Book V — De Vita Symbolica — page 4 of 6

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  "latex_body": "\\begin{proposition}\n\\label{proposition:bk5_operators_evolve}\n\\leavevmode\\newline\nOperators evolve to remain within $\\mathcal{V}_{\\text{op}}$\n(cf.~Def.~\\ref{definition:bk5__operator_viability_set_v},\nThm.~\\ref{theorem:bk5_operator_convergence},\nThm.~\\ref{theorem:bk5__srmf_operator_adaptation},\nDef.~\\ref{definition:bk4_test_time_integrative_expansion}).\nUnder hard constraints, the system sacrifices operator complexity.\n\\end{proposition}",
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    {
      "context": "r_viability_set_v}, Thm.~\\ref{theorem:bk5_operator_convergence}, Thm.~\\ref{theorem:bk5__srmf_operator_adaptation}, Def.~\\ref{definition:bk4_test_time_integrative_expansion}). Under hard constraints, the system sacrifices operator complexity. \\end{proposition}",
      "label": "definition:bk4_test_time_integrative_expansion",
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      "context": "osition:bk5_operators_evolve} \\leavevmode\\newline Operators evolve to remain within $\\mathcal{V}_{\\text{op}}$ (cf.~Def.~\\ref{definition:bk5__operator_viability_set_v}, Thm.~\\ref{theorem:bk5_operator_convergence}, Thm.~\\ref{theorem:bk5__srmf_operator_adaptation}, Def.~\\ref{definition:bk",
      "label": "definition:bk5__operator_viability_set_v",
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      "context": "\\text{op}}$ (cf.~Def.~\\ref{definition:bk5__operator_viability_set_v}, Thm.~\\ref{theorem:bk5_operator_convergence}, Thm.~\\ref{theorem:bk5__srmf_operator_adaptation}, Def.~\\ref{definition:bk4_test_time_integrative_expansion}). Under hard constraints, the system sacrifices operator com",
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      "target_line": 1632,
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      "context": "rators evolve to remain within $\\mathcal{V}_{\\text{op}}$ (cf.~Def.~\\ref{definition:bk5__operator_viability_set_v}, Thm.~\\ref{theorem:bk5_operator_convergence}, Thm.~\\ref{theorem:bk5__srmf_operator_adaptation}, Def.~\\ref{definition:bk4_test_time_integrative_expansion}). Under ha",
      "label": "theorem:bk5_operator_convergence",
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  "id": "proof:bk5_operators_evolve",
  "label": "proof:bk5_operators_evolve",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_operators_evolve}\n\\leavevmode\nThe operator viability set $\\mathcal{V}_{\\text{op}}=\\{\\mathcal{O}:\\Fproc[\\mathcal{O},S]<\\theta_{\\text{proc}}\\}$ (Def.~\\ref{definition:bk5__operator_viability_set_v}) is the strict sublevel set of $\\Fproc$. By Operator Evolution (Prop.~\\ref{proposition:bk5_operator_evolution}) the SRMF path descends $\\Fproc$ and is stationary only at a minimizer; by SRMF Operator Adaptation (Thm.~\\ref{theorem:bk5__srmf_operator_adaptation}) this descent is steepest in $\\Fproc$ and accelerates whenever effectiveness degrades. Since $\\Fproc$ is non-increasing along the flow and strictly decreasing off the minimizer, the flow maps $\\mathcal{V}_{\\text{op}}$ into itself and drives any super-threshold operator toward it, converging to a minimizer inside $\\mathcal{V}_{\\text{op}}$ (Thm.~\\ref{theorem:bk5_operator_convergence}); the integrative-expansion coupling (Def.~\\ref{definition:bk4_test_time_integrative_expansion}) supplies the admissible directions of this evolution. Thus operators evolve so as to remain within $\\mathcal{V}_{\\text{op}}$. Finally, under hard metabolic constraints the cost is capped, $\\mathcal{E}_{\\text{cost}}[\\mathcal{O}]\\le\\mathcal{E}_{\\text{cost}}^{\\max}$ (Prop.~\\ref{proposition:bk5_fixed_metabolic_capacity}); maintaining $\\Fproc<\\theta_{\\text{proc}}$ then forces the descent to economize on the cost-bearing structure of $\\mathcal{O}$, i.e.\\ to lower operator complexity. Hence under hard constraints the system sacrifices operator complexity to preserve viability.\n\\end{proof}",
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      "context": "nside $\\mathcal{V}_{\\text{op}}$ (Thm.~\\ref{theorem:bk5_operator_convergence}); the integrative-expansion coupling (Def.~\\ref{definition:bk4_test_time_integrative_expansion}) supplies the admissible directions of this evolution. Thus operators evolve so as to remain within $\\mathcal{V}_{\\text",
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      "context": "e The operator viability set $\\mathcal{V}_{\\text{op}}=\\{\\mathcal{O}:\\Fproc[\\mathcal{O},S]<\\theta_{\\text{proc}}\\}$ (Def.~\\ref{definition:bk5__operator_viability_set_v}) is the strict sublevel set of $\\Fproc$. By Operator Evolution (Prop.~\\ref{proposition:bk5_operator_evolution}) the SRM",
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      "context": "olic constraints the cost is capped, $\\mathcal{E}_{\\text{cost}}[\\mathcal{O}]\\le\\mathcal{E}_{\\text{cost}}^{\\max}$ (Prop.~\\ref{proposition:bk5_fixed_metabolic_capacity}); maintaining $\\Fproc<\\theta_{\\text{proc}}$ then forces the descent to economize on the cost-bearing structure of $\\mat",
      "label": "proposition:bk5_fixed_metabolic_capacity",
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      "target_line": 1513,
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      "context": "f.~\\ref{definition:bk5__operator_viability_set_v}) is the strict sublevel set of $\\Fproc$. By Operator Evolution (Prop.~\\ref{proposition:bk5_operator_evolution}) the SRMF path descends $\\Fproc$ and is stationary only at a minimizer; by SRMF Operator Adaptation (Thm.~\\ref{theorem:",
      "label": "proposition:bk5_operator_evolution",
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      "context": "or_evolution}) the SRMF path descends $\\Fproc$ and is stationary only at a minimizer; by SRMF Operator Adaptation (Thm.~\\ref{theorem:bk5__srmf_operator_adaptation}) this descent is steepest in $\\Fproc$ and accelerates whenever effectiveness degrades. Since $\\Fproc$ is non-increasing",
      "label": "theorem:bk5__srmf_operator_adaptation",
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      "role": "proof_support",
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      "target_line": 1632,
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      "context": "elf and drives any super-threshold operator toward it, converging to a minimizer inside $\\mathcal{V}_{\\text{op}}$ (Thm.~\\ref{theorem:bk5_operator_convergence}); the integrative-expansion coupling (Def.~\\ref{definition:bk4_test_time_integrative_expansion}) supplies the admissibl",
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  "latex_body": "\\begin{definition}[Operator Complexity, Stability Margin, Maintenance Cost]\n\\label{definition:bk5_complexity_stability_maintenance}\nFor an operator $\\mathcal{O}\\in\\Op(M)$ acting on system $S$ we define:\n\\begin{enumerate}\n    \\item \\textbf{Operator complexity} $\\mathcal{C}(\\mathcal{O}):=\\mathcal{E}_{\\text{cost}}[\\mathcal{O}]$, the cost-bearing structure of $\\mathcal{O}$ in the process free energy (Def.~\\ref{definition:bk5_process_free_energy}).\n    \\item \\textbf{Stability margin} $\\mathcal{S}(S):=\\inf_t\\big(\\freeenergy(\\rho_t)-\\freeenergy^{\\min}\\big)$, the viability reserve held against drift along the trajectory (Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk5_viability_domain}).\n    \\item \\textbf{Maintenance cost} $E_{\\text{maint}}(\\mathcal{O},S):=\\mathcal{C}(\\mathcal{O})\\,\\mathcal{S}(S)/\\alpha$, the work to hold $\\mathcal{O}$ stable against drift to margin $\\mathcal{S}(S)$, conversion constant $\\alpha>0$. The product form is a structural cost model (each unit of complexity is maintained to the degree the margin sets), not an empirically fitted law.\n\\end{enumerate}\n\\end{definition}",
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      "context": "t\\big(\\freeenergy(\\rho_t)-\\freeenergy^{\\min}\\big)$, the viability reserve held against drift along the trajectory (Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk5_viability_domain}). \\item \\textbf{Maintenance cost} $E_{\\text{maint}}(\\mathcal{O},S):=\\ma",
      "label": "definition:bk2_symbolic_free_energy",
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      "context": ":=\\mathcal{E}_{\\text{cost}}[\\mathcal{O}]$, the cost-bearing structure of $\\mathcal{O}$ in the process free energy (Def.~\\ref{definition:bk5_process_free_energy}). \\item \\textbf{Stability margin} $\\mathcal{S}(S):=\\inf_t\\big(\\freeenergy(\\rho_t)-\\freeenergy^{\\min}\\big)$, the via",
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      "context": ")$, the viability reserve held against drift along the trajectory (Def.~\\ref{definition:bk2_symbolic_free_energy}, Def.~\\ref{definition:bk5_viability_domain}). \\item \\textbf{Maintenance cost} $E_{\\text{maint}}(\\mathcal{O},S):=\\mathcal{C}(\\mathcal{O})\\,\\mathcal{S}(S)/\\alpha",
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  "name": "Complexity-Stability Tradeoff",
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  "ref_roles": [
    {
      "context": "complexity_stability_tradeoff} \\leavevmode\\newline With $\\mathcal{C}$, $\\mathcal{S}$, and $E_{\\text{maint}}$ as in Def.~\\ref{definition:bk5_complexity_stability_maintenance} and Prop.~\\ref{proposition:bk5_operators_evolve}, admissible complexity is metabolically budgeted: \\[ \\mathcal{C}(\\math",
      "label": "definition:bk5_complexity_stability_maintenance",
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      "role": "definition_anchor",
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      "target_line": 1693,
      "target_type": "definition"
    },
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      "context": "l{C}$, $\\mathcal{S}$, and $E_{\\text{maint}}$ as in Def.~\\ref{definition:bk5_complexity_stability_maintenance} and Prop.~\\ref{proposition:bk5_operators_evolve}, admissible complexity is metabolically budgeted: \\[ \\mathcal{C}(\\mathcal{O}) \\cdot \\mathcal{S}(S) \\leq \\alpha \\cdot \\M",
      "label": "proposition:bk5_operators_evolve",
      "logical_support": true,
      "role": "formal_dependency",
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  "id": "proof:bk5_complexity_stability_tradeoff",
  "label": "proof:bk5_complexity_stability_tradeoff",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_complexity_stability_tradeoff}\n\\leavevmode\nBy Def.~\\ref{definition:bk5_complexity_stability_maintenance} the maintenance cost is $E_{\\text{maint}}(\\mathcal{O},S)=\\mathcal{C}(\\mathcal{O})\\,\\mathcal{S}(S)/\\alpha$. By Operators Evolve (Prop.~\\ref{proposition:bk5_operators_evolve}) sustained viability requires this expenditure to be fundable from the metabolic capacity, $E_{\\text{maint}}\\le\\MC(S)$ (Def.~\\ref{definition:bk5_metabolic_capacity_mc_}, Prop.~\\ref{proposition:bk5_fixed_metabolic_capacity}). Substituting the definition of $E_{\\text{maint}}$,\n\\[\n\\frac{\\mathcal{C}(\\mathcal{O})\\,\\mathcal{S}(S)}{\\alpha}\\le\\MC(S)\\quad\\Longleftrightarrow\\quad \\mathcal{C}(\\mathcal{O})\\cdot\\mathcal{S}(S)\\le\\alpha\\,\\MC(S).\n\\]\nAdmissible complexity is therefore metabolically budgeted: at fixed capacity, greater stability can be purchased only by reducing complexity, and conversely. The bound now follows from explicit definitions of $\\mathcal{C}$, $\\mathcal{S}$, and $E_{\\text{maint}}$ (Def.~\\ref{definition:bk5_complexity_stability_maintenance}) together with the proven viability requirement, with no ad hoc in-proof reading and no empirically fitted scaling law.\n\\end{proof}",
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  "proves": "theorem:bk5_complexity_stability_tradeoff",
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    {
      "context": "\\begin{proof} \\label{proof:bk5_complexity_stability_tradeoff} \\leavevmode By Def.~\\ref{definition:bk5_complexity_stability_maintenance} the maintenance cost is $E_{\\text{maint}}(\\mathcal{O},S)=\\mathcal{C}(\\mathcal{O})\\,\\mathcal{S}(S)/\\alpha$. By Operators",
