Book V — De Vita Symbolica — page 5 of 6

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      "context": "e memory (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}; cf.~Thm.~\\ref{theorem:appC_phi_from_lagrangian}, Thm.~\\ref{theorem:appC_phi_as_spectral_radius}). \\end{remark}",
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      "context": "s the Perron fixed ratio of balanced recursive memory (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}; cf.~Thm.~\\ref{theorem:appC_phi_from_lagrangian}, Thm.~\\ref{theorem:appC_phi_as_spectral_radius}). \\end{remark}",
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  "latex_body": "\\begin{remark}[Scale-Resonant Curvature vs Symbolic Chaos]\n\\label{remark:bk5_curvature_vs_chaos}\nManifolds whose holonomy and curvature amplitudes realize the balanced memory\nclosure exhibit \\textbf{scale-resonant curvature}: the holonomy-to-curvature\nratio remains stable at $\\varphi$ across scales\n(cf.~Thm.~\\ref{theorem:bk5_golden_ratio_curvature_scalar}).  Manifolds\nencountering $\\sqrt{2}$ transitions exhibit \\textbf{symbolic fracture} when the\naxis-generated representation must pay the elementary diagonal gap\n(cf.~Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}).  That $\\varphi$\nplays the resonant role is not accidental: it is the Perron fixed ratio of\nbalanced recursive memory (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant};\ncf.~Thm.~\\ref{theorem:appC_phi_from_lagrangian},\nThm.~\\ref{theorem:appC_phi_as_spectral_radius}).\n\\end{remark}",
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      "context": "s the Perron fixed ratio of balanced recursive memory (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}; cf.~Thm.~\\ref{theorem:appC_phi_from_lagrangian}, Thm.~\\ref{theorem:appC_phi_as_spectral_radius}). \\end{remark}",
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      "context": "\\textbf{scale-resonant curvature}: the holonomy-to-curvature ratio remains stable at $\\varphi$ across scales (cf.~Thm.~\\ref{theorem:bk5_golden_ratio_curvature_scalar}). Manifolds encountering $\\sqrt{2}$ transitions exhibit \\textbf{symbolic fracture} when the axis-generated representat",
      "label": "theorem:bk5_golden_ratio_curvature_scalar",
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      "context": "at $\\varphi$ plays the resonant role is not accidental: it is the Perron fixed ratio of balanced recursive memory (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}; cf.~Thm.~\\ref{theorem:appC_phi_from_lagrangian}, Thm.~\\ref{theorem:appC_phi_as_spectral_radius}). \\end{remark}",
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      "context": "xhibit \\textbf{symbolic fracture} when the axis-generated representation must pay the elementary diagonal gap (cf.~Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}). That $\\varphi$ plays the resonant role is not accidental: it is the Perron fixed ratio of balanced recursive memory",
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  "latex_body": "\\begin{definition}[Symbolic Collapse Resilience Test]\n\\label{definition:bk5_collapse_resilience_test}\nGiven a fuzzy symbolic system with resolution parameter $\\epsilon$, \\textbf{collapse resilience} measures how long symbolic coherence persists under iterative approximation errors when representing an irrational constant (cf.~Def.~\\ref{definition:bk5_symbolic_torsion}, Def.~\\ref{definition:bk4_collapse_of_symbolic_ide}, Thm.~\\ref{theorem:bk4_test_time_differentiation_c}, Def.~\\ref{definition:bk4_test_time_coherent_sampling}, Thm.~\\ref{theorem:bk4_ttpr_symbolic_stability}, Rem.~\\ref{remark:bk4_ttpr_entropy}).\n\\end{definition}",
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      "context": "ive approximation errors when representing an irrational constant (cf.~Def.~\\ref{definition:bk5_symbolic_torsion}, Def.~\\ref{definition:bk4_collapse_of_symbolic_ide}, Thm.~\\ref{theorem:bk4_test_time_differentiation_c}, Def.~\\ref{definition:bk4_test_time_coherent_sampling}, Thm.~\\ref{t",
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      "target_line": 2218,
      "target_type": "definition"
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      "context": "ong symbolic coherence persists under iterative approximation errors when representing an irrational constant (cf.~Def.~\\ref{definition:bk5_symbolic_torsion}, Def.~\\ref{definition:bk4_collapse_of_symbolic_ide}, Thm.~\\ref{theorem:bk4_test_time_differentiation_c}, Def.~\\ref{defi",
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      "label": "remark:bk4_ttpr_entropy",
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      "role": "formal_dependency",
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      "target_line": 2116,
