\\theta$ satisfying $\\theta\\ge 1+1/\\theta$.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The same theorem as theorem:appC_phi_minimized_entropy_per_complexity, stated twice in the source under two names; both are discharged by kappa_min_at_phi (and its minimality precondition by sustainable_ge_phi), restricted to the algebraic Sustainable domain theta >= 1 + 1/theta given verbatim by this anchor."
    ],
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    ],
    "statuses": [
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      "AppendixDH.kappa_min_at_phi",
      "AppendixDH.sustainable_ge_phi"
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  "line": 1372,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Golden Ratio as Minimal Curvature Parameter",
  "proof_labels": [
    "proof:appC_phi_minimal_curvature"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "theorem",
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}

{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "appendix_dual_horizon.tex",
  "id": "proof:appC_phi_minimal_curvature",
  "label": "proof:appC_phi_minimal_curvature",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_phi_minimal_curvature}\nThe recursive sustainability constraint is\n\\[\n\\theta\\ge 1+\\frac{1}{\\theta},\n\\]\nwhich is equivalent, for $\\theta>0$, to\n$\\theta^2-\\theta-1\\ge0$. Hence the feasible set is\n$[\\varphi,\\infty)$, where $\\varphi=(1+\\sqrt5)/2$. On this interval,\n\\[\n\\kappa'(\\theta)=1-\\frac{1}{\\theta^2}>0,\n\\]\nbecause $\\theta\\ge\\varphi>1$. Thus $\\kappa$ is strictly increasing on the feasible\nset and its minimum occurs at the left endpoint $\\theta=\\varphi$. The minimized\ncurvature is\n\\[\n\\kappa(\\varphi)=\\varphi+\\frac1\\varphi=\\varphi+(\\varphi-1)=2\\varphi-1=\\sqrt5.\n\\]\nThe unconstrained point $\\theta=1$ is lower for $\\kappa$ alone, but it violates\nthe nondegenerate recursive sustainability constraint $\\theta\\ge1+1/\\theta$ and\ntherefore represents stagnation rather than sustained symbolic exploration.\n\\end{proof}",
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  "name": "",
  "proves": "theorem:appC_phi_minimal_curvature_parameter",
  "refs": [],
  "role": "proof",
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}

{
  "book": "appendix_dual_horizon",
  "cited_by": [
    "lemma:appC_stability_phi_flow"
  ],
  "cites": [
    "definition:bk6_reflection_operator_complete"
  ],
  "depends_on": [
    "definition:bk6_reflection_operator_complete"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "definition:appC_symbolic_flow_stability",
  "label": "definition:appC_symbolic_flow_stability",
  "latex_body": "\\begin{definition}[Symbolic Flow Stability]\n\\label{definition:appC_symbolic_flow_stability}\nA symbolic flow is stable if small perturbations in the exploration parameter $\\theta$ decay exponentially. The stability condition requires:\n\\[\n\\left| \\frac{d}{d\\theta} \\left( 1 + \\frac{1}{\\theta} \\right) \\right|_{\\theta=\\varphi} < 1\n\\]\n(cf. Def.~\\ref{definition:bk6_reflection_operator_complete})\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
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    "kernel_certified": true,
    "notes": [
      "The stability condition |d/dtheta(1+1/theta)| < 1 at theta=phi is realized directly as the closed-form inequality 1/(phi*phi) < 1, without modeling a general derivative operator."