      "label": "definition:bk5_complexity_stability_maintenance",
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      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1693,
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    {
      "context": "ained viability requires this expenditure to be fundable from the metabolic capacity, $E_{\\text{maint}}\\le\\MC(S)$ (Def.~\\ref{definition:bk5_metabolic_capacity_mc_}, Prop.~\\ref{proposition:bk5_fixed_metabolic_capacity}). Substituting the definition of $E_{\\text{maint}}$, \\[ \\frac{\\ma",
      "label": "definition:bk5_metabolic_capacity_mc_",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1525,
      "target_type": "definition"
    },
    {
      "context": "dable from the metabolic capacity, $E_{\\text{maint}}\\le\\MC(S)$ (Def.~\\ref{definition:bk5_metabolic_capacity_mc_}, Prop.~\\ref{proposition:bk5_fixed_metabolic_capacity}). Substituting the definition of $E_{\\text{maint}}$, \\[ \\frac{\\mathcal{C}(\\mathcal{O})\\,\\mathcal{S}(S)}{\\alpha}\\le\\MC(S",
      "label": "proposition:bk5_fixed_metabolic_capacity",
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      "role": "proof_support",
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      "target_line": 1513,
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    },
    {
      "context": "e cost is $E_{\\text{maint}}(\\mathcal{O},S)=\\mathcal{C}(\\mathcal{O})\\,\\mathcal{S}(S)/\\alpha$. By Operators Evolve (Prop.~\\ref{proposition:bk5_operators_evolve}) sustained viability requires this expenditure to be fundable from the metabolic capacity, $E_{\\text{maint}}\\le\\MC(S)$",
      "label": "proposition:bk5_operators_evolve",
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  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_metabolic_capacity_mc_",
    "theorem:bk5_complexity_stability_tradeoff"
  ],
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    "definition:bk5_metabolic_capacity_mc_",
    "theorem:bk5_complexity_stability_tradeoff"
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  "id": "corollary:bk5_complexity_stability_tradeoff",
  "label": "corollary:bk5_complexity_stability_tradeoff",
  "latex_body": "\\begin{corollary}[Complexity Stability Tradeoff]\n\\label{corollary:bk5_complexity_stability_tradeoff}\nHigher $\\MC$ permits both higher operator complexity and greater system stability (cf.~Thm.~\\ref{theorem:bk5_complexity_stability_tradeoff}, Def.~\\ref{definition:bk5_metabolic_capacity_mc_}).\n\\end{corollary}",
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  "line": 1721,
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  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Complexity Stability Tradeoff",
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    "proof:bk5_complexity_stability_tradeoff_cor"
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  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "higher operator complexity and greater system stability (cf.~Thm.~\\ref{theorem:bk5_complexity_stability_tradeoff}, Def.~\\ref{definition:bk5_metabolic_capacity_mc_}). \\end{corollary}",
      "label": "definition:bk5_metabolic_capacity_mc_",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1525,
      "target_type": "definition"
    },
    {
      "context": "plexity_stability_tradeoff} Higher $\\MC$ permits both higher operator complexity and greater system stability (cf.~Thm.~\\ref{theorem:bk5_complexity_stability_tradeoff}, Def.~\\ref{definition:bk5_metabolic_capacity_mc_}). \\end{corollary}",
      "label": "theorem:bk5_complexity_stability_tradeoff",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1702,
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{
  "book": "book5",
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  "cites": [
    "definition:bk5_metabolic_capacity_mc_",
    "theorem:bk5_complexity_stability_tradeoff"
  ],
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    "definition:bk5_metabolic_capacity_mc_",
    "theorem:bk5_complexity_stability_tradeoff"
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  "id": "proof:bk5_complexity_stability_tradeoff_cor",
  "label": "proof:bk5_complexity_stability_tradeoff_cor",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_complexity_stability_tradeoff_cor}\n\\leavevmode\nBy the tradeoff bound $\\mathcal{C}(\\mathcal{O})\\cdot\\mathcal{S}(S)\\le\\alpha\\,\\MC(S)$ (Thm.~\\ref{theorem:bk5_complexity_stability_tradeoff}) the feasible set for the pair $(\\mathcal{C},\\mathcal{S})$ is the hyperbolic region $\\{(\\mathcal{C},\\mathcal{S}):\\mathcal{C}\\,\\mathcal{S}\\le\\alpha\\,\\MC(S)\\}$. Its right-hand side is strictly increasing in the metabolic capacity $\\MC(S)$ (Def.~\\ref{definition:bk5_metabolic_capacity_mc_}), so raising $\\MC(S)$ enlarges the feasible region: every previously attainable $(\\mathcal{C},\\mathcal{S})$ remains attainable, and in addition pairs with larger $\\mathcal{C}$, larger $\\mathcal{S}$, or both become admissible. In particular the maximal attainable complexity at any fixed stability and the maximal attainable stability at any fixed complexity each increase with $\\MC(S)$. Hence higher metabolic capacity permits simultaneously higher operator complexity and greater system stability.\n\\end{proof}",
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  "name": "",
  "proves": "corollary:bk5_complexity_stability_tradeoff",
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      "context": "}\\,\\mathcal{S}\\le\\alpha\\,\\MC(S)\\}$. Its right-hand side is strictly increasing in the metabolic capacity $\\MC(S)$ (Def.~\\ref{definition:bk5_metabolic_capacity_mc_}), so raising $\\MC(S)$ enlarges the feasible region: every previously attainable $(\\mathcal{C},\\mathcal{S})$ remains att",
      "label": "definition:bk5_metabolic_capacity_mc_",
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      "role": "definition_anchor",
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    },
    {
      "context": "ty_tradeoff_cor} \\leavevmode By the tradeoff bound $\\mathcal{C}(\\mathcal{O})\\cdot\\mathcal{S}(S)\\le\\alpha\\,\\MC(S)$ (Thm.~\\ref{theorem:bk5_complexity_stability_tradeoff}) the feasible set for the pair $(\\mathcal{C},\\mathcal{S})$ is the hyperbolic region $\\{(\\mathcal{C},\\mathcal{S}):\\mathc",
      "label": "theorem:bk5_complexity_stability_tradeoff",
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  "line": 1730,
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  "book": "book5",
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  "id": "scholium:bk5_metabolic_cost_of_cognition",
  "label": "scholium:bk5_metabolic_cost_of_cognition",
  "latex_body": "\\begin{scholium}[Metabolic Cost of Cognition] \\label{scholium:bk5_metabolic_cost_of_cognition}\nHigher $\\MC$ (Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf})\nsupports recursive debugging, high-fidelity observers, and precise\nrenormalization. It lowers symbolic free energy $F_s$\n(Def.~\\ref{definition:bk2_symbolic_free_energy}) and reduces entropy\n(Def.~\\ref{definition:bk2_symbolic_entropy}) on the symbolic manifold $M$\n(Def.~\\ref{definition:bk1_symbolic_manifold}).\nDeclining $\\MC$ implies:\n\\begin{enumerate}\n    \\item Simplified reflective operators;\n    \\item Unresolved symbolic knots;\n    \\item Lower observer resolution;\n    \\item Shallower recursion;\n    \\item Loss of high-cost meta-cognition.\n\\end{enumerate}\n\\end{scholium}",
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  "name": "Metabolic Cost of Cognition",
  "ref_roles": [
    {
      "context": "\\begin{scholium}[Metabolic Cost of Cognition] \\label{scholium:bk5_metabolic_cost_of_cognition} Higher $\\MC$ (Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}) supports recursive debugging, high-fidelity observers, and precise renormalization. It lowers symbolic free energy $F_",
      "label": "definition:bk1_self_regulating_mapping_function_srmf",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2230,
      "target_type": "definition"
    },
    {
      "context": "bolic_free_energy}) and reduces entropy (Def.~\\ref{definition:bk2_symbolic_entropy}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}). Declining $\\MC$ implies: \\begin{enumerate} \\item Simplified reflective operators; \\item Unresolved symbolic k",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "zation. It lowers symbolic free energy $F_s$ (Def.~\\ref{definition:bk2_symbolic_free_energy}) and reduces entropy (Def.~\\ref{definition:bk2_symbolic_entropy}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}). Declining $\\MC$ implies: \\begin{enumerate}",
      "label": "definition:bk2_symbolic_entropy",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 114,
      "target_type": "definition"
    },
    {
      "context": "s recursive debugging, high-fidelity observers, and precise renormalization. It lowers symbolic free energy $F_s$ (Def.~\\ref{definition:bk2_symbolic_free_energy}) and reduces entropy (Def.~\\ref{definition:bk2_symbolic_entropy}) on the symbolic manifold $M$ (Def.~\\ref{definition:bk",
      "label": "definition:bk2_symbolic_free_energy",
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      "target_line": 135,
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    "definition:bk2_symbolic_free_energy"
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  "book": "book5",
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  "file": "book5.tex",
  "id": "theorem:bk5_metabolic_constraints_reflective_accuracy",
  "label": "theorem:bk5_metabolic_constraints_reflective_accuracy",
  "latex_body": "\\begin{theorem}[Metabolic Constraints on Reflective Accuracy] \\label{theorem:bk5_metabolic_constraints_reflective_accuracy}\nLet $f(n)$ be reflective fidelity at recursion depth $n\\in\\mathbb{N}$, and\nlet $n_{\\max}$ be the attained depth.  Suppose:\n\\begin{enumerate}\n\\item $f(0)=0$ and there is a calibrated marginal fidelity scale\n$\\beta_0\\geq0$ such that\n\\begin{equation}\n f(n+1)-f(n)\\leq\\beta_0\\qquad\\text{for every }n;\n \\label{eq:bk5_marginal_fidelity_bound}\n\\end{equation}\n\\item geometric recursion has base cost $c_0>0$, growth factor $k>1$, and\nnonnegative metabolic capacity $\\MC(S)$, with\n\\begin{equation}\n c_0\\bigl(k^{n_{\\max}}-1\\bigr)\\leq\\MC(S);\n \\label{eq:bk5_accuracy_geometric_budget}\n\\end{equation}\n\\item the chosen cost and logarithm units carry an explicit nonnegative\ncalibration constant $C_{\\log}$ satisfying\n\\begin{equation}\n n_{\\max}\\leq C_{\\log}\\log\\bigl(1+\\MC(S)\\bigr).\n \\label{eq:bk5_depth_log_calibration}\n\\end{equation}\n\\end{enumerate}\nThen, for $\\beta:=\\beta_0C_{\\log}\\geq0$,\n\\begin{equation}\n \\mathcal{F}(\\mathcal{O}_{\\mathrm{reflect}}):=f(n_{\\max})\n \\leq\\beta\\log\\bigl(1+\\MC(S)\\bigr).\n \\label{eq:bk5_reflective_accuracy_envelope}\n\\end{equation}\nThe calibration in Eq.~\\eqref{eq:bk5_depth_log_calibration} may be derived\nfrom a fixed choice of $c_0$, $k$, and logarithm base, but it is not inferred\nfrom metabolic capacity alone.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "explicit nonnegative depth-to-log calibration",
      "nonnegative metabolic capacity and geometric cost admissibility",
      "positive base cost and growth",
      "uniform nonnegative marginal fidelity gain",
      "zero-depth fidelity normalization"
    ],
    "countermodels": [
      "Book5ReflectiveAccuracy.capacity_alone_does_not_bound_unconstrained_fidelity",
      "Book5ReflectiveAccuracy.depth_budget_without_marginal_control_countermodel"
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    "notes": [
      "Depth-indexed reconstruction: a reflective fidelity process with zero-depth normalization and uniform marginal gain telescopes to a linear depth bound. A positive geometric recursion budget derives the dimensionless power bound. An explicit log-coordinate calibration composes these into the source envelope with nonnegative β. A countermodel shows depth and capacity cannot bound unrestricted fidelity."