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      "context": "onal constant (cf.~Def.~\\ref{definition:bk5_symbolic_torsion}, Def.~\\ref{definition:bk4_collapse_of_symbolic_ide}, Thm.~\\ref{theorem:bk4_test_time_differentiation_c}, Def.~\\ref{definition:bk4_test_time_coherent_sampling}, Thm.~\\ref{theorem:bk4_ttpr_symbolic_stability}, Rem.~\\ref{remar",
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  "id": "scholium:bk5_experimental_predictions",
  "label": "scholium:bk5_experimental_predictions",
  "latex_body": "\\begin{scholium}[Experimental Predictions]\n\\label{scholium:bk5_experimental_predictions}\nThe simulation should demonstrate the following behaviors.\nSee Def.~\\ref{definition:bk5_collapse_resilience_test},\nThm.~\\ref{theorem:bk5_sqrt2_maximal_fracture},\nProp.~\\ref{proposition:bk5_golden_ratio_thermodynamic_optimum}, and\nScholium~\\ref{scholium:bk4_ttdc_impulse_collapse}.\n\\begin{enumerate}\n  \\item $\\sqrt{2}$ appears as the first nonzero representability ratio for elementary diagonal transitions.\n  \\item $\\varphi$ maintains stable balanced-memory ratios across multiple scales.\n  \\item Systems mixing diagonal fracture with balanced memory should show a measurable separation between spatial representability cost and recursive memory resonance.\n\\end{enumerate}\n\\end{scholium}",
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  "name": "Experimental Predictions",
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      "context": "ions] \\label{scholium:bk5_experimental_predictions} The simulation should demonstrate the following behaviors. See Def.~\\ref{definition:bk5_collapse_resilience_test}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}, Prop.~\\ref{proposition:bk5_golden_ratio_thermodynamic_optimum}, and Sch",
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      "context": "behaviors. See Def.~\\ref{definition:bk5_collapse_resilience_test}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}, Prop.~\\ref{proposition:bk5_golden_ratio_thermodynamic_optimum}, and Scholium~\\ref{scholium:bk4_ttdc_impulse_collapse}. \\begin{enumerate} \\item $\\sqrt{2}$ appears as the first nonze",
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      "context": "~\\ref{theorem:bk5_sqrt2_maximal_fracture}, Prop.~\\ref{proposition:bk5_golden_ratio_thermodynamic_optimum}, and Scholium~\\ref{scholium:bk4_ttdc_impulse_collapse}. \\begin{enumerate} \\item $\\sqrt{2}$ appears as the first nonzero representability ratio for elementary diagonal trans",
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      "context": "The simulation should demonstrate the following behaviors. See Def.~\\ref{definition:bk5_collapse_resilience_test}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}, Prop.~\\ref{proposition:bk5_golden_ratio_thermodynamic_optimum}, and Scholium~\\ref{scholium:bk4_ttdc_impulse_collapse}.",
      "label": "theorem:bk5_sqrt2_maximal_fracture",
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  "latex_body": "\\begin{theorem}[Fundamental Dichotomy of Symbolic Constants]\n\\label{theorem:bk5_fundamental_dichotomy}\nIn fuzzy symbolic calculus, the constants $\\varphi$ and $\\sqrt{2}$ instantiate two fundamental mechanisms:\n- \\textbf{Resonant constants} (exemplified by $\\varphi$) selected by balanced recursive memory\n- \\textbf{Fracture constants} (exemplified by $\\sqrt{2}$) forced by irreducible orthogonal incompatibility\n\nThis dichotomy reflects the deep structure of symbolic representation under bounded observation (cf.~Def.~\\ref{definition:bk5_collapse_resilience_test}, Prop.~\\ref{proposition:bk5_complementary_constants}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}).\n\\end{theorem}",
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  "name": "Fundamental Dichotomy of Symbolic Constants",
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      "context": "ompatibility This dichotomy reflects the deep structure of symbolic representation under bounded observation (cf.~Def.~\\ref{definition:bk5_collapse_resilience_test}, Prop.~\\ref{proposition:bk5_complementary_constants}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref",
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      "context": "ure of symbolic representation under bounded observation (cf.~Def.~\\ref{definition:bk5_collapse_resilience_test}, Prop.~\\ref{proposition:bk5_complementary_constants}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}). \\end{theorem}",
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      "context": "tion (cf.~Def.~\\ref{definition:bk5_collapse_resilience_test}, Prop.~\\ref{proposition:bk5_complementary_constants}, Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}). \\end{theorem}",