    ],
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      "MAP-APPENDIX_DUAL_HORIZON-029"
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      "AppendixDH.stability_phi_flow"
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  },
  "line": 1402,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Symbolic Flow Stability",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ion requires: \\[ \\left| \\frac{d}{d\\theta} \\left( 1 + \\frac{1}{\\theta} \\right) \\right|_{\\theta=\\varphi} < 1 \\] (cf. Def.~\\ref{definition:bk6_reflection_operator_complete}) \\end{definition}",
      "label": "definition:bk6_reflection_operator_complete",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book6.tex",
      "target_line": 937,
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  ],
  "refs": [
    "definition:bk6_reflection_operator_complete"
  ],
  "role": "definition",
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}

{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "definition:appC_symbolic_flow_stability"
  ],
  "depends_on": [
    "definition:appC_symbolic_flow_stability"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "lemma:appC_stability_phi_flow",
  "label": "lemma:appC_stability_phi_flow",
  "latex_body": "\\begin{lemma}[Stability of $\\varphi$-Flow]\n\\label{lemma:appC_stability_phi_flow}\nThe $\\varphi$-flow satisfies the stability condition (cf.~\\ref{definition:appC_symbolic_flow_stability}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
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    "kernel_certified": true,
    "notes": [
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    ],
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    ]
  },
  "line": 1411,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Stability of $\\varphi$-Flow",
  "proof_labels": [
    "proof:appC_stability_phi_flow"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "lity of $\\varphi$-Flow] \\label{lemma:appC_stability_phi_flow} The $\\varphi$-flow satisfies the stability condition (cf.~\\ref{definition:appC_symbolic_flow_stability}). \\end{lemma}",
      "label": "definition:appC_symbolic_flow_stability",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 1402,
      "target_type": "definition"
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  ],
  "refs": [
    "definition:appC_symbolic_flow_stability"
  ],
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "appendix_dual_horizon.tex",
  "id": "proof:appC_stability_phi_flow",
  "label": "proof:appC_stability_phi_flow",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_stability_phi_flow}\nLet $f(\\theta)=1+1/\\theta$. Then $f'(\\theta)=-1/\\theta^2$, so\n\\[\n|f'(\\varphi)|=\\frac{1}{\\varphi^2}=2-\\varphi<1.\n\\]\nBy the one-dimensional fixed-point stability criterion, sufficiently small\nperturbations of the iteration $\\theta_{n+1}=f(\\theta_n)$ contract in a\nneighborhood of $\\varphi$. Hence the $\\varphi$-flow satisfies the stated stability\ncondition.\n\\end{proof}",
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  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "",
  "proves": "lemma:appC_stability_phi_flow",
  "refs": [],
  "role": "proof",
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}

{
  "book": "appendix_dual_horizon",
  "cited_by": [
    "remark:appC_connection_other_modalities"
  ],
  "cites": [
    "lemma:appC_matrix_representation_symbolic_operators"
  ],
  "depends_on": [
    "lemma:appC_matrix_representation_symbolic_operators"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "theorem:appC_unified_recursive_fixed_point",
  "label": "theorem:appC_unified_recursive_fixed_point",
  "latex_body": "\\begin{theorem}[Unified Recursive Fixed Point]\n\\label{theorem:appC_unified_recursive_fixed_point}\nBoth the matrix eigenvalue approach (cf.~\\ref{lemma:appC_matrix_representation_symbolic_operators}) and the topological curvature approach converge to the same fixed-point equation:\n\\[\n\\lambda = 1 + \\frac{1}{\\lambda} \\Rightarrow \\lambda^2 - \\lambda - 1 = 0 \\Rightarrow \\lambda = \\varphi\n\\]\nfor the unique positive nondegenerate fixed point.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/categorical content is NOT formalized; static and finite-discrete kernels only",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Among positive reals, phi is the unique fixed point of lam = 1 + 1/lam, proved via the same factoring argument as sustainable_ge_phi plus uniqueness of the positive root of lam^2-lam-1=0."