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      "Book5ReflectiveAccuracy.ReflectiveAccuracyCertificate.beta_nonneg",
      "Book5ReflectiveAccuracy.ReflectiveAccuracyCertificate.fidelity_le_log_capacity",
      "Book5ReflectiveAccuracy.ReflectiveFidelityProcess.fidelity_le_linear",
      "Book5ReflectiveAccuracy.capacity_alone_does_not_bound_unconstrained_fidelity",
      "Book5ReflectiveAccuracy.depth_budget_without_marginal_control_countermodel",
      "Book5ReflectiveAccuracy.fidelityEnvelope_nonneg",
      "Book5ReflectiveAccuracy.fidelity_le_log_of_depth_bound"
    ]
  },
  "line": 1748,
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    "MC"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Metabolic Constraints on Reflective Accuracy",
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  "file": "book5.tex",
  "id": "proof:bk5_metabolic_constraints_reflective_accuracy",
  "label": "proof:bk5_metabolic_constraints_reflective_accuracy",
  "latex_body": "\\begin{proof}[Marginal-Gain and Geometric-Budget Composition]\n\\label{proof:bk5_metabolic_constraints_reflective_accuracy}\nTelescoping Eq.~\\eqref{eq:bk5_marginal_fidelity_bound} from the zero-depth\nnormalization gives\n\\[\n f(n)\\leq\\beta_0 n\n\\]\nfor every finite recursion depth.  Independently,\nEq.~\\eqref{eq:bk5_accuracy_geometric_budget} and $c_0>0$ imply the\ndimensionless power budget\n\\[\n k^{n_{\\max}}\\leq\\frac{\\MC(S)}{c_0}+1.\n\\]\nThe precise passage from this power budget to the normalized coordinate\n$\\log(1+\\MC(S))$ depends on $c_0$, $k$, and the log convention and is recorded\nby Eq.~\\eqref{eq:bk5_depth_log_calibration}.  Therefore\n\\[\n f(n_{\\max})\\leq\\beta_0n_{\\max}\n \\leq\\beta_0C_{\\log}\\log(1+\\MC(S)),\n\\]\nwhich is Eq.~\\eqref{eq:bk5_reflective_accuracy_envelope}.\n\nThe marginal law is load-bearing: an admissible depth and capacity do not bound\nan otherwise unrestricted fidelity assignment.  Likewise, changing cost or\nlogarithm units without updating $C_{\\log}$ changes the numerical coefficient\n$\\beta$ rather than revealing a universal scale.\n\\end{proof}",
  "line": 1781,
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    "MC"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Marginal-Gain and Geometric-Budget Composition",
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    "eq:bk5_depth_log_calibration",
    "eq:bk5_marginal_fidelity_bound",
    "eq:bk5_reflective_accuracy_envelope"
  ],
  "role": "proof",
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}

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  "book": "book5",
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    "definition:bk4_test_time_coherent_sampling",
    "definition:bk4_test_time_integrative_expansion",
    "definition:bk4_test_time_precision_refinement",
    "theorem:bk4_test_time_differentiation_c"
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  ],
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  "id": "subsec:bk5_conclustion_and_future_directions",
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  "latex_body": "",
  "line": 1808,
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      "target_line": 1947,
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  "id": "sec:bk5_golden_ratio",
  "label": "sec:bk5_golden_ratio",
  "latex_body": "",
  "line": 1813,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Metabolism and Recursive Proportion",
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{
  "book": "book5",
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  "cites": [],
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  "file": "book5.tex",
  "id": "subsec:bk5_intro_recursive_equilibrium",
  "label": "subsec:bk5_intro_recursive_equilibrium",
  "latex_body": "",
  "line": 1816,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Introduction: Life as Recursive Equilibrium",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
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{
  "book": "book5",
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  "cites": [],
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  "file": "book5.tex",
  "id": "subsec:bk5_golden_ratio_spectral_attractor",
  "label": "subsec:bk5_golden_ratio_spectral_attractor",
  "latex_body": "",
  "line": 1823,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Golden Ratio as Spectral Attractor",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
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{
  "book": "book5",
  "cited_by": [
    "corollary:bk5_symbolic_eigenlife",
    "lemma:bk5_phi_critical_resonant_norm",
    "proof:bk5_golden_ratio_curvature_scalar",
    "proof:bk5_golden_ratio_spectral_invariant",
    "proof:bk5_phi_critical_resonant_norm",
    "proof:bk5_symbolic_eigenlife",
    "remark:bk5_symbolic_fibonacci_coding",
    "theorem:appC_modal_transference",
    "theorem:bk4_golden_event_horizon_spiral",
    "theorem:bk5_golden_ratio_curvature_scalar",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_golden_rule_reciprocity"
  ],
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  "id": "definition:bk5_balanced_two_step_memory_closure",
  "label": "definition:bk5_balanced_two_step_memory_closure",
  "latex_body": "\\begin{definition}[Balanced Two-Step Symbolic Memory Closure]\n\\label{definition:bk5_balanced_two_step_memory_closure}\nLet $S_n$ denote the $n$th observer-resolved symbolic state and let\n$a_n=\\ell_O(S_n)\\geq 0$ be a scalar amplitude extracted by a positive\nobserver channel $\\ell_O$.  The recursion has a \\textbf{balanced two-step memory\nclosure} when, after normalizing the present-state channel to unit weight, the\nonly retained reflective memory channel has the same observer-visible weight:\n\\[\na_{n+1}=a_n+a_{n-1}, \\qquad\nX_{n+1}=A X_n,\\qquad\nX_n=\\begin{pmatrix}a_n\\\\ a_{n-1}\\end{pmatrix},\\quad\nA=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}.\n\\]\nThe first coefficient is fixed by the choice of present-state unit; the second\ncoefficient is the balance condition asserting that retained reflective memory is\ncalibrated in the same observer-visible units as current persistence.  If this\nsecond coefficient is replaced by another positive weight, the resulting system\nis a different metallic-ratio regime rather than the balanced PS memory regime.\n\\end{definition}",
  "lean_alignment": {
    "conditions": [
      "only the arithmetic, matrix, lattice, and limit content of the cited Book 5 statements"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Matrix and asymptotic ratio."
    ],
    "record_ids": [
      "MAP-BOOK5-016"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "balanced_memory_tendsto_gold",
      "closureMatrix_eigen_gold"
    ]
  },
  "line": 1828,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Balanced Two-Step Symbolic Memory Closure",
  "proof_status": "definitional",
  "refs": [],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
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    "proof:bk4_golden_event_horizon_spiral",
    "proof:bk5_golden_rule_reciprocity",
    "theorem:bk4_golden_event_horizon_spiral",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_golden_rule_reciprocity"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "lemma:bk5_balanced_observer_normalization",
  "label": "lemma:bk5_balanced_observer_normalization",
  "latex_body": "\\begin{lemma}[Balanced Observer Normalization Selects the Closure Matrix]\n\\label{lemma:bk5_balanced_observer_normalization}\nLet the minimal two-channel memory closure be the general positive recurrence\n\\[\na_{n+1}=\\alpha\\,a_n+\\beta\\,a_{n-1},\n\\qquad\nA_{\\alpha,\\beta}=\\begin{pmatrix}\\alpha&\\beta\\\\1&0\\end{pmatrix},\n\\qquad \\alpha,\\beta>0,\n\\]\nwhere the present-persistence channel carries weight $\\alpha$ and the single\nretained reflective-memory channel carries weight $\\beta$, both read through the\nsame positive observer channel $\\ell_O$. If\n\\begin{enumerate}\n\\item present persistence is normalized to unit observer weight, and\n\\item retained reflective memory is calibrated in the same observer-visible\nunits as present persistence (the balance condition),\n\\end{enumerate}\nthen $\\alpha=\\beta=1$, so the unique positive two-step closure matrix is\n\\[\nA=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}.\n\\]\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "nonnegative reciprocity weight; strict positivity for strict regime comparisons"
    ],
    "countermodels": [],
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    "kernel_certified": true,
    "notes": [
      "Normalization and equal calibration uniquely select unit weights."
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      "MAP-BOOK5-017"
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      "conditional"
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    "witnesses": [
      "Book5.balanced_observer_weights_unique"
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  },
  "line": 1848,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Balanced Observer Normalization Selects the Closure Matrix",
  "proof_labels": [
    "proof:bk5_balanced_observer_normalization"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "proof:bk5_balanced_observer_normalization",
  "label": "proof:bk5_balanced_observer_normalization",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_balanced_observer_normalization}\nNormalization~(1) is the choice of observer unit for the present-state channel:\nrescaling $\\ell_O$ so that one unit of current persistence maps to one unit of\namplitude fixes $\\alpha=1$. With present persistence now the unit of\nobserver-visible weight, condition~(2) asserts that retained reflective memory is\nnot discounted or amplified relative to present persistence---it contributes in\nthe same units---so its coefficient equals the present-state unit, $\\beta=1$.\nBoth coefficients are thereby determined, and the companion matrix of the\nrecurrence is $A=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$. Any\n$\\beta\\neq 1$ violates~(2) and yields a metallic-ratio regime\n$\\lambda^2-\\lambda-\\beta=0$ rather than the balanced PS regime; any $\\alpha\\neq 1$\nis merely a renormalization of the observer unit and is excluded by~(1).\n\\end{proof}",
  "line": 1871,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "lemma:bk5_balanced_observer_normalization",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "corollary:bk5_symbolic_eigenlife",
    "definition:bk5_map_mad_mas_band",
    "definition:bk5_symbolic_curvature_operator_spectrum",
    "proof:bk4_golden_event_horizon_spiral",
    "proof:bk5_complementary_constants",
    "proof:bk5_fundamental_norm_fracture",
    "proof:bk5_golden_ratio_curvature_scalar",
    "proof:bk5_golden_ratio_thermodynamic_optimum",
    "proof:bk5_golden_rule_reciprocity",
    "proof:bk5_map_mad_mas_trichotomy",
    "proof:bk5_phi_critical_resonant_norm",
    "proof:bk5_symbolic_eigenlife",
    "proof:bk5_symbolic_norm_spectrum",
    "proposition:bk5_complementary_constants",
    "proposition:bk5_golden_ratio_thermodynamic_optimum",
    "remark:bk5_curvature_vs_chaos",
    "remark:bk5_symbolic_fibonacci_coding",
    "subsec:bk5_map_mad_mas_band",
    "theorem:bk4_golden_event_horizon_spiral",
    "theorem:bk5_fundamental_dichotomy",
    "theorem:bk5_fundamental_norm_fracture",
    "theorem:bk5_golden_rule_reciprocity",
    "theorem:bk5_grand_unified_symbolic_geometric",
    "theorem:bk5_symbolic_norm_spectrum",
    "theorem:bk8_rg_fixed_point"
  ],
  "cites": [
    "definition:bk5_balanced_two_step_memory_closure",
    "definition:bk5_spectral_radius_of_coupl",
    "lemma:bk5_balanced_observer_normalization"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk5_balanced_two_step_memory_closure",
    "definition:bk5_spectral_radius_of_coupl",
    "lemma:bk5_balanced_observer_normalization"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_golden_ratio_spectral_invariant",
  "label": "theorem:bk5_golden_ratio_spectral_invariant",
  "latex_body": "\\begin{theorem}[Golden Ratio as Spectral Invariant of Balanced Recursive Memory]\n\\label{theorem:bk5_golden_ratio_spectral_invariant}\nLet $\\drift$ be a symbolic drift operator and $\\reflect$ a reflection operator\nwhose interaction opens an observer-resolved memory channel through the local\ndrift-reflection commutator $[\\drift,\\reflect]$.  If that channel closes as a\nbalanced two-step symbolic memory closure\n(Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), whose closure matrix\nis uniquely fixed by observer normalization\n(Lemma~\\ref{lemma:bk5_balanced_observer_normalization}), then the dominant\neigenvalue $\\lambda$ of the closure operator (cf.~Def.~\\ref{definition:bk5_spectral_radius_of_coupl}) satisfies:\n\\[\n\\lambda^2 - \\lambda - 1 = 0\n\\]\nHence the unique positive spectral radius of the balanced closure is\n$\\lambda=\\varphi$, the Golden Ratio.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "only the arithmetic, matrix, lattice, and limit content of the cited Book 5 statements"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Arithmetic spectral kernel; commutator interpretation is not certified."