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  "latex_body": "\\begin{demonstratio}[The Diagonal Dissociation Principle]\n\\label{demonstratio:bk5_diagonal_dissociation}\nWhere $\\varphi$ emerges from the recursive equation $x = 1 + 1/x$ (self-similarity), $\\sqrt{2}$ emerges from the Pythagorean equation $x^2 = 1^2 + 1^2$ (orthogonal combination). This geometric distinction translates directly into symbolic behavior: recursion enables compression, while orthogonality demands expansion (cf.~Thm.~\\ref{theorem:bk5_fundamental_dichotomy}).\n\nThe irrationality of $\\sqrt{2}$ is not merely a number-theoretic accident; in an orthonormal symbolic frame, it is the \\textbf{symbolic signature} of the first dimensional incommensurability.\n\\end{demonstratio}",
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  "name": "Life on the Edge of Chaos",
  "ref_roles": [
    {
      "context": "ime, the minimization of Symbolic Free Energy selects the Perron ratio of the drift-reflection memory closure (cf.~Def.~\\ref{definition:bk2_symbolic_free_energy}, Prop.~\\ref{proposition:bk5_golden_ratio_thermodynamic_optimum}). The Golden Ratio, $\\varphi$, is therefore the edge-of",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "ects the Perron ratio of the drift-reflection memory closure (cf.~Def.~\\ref{definition:bk2_symbolic_free_energy}, Prop.~\\ref{proposition:bk5_golden_ratio_thermodynamic_optimum}). The Golden Ratio, $\\varphi$, is therefore the edge-of-chaos ratio for that regime: it is the proportion at which nove",
      "label": "proposition:bk5_golden_ratio_thermodynamic_optimum",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2102,
      "target_type": "proposition"
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    "proposition:bk5_golden_ratio_thermodynamic_optimum"
  ],
  "role": "scholium",
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}

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  "latex_body": "",
  "line": 2277,
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  "matter_role": "canonical_book",
  "name": "Norm-Induced Fracture and \\texorpdfstring{$\\ell_p$",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_fuzzy_symbolic_manifold"
  ],
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    "definition:bk5_fuzzy_symbolic_manifold"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_integrability_class",
  "label": "definition:bk5_symbolic_integrability_class",
  "latex_body": "\\begin{definition}[Symbolic Integrability Class]\n\\label{definition:bk5_symbolic_integrability_class}\nA fuzzy symbolic manifold $\\tilde{M}$ belongs to \\textbf{symbolic integrability class} $\\mathcal{I}_p$ if its dominant geometric transitions are governed by the $\\ell_p$ norm (cf.~Def.~\\ref{definition:bk5_fuzzy_symbolic_manifold}), where symbolic paths of length $\\delta$ satisfy:\n$$\\|\\vec{v}\\|_p = \\left(\\sum_{i=1}^n |v_i|^p\\right)^{1/p} \\leq \\delta + \\epsilon_\\mathcal{O}$$\nfor observer resolution $\\epsilon_\\mathcal{O}$. The class $\\mathcal{I}_p$ determines the \\textbf{symbolic decomposability} of transitions within $\\tilde{M}$.\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"
    ],
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  "line": 2280,
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  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Integrability Class",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "integrability class} $\\mathcal{I}_p$ if its dominant geometric transitions are governed by the $\\ell_p$ norm (cf.~Def.~\\ref{definition:bk5_fuzzy_symbolic_manifold}), where symbolic paths of length $\\delta$ satisfy: $$\\|\\vec{v}\\|_p = \\left(\\sum_{i=1}^n |v_i|^p\\right)^{1/p} \\leq \\delt",
      "label": "definition:bk5_fuzzy_symbolic_manifold",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2136,
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    "definition:bk5_fuzzy_symbolic_manifold"
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  "role": "definition",
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  "book": "book5",
  "cited_by": [
    "definition:bk5_symbolic_curvature_operator_spectrum"
  ],
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  "file": "book5.tex",
  "id": "definition:bk5_symbolic_curvature_control",
  "label": "definition:bk5_symbolic_curvature_control",
  "latex_body": "\\begin{definition}[Symbolic Curvature Control Parameter]\n\\label{definition:bk5_symbolic_curvature_control}\nFor a nonzero transition $\\vec{v}$ with support\n$\\operatorname{supp}(\\vec{v})=\\{i:v_i\\neq0\\}$ and\n$s(\\vec{v})=|\\operatorname{supp}(\\vec{v})|$, the \\textbf{symbolic curvature\ncontrol parameter} in class $\\mathcal{I}_p$ is\n\\[\n\\kappa_p(\\vec{v})=\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p}-1,\\qquad 1\\leq p\\leq\\infty.\n\\]\nHere $\\|\\vec{v}\\|_1$ is the axis-generated symbolic length and $\\|\\vec{v}\\|_p$\nis the geometric length in the dominant norm.  Thus $\\kappa_p$ measures the\nextra axis-symbolic cost paid to represent a geometrically shorter transition.\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,