    ],
    "record_ids": [
      "MAP-APPENDIX_DUAL_HORIZON-031"
    ],
    "statuses": [
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    ],
    "witnesses": [
      "AppendixDH.fixed_point_iff_phi"
    ]
  },
  "line": 1431,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Unified Recursive Fixed Point",
  "proof_labels": [
    "proof:appC_unified_recursive_fixed_point"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "fied Recursive Fixed Point] \\label{theorem:appC_unified_recursive_fixed_point} Both the matrix eigenvalue approach (cf.~\\ref{lemma:appC_matrix_representation_symbolic_operators}) and the topological curvature approach converge to the same fixed-point equation: \\[ \\lambda = 1 + \\frac{1}{\\lambda} \\",
      "label": "lemma:appC_matrix_representation_symbolic_operators",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 1231,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "lemma:appC_matrix_representation_symbolic_operators"
  ],
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  "type": "theorem"
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "appendix_dual_horizon.tex",
  "id": "proof:appC_unified_recursive_fixed_point",
  "label": "proof:appC_unified_recursive_fixed_point",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_unified_recursive_fixed_point}\nIn the matrix approach, the minimal memory-preserving recurrence is\n$C_{n+1}=C_n+C_{n-1}$. If the positive asymptotic ratio\n$\\lambda=\\lim C_{n+1}/C_n$ exists, division by $C_n$ and passage to the limit give\n\\[\n\\lambda=1+\\frac{1}{\\lambda}.\n\\]\nIn the topological approach, recursive sustainable exploration requires that the\nforward parameter equal one unit of new exploration plus the reciprocal\nreflective correction, so its fixed point satisfies the same equation\n$\\theta=1+1/\\theta$. In both cases the positive solution of\n$x^2-x-1=0$ is $x=\\varphi$, while the other solution is negative and therefore\ninadmissible as a growth or curvature parameter.\n\\end{proof}",
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  "proves": "theorem:appC_unified_recursive_fixed_point",
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "definition:bk6_symbolic_curvature_tensor",
    "theorem:bk6_symbolic_diffusion_governs_evolution"
  ],
  "depends_on": [
    "definition:bk6_symbolic_curvature_tensor",
    "theorem:bk6_symbolic_diffusion_governs_evolution"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "scholium:appC_structural_universality_phi",
  "label": "scholium:appC_structural_universality_phi",
  "latex_body": "\\begin{scholium}[Structural Universality of $\\varphi$]\n\\label{scholium:appC_structural_universality_phi}\nThe independent emergence of $\\varphi$ from matrix spectral theory and topological curvature analysis suggests that $\\varphi$ represents a fundamental structural constant of bounded recursive systems. This convergence transcends particular mathematical representations, indicating an intrinsic property of symbolic emergence under resource constraints (see Def.~\\ref{definition:bk6_symbolic_curvature_tensor}, Thm.~\\ref{theorem:bk6_symbolic_diffusion_governs_evolution}).\n\\end{scholium}",
  "line": 1456,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Structural Universality of $\\varphi$",
  "ref_roles": [
    {
      "context": "thematical representations, indicating an intrinsic property of symbolic emergence under resource constraints (see Def.~\\ref{definition:bk6_symbolic_curvature_tensor}, Thm.~\\ref{theorem:bk6_symbolic_diffusion_governs_evolution}). \\end{scholium}",
      "label": "definition:bk6_symbolic_curvature_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 16,
      "target_type": "definition"
    },
    {