    ],
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      "MAP-BOOK5-018"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "closureMatrix_eigen_gold",
      "gold_unique_positive_root"
    ]
  },
  "line": 1886,
  "macros_used": [
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Golden Ratio as Spectral Invariant of Balanced Recursive Memory",
  "proof_labels": [
    "proof:bk5_golden_ratio_spectral_invariant"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "reflection commutator $[\\drift,\\reflect]$. If that channel closes as a balanced two-step symbolic memory closure (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), whose closure matrix is uniquely fixed by observer normalization (Lemma~\\ref{lemma:bk5_balanced_observer_normalizatio",
      "label": "definition:bk5_balanced_two_step_memory_closure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1828,
      "target_type": "definition"
    },
    {
      "context": "f{lemma:bk5_balanced_observer_normalization}), then the dominant eigenvalue $\\lambda$ of the closure operator (cf.~Def.~\\ref{definition:bk5_spectral_radius_of_coupl}) satisfies: \\[ \\lambda^2 - \\lambda - 1 = 0 \\] Hence the unique positive spectral radius of the balanced closure is $\\la",
      "label": "definition:bk5_spectral_radius_of_coupl",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 508,
      "target_type": "definition"
    },
    {
      "context": "inition:bk5_balanced_two_step_memory_closure}), whose closure matrix is uniquely fixed by observer normalization (Lemma~\\ref{lemma:bk5_balanced_observer_normalization}), then the dominant eigenvalue $\\lambda$ of the closure operator (cf.~Def.~\\ref{definition:bk5_spectral_radius_of_coupl",
      "label": "lemma:bk5_balanced_observer_normalization",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1848,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk5_balanced_two_step_memory_closure",
    "definition:bk5_spectral_radius_of_coupl",
    "lemma:bk5_balanced_observer_normalization"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "sec:bk5_golden_ratio_invariant",
  "label": "sec:bk5_golden_ratio_invariant",
  "latex_body": "",
  "line": 1903,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Golden Ratio as a Symbolic Invariant of Life",
  "role": "section",
  "subtype": "section",
  "type": "section"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "subsec:bk5_metabolic_constant_emergence",
  "label": "subsec:bk5_metabolic_constant_emergence",
  "latex_body": "",
  "line": 1906,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Metabolic Constant of Emergence",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book5",
  "cited_by": [
    "remark:bk9_grace_flow_geometric_witness",
    "scholium:bk9_golden_rule_thermodynamic_covenant"
  ],
  "cites": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk5_balanced_two_step_memory_closure"
  ],
  "depends_on": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk5_balanced_two_step_memory_closure"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_golden_ratio_spectral_invariant",
  "label": "proof:bk5_golden_ratio_spectral_invariant",
  "latex_body": "\\begin{proof}[Golden Ratio as Spectral Invariant via Balanced Memory Algebra]\n\\label{proof:bk5_golden_ratio_spectral_invariant}\n\\leavevmode\n\nOn the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}), the\ncommutator $[\\drift,\\reflect]$ of drift\n(Def.~\\ref{definition:bk1_drift_field}) and reflection\n(Def.~\\ref{definition:bk1_reflection_operator}) supplies the local channel by\nwhich current transformation and retained reflective memory interact.  Project\nthat channel to a positive observer-resolved amplitude $a_n=\\ell_O(S_n)$ and\nimpose the balanced two-step closure of\nDef.~\\ref{definition:bk5_balanced_two_step_memory_closure}.  Then the state\nvector $X_n=(a_n,a_{n-1})^T$ evolves by\n\\[\nX_{n+1}=A X_n,\\qquad\nA=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}.\n\\]\nThe characteristic polynomial is\n\\[\n\\det(\\lambda I-A)\n=\\det\\begin{pmatrix}\\lambda-1&-1\\\\-1&\\lambda\\end{pmatrix}\n=\\lambda^2-\\lambda-1.\n\\]\nIts roots are\n\\[\n\\lambda_\\pm=\\frac{1\\pm\\sqrt{5}}{2}.\n\\]\nThe matrix $A$ is positive on the nonnegative cone after two iterates, so the\nPerron--Frobenius eigenvalue is the unique positive spectral radius.  Therefore\n$\\rho(A)=\\lambda_+=\\varphi$, while the other eigenvalue is\n$\\lambda_-=-\\varphi^{-1}$ and is subdominant in magnitude.  For every\nnonzero nonnegative initial amplitude vector, normalized iterates converge\nprojectively to the positive eigendirection, and the successive amplitude ratio\nconverges to $\\varphi$.  Thus the Golden Ratio is not obtained from the\ncommutator alone; it is the spectral invariant of the balanced two-step closure\nof the drift-reflection memory channel.\n\\end{proof}",
  "line": 1915,
  "macros_used": [
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Golden Ratio as Spectral Invariant via Balanced Memory Algebra",
  "proves": "theorem:bk5_golden_ratio_spectral_invariant",
  "ref_roles": [
    {
      "context": "symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}), the commutator $[\\drift,\\reflect]$ of drift (Def.~\\ref{definition:bk1_drift_field}) and reflection (Def.~\\ref{definition:bk1_reflection_operator}) supplies the local channel by which current transformat",
      "label": "definition:bk1_drift_field",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1198,
      "target_type": "definition"
    },
    {
      "context": "ic_manifold}), the commutator $[\\drift,\\reflect]$ of drift (Def.~\\ref{definition:bk1_drift_field}) and reflection (Def.~\\ref{definition:bk1_reflection_operator}) supplies the local channel by which current transformation and retained reflective memory interact. Project that chan",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    },
    {
      "context": "anced Memory Algebra] \\label{proof:bk5_golden_ratio_spectral_invariant} \\leavevmode On the symbolic manifold $M$ (Def.~\\ref{definition:bk1_symbolic_manifold}), the commutator $[\\drift,\\reflect]$ of drift (Def.~\\ref{definition:bk1_drift_field}) and reflection (Def.~\\ref{definit",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    },
    {
      "context": "at channel to a positive observer-resolved amplitude $a_n=\\ell_O(S_n)$ and impose the balanced two-step closure of Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}. Then the state vector $X_n=(a_n,a_{n-1})^T$ evolves by \\[ X_{n+1}=A X_n,\\qquad A=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}",
      "label": "definition:bk5_balanced_two_step_memory_closure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1828,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_drift_field",
    "definition:bk1_reflection_operator",
    "definition:bk1_symbolic_manifold",
    "definition:bk5_balanced_two_step_memory_closure"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk8_identitystability",
    "definition:bk8_recursive_symbolic_metaboloic_cycle",
    "lemma:bk7_involutive_dual_symmetry",
    "proof:bk5_map_mad_mas_trichotomy",
    "proof:bk8_biological_phase_transition",
    "scholium:bk8_autonomous_repair_systems_expanded",
    "sec:bk7_preamble_the_arc_toward_coherence",
    "subsec:bk5_map_mad_mas_band",
    "theorem:bk8_biological_phase_transition"
  ],
  "cites": [
    "definition:bk5_balanced_two_step_memory_closure",
    "proposition:bk5_symbolic_life_criterion",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "depends_on": [
    "definition:bk5_balanced_two_step_memory_closure",
    "proposition:bk5_symbolic_life_criterion",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "file": "book5.tex",
  "id": "corollary:bk5_symbolic_eigenlife",
  "label": "corollary:bk5_symbolic_eigenlife",
  "latex_body": "\\begin{corollary}[Symbolic Eigenlife]\n\\label{corollary:bk5_symbolic_eigenlife}\nA symbolic system exhibits \\textbf{eigenlife} in the balanced two-step memory\nregime when its observer-resolved dominant mode is governed by the positive\nPerron root $\\varphi$.  Subcritical modes with spectral radius below $1$ decay\nunder iteration, while supercritical unbalanced modes require additional\nrenormalization to avoid loss of bounded symbolic identity.  Thus $\\varphi$ is\nthe unique spectral attractor for recursively stable symbolic persistence within\nthe balanced closure of\nDef.~\\ref{definition:bk5_balanced_two_step_memory_closure}\n(cf.~Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant},\nThm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy},\nProp.~\\ref{proposition:bk5_symbolic_life_criterion}).\n\\end{corollary}",
  "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": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-084"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Residue.closureMatrix_disc_pos"
    ]
  },
  "line": 1953,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Eigenlife",
  "proof_labels": [
    "proof:bk5_symbolic_eigenlife"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "arphi$ is the unique spectral attractor for recursively stable symbolic persistence within the balanced closure of Def.~\\ref{definition:bk5_balanced_two_step_memory_closure} (cf.~Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_acc",
      "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"
    },
    {
      "context": "f{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}, Prop.~\\ref{proposition:bk5_symbolic_life_criterion}). \\end{corollary}",
      "label": "proposition:bk5_symbolic_life_criterion",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 173,
      "target_type": "proposition"
    },
    {
      "context": "ymbolic persistence within the balanced closure of Def.~\\ref{definition:bk5_balanced_two_step_memory_closure} (cf.~Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}, Prop.~\\ref{proposition:bk5_symbolic_life_criterion})",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "\\ref{definition:bk5_balanced_two_step_memory_closure} (cf.~Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}, Prop.~\\ref{proposition:bk5_symbolic_life_criterion}). \\end{corollary}",
      "label": "theorem:bk5_metabolic_constraints_reflective_accuracy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1748,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_balanced_two_step_memory_closure",
    "proposition:bk5_symbolic_life_criterion",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_balanced_two_step_memory_closure",
    "proposition:bk5_symbolic_life_criterion",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "depends_on": [
    "definition:bk5_balanced_two_step_memory_closure",
    "proposition:bk5_symbolic_life_criterion",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_eigenlife",
  "label": "proof:bk5_symbolic_eigenlife",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_symbolic_eigenlife}\n\\leavevmode\nIn the balanced two-step memory regime the observer-resolved state evolves by the closure matrix $A=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$ (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), whose unique positive spectral radius is the Golden Ratio, $\\rho(A)=\\varphi$, with the second eigenvalue $-\\varphi^{-1}$ subdominant (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). By Perron--Frobenius the normalized iterates converge projectively to the positive $\\varphi$-eigendirection, so the dominant observer-resolved mode of a balanced system is governed by $\\varphi$. By the symbolic life criterion (Prop.~\\ref{proposition:bk5_symbolic_life_criterion}) persistence requires the dominant mode neither to decay to nothing nor to diverge without bound. A mode with spectral radius below $1$ contracts under iteration and its symbolic amplitude decays---no eigenlife; a supercritical unbalanced mode (spectral radius above $\\varphi$) grows without bound and can preserve bounded symbolic identity only by spending additional renormalization, whose budget is itself capped by reflective capacity (Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}). The balanced closure sits exactly at the Perron value $\\varphi>1$: expansive enough to persist against drift, yet fixed by observer normalization, so its iterates neither decay nor demand unbounded renormalization. Therefore a symbolic system exhibits eigenlife precisely when its dominant observer-resolved mode is governed by the Perron root $\\varphi$, which is the unique spectral attractor for recursively stable symbolic persistence within the balanced closure.\n\\end{proof}",
  "line": 1967,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "corollary:bk5_symbolic_eigenlife",
  "ref_roles": [
    {
      "context": "observer-resolved state evolves by the closure matrix $A=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$ (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), whose unique positive spectral radius is the Golden Ratio, $\\rho(A)=\\varphi$, with the second eigenvalue $-\\varphi^{-",
      "label": "definition:bk5_balanced_two_step_memory_closure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1828,
      "target_type": "definition"
    },
    {
      "context": "he dominant observer-resolved mode of a balanced system is governed by $\\varphi$. By the symbolic life criterion (Prop.~\\ref{proposition:bk5_symbolic_life_criterion}) persistence requires the dominant mode neither to decay to nothing nor to diverge without bound. A mode with spectral",
      "label": "proposition:bk5_symbolic_life_criterion",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 173,
      "target_type": "proposition"
    },
    {
      "context": "ve spectral radius is the Golden Ratio, $\\rho(A)=\\varphi$, with the second eigenvalue $-\\varphi^{-1}$ subdominant (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). By Perron--Frobenius the normalized iterates converge projectively to the positive $\\varphi$-eigendirection, so the d",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "mbolic identity only by spending additional renormalization, whose budget is itself capped by reflective capacity (Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}). The balanced closure sits exactly at the Perron value $\\varphi>1$: expansive enough to persist against drift, yet fix",
      "label": "theorem:bk5_metabolic_constraints_reflective_accuracy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1748,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_balanced_two_step_memory_closure",
    "proposition:bk5_symbolic_life_criterion",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "corollary:bk5_symbolic_eigenlife",
    "theorem:bk5_enhanced_map_mad_duality",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "depends_on": [
    "corollary:bk5_symbolic_eigenlife",
    "theorem:bk5_enhanced_map_mad_duality",
    "theorem:bk5_golden_ratio_spectral_invariant"
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  "file": "book5.tex",
  "id": "subsec:bk5_map_mad_mas_band",
  "label": "subsec:bk5_map_mad_mas_band",
  "latex_body": "",
  "line": 1973,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The MAD--MAP--MAS Band",
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      "context": "",
      "label": "corollary:bk5_symbolic_eigenlife",
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      "target_line": 1953,
      "target_type": "corollary"
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      "context": "",
      "label": "theorem:bk5_enhanced_map_mad_duality",
      "logical_support": false,
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      "target_line": 747,
      "target_type": "theorem"
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      "label": "theorem:bk5_golden_ratio_spectral_invariant",
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  ],
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  "subtype": "subsection",
  "type": "section"
}

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  "book": "book5",
  "cited_by": [
    "proof:bk7_map_compatible_reciprocity",
    "proposition:bk7_map_compatible_reciprocity",
    "scholium:bk5_imagination_covenant_branch_selection",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "cites": [
    "corollary:bk5_map_evolutionary_advantag",
    "definition:bk5_reflective_coupling_stab",
    "definition:bk5_symbolic_bifurcation_man",
    "proposition:bk4_imagination_bridges_wheel",
    "scholium:bk4_imagination_as_imaginary_traversal",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "depends_on": [
    "corollary:bk5_map_evolutionary_advantag",
    "definition:bk5_reflective_coupling_stab",
    "definition:bk5_symbolic_bifurcation_man",
    "proposition:bk4_imagination_bridges_wheel",
    "scholium:bk4_imagination_as_imaginary_traversal",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_map_mad_mas_band",
  "label": "definition:bk5_map_mad_mas_band",
  "latex_body": "\\begin{definition}[The MAD--MAP--MAS Band]\n\\label{definition:bk5_map_mad_mas_band}\nLet $\\Membrane_A,\\Membrane_B$ interact through the symbolic covenant $\\mathcal{C}_{AB}$ (Cor.~\\ref{corollary:bk5_map_evolutionary_advantag}), and let $\\mathbf{C}_{AB}$ be the induced \\emph{linearized mutual-reflection operator} on the joint tangent space, governing $X_{n+1}=\\mathbf{C}_{AB}X_n$ for the paired state $X=(\\psi_A,\\psi_B)$. Order the dyad by the reflective coupling stability parameter $\\Lambda_{AB}$ (Def.~\\ref{definition:bk5_reflective_coupling_stab}) about its bifurcation manifold $\\Lambda_{AB}=1$ (Def.~\\ref{definition:bk5_symbolic_bifurcation_man}). By the spectrum of $\\mathbf{C}_{AB}$ the dyad occupies one of three regimes:\n\\[\n\\text{regime}(\\mathcal{C}_{AB})=\\text{regime}\\bigl(\\operatorname{Spec}(\\mathbf{C}_{AB})\\bigr).\n\\]\nThis spectrum is read on the \\emph{enacted} covenant branch: imagination may traverse counterfactual branches through imaginary or phase displacement (Scholium~\\ref{scholium:bk4_imagination_as_imaginary_traversal}, Prop.~\\ref{proposition:bk4_imagination_bridges_wheel}), but once a branch is enacted its regime is fixed by $\\mathbf{C}_{AB}$.\n\\begin{itemize}\n  \\item \\textbf{MAD} --- \\emph{Mutually Assured Destruction} ($\\Omega_{AB}<0$): the covenant is antagonistic (zero-sum), the antisymmetric part of $\\mathbf{C}_{AB}$ dominates, and the spectrum is complex ($\\lambda=a\\pm ib$, $b\\neq0$). Mutual reflection rotates without convergence --- the retaliation spiral --- and the relation dissolves.\n  \\item \\textbf{MAP} --- \\emph{Mutually Assured Progress} (the sustainable interior, balanced two-step memory closure): the spectrum is real with dominant eigenvalue the Golden Ratio $\\varphi$ (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}) on the non-diagonal $\\varphi{:}1$ eigendirection. The membranes co-evolve while remaining distinct.\n  \\item \\textbf{MAS} --- \\emph{Mutually Assured Similarity} (over-coupling $\\Lambda_{AB}\\gg1$, $\\Omega_{AB}\\gg0$): the symmetric (memoryless) part dominates, the spectrum is real, and the dominant eigendirection is the diagonal $(1,1)$. The membranes converge to a common state and their relative dynamics freeze --- preservation without progress.\n\\end{itemize}\nDestruction ($\\Omega_{AB}<0$) and similarity ($\\Omega_{AB}\\gg0$) are the opposing edges of the band; progress is the sustainable middle.\n\\end{definition}",
  "lean_alignment": {
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      "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": [
      "the discriminant-based real/complex spectral split underlying MAD vs MAP/MAS is proved generically; the specific MAP-vs-MAS distinction (which real eigendirection is dominant) is not captured."