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    ],
    "record_ids": [
      "MAP-BOOK5-091"
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      "exact"
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      "Book5Residue.l1_ge_l2"
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  "line": 2287,
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  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Curvature Control Parameter",
  "proof_status": "definitional",
  "refs": [],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk5_symbolic_compression_experiment",
    "proof:bk5_complete_symbolic_regime_classification",
    "proof:bk5_fractal_dimension_connection",
    "proof:bk5_fundamental_norm_fracture",
    "proof:bk5_shortest_path_representability",
    "proof:bk5_symbolic_integrability_classes",
    "proof:bk5_symbolic_norm_spectrum",
    "proof:bk5_symbolic_torsion_phase_diagram",
    "proposition:bk5_symbolic_integrability_classes",
    "remark:bk5_lattice_field_theory_analogy",
    "theorem:bk5_fundamental_norm_fracture",
    "theorem:bk5_symbolic_norm_spectrum"
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  "file": "book5.tex",
  "id": "theorem:bk5_lp_norm_fracture_hierarchy",
  "label": "theorem:bk5_lp_norm_fracture_hierarchy",
  "latex_body": "\\begin{theorem}[\\(\\ell_p\\)-Norm Fracture Hierarchy]\n\\label{theorem:bk5_lp_norm_fracture_hierarchy}\nFor every nonzero transition $\\vec{v}$ in a fuzzy symbolic manifold and every\n$1\\leq p\\leq\\infty$,\n\\[\n1\\leq \\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p}\n\\leq s(\\vec{v})^{1-1/p}.\n\\]\nThe upper bound is attained exactly when all nonzero coordinates of\n$\\vec{v}$ have equal magnitude.  Consequently, for the elementary diagonal\n$\\vec{v}=e_i+e_j$,\n\\[\n\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_1}=1,\\qquad\n\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_2}=\\sqrt{2},\\qquad\n\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_\\infty}=2.\n\\]\nThus $\\ell_1$ is perfectly axis-integrable, $\\ell_2$ introduces the first\northogonal fracture ratio $\\sqrt{2}$, and $\\ell_\\infty$ collapses all\ncoordinate distribution inside the support to its largest coordinate.\n\\end{theorem}",
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    "conditions": [
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      "finite coordinate set",
      "finite nonempty support",
      "named nonempty finite support",
      "nonzero common magnitude for equality witness",
      "nonzero diagonal amplitude",
      "positive supremum cost for division-form ratio bounds",
      "real exponent p > 1",
      "real exponent p >= 1",
      "real-valued coordinates",
      "support size at least two for logarithmic dimension",
      "two-coordinate special case"
    ],
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    "kernel_certified": true,
    "notes": [
      "Complete finite-support kernel: lower and cardinality upper bounds, finite p>1 and Linfinity equality characterizations, and the elementary diagonal values."
    ],
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      "MAP-BOOK5-021"
    ],
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      "exact"
    ],
    "witnesses": [
      "Book5.axisCostOn_le_card_rpow_mul_lpCostOn",
      "Book5.diagonal_l1_l2_ratio",
      "Book5.infinity_ratio_bounds",
      "Book5.infinity_sharp_iff_equal_magnitudes",
      "Book5.lpCostOn_le_axisCostOn",
      "Book5.sharp_norm_bound_iff_equal_magnitudes"
    ]
  },
  "line": 2301,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "\\(\\ell_p\\)-Norm Fracture Hierarchy",
  "proof_labels": [
    "proof:bk5_lp_norm_fracture_hierarchy"
  ],
  "proof_status": "proven",
  "refs": [],
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{
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  "cites": [],
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  "file": "book5.tex",
  "id": "proof:bk5_lp_norm_fracture_hierarchy",
  "label": "proof:bk5_lp_norm_fracture_hierarchy",
  "latex_body": "\\begin{proof}[Norm inequality proof]\n\\label{proof:bk5_lp_norm_fracture_hierarchy}\n\\leavevmode\n\nThe lower bound follows from the standard monotonicity\n$\\|\\vec{v}\\|_p\\leq\\|\\vec{v}\\|_1$ for $p\\geq1$.  For the upper bound, restrict\nto the support of $\\vec{v}$ and apply Hölder's inequality:\n\\[\n\\|\\vec{v}\\|_1\n=\\sum_{i\\in\\operatorname{supp}(\\vec{v})}|v_i|\n\\leq s(\\vec{v})^{1-1/p}\\left(\\sum_i |v_i|^p\\right)^{1/p}\n=s(\\vec{v})^{1-1/p}\\|\\vec{v}\\|_p.\n\\]\nDividing by $\\|\\vec{v}\\|_p$ gives the claimed bound.  Equality in Hölder\noccurs exactly when the nonzero magnitudes $|v_i|$ are all equal.  Substituting\n$\\vec{v}=e_i+e_j$ gives $\\|\\vec{v}\\|_1=2$, $\\|\\vec{v}\\|_2=\\sqrt{2}$, and\n$\\|\\vec{v}\\|_\\infty=1$, hence the three displayed ratios.\n\\end{proof}",