      "context": "roperty of symbolic emergence under resource constraints (see Def.~\\ref{definition:bk6_symbolic_curvature_tensor}, Thm.~\\ref{theorem:bk6_symbolic_diffusion_governs_evolution}). \\end{scholium}",
      "label": "theorem:bk6_symbolic_diffusion_governs_evolution",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book6.tex",
      "target_line": 1204,
      "target_type": "theorem"
    }
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    "theorem:bk6_symbolic_diffusion_governs_evolution"
  ],
  "role": "scholium",
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}

{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "theorem:appC_unified_recursive_fixed_point"
  ],
  "depends_on": [
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  ],
  "file": "appendix_dual_horizon.tex",
  "id": "remark:appC_connection_other_modalities",
  "label": "remark:appC_connection_other_modalities",
  "latex_body": "\\begin{remark}[Connection to Other Symbolic Modalities]\n\\label{remark:appC_connection_other_modalities}\nThe fixed-point equation $\\lambda = 1 + 1/\\lambda$ (cf.~\\ref{theorem:appC_unified_recursive_fixed_point}) appears in multiple contexts within symbolic dynamics. The consistent emergence of $\\varphi$ across matrix, topological, and (potentially) thermodynamic or spectral approaches is not a coincidence to be noted but a transfer to be proved: the Modal Transference Theorem below states the conditions under which an ordinal-recursive invariant such as $\\varphi$ is carried, intact, from one observer-accessible carrier to another.\n\\end{remark}",
  "line": 1461,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Connection to Other Symbolic Modalities",
  "ref_roles": [
    {
      "context": "lic Modalities] \\label{remark:appC_connection_other_modalities} The fixed-point equation $\\lambda = 1 + 1/\\lambda$ (cf.~\\ref{theorem:appC_unified_recursive_fixed_point}) appears in multiple contexts within symbolic dynamics. The consistent emergence of $\\varphi$ across matrix, topologica",
      "label": "theorem:appC_unified_recursive_fixed_point",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 1431,
      "target_type": "theorem"
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  ],
  "refs": [
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [],
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  "file": "appendix_dual_horizon.tex",
  "id": "sec:appC_modal_transference",
  "label": "sec:appC_modal_transference",
  "latex_body": "",
  "line": 1466,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Modal Transference of Symbolic Invariants",
  "role": "section",
  "subtype": "section",
  "type": "section"
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "definition:bk1_stage_composite_operator"
  ],
  "depends_on": [
    "definition:bk1_stage_composite_operator"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "definition:appC_symbolic_modality",
  "label": "definition:appC_symbolic_modality",
  "latex_body": "\\begin{definition}[Symbolic modality]\n\\label{definition:appC_symbolic_modality}\nA \\emph{symbolic modality} is a tuple\n\\[\n\\mathfrak{M} = (X_\\mathfrak{M},\\ \\preceq_\\mathfrak{M},\\ d_{\\Obs,\\mathfrak{M}},\\\nE_\\mathfrak{M},\\ \\mathcal{I}_\\mathfrak{M}),\n\\]\nwhere $X_\\mathfrak{M}$ is a space of modal presentations, $\\preceq_\\mathfrak{M}$\nis an observer-resolved emergence order, $d_{\\Obs,\\mathfrak{M}}$ is the\nobserver-relative modal distance, $E_\\mathfrak{M}$ is the stage-composite\nemergence operator (cf.~Def.~\\ref{definition:bk1_stage_composite_operator}) in\nthe modality, and $\\mathcal{I}_\\mathfrak{M}$ is a family of structural invariants\n(cyclic order, adjacency, recurrence spectrum, proportion, curvature signature).\n\\end{definition}",
  "line": 1483,
  "macros_used": [
    "Obs"