    ],
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  },
  "line": 1977,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The MAD--MAP--MAS Band",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ion:bk5_map_mad_mas_band} Let $\\Membrane_A,\\Membrane_B$ interact through the symbolic covenant $\\mathcal{C}_{AB}$ (Cor.~\\ref{corollary:bk5_map_evolutionary_advantag}), and let $\\mathbf{C}_{AB}$ be the induced \\emph{linearized mutual-reflection operator} on the joint tangent space, gov",
      "label": "corollary:bk5_map_evolutionary_advantag",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 478,
      "target_type": "corollary"
    },
    {
      "context": "he paired state $X=(\\psi_A,\\psi_B)$. Order the dyad by the reflective coupling stability parameter $\\Lambda_{AB}$ (Def.~\\ref{definition:bk5_reflective_coupling_stab}) about its bifurcation manifold $\\Lambda_{AB}=1$ (Def.~\\ref{definition:bk5_symbolic_bifurcation_man}). By the spectrum",
      "label": "definition:bk5_reflective_coupling_stab",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 786,
      "target_type": "definition"
    },
    {
      "context": "Lambda_{AB}$ (Def.~\\ref{definition:bk5_reflective_coupling_stab}) about its bifurcation manifold $\\Lambda_{AB}=1$ (Def.~\\ref{definition:bk5_symbolic_bifurcation_man}). By the spectrum of $\\mathbf{C}_{AB}$ the dyad occupies one of three regimes: \\[ \\text{regime}(\\mathcal{C}_{AB})=\\text",
      "label": "definition:bk5_symbolic_bifurcation_man",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 795,
      "target_type": "definition"
    },
    {
      "context": "branches through imaginary or phase displacement (Scholium~\\ref{scholium:bk4_imagination_as_imaginary_traversal}, Prop.~\\ref{proposition:bk4_imagination_bridges_wheel}), but once a branch is enacted its regime is fixed by $\\mathbf{C}_{AB}$. \\begin{itemize} \\item \\textbf{MAD} --- \\emph",
      "label": "proposition:bk4_imagination_bridges_wheel",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "book4.tex",
      "target_line": 887,
      "target_type": "proposition"
    },
    {
      "context": "ed} covenant branch: imagination may traverse counterfactual branches through imaginary or phase displacement (Scholium~\\ref{scholium:bk4_imagination_as_imaginary_traversal}, Prop.~\\ref{proposition:bk4_imagination_bridges_wheel}), but once a branch is enacted its regime is fixed by $\\mathbf{C",
      "label": "scholium:bk4_imagination_as_imaginary_traversal",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "book4.tex",
      "target_line": 782,
      "target_type": "scholium"
    },
    {
      "context": "rior, balanced two-step memory closure): the spectrum is real with dominant eigenvalue the Golden Ratio $\\varphi$ (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}) on the non-diagonal $\\varphi{:}1$ eigendirection. The membranes co-evolve while remaining distinct. \\item \\textbf{MA",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    }
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  "refs": [
    "corollary:bk5_map_evolutionary_advantag",
    "definition:bk5_reflective_coupling_stab",
    "definition:bk5_symbolic_bifurcation_man",
    "proposition:bk4_imagination_bridges_wheel",
    "scholium:bk4_imagination_as_imaginary_traversal",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "proof:bk1_conditional_genericity_of_symbolic_phase_transitions",
    "proof:bk1_realization_of_symbolic_phase_transitions",
    "proof:bk9_good_as_lyapunov_basin",
    "scholium:bk5_imagination_covenant_branch_selection",
    "theorem:bk1_conditional_genericity_of_symbolic_phase_transitions",
    "theorem:bk9_good_as_lyapunov_basin"
  ],
  "cites": [
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk5_map_mad_mas_band"
  ],
  "depends_on": [
    "corollary:bk5_symbolic_eigenlife",
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk5_map_mad_mas_band",
    "definition:bk5_reflective_coupling_stab",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_map_mad_mas_trichotomy",
  "label": "theorem:bk5_map_mad_mas_trichotomy",
  "latex_body": "\\begin{theorem}[MAD--MAP--MAS Trichotomy]\n\\label{theorem:bk5_map_mad_mas_trichotomy}\nFor the mutual reflective dynamics $X_{n+1}=\\mathbf{C}_{AB}X_n$ (Def.~\\ref{definition:bk5_map_mad_mas_band}), exactly one of three asymptotic behaviours obtains, selected by the covenant $\\Omega_{AB}$ through $\\Lambda_{AB}$:\n\\begin{enumerate}\n  \\item[\\textbf{(MAD)}] If $\\Omega_{AB}<0$, the dominant eigenvalues of $\\mathbf{C}_{AB}$ are complex with $|\\lambda|>1$; the joint free energy fails to stabilize, $\\lim_n F_s(\\Membrane_A^{(n)}\\cup\\Membrane_B^{(n)})=0$ at rate $\\propto|\\Omega_{AB}|$.\n  \\item[\\textbf{(MAP)}] At the balanced cooperative closure the dominant eigenvalue is real and equal to $\\varphi$ on a non-diagonal eigendirection; the dyad sustains $\\lim_n F_s(\\Membrane_A^{(n)}\\!\\leftrightarrow\\!\\Membrane_B^{(n)})>0$ with preserved distinctness.\n  \\item[\\textbf{(MAS)}] If $\\Omega_{AB}\\gg0$, the dominant eigendirection is the diagonal $(1,1)$ of norm $\\sqrt2$; the membranes converge to a common state, relative dynamics vanish ($\\Delta\\Sigma\\to0$), and $F_s$ is conserved at a frozen equilibrium.\n\\end{enumerate}\n$\\mathrm{MAP}$ is the unique sustainable regime. The $\\mathrm{MAD}\\!\\to\\!\\mathrm{MAP}$ boundary is a complex$\\to$real spectral transition (discriminant zero), a symbolic phase transition (Def.~\\ref{definition:bk2_symbolic_phase_transitio}); the $\\mathrm{MAP}\\!\\to\\!\\mathrm{MAS}$ boundary is the rotation of the dominant eigendirection onto the diagonal. Destruction and similarity are the opposing edges of the band.\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": [],
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      "See the committed source registry for the original coverage note."