  "line": 2322,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Norm inequality proof",
  "proves": "theorem:bk5_lp_norm_fracture_hierarchy",
  "refs": [],
  "role": "proof",
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}

{
  "book": "book5",
  "cited_by": [
    "proof:bk5_complete_symbolic_regime_classification",
    "theorem:bk5_fundamental_norm_fracture"
  ],
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  "id": "proposition:bk5_symbolic_integrability_classes",
  "label": "proposition:bk5_symbolic_integrability_classes",
  "latex_body": "\\begin{proposition}[Symbolic Integrability Classes]\n\\label{proposition:bk5_symbolic_integrability_classes}\nFuzzy symbolic manifolds can be rigorously classified into three fundamental integrability classes (cf.~Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}):\n\n- \\textbf{Class $\\mathcal{I}_1$}: \\textbf{Symbolically Reducible} - geometric and axis-symbolic lengths agree.\n- \\textbf{Class $\\mathcal{I}_2$}: \\textbf{Symbolically Fractured} - elementary orthogonal diagonals carry ratio $\\sqrt{2}$.\n- \\textbf{Class $\\mathcal{I}_\\infty$}: \\textbf{Support-Collapsed} - length depends only on the largest coordinate, so distribution inside the support is lost.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "balanced two-step memory characteristic equation",
      "finite support represented by Fin n",
      "finite-support norm kernel from LPS-P20 through LPS-P24",
      "nonzero coordinate magnitude and positive support for the ratio",
      "positive finite real p",
      "support size at least two for logarithmic effective dimension",
      "transition diagnostics are local classifications, not agent identities"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Typed transition diagnostics certify axis integrability, Euclidean fracture, and support collapse, including a concrete loss-of-distribution witness."
    ],
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      "MAP-BOOK5-037"
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    "statuses": [
      "exact"
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    "witnesses": [
      "Book5.elementary_spectrum_values",
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      "Book5.symbolicTorsion_two_two"
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  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Integrability Classes",
  "proof_labels": [
    "proof:bk5_symbolic_integrability_classes"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "y_classes} Fuzzy symbolic manifolds can be rigorously classified into three fundamental integrability classes (cf.~Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}): - \\textbf{Class $\\mathcal{I}_1$}: \\textbf{Symbolically Reducible} - geometric and axis-symbolic lengths agree. - \\te",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
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  "refs": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
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}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_integrability_classes",
  "label": "proof:bk5_symbolic_integrability_classes",
  "latex_body": "\\begin{proof}[Classification Proof]\n\\label{proof:bk5_symbolic_integrability_classes}\n\\leavevmode\n\nIn $\\mathcal{I}_1$, $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_1=1$, so\n$\\kappa_1(\\vec{v})=0$ for every nonzero transition.  In $\\mathcal{I}_2$,\nThm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy} gives the elementary\ndiagonal ratio $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_2=\\sqrt{2}$, so\n$\\kappa_2(e_i+e_j)=\\sqrt{2}-1>0$.  In $\\mathcal{I}_\\infty$,\n$\\|\\vec{v}\\|_\\infty=\\max_i |v_i|$; therefore transitions with different\ncoordinate distributions can share the same geometric length.  The class is\nsupport-collapsed because only the largest coordinate survives in the norm,\nwhile the axis-symbolic cost remains sensitive to the whole support.\n\\end{proof}",
  "line": 2350,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Classification Proof",
  "proves": "proposition:bk5_symbolic_integrability_classes",
  "ref_roles": [
    {
      "context": "}_1$, $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_1=1$, so $\\kappa_1(\\vec{v})=0$ for every nonzero transition. In $\\mathcal{I}_2$, Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy} gives the elementary diagonal ratio $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_2=\\sqrt{2}$, so $\\kappa_2(e_i+e_j)=\\sqrt{2}-1>0$. In $\\",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2301,