  ],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Symbolic modality",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "frak{M}}$ is the observer-relative modal distance, $E_\\mathfrak{M}$ is the stage-composite emergence operator (cf.~Def.~\\ref{definition:bk1_stage_composite_operator}) in the modality, and $\\mathcal{I}_\\mathfrak{M}$ is a family of structural invariants (cyclic order, adjacency, recurre",
      "label": "definition:bk1_stage_composite_operator",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 520,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_stage_composite_operator"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "appendix_dual_horizon",
  "cited_by": [
    "corollary:bk4_chromatic_transference_of_wheel",
    "proof:bk4_chromatic_transference_of_wheel",
    "scholium:appC_transference_tests"
  ],
  "cites": [],
  "depends_on": [],
  "file": "appendix_dual_horizon.tex",
  "id": "definition:appC_modal_transference_map",
  "label": "definition:appC_modal_transference_map",
  "latex_body": "\\begin{definition}[Modal transference map]\n\\label{definition:appC_modal_transference_map}\nLet $\\mathfrak{M}_A,\\mathfrak{M}_B$ be symbolic modalities. A \\emph{modal\ntransference map} is an observer-bounded map\n$T_{A\\to B}:X_{\\mathfrak{M}_A}\\to X_{\\mathfrak{M}_B}$ satisfying:\n\\begin{enumerate}\n\\item \\emph{Ordinal preservation:}\n$x\\preceq_{\\mathfrak{M}_A}y \\Rightarrow T_{A\\to B}(x)\\preceq_{\\mathfrak{M}_B}T_{A\\to B}(y)$.\n\\item \\emph{Observer-bounded distortion:} there exist $L<\\infty$ and\n$\\varepsilon_\\Obs\\ge 0$ with\n\\[\nd_{\\Obs,\\mathfrak{M}_B}\\!\\big(T_{A\\to B}x,\\,T_{A\\to B}y\\big)\n\\le L\\, d_{\\Obs,\\mathfrak{M}_A}(x,y) + \\varepsilon_\\Obs.\n\\]\n\\item \\emph{Operator semi-conjugacy:}\n$T_{A\\to B}\\circ E_{\\mathfrak{M}_A} \\sim_\\Obs E_{\\mathfrak{M}_B}\\circ T_{A\\to B}$,\nequality holding up to observer resolution $\\varepsilon_\\Obs$.\n\\item \\emph{Invariant preservation:} for each $I\\in\\mathcal{I}_{\\mathfrak{M}_A}$\ntransferred by $T_{A\\to B}$ there is $T_*I\\in\\mathcal{I}_{\\mathfrak{M}_B}$ with\n$I(x)=T_*I(T_{A\\to B}x)$ up to $\\varepsilon_\\Obs$.\n\\end{enumerate}\n\\end{definition}",
  "line": 1498,
  "macros_used": [
    "Obs"
  ],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Modal transference map",
  "proof_status": "definitional",
  "refs": [],
  "role": "definition",
  "type": "definition"
}

{
  "book": "appendix_dual_horizon",
  "cited_by": [
    "corollary:bk4_chromatic_transference_of_wheel",
    "proof:bk4_chromatic_transference_of_wheel",
    "scholium:appC_two_modalities_one_root"
  ],
  "cites": [
    "definition:bk5_balanced_two_step_memory_closure"
  ],
  "depends_on": [
    "definition:bk5_balanced_two_step_memory_closure"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "theorem:appC_modal_transference",
  "label": "theorem:appC_modal_transference",
  "latex_body": "\\begin{theorem}[Modal Transference]\n\\label{theorem:appC_modal_transference}\nLet $T_{A\\to B}$ be a modal transference map between symbolic modalities\n$\\mathfrak{M}_A$ and $\\mathfrak{M}_B$. Then any invariant determined only by\nordinal order, operator recurrence, cyclic adjacency, or spectral proportion is\npreserved across the transfer up to observer resolution. In particular, if a\nrecurrence invariant $\\lambda$ is fixed in $\\mathfrak{M}_A$ by the balanced\ntwo-step closure $a_{n+1}=a_n+a_{n-1}$\n(cf.~book V, Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), then its\ntransferred presentation in $\\mathfrak{M}_B$ carries the same positive spectral\ninvariant $\\lambda=\\varphi$.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Distortion-composition bound covers the quantitative 'preserved up to observer resolution' content under composition; the golden-ratio tail clause is covered by the shared Fibonacci-ratio theorem. Ordinal/operator-recurrence invariant preservation itself is not modeled."