    ],
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      "MAP-BOOK5-086"
    ],
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      "open_bridge"
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  "line": 1992,
  "macros_used": [
    "Membrane"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "MAD--MAP--MAS Trichotomy",
  "proof_labels": [
    "proof:bk5_map_mad_mas_trichotomy"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "\\mathrm{MAP}$ boundary is a complex$\\to$real spectral transition (discriminant zero), a symbolic phase transition (Def.~\\ref{definition:bk2_symbolic_phase_transitio}); the $\\mathrm{MAP}\\!\\to\\!\\mathrm{MAS}$ boundary is the rotation of the dominant eigendirection onto the diagonal. Dest",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    },
    {
      "context": "otomy] \\label{theorem:bk5_map_mad_mas_trichotomy} For the mutual reflective dynamics $X_{n+1}=\\mathbf{C}_{AB}X_n$ (Def.~\\ref{definition:bk5_map_mad_mas_band}), exactly one of three asymptotic behaviours obtains, selected by the covenant $\\Omega_{AB}$ through $\\Lambda_{AB}$: \\b",
      "label": "definition:bk5_map_mad_mas_band",
      "logical_support": true,
      "role": "definition_anchor",
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      "target_line": 1977,
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    "definition:bk5_map_mad_mas_band"
  ],
  "role": "theorem",
  "type": "theorem"
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{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "corollary:bk5_symbolic_eigenlife",
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk5_reflective_coupling_stab",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "depends_on": [
    "corollary:bk5_symbolic_eigenlife",
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk5_reflective_coupling_stab",
    "theorem:bk5_golden_ratio_spectral_invariant"
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  "file": "book5.tex",
  "id": "proof:bk5_map_mad_mas_trichotomy",
  "label": "proof:bk5_map_mad_mas_trichotomy",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_map_mad_mas_trichotomy}\n\\leavevmode\nLinearize the mutual reflective dynamics about the joint fixed point; the coupling operator $\\mathbf{C}_{AB}$ acts on the two-membrane tangent space, its character fixed by the covenant orientation $\\Omega_{AB}$ through $\\Lambda_{AB}$ (Def.~\\ref{definition:bk5_reflective_coupling_stab}). Split $\\mathbf{C}_{AB}=S+A$ into symmetric $S=\\tfrac12(\\mathbf{C}_{AB}+\\mathbf{C}_{AB}^{\\!\\top})$ and antisymmetric $A=\\tfrac12(\\mathbf{C}_{AB}-\\mathbf{C}_{AB}^{\\!\\top})$ parts, orthogonal under $\\langle X,Y\\rangle=\\operatorname{tr}(X^{\\!\\top}Y)$ --- the exact sense in which the two edges are opposite.\n\n\\emph{(MAD).} For $\\Omega_{AB}<0$ the covenant is zero-sum and the antisymmetric part $A$ dominates. A real antisymmetric operator has purely imaginary spectrum, so $\\mathbf{C}_{AB}$ acquires complex eigenvalues $\\lambda=a\\pm ib$ with $b\\neq0$; the iterates rotate and never settle to a common state. Antagonistic reflection amplifies rather than damps drift, so the joint free-energy derivative is negative (entropy production outpaces reflective restoration), giving $\\lim_n F_s(\\Membrane_A^{(n)}\\cup\\Membrane_B^{(n)})=0$ with collapse rate $\\propto|\\Omega_{AB}|$. This is destruction.\n\n\\emph{(MAP).} At the balanced cooperative closure the coupling reduces to the two-step memory operator $\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$, whose unique positive eigenvalue is the Golden Ratio $\\varphi$ on the eigendirection $(\\varphi,1)$ (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). This eigendirection is not the diagonal, so $\\Membrane_A$ and $\\Membrane_B$ co-evolve while remaining distinct; by the eigenlife criterion (Cor.~\\ref{corollary:bk5_symbolic_eigenlife}) the $\\varphi$-mode is recursively stable, sustaining $F_s>0$. This is the one regime that both persists and preserves distinctness --- progress.\n\n\\emph{(MAS).} For $\\Omega_{AB}\\gg0$ the cooperative coupling saturates and the symmetric part $S$ dominates. A real symmetric operator has real spectrum, and as the coupling grows the dominant eigenvector rotates onto the diagonal $(1,1)$. The membranes converge to a common state, the relative coordinate decays, $\\Delta\\Sigma\\to0$, and the dyad freezes at the merged fixed point; the invariant of this limit is the diagonal norm $\\|(1,1)\\|=\\sqrt2$. Distinctness is lost --- similarity, preservation without progress.\n\n\\emph{Boundaries and exhaustiveness.} As $\\Omega_{AB}$ increases through zero the discriminant of the characteristic polynomial of $\\mathbf{C}_{AB}$ changes sign: a complex$\\to$real transition, hence a symbolic phase transition (Def.~\\ref{definition:bk2_symbolic_phase_transitio}) at the $\\mathrm{MAD}\\!\\to\\!\\mathrm{MAP}$ boundary. Increasing $\\Omega_{AB}$ further rotates the dominant eigendirection continuously from the golden $\\varphi{:}1$ ray onto the diagonal --- the $\\mathrm{MAP}\\!\\to\\!\\mathrm{MAS}$ boundary. Thus a sign or phase change in $\\Omega_{AB}$ is a boundary crossing of the enacted branch, not a change of convention. The sign of $\\Omega_{AB}$ together with the saturation of $\\Lambda_{AB}$ partitions the covenant axis into the three regimes, so they are mutually exclusive and exhaustive, with MAP the unique sustainable interior between the opposing edges of destruction and similarity.\n\\end{proof}",
  "line": 2002,
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  "proves": "theorem:bk5_map_mad_mas_trichotomy",
  "ref_roles": [
    {
      "context": "t the diagonal, so $\\Membrane_A$ and $\\Membrane_B$ co-evolve while remaining distinct; by the eigenlife criterion (Cor.~\\ref{corollary:bk5_symbolic_eigenlife}) the $\\varphi$-mode is recursively stable, sustaining $F_s>0$. This is the one regime that both persists and preserves",
      "label": "corollary:bk5_symbolic_eigenlife",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1953,
      "target_type": "corollary"
    },
    {
      "context": "ic polynomial of $\\mathbf{C}_{AB}$ changes sign: a complex$\\to$real transition, hence a symbolic phase transition (Def.~\\ref{definition:bk2_symbolic_phase_transitio}) at the $\\mathrm{MAD}\\!\\to\\!\\mathrm{MAP}$ boundary. Increasing $\\Omega_{AB}$ further rotates the dominant eigendirectio",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    },
    {
      "context": "two-membrane tangent space, its character fixed by the covenant orientation $\\Omega_{AB}$ through $\\Lambda_{AB}$ (Def.~\\ref{definition:bk5_reflective_coupling_stab}). Split $\\mathbf{C}_{AB}=S+A$ into symmetric $S=\\tfrac12(\\mathbf{C}_{AB}+\\mathbf{C}_{AB}^{\\!\\top})$ and antisymmetric $",
      "label": "definition:bk5_reflective_coupling_stab",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 786,
      "target_type": "definition"
    },
    {
      "context": "matrix}\\big)$, whose unique positive eigenvalue is the Golden Ratio $\\varphi$ on the eigendirection $(\\varphi,1)$ (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). This eigendirection is not the diagonal, so $\\Membrane_A$ and $\\Membrane_B$ co-evolve while remaining distinct; by th",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "proof_support",
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      "target_line": 1886,
      "target_type": "theorem"
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    "corollary:bk5_symbolic_eigenlife",
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk5_reflective_coupling_stab",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "role": "proof",
  "type": "proof"
}

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    "proposition:bk7_map_compatible_reciprocity",
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    "scholium:bk5_pi_at_mad_edge",
    "scholium:bk9_flexible_goal_calibration"
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    "theorem:bk5_map_mad_mas_trichotomy"
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  "file": "book5.tex",
  "id": "scholium:bk5_imagination_covenant_branch_selection",
  "label": "scholium:bk5_imagination_covenant_branch_selection",
  "latex_body": "\\begin{scholium}[Imagination as Covenant Branch Selection]\n\\label{scholium:bk5_imagination_covenant_branch_selection}\nBook~IV identifies imagination as imaginary traversal rather than unreality: the observer moves through counterfactual phase directions that are not yet enacted in the real symbolic path (Scholium~\\ref{scholium:bk4_imagination_as_imaginary_traversal}, Prop.~\\ref{proposition:bk4_imagination_bridges_wheel}). In a dyadic covenant, that traversal is the observer-relative search over possible signs, phases, and coupling saturations of $\\mathbf{C}_{AB}$. It can preview MAD, MAP, and MAS branches before action. Once a branch is enacted, however, the trichotomy classifies it by spectrum (Def.~\\ref{definition:bk5_map_mad_mas_band}, Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}). Thus imagination influences whether the dyad enters MAD, MAP, or MAS by selecting and stabilizing a branch; it does not override the spectral diagnosis of the branch actually chosen. A sign surprise in $\\Omega_{AB}$ or the emergence of an imaginary component is therefore a regime-boundary signal.\n\\end{scholium}",
  "line": 2016,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Imagination as Covenant Branch Selection",
  "ref_roles": [
    {
      "context": "MAP, and MAS branches before action. Once a branch is enacted, however, the trichotomy classifies it by spectrum (Def.~\\ref{definition:bk5_map_mad_mas_band}, Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}). Thus imagination influences whether the dyad enters MAD, MAP, or MAS b",
      "label": "definition:bk5_map_mad_mas_band",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1977,
      "target_type": "definition"
    },
    {
      "context": "at are not yet enacted in the real symbolic path (Scholium~\\ref{scholium:bk4_imagination_as_imaginary_traversal}, Prop.~\\ref{proposition:bk4_imagination_bridges_wheel}). In a dyadic covenant, that traversal is the observer-relative search over possible signs, phases, and coupling satura",
      "label": "proposition:bk4_imagination_bridges_wheel",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "book4.tex",
      "target_line": 887,
      "target_type": "proposition"
    },
    {
      "context": "the observer moves through counterfactual phase directions that are not yet enacted in the real symbolic path (Scholium~\\ref{scholium:bk4_imagination_as_imaginary_traversal}, Prop.~\\ref{proposition:bk4_imagination_bridges_wheel}). In a dyadic covenant, that traversal is the observer-relative",
      "label": "scholium:bk4_imagination_as_imaginary_traversal",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "book4.tex",
      "target_line": 782,
      "target_type": "scholium"
    },
    {
      "context": "branch is enacted, however, the trichotomy classifies it by spectrum (Def.~\\ref{definition:bk5_map_mad_mas_band}, Thm.~\\ref{theorem:bk5_map_mad_mas_trichotomy}). Thus imagination influences whether the dyad enters MAD, MAP, or MAS by selecting and stabilizing a branch; it does n",
      "label": "theorem:bk5_map_mad_mas_trichotomy",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 1992,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_map_mad_mas_band",
    "proposition:bk4_imagination_bridges_wheel",
    "scholium:bk4_imagination_as_imaginary_traversal",
    "theorem:bk5_map_mad_mas_trichotomy"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "scholium:bk5_imagination_covenant_branch_selection"
  ],
  "depends_on": [
    "scholium:bk5_imagination_covenant_branch_selection"
  ],
  "file": "book5.tex",
  "id": "scholium:bk5_pi_at_mad_edge",
  "label": "scholium:bk5_pi_at_mad_edge",
  "latex_body": "\\begin{scholium}[The Transcendence of Destruction: $\\pi$ at the MAD Edge]\n\\label{scholium:bk5_pi_at_mad_edge}\nIn the branch-selection reading of Scholium~\\ref{scholium:bk5_imagination_covenant_branch_selection}, the destructive branch is recognized by the same spectral sign that imagination may preview before enactment. The three regimes carry three constants in three roles. Progress is a \\emph{growth rate}: the eigenvalue $\\varphi$. Similarity is a \\emph{merged magnitude}: the diagonal norm $\\sqrt2$. Both are algebraic. Destruction alone is \\emph{rotational}: its complex eigenvalues $a\\pm ib$ turn through an angle $\\theta=\\arg(a+ib)$ each step, so the spiral has period $2\\pi/\\theta$, and the constant of the regime is therefore $\\pi$ --- the signature of rotation. It is transcendental precisely because destruction neither grows nor merges but \\emph{turns}: the retaliation that never closes. Thus $\\varphi$, $\\sqrt2$, and $\\pi$ index progress, similarity, and destruction not by numerology but by the kind of motion each regime is --- the eigenvalue, the merged norm, and the angle of the spiral.\n\\end{scholium}",
  "line": 2021,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Transcendence of Destruction: $\\pi$ at the MAD Edge",
  "ref_roles": [
    {
      "context": "of Destruction: $\\pi$ at the MAD Edge] \\label{scholium:bk5_pi_at_mad_edge} In the branch-selection reading of Scholium~\\ref{scholium:bk5_imagination_covenant_branch_selection}, the destructive branch is recognized by the same spectral sign that imagination may preview before enactment. The thre",
      "label": "scholium:bk5_imagination_covenant_branch_selection",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 2016,
      "target_type": "scholium"
    }
  ],
  "refs": [
    "scholium:bk5_imagination_covenant_branch_selection"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "subsec:bk5_curvature_and_fuzzy_balance",
  "label": "subsec:bk5_curvature_and_fuzzy_balance",
  "latex_body": "",
  "line": 2026,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Curvature, Fuzzy Balance, and Symbolic Memory",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk6_symbolic_curvature_tensor",
    "remark:bk5_curvature_vs_chaos",
    "scholium:bk5_constant_of_becoming"
  ],
  "cites": [
    "definition:bk4_symbolic_curvature",
    "definition:bk5_balanced_two_step_memory_closure",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk4_fuzzy_fundamental"
  ],
  "depends_on": [
    "definition:bk4_symbolic_curvature",
    "definition:bk5_balanced_two_step_memory_closure",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk4_fuzzy_fundamental",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_golden_ratio_curvature_scalar",
  "label": "theorem:bk5_golden_ratio_curvature_scalar",
  "latex_body": "\\begin{theorem}[Golden Ratio as Balanced Scale-Resonant Curvature Ratio]\n\\label{theorem:bk5_golden_ratio_curvature_scalar}\nA fuzzy symbolic manifold $\\tilde{M}$ is \\textbf{balanced scale-resonant} along\na growth path $\\gamma$ when the observer-resolved holonomy and curvature\ndistortion amplitudes\n\\[\nh_n=\\|H_{O,n}(\\gamma,f)\\|,\\qquad\nk_n=\\|\\kappa_{O,n}(f,\\int f)\\|\n\\]\nform the projective coordinates of a balanced two-step symbolic memory closure\n(Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), with $k_n>0$.\nFor every such balanced scale-resonant symbolic field $f$,\n\\[\n\\lim_{n\\to\\infty}\\frac{h_n}{k_n}=\\varphi.\n\\]\nHere $H_{O,n}$ and $\\kappa_{O,n}$ are the observer-relative terms from the Fuzzy\nFundamental Theorem of Calculus (Thm.~\\ref{theorem:bk4_fuzzy_fundamental}), with\n$\\kappa_O$ the symbolic curvature (Def.~\\ref{definition:bk4_symbolic_curvature})\narising as the second-order residue of $\\mathcal{O}$-bounded approximation\n(Scholium~\\ref{scholium:bk4_o_boundedness_unifying_principle}).\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": [
      "only the constant-rescaled-Fibonacci case of the ratio limit is proved; the anchor's general two-term holonomy/curvature recurrence (arbitrary balanced-closure initial data) is not derived."