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    }
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    "theorem:bk5_lp_norm_fracture_hierarchy"
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  "id": "theorem:bk5_symbolic_torsion_phase_diagram",
  "label": "theorem:bk5_symbolic_torsion_phase_diagram",
  "latex_body": "\\begin{theorem}[Symbolic Torsion Phase Diagram]\n\\label{theorem:bk5_symbolic_torsion_phase_diagram}\nFor an equal-magnitude transition with support size $s\\geq1$, the symbolic\nfracture parameter is\n\\[\n\\kappa_p(s)=s^{1-1/p}-1,\\qquad 1\\leq p\\leq\\infty.\n\\]\nThus\n\\[\n\\kappa_1(s)=0,\\qquad\n\\kappa_2(2)=\\sqrt{2}-1,\\qquad\n\\kappa_\\infty(s)=s-1.\n\\]\nFor fixed finite support, fracture is finite and monotone in $p$; divergence\noccurs only along unbounded support size $s\\to\\infty$.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "finite support represented by Fin n",
      "named nonempty finite support",
      "nonzero common magnitude for equality witness",
      "nonzero coordinate magnitude and positive support for the ratio",
      "positive finite real p",
      "positive supremum cost for division-form ratio bounds",
      "support size at least two for logarithmic dimension",
      "support size at least two for logarithmic effective dimension"
    ],
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    "notes": [
      "Exact finite-positive-p and Linfinity formulas, special values, monotonicity in positive finite p, and divergence of Linfinity torsion along unbounded support."
    ],
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      "MAP-BOOK5-039"
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      "exact"
    ],
    "witnesses": [
      "Book5.equalMagnitude_ratio",
      "Book5.symbolicTorsionInfinity_eq_ratio_sub_one",
      "Book5.symbolicTorsionInfinity_tendsto_atTop",
      "Book5.symbolicTorsion_eq_equalMagnitude_ratio_sub_one",
      "Book5.symbolicTorsion_mono_exponent",
      "Book5.symbolicTorsion_one",
      "Book5.symbolicTorsion_two_two"
    ]
  },
  "line": 2366,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Torsion Phase Diagram",
  "proof_labels": [
    "proof:bk5_symbolic_torsion_phase_diagram"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_torsion_phase_diagram",
  "label": "proof:bk5_symbolic_torsion_phase_diagram",
  "latex_body": "\\begin{proof}[Phase diagram from the fracture ratio]\n\\label{proof:bk5_symbolic_torsion_phase_diagram}\n\\leavevmode\n\nFor an equal-magnitude transition with support size $s$, equality holds in\nThm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, so\n$\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p=s^{1-1/p}$.  Subtracting $1$ gives\n$\\kappa_p(s)=s^{1-1/p}-1$.  The displayed special cases follow by substituting\n$p=1$, $(p,s)=(2,2)$, and $p=\\infty$.  Since $1-1/p$ is monotone increasing\nin $p$, $\\kappa_p(s)$ is monotone in $p$ for fixed $s$; since\n$\\kappa_\\infty(s)=s-1$, unbounded growth requires $s\\to\\infty$.\n\\end{proof}",
  "line": 2383,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Phase diagram from the fracture ratio",
  "proves": "theorem:bk5_symbolic_torsion_phase_diagram",
  "ref_roles": [
    {
      "context": "lic_torsion_phase_diagram} \\leavevmode For an equal-magnitude transition with support size $s$, equality holds in Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, so $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p=s^{1-1/p}$. Subtracting $1$ gives $\\kappa_p(s)=s^{1-1/p}-1$. The displayed special c",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "proof",
  "type": "proof"
}

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  "book": "book5",
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  "cites": [
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  "file": "book5.tex",
  "id": "scholium:bk5_critical_point_p2",
  "label": "scholium:bk5_critical_point_p2",
  "latex_body": "\\begin{scholium}[Critical Point at p=2]\n\\label{scholium:bk5_critical_point_p2}\nThe Euclidean norm $p = 2$ is the first familiar geometric regime in which an\nelementary orthogonal diagonal has nonzero axis-symbolic fracture\n(cf.~Thm.~\\ref{theorem:bk5_symbolic_torsion_phase_diagram}). This is not\ncoincidental: it reflects the fundamental role of orthogonality in geometric\nrepresentation. The emergence of $\\sqrt{2}$ at this point marks the first\nunit-square diagonal representability ratio.\n\\end{scholium}",
  "line": 2396,
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  "matter_region": "mainmatter",
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  "name": "Critical Point at p=2",