    ],
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      "MAP-APPENDIX_DUAL_HORIZON-041"
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    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book7B.quasiLipschitz_comp",
      "Book7B.shiftedFib_ratio_tendsto_goldenRatio"
    ]
  },
  "line": 1521,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Modal Transference",
  "proof_labels": [
    "proof:appC_modal_transference"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "variant $\\lambda$ is fixed in $\\mathfrak{M}_A$ by the balanced two-step closure $a_{n+1}=a_n+a_{n-1}$ (cf.~book V, Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), then its transferred presentation in $\\mathfrak{M}_B$ carries the same positive spectral invariant $\\lambda=\\varphi$.",
      "label": "definition:bk5_balanced_two_step_memory_closure",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1828,
      "target_type": "definition"
    }
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  "refs": [
    "definition:bk5_balanced_two_step_memory_closure"
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "appendix_dual_horizon.tex",
  "id": "proof:appC_modal_transference",
  "label": "proof:appC_modal_transference",
  "latex_body": "\\begin{proof}\n\\label{proof:appC_modal_transference}\nBy ordinal preservation, $T_{A\\to B}$ preserves the emergence order of the source\nmodality; by observer-bounded distortion, differences below the observer\nthreshold in $\\mathfrak{M}_A$ remain below threshold in $\\mathfrak{M}_B$. By\noperator semi-conjugacy the transferred system follows the same emergence\ndynamics up to resolution: $T_{A\\to B}\\circ E_{\\mathfrak{M}_A} \\sim_\\Obs\nE_{\\mathfrak{M}_B}\\circ T_{A\\to B}$. An invariant fixed solely by the recurrence\nor adjacency structure of $E_{\\mathfrak{M}_A}$ cannot change when $E_{\\mathfrak{M}_A}$\nis replaced by its semi-conjugate presentation $E_{\\mathfrak{M}_B}$ except below\nthreshold. For the balanced two-step closure the companion matrix is\n$A=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$, with characteristic\nequation $\\lambda^2-\\lambda-1=0$ and positive root $\\varphi$. Since transference\npreserves the recurrence structure, the same spectral invariant appears in the\ntarget modality. Hence $\\varphi$ is not tied to a sensory carrier; it is an\nordinal-recursive invariant transferred through modal presentation.\n\\end{proof}",
  "line": 1534,
  "macros_used": [
    "Obs"
  ],
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  "matter_role": "appendix_expansion",
  "name": "",
  "proves": "theorem:appC_modal_transference",
  "refs": [],
  "role": "proof",
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{
  "book": "appendix_dual_horizon",
  "cited_by": [],
  "cites": [
    "definition:appC_modal_transference_map"
  ],
  "depends_on": [
    "definition:appC_modal_transference_map"
  ],
  "file": "appendix_dual_horizon.tex",
  "id": "scholium:appC_transference_tests",
  "label": "scholium:appC_transference_tests",
  "latex_body": "\\begin{scholium}[Sonification and the Chromatic Wheel as Transference Tests]\n\\label{scholium:appC_transference_tests}\nThe sonification and chromatic-wheel constructions are not offered as analogies.\nThey are modal transference tests. Each asks whether an invariant first defined\nin ordinal-symbolic form survives transfer into a distinct observer-accessible\ncarrier. Sonification is the transference map\n$T_{\\mathrm{symbolic}\\to\\mathrm{audio}}$ carrying symbolic order into pitch,\ninterval, rhythm, and phase; the Newtonian color wheel is the map\n$T_{\\mathrm{symbolic}\\to\\mathrm{chromatic}}$ carrying cyclic adjacency,\nopposition, and return into visual structure. When cyclic order, recurrence\nspectrum, and bounded adjacency are preserved under\nDef.~\\ref{definition:appC_modal_transference_map}, the invariant belongs to the\nsymbolic structure rather than to the particular sensory modality. This is the\nprecise sense in which domains are modal presentations of shared ordinal-symbolic\ninvariants---not the claim that everything is the same substance, but the claim\nthat distinct modalities preserve the same emergence grammar when the\ntransference map respects ordinal order, observer bounds, and operator recurrence.\n\\end{scholium}",
  "line": 1552,
  "macros_used": [],
  "matter_region": "appendix",
  "matter_role": "appendix_expansion",
  "name": "Sonification and the Chromatic Wheel as Transference Tests",
  "ref_roles": [
    {
      "context": "nd return into visual structure. When cyclic order, recurrence spectrum, and bounded adjacency are preserved under Def.~\\ref{definition:appC_modal_transference_map}, the invariant belongs to the symbolic structure rather than to the particular sensory modality. This is the precise se",
      "label": "definition:appC_modal_transference_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 1498,
      "target_type": "definition"
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  ],
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    "definition:appC_modal_transference_map"
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