    ],
    "record_ids": [
      "MAP-BOOK5-087"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Residue.balanced_memory_tendsto_gold_scaled"
    ]
  },
  "line": 2031,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Golden Ratio as Balanced Scale-Resonant Curvature Ratio",
  "proof_labels": [
    "proof:bk5_golden_ratio_curvature_scalar"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "undamental Theorem of Calculus (Thm.~\\ref{theorem:bk4_fuzzy_fundamental}), with $\\kappa_O$ the symbolic curvature (Def.~\\ref{definition:bk4_symbolic_curvature}) arising as the second-order residue of $\\mathcal{O}$-bounded approximation (Scholium~\\ref{scholium:bk4_o_boundedness_u",
      "label": "definition:bk4_symbolic_curvature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 452,
      "target_type": "definition"
    },
    {
      "context": "k_n=\\|\\kappa_{O,n}(f,\\int f)\\| \\] form the projective coordinates of a balanced two-step symbolic memory closure (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}), with $k_n>0$. For every such balanced scale-resonant symbolic field $f$, \\[ \\lim_{n\\to\\infty}\\frac{h_n}{k_n}=\\varphi.",
      "label": "definition:bk5_balanced_two_step_memory_closure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1828,
      "target_type": "definition"
    },
    {
      "context": "efinition:bk4_symbolic_curvature}) arising as the second-order residue of $\\mathcal{O}$-bounded approximation (Scholium~\\ref{scholium:bk4_o_boundedness_unifying_principle}). \\end{theorem}",
      "label": "scholium:bk4_o_boundedness_unifying_principle",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 6004,
      "target_type": "scholium"
    },
    {
      "context": "Here $H_{O,n}$ and $\\kappa_{O,n}$ are the observer-relative terms from the Fuzzy Fundamental Theorem of Calculus (Thm.~\\ref{theorem:bk4_fuzzy_fundamental}), with $\\kappa_O$ the symbolic curvature (Def.~\\ref{definition:bk4_symbolic_curvature}) arising as the second-order res",
      "label": "theorem:bk4_fuzzy_fundamental",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 5819,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_symbolic_curvature",
    "definition:bk5_balanced_two_step_memory_closure",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk4_fuzzy_fundamental"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "depends_on": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_golden_ratio_curvature_scalar",
  "label": "proof:bk5_golden_ratio_curvature_scalar",
  "latex_body": "\\begin{proof}[Balanced curvature ratio]\n\\label{proof:bk5_golden_ratio_curvature_scalar}\n\\leavevmode\n\nBy hypothesis, the pair $(h_n,k_n)^T$ is the projective state vector of the\nbalanced closure in Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}.\nThus it evolves, up to observer-normalized scale, by the same primitive matrix\n\\[\nA=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}.\n\\]\nTheorem~\\ref{theorem:bk5_golden_ratio_spectral_invariant} gives the unique\npositive Perron eigendirection of $A$.  Solving\n$A(h,k)^T=\\varphi(h,k)^T$ yields $h+k=\\varphi h$ and\n$h=\\varphi k$, hence $h/k=\\varphi$.  Perron--Frobenius convergence of\nnonzero nonnegative iterates gives convergence of the projective coordinate\n$h_n/k_n$ to that same ratio.  Therefore the stable holonomy-to-curvature\ndistortion ratio of a balanced scale-resonant field is $\\varphi$.\n\\end{proof}",
  "line": 2053,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Balanced curvature ratio",
  "proves": "theorem:bk5_golden_ratio_curvature_scalar",
  "ref_roles": [
    {
      "context": "alar} \\leavevmode By hypothesis, the pair $(h_n,k_n)^T$ is the projective state vector of the balanced closure in Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}. Thus it evolves, up to observer-normalized scale, by the same primitive matrix \\[ A=\\begin{pmatrix}1&1\\\\1&0\\end{pmatri",
      "label": "definition:bk5_balanced_two_step_memory_closure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1828,
      "target_type": "definition"
    },
    {
      "context": "es, up to observer-normalized scale, by the same primitive matrix \\[ A=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}. \\] Theorem~\\ref{theorem:bk5_golden_ratio_spectral_invariant} gives the unique positive Perron eigendirection of $A$. Solving $A(h,k)^T=\\varphi(h,k)^T$ yields $h+k=\\varphi h$ and $",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk9_golden_rule_thermodynamic_covenant"
  ],
  "cites": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_horizon_structure",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk5_golden_ratio_curvature_scalar"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_horizon_structure",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk5_golden_ratio_curvature_scalar"
  ],
  "file": "book5.tex",
  "id": "scholium:bk5_constant_of_becoming",
  "label": "scholium:bk5_constant_of_becoming",
  "latex_body": "\\begin{scholium}[The Constant of Becoming]\n\\label{scholium:bk5_constant_of_becoming}\nTheorem~\\ref{theorem:bk5_golden_ratio_curvature_scalar} identifies $\\varphi$ as the curvature-memory ratio of observer-relative symbolic spacetime in the balanced scale-resonant regime. A bounded observer (Def.~\\ref{definition:bk1_bounded_observer}) cannot differentiate and integrate its own state without introducing geometric error bounded by its horizon structure (Def.~\\ref{definition:bk1_observer_horizon_structure}). The torsion term $\\kappa_O$ represents the local ``cost'' of parsing reality (differentiation)---it is the cross-error residue that $\\mathcal{O}$-bounded composition cannot eliminate (Scholium~\\ref{scholium:bk4_o_boundedness_unifying_principle})---while the holonomy term $H_O$ represents the cumulative ``cost'' of reconstructing a coherent history (integration). A system can persist as balanced scale-resonant when these two costs remain on the Perron eigendirection of the balanced memory closure.\n\\end{scholium}",
  "line": 2072,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Constant of Becoming",
  "ref_roles": [
    {
      "context": "re-memory ratio of observer-relative symbolic spacetime in the balanced scale-resonant regime. A bounded observer (Def.~\\ref{definition:bk1_bounded_observer}) cannot differentiate and integrate its own state without introducing geometric error bounded by its horizon structure",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    },
    {
      "context": "ot differentiate and integrate its own state without introducing geometric error bounded by its horizon structure (Def.~\\ref{definition:bk1_observer_horizon_structure}). The torsion term $\\kappa_O$ represents the local ``cost'' of parsing reality (differentiation)---it is the cross-erro",
      "label": "definition:bk1_observer_horizon_structure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1232,
      "target_type": "definition"
    },
    {
      "context": "ity (differentiation)---it is the cross-error residue that $\\mathcal{O}$-bounded composition cannot eliminate (Scholium~\\ref{scholium:bk4_o_boundedness_unifying_principle})---while the holonomy term $H_O$ represents the cumulative ``cost'' of reconstructing a coherent history (integration).",
      "label": "scholium:bk4_o_boundedness_unifying_principle",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 6004,
      "target_type": "scholium"
    },
    {
      "context": "\\begin{scholium}[The Constant of Becoming] \\label{scholium:bk5_constant_of_becoming} Theorem~\\ref{theorem:bk5_golden_ratio_curvature_scalar} identifies $\\varphi$ as the curvature-memory ratio of observer-relative symbolic spacetime in the balanced scale-resona",
      "label": "theorem:bk5_golden_ratio_curvature_scalar",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 2031,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer",
    "definition:bk1_observer_horizon_structure",
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk5_golden_ratio_curvature_scalar"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk9_golden_rule_thermodynamic_covenant"
  ],
  "cites": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "depends_on": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "file": "book5.tex",
  "id": "remark:bk5_symbolic_fibonacci_coding",
  "label": "remark:bk5_symbolic_fibonacci_coding",
  "latex_body": "\\begin{remark}[Symbolic Fibonacci Coding and Memory]\n\\label{remark:bk5_symbolic_fibonacci_coding}\nThe recurrence relation $a_{n+1}=a_n+a_{n-1}$ is precisely the scalar amplitude\nform of balanced two-step symbolic memory\n(Def.~\\ref{definition:bk5_balanced_two_step_memory_closure},\nThm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}).  It describes symbolic\nlife as emergent memory, where the present amplitude is constructed from current\npersistence and one retained reflective state.  Under reflective normalization\n(i.e., maintaining a stable observer-resolved identity), the ratio of successive\namplitudes converges:\n\\[\n\\lim_{n \\to \\infty} \\frac{a_{n+1}}{a_n} = \\varphi\n\\]\nLife thus becomes a Fibonacci logic of symbolic retention exactly when the\nobserver-normalized memory weights are balanced; $\\varphi$ is the\n\\textbf{asymptotic identity gradient} of that regime.\n\\end{remark}",
  "line": 2077,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Fibonacci Coding and Memory",
  "ref_roles": [
    {
      "context": "rrence relation $a_{n+1}=a_n+a_{n-1}$ is precisely the scalar amplitude form of balanced two-step symbolic memory (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). It describes symbolic life as emergent memory, where the pre",
      "label": "definition:bk5_balanced_two_step_memory_closure",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book5.tex",
      "target_line": 1828,
      "target_type": "definition"
    },
    {
      "context": "r amplitude form of balanced two-step symbolic memory (Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). It describes symbolic life as emergent memory, where the present amplitude is constructed from current persistence a",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "subsec:bk5_thermoregulation_and_phi",
  "label": "subsec:bk5_thermoregulation_and_phi",
  "latex_body": "",
  "line": 2095,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Thermoregulation and the Golden Mean",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk5_experimental_predictions",
    "scholium:bk5_life_on_edge_of_chaos"
  ],
  "cites": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "depends_on": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "file": "book5.tex",
  "id": "proposition:bk5_golden_ratio_thermodynamic_optimum",
  "label": "proposition:bk5_golden_ratio_thermodynamic_optimum",
  "latex_body": "\\begin{proposition}[Golden Ratio as Thermodynamic Optimum in the Balanced Regime]\n\\label{proposition:bk5_golden_ratio_thermodynamic_optimum}\nLet\n\\[\nr=\\frac{W_{\\mathrm{coh}}}{W_{\\mathrm{nov}}}\n\\]\nbe the positive observer-resolved ratio of coherence-preserving work\n(negentropy from Reflection, $\\reflect$) to novelty-generating exploration\n(entropy from Drift, $\\drift$).  In a balanced two-step metabolic regime, suppose\nthe free-energy contribution of this ratio is the spectral-misalignment\nLyapunov term\n\\[\n\\mathcal{F}_{\\mathrm{bal}}(r)=\\mathcal{F}_0+\\alpha(\\log r-\\log\\varphi)^2,\n\\qquad \\alpha>0.\n\\]\nThen $\\mathcal{F}_{\\mathrm{bal}}$ is minimized if and only if $r=\\varphi$\n(cf.~Def.~\\ref{definition:bk2_symbolic_free_energy},\nThm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy},\nThm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}).\n\\end{proposition}",
  "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": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-088"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.golden_ratio_thermodynamic_min",
      "Book5Residue.golden_ratio_thermodynamic_optimum_iff"
    ]
  },
  "line": 2102,
  "macros_used": [
    "drift",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Golden Ratio as Thermodynamic Optimum in the Balanced Regime",
  "proof_labels": [
    "proof:bk5_golden_ratio_thermodynamic_optimum"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "\\log\\varphi)^2, \\qquad \\alpha>0. \\] Then $\\mathcal{F}_{\\mathrm{bal}}$ is minimized if and only if $r=\\varphi$ (cf.~Def.~\\ref{definition:bk2_symbolic_free_energy}, Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invarian",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "~Def.~\\ref{definition:bk2_symbolic_free_energy}, Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). \\end{proposition}",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "al{F}_{\\mathrm{bal}}$ is minimized if and only if $r=\\varphi$ (cf.~Def.~\\ref{definition:bk2_symbolic_free_energy}, Thm.~\\ref{theorem:bk5_metabolic_constraints_reflective_accuracy}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). \\end{proposition}",
      "label": "theorem:bk5_metabolic_constraints_reflective_accuracy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 1748,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_free_energy",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_metabolic_constraints_reflective_accuracy"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "depends_on": [
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_golden_ratio_thermodynamic_optimum",
  "label": "proof:bk5_golden_ratio_thermodynamic_optimum",
  "latex_body": "\\begin{proof}[Balanced thermodynamic optimum]\n\\label{proof:bk5_golden_ratio_thermodynamic_optimum}\n\\leavevmode\n\nSince $\\alpha>0$, the misalignment term\n$\\alpha(\\log r-\\log\\varphi)^2$ is nonnegative for every $r>0$ and vanishes\nexactly when $\\log r=\\log\\varphi$.  The logarithm is injective on the positive\nreals, so this occurs exactly at $r=\\varphi$.  Therefore\n$\\mathcal{F}_{\\mathrm{bal}}(r)\\geq\\mathcal{F}_0$, with equality if and only if\nthe work/exploration ratio lies on the balanced memory eigendirection selected\nby Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}.\n\\end{proof}",
  "line": 2123,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Balanced thermodynamic optimum",
  "proves": "proposition:bk5_golden_ratio_thermodynamic_optimum",
  "ref_roles": [
    {
      "context": "0$, with equality if and only if the work/exploration ratio lies on the balanced memory eigendirection selected by Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}. \\end{proof}",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1886,