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    {
      "context": "first familiar geometric regime in which an elementary orthogonal diagonal has nonzero axis-symbolic fracture (cf.~Thm.~\\ref{theorem:bk5_symbolic_torsion_phase_diagram}). This is not coincidental: it reflects the fundamental role of orthogonality in geometric representation. The emergenc",
      "label": "theorem:bk5_symbolic_torsion_phase_diagram",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2366,
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    "theorem:bk5_symbolic_torsion_phase_diagram"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book5",
  "cited_by": [
    "proof:bk5_shortest_path_representability"
  ],
  "cites": [
    "definition:bk5_diagonal_transition",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "depends_on": [
    "definition:bk5_diagonal_transition",
    "theorem:bk5_lp_norm_fracture_hierarchy",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "file": "book5.tex",
  "id": "lemma:bk5_shortest_path_representability",
  "label": "lemma:bk5_shortest_path_representability",
  "latex_body": "\\begin{lemma}[Shortest Path Representability Criterion]\n\\label{lemma:bk5_shortest_path_representability}\nSymbolic fracture arises precisely when the geometrically shortest transition\nhas smaller $\\ell_p$ length than every axis-generated symbolic path realizing\nthe same endpoint (cf.~Def.~\\ref{definition:bk5_diagonal_transition},\nThm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty geometric and symbolic candidate sets",
      "positive integer n for strict diagonal gap",
      "shared real-valued path cost",
      "symbolic candidates included in geometric candidates"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite admissible-path kernel with symbolic candidates explicitly included in geometric candidates; proves the exact minimum and representability criteria."
    ],
    "record_ids": [
      "MAP-BOOK5-041"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5.diagonal_symbolic_longer",
      "Book5.shortestGeometric_le_shortestSymbolic",
      "Book5.symbolicDecoherence_eq_zero_iff_has_symbolic_geodesic",
      "Book5.symbolicDecoherence_pos_iff"
    ]
  },
  "line": 2406,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Shortest Path Representability Criterion",
  "proof_labels": [
    "proof:bk5_shortest_path_representability"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "st transition has smaller $\\ell_p$ length than every axis-generated symbolic path realizing the same endpoint (cf.~Def.~\\ref{definition:bk5_diagonal_transition}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}). \\end{lemma}",
      "label": "definition:bk5_diagonal_transition",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2147,
      "target_type": "definition"
    },
    {
      "context": "every axis-generated symbolic path realizing the same endpoint (cf.~Def.~\\ref{definition:bk5_diagonal_transition}, Thm.~\\ref{theorem:bk5_sqrt2_maximal_fracture}). \\end{lemma}",
      "label": "theorem:bk5_sqrt2_maximal_fracture",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2152,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_diagonal_transition",
    "theorem:bk5_sqrt2_maximal_fracture"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book5",
  "cited_by": [
    "corollary:bk5_symbolic_decoherence_theory",
    "proof:bk5_symbolic_decoherence_theory"
  ],
  "cites": [
    "lemma:bk5_shortest_path_representability",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "lemma:bk5_shortest_path_representability",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_shortest_path_representability",
  "label": "proof:bk5_shortest_path_representability",
  "latex_body": "\\begin{proof}[Representability Proof]\n\\label{proof:bk5_shortest_path_representability}\n\\leavevmode\n\nConsider transition $(0,0) \\to (n,n)$ for integer $n$.\nSee Lem.~\\ref{lemma:bk5_shortest_path_representability} and Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}.\n\n\\textbf{Axis-aligned path}: $(0,0) \\to (n,0) \\to (n,n)$\n\\begin{itemize}\n  \\item Length: $\\ell_1 = 2n$\n  \\item Symbolic representation: $n \\cdot (1,0) + n \\cdot (0,1)$ \\quad \\checkmark\\ Representable\n\\end{itemize}\n\n\\textbf{Diagonal path}: $(0,0) \\to (n,n)$ directly\n\\begin{itemize}\n  \\item Length: $\\ell_2 = n\\sqrt{2}$\n  \\item Symbolic representation: $n \\cdot \\left(\\frac{1}{\\sqrt{2}}, \\frac{1}{\\sqrt{2}}\\right)$ \\quad $\\times$ Non-representable\n\\end{itemize}\n\nThe \\textbf{representability gap} is:\n\\[\n\\Delta = \\ell_1 - \\ell_2 = 2n - n\\sqrt{2} = n(2 - \\sqrt{2}) \\approx 0.586n\n\\]\n\nThis gap quantifies the \\textbf{symbolic decoherence} — the cost of forcing geometric optimality into symbolic constraints.\n\\end{proof}",