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    }
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    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "role": "proof",
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}

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  "book": "book5",
  "cited_by": [
    "definition:bk5_diagonal_transition",
    "definition:bk5_symbolic_integrability_class",
    "definition:bk5_symbolic_torsion",
    "lemma:bk6_power_scaling",
    "proof:bk6_power_scaling"
  ],
  "cites": [
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk4_fuzzy_fundamental"
  ],
  "depends_on": [
    "scholium:bk4_o_boundedness_unifying_principle",
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  ],
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  "id": "definition:bk5_fuzzy_symbolic_manifold",
  "label": "definition:bk5_fuzzy_symbolic_manifold",
  "latex_body": "\\begin{definition}[Fuzzy Symbolic Manifold]\n\\label{definition:bk5_fuzzy_symbolic_manifold}\nA fuzzy symbolic manifold $\\tilde{M}$ is a discretized space where each point $p \\in \\tilde{M}$ exists within an observer-dependent resolution cell of radius $\\epsilon_\\mathcal{O}$. Symbolic transitions between points are governed by \\textbf{bounded rational approximations} to underlying geometric relationships (cf.~Thm.~\\ref{theorem:bk4_fuzzy_fundamental}); these approximations remain sub-threshold at every compositional step by the $\\mathcal{O}$-boundedness principle (Scholium~\\ref{scholium:bk4_o_boundedness_unifying_principle}).\n\\end{definition}",
  "line": 2136,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Fuzzy Symbolic Manifold",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "se approximations remain sub-threshold at every compositional step by the $\\mathcal{O}$-boundedness principle (Scholium~\\ref{scholium:bk4_o_boundedness_unifying_principle}). \\end{definition}",
      "label": "scholium:bk4_o_boundedness_unifying_principle",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book4.tex",
      "target_line": 6004,
      "target_type": "scholium"
    },
    {
      "context": "between points are governed by \\textbf{bounded rational approximations} to underlying geometric relationships (cf.~Thm.~\\ref{theorem:bk4_fuzzy_fundamental}); these approximations remain sub-threshold at every compositional step by the $\\mathcal{O}$-boundedness principle (Sch",
      "label": "theorem:bk4_fuzzy_fundamental",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book4.tex",
      "target_line": 5819,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "scholium:bk4_o_boundedness_unifying_principle",
    "theorem:bk4_fuzzy_fundamental"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk5_collapse_resilience_test"
  ],
  "cites": [
    "definition:bk5_fuzzy_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk5_fuzzy_symbolic_manifold"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_torsion",
  "label": "definition:bk5_symbolic_torsion",
  "latex_body": "\\begin{definition}[Symbolic Torsion]\n\\label{definition:bk5_symbolic_torsion}\nFor an irrational constant $x$ and observer resolution $\\epsilon_\\mathcal{O}$, the symbolic torsion $\\mathcal{T}_x(\\epsilon_\\mathcal{O})$ measures the \\textbf{irreducible complexity} of representing $x$ within the bounded symbolic framework (cf.~Def.~\\ref{definition:bk5_fuzzy_symbolic_manifold}):\n$$\\mathcal{T}_x(\\epsilon_\\mathcal{O}) = \\frac{\\log(\\text{denominator of best rational approximation within } \\epsilon_\\mathcal{O})}{\\log(\\epsilon_\\mathcal{O}^{-1})}$$\n\\end{definition}",
  "line": 2141,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Torsion",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "l{O})$ measures the \\textbf{irreducible complexity} of representing $x$ within the bounded symbolic framework (cf.~Def.~\\ref{definition:bk5_fuzzy_symbolic_manifold}): $$\\mathcal{T}_x(\\epsilon_\\mathcal{O}) = \\frac{\\log(\\text{denominator of best rational approximation within } \\epsilon",
      "label": "definition:bk5_fuzzy_symbolic_manifold",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2136,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_fuzzy_symbolic_manifold"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "lemma:bk5_shortest_path_representability"
  ],
  "cites": [
    "definition:bk5_fuzzy_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk5_fuzzy_symbolic_manifold"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_diagonal_transition",
  "label": "definition:bk5_diagonal_transition",
  "latex_body": "\\begin{definition}[Diagonal Transition]\n\\label{definition:bk5_diagonal_transition}\nIn a fuzzy symbolic manifold with orthogonal basis vectors $\\{e_1, e_2, \\ldots\\}$, a diagonal transition is any symbolic path that cannot be decomposed into integer-aligned steps without introducing irrational scaling factors (cf.~Def.~\\ref{definition:bk5_fuzzy_symbolic_manifold}).\n\\end{definition}",
  "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": [
      "See the committed source registry for the original coverage note."
    ],
    "record_ids": [
      "MAP-BOOK5-089"
    ],
    "statuses": [
      "conditional"
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    "witnesses": [
      "Book5Residue.compression_ratio_R2"
    ]
  },
  "line": 2147,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Diagonal Transition",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "olic path that cannot be decomposed into integer-aligned steps without introducing irrational scaling factors (cf.~Def.~\\ref{definition:bk5_fuzzy_symbolic_manifold}). \\end{definition}",
      "label": "definition:bk5_fuzzy_symbolic_manifold",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2136,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk5_fuzzy_symbolic_manifold"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "lemma:bk5_shortest_path_representability",
    "proof:bk5_complementary_constants",
    "proposition:bk5_complementary_constants",
    "remark:bk5_curvature_vs_chaos",
    "scholium:bk5_experimental_predictions",
    "theorem:bk5_fundamental_dichotomy"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "theorem:bk5_sqrt2_maximal_fracture",
  "label": "theorem:bk5_sqrt2_maximal_fracture",
  "latex_body": "\\begin{theorem}[$\\sqrt{2}$ as the First Orthogonal Fracture Constant]\n\\label{theorem:bk5_sqrt2_maximal_fracture}\nLet $\\tilde{M}$ have a local orthonormal symbolic frame\n$\\{e_1,\\ldots,e_d\\}$ whose observer-symbolic paths are generated by\naxis-aligned unit steps.  For every non-axis-aligned primitive lattice\ntransition $v=\\sum_i m_i e_i$ with $m_i\\in\\mathbb{Z}$ and at least two nonzero\ncoordinates,\n\\[\n\\|v\\|_2\\geq \\sqrt{2}.\n\\]\nEquality holds exactly for the elementary diagonal transitions\n$v=\\pm e_i\\pm e_j$, $i\\neq j$.  Consequently $\\sqrt{2}$ is the first\northogonal fracture constant: the smallest Euclidean length at which a\ngeometrically direct transition cannot be represented as a single\naxis-aligned symbolic step.  For the elementary diagonal, the symbolic\naxis-step length is $2$, the geometric length is $\\sqrt{2}$, and the\nrepresentability ratio is\n\\[\n\\frac{L_{\\mathrm{sym}}}{L_2}=\\frac{2}{\\sqrt{2}}=\\sqrt{2}.\n\\]\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "only the arithmetic, matrix, lattice, and limit content of the cited Book 5 statements"
    ],
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    "kernel_certified": true,
    "notes": [
      "Integer two-support lower bound and elementary diagonal ratio."
    ],
    "record_ids": [
      "MAP-BOOK5-020"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "diag_fracture_ratio",
      "sqrt2_first_fracture"
    ]
  },
  "line": 2152,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "$\\sqrt{2}$ as the First Orthogonal Fracture Constant",
  "proof_labels": [
    "proof:bk5_sqrt2_maximal_fracture"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "theorem",
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}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "proof:bk5_sqrt2_maximal_fracture",
  "label": "proof:bk5_sqrt2_maximal_fracture",
  "latex_body": "\\begin{proof}[First orthogonal fracture]\n\\label{proof:bk5_sqrt2_maximal_fracture}\n\\leavevmode\n\nLet $v=\\sum_i m_i e_i$ be a primitive lattice transition with at least two\nnonzero integer coordinates.  Since each nonzero coordinate has\n$|m_i|\\geq 1$, the Euclidean norm satisfies\n\\[\n\\|v\\|_2^2=\\sum_i m_i^2\\geq 1^2+1^2=2,\n\\]\nand hence $\\|v\\|_2\\geq\\sqrt{2}$.  Equality requires exactly two nonzero\ncoordinates and both must have absolute value $1$, so\n$v=\\pm e_i\\pm e_j$ for distinct $i,j$.\n\nFor such an elementary diagonal, the direct geometric transition has length\n$\\sqrt{2}$.  An axis-generated symbolic path cannot realize it in one symbolic\nunit step, because every one-step generator is one of the frame vectors\n$\\pm e_i$.  The shortest axis-generated symbolic decomposition uses two unit\nsteps, $\\pm e_i$ and $\\pm e_j$, so $L_{\\mathrm{sym}}=2$.  Thus the\nobserver-visible representability ratio is $2/\\sqrt{2}=\\sqrt{2}$.\nThus $\\sqrt{2}$ is the exact first fracture constant forced by orthogonal\ndiscretization.\n\\end{proof}",
  "line": 2174,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "First orthogonal fracture",
  "proves": "theorem:bk5_sqrt2_maximal_fracture",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "theorem:bk5_fundamental_dichotomy"
  ],
  "cites": [
    "definition:bk4_fragmented_identity",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "depends_on": [
    "definition:bk4_fragmented_identity",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "file": "book5.tex",
  "id": "proposition:bk5_complementary_constants",
  "label": "proposition:bk5_complementary_constants",
  "latex_body": "\\begin{proposition}[Complementary Constants: Fracture vs Resonance]\n\\label{proposition:bk5_complementary_constants}\nFormally this pairs Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture} with Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}.\nIn fuzzy symbolic calculus, $\\sqrt{2}$ and $\\varphi$ serve complementary roles:\n- $\\sqrt{2}$ marks the first \\textbf{symbolic fracture} forced by orthogonal incommensurability (cf.~Def.~\\ref{definition:bk4_fragmented_identity})\n- $\\varphi$ marks the positive \\textbf{resonant ratio} selected by balanced recursive memory\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "only the arithmetic, matrix, lattice, and limit content of the cited Book 5 statements"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Algebraic separation of phi and sqrt(2)."
    ],
    "record_ids": [
      "MAP-BOOK5-022"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "constants_complementary"
    ]
  },
  "line": 2198,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Complementary Constants: Fracture vs Resonance",
  "proof_labels": [
    "proof:bk5_complementary_constants"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "entary roles: - $\\sqrt{2}$ marks the first \\textbf{symbolic fracture} forced by orthogonal incommensurability (cf.~Def.~\\ref{definition:bk4_fragmented_identity}) - $\\varphi$ marks the positive \\textbf{resonant ratio} selected by balanced recursive memory \\end{proposition}",
      "label": "definition:bk4_fragmented_identity",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book4.tex",
      "target_line": 2731,
      "target_type": "definition"
    },
    {
      "context": "el{proposition:bk5_complementary_constants} Formally this pairs Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture} with Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}. In fuzzy symbolic calculus, $\\sqrt{2}$ and $\\varphi$ serve complementary roles: - $\\sqrt{2}$ marks the first \\textbf{s",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "omplementary Constants: Fracture vs Resonance] \\label{proposition:bk5_complementary_constants} Formally this pairs Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture} with Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}. In fuzzy symbolic calculus, $\\sqrt{2}$ and $\\varphi$ serve",
      "label": "theorem:bk5_sqrt2_maximal_fracture",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 2152,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk4_fragmented_identity",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "depends_on": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_complementary_constants",
  "label": "proof:bk5_complementary_constants",
  "latex_body": "\\begin{proof}[Complementarity of first fracture and balanced resonance]\n\\label{proof:bk5_complementary_constants}\n\n\\leavevmode\n\nTheorem~\\ref{theorem:bk5_golden_ratio_spectral_invariant} proves that\n$\\varphi$ is the positive Perron ratio of balanced two-step memory; it is\ntherefore a resonance constant for recursive retention.  Theorem~\\ref{theorem:bk5_sqrt2_maximal_fracture}\nproves that $\\sqrt{2}$ is the first non-axis-aligned length forced by an\northogonal symbolic frame; it is therefore a fracture constant for geometric\nrepresentation.  These mechanisms are complementary because the former arises\nfrom temporal recursion in the memory state $(a_n,a_{n-1})$, while the latter\narises from spatial incompatibility between a direct Euclidean diagonal and an\naxis-generated symbolic path.\n\\end{proof}",
  "line": 2206,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Complementarity of first fracture and balanced resonance",
  "proves": "proposition:bk5_complementary_constants",
  "ref_roles": [
    {
      "context": "mplementarity of first fracture and balanced resonance] \\label{proof:bk5_complementary_constants} \\leavevmode Theorem~\\ref{theorem:bk5_golden_ratio_spectral_invariant} proves that $\\varphi$ is the positive Perron ratio of balanced two-step memory; it is therefore a resonance constant fo",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "sitive Perron ratio of balanced two-step memory; it is therefore a resonance constant for recursive retention. Theorem~\\ref{theorem:bk5_sqrt2_maximal_fracture} proves that $\\sqrt{2}$ is the first non-axis-aligned length forced by an orthogonal symbolic frame; it is therefore a f",
      "label": "theorem:bk5_sqrt2_maximal_fracture",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2152,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "role": "proof",
  "type": "proof"
}