  "line": 2414,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Representability Proof",
  "proves": "lemma:bk5_shortest_path_representability",
  "ref_roles": [
    {
      "context": "{proof:bk5_shortest_path_representability} \\leavevmode Consider transition $(0,0) \\to (n,n)$ for integer $n$. See Lem.~\\ref{lemma:bk5_shortest_path_representability} and Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}. \\textbf{Axis-aligned path}: $(0,0) \\to (n,0) \\to (n,n)$ \\begin{",
      "label": "lemma:bk5_shortest_path_representability",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2406,
      "target_type": "lemma"
    },
    {
      "context": "Consider transition $(0,0) \\to (n,n)$ for integer $n$. See Lem.~\\ref{lemma:bk5_shortest_path_representability} and Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}. \\textbf{Axis-aligned path}: $(0,0) \\to (n,0) \\to (n,n)$ \\begin{itemize} \\item Length: $\\ell_1 = 2n$ \\item Symboli",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "lemma:bk5_shortest_path_representability",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "theorem:bk5_fundamental_norm_fracture"
  ],
  "cites": [
    "proof:bk5_shortest_path_representability"
  ],
  "depends_on": [
    "proof:bk5_shortest_path_representability"
  ],
  "file": "book5.tex",
  "id": "corollary:bk5_symbolic_decoherence_theory",
  "label": "corollary:bk5_symbolic_decoherence_theory",
  "latex_body": "\\begin{corollary}[Symbolic Decoherence Theory]\n\\label{corollary:bk5_symbolic_decoherence_theory}\nThe \\textbf{symbolic decoherence} $\\drift$ of a transition is the excess\naxis-symbolic cost over geometric optimality (cf.~Prf.~\\ref{proof:bk5_shortest_path_representability}):\n$$\\drift(\\vec{v}) = \\|\\vec{v}\\|_{\\text{symbolic}} - \\|\\vec{v}\\|_{\\text{geometric}}$$\nwhere $\\|\\vec{v}\\|_{\\text{symbolic}}$ is the length of the shortest symbolically representable path.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "finite nonempty geometric and symbolic candidate sets",
      "positive integer n for strict diagonal gap",
      "shared real-valued path cost",
      "symbolic candidates included in geometric candidates"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "For finite path families, decoherence is the nonnegative excess symbolic minimum, vanishing exactly when a symbolic geodesic exists; the diagonal gap is exact."
    ],
    "record_ids": [
      "MAP-BOOK5-042"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5.diagonalDecoherence_formula",
      "Book5.diagonalDecoherence_pos",
      "Book5.symbolicDecoherence_eq_zero_iff",
      "Book5.symbolicDecoherence_eq_zero_iff_has_symbolic_geodesic",
      "Book5.symbolicDecoherence_nonneg"
    ]
  },
  "line": 2441,
  "macros_used": [
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Decoherence Theory",
  "proof_labels": [
    "proof:bk5_symbolic_decoherence_theory"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "tbf{symbolic decoherence} $\\drift$ of a transition is the excess axis-symbolic cost over geometric optimality (cf.~Prf.~\\ref{proof:bk5_shortest_path_representability}): $$\\drift(\\vec{v}) = \\|\\vec{v}\\|_{\\text{symbolic}} - \\|\\vec{v}\\|_{\\text{geometric}}$$ where $\\|\\vec{v}\\|_{\\text{symbol",
      "label": "proof:bk5_shortest_path_representability",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2414,
      "target_type": "proof"
    }
  ],
  "refs": [
    "proof:bk5_shortest_path_representability"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "proof:bk5_shortest_path_representability"
  ],
  "depends_on": [
    "proof:bk5_shortest_path_representability"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_decoherence_theory",
  "label": "proof:bk5_symbolic_decoherence_theory",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_symbolic_decoherence_theory}\n\\leavevmode\nThe symbolic decoherence is well-defined and nonnegative. By the shortest-path representability analysis (Prf.~\\ref{proof:bk5_shortest_path_representability}) the symbolically representable paths between two configurations are a proper subset of all geometric paths: a symbolic path is constrained to axis-representable steps, whereas the geometric optimum (the Euclidean geodesic) need not be representable. Minimizing a length functional over a smaller admissible set cannot yield a shorter optimum, so $\\|\\vec v\\|_{\\text{symbolic}}\\ge\\|\\vec v\\|_{\\text{geometric}}$, and therefore\n\\[\n\\drift(\\vec v)=\\|\\vec v\\|_{\\text{symbolic}}-\\|\\vec v\\|_{\\text{geometric}}\\ge 0\n\\]\nis a well-defined, nonnegative excess. It vanishes exactly when the geometric optimum is itself symbolically representable, and is otherwise strictly positive --- the representability gap computed there, e.g.\\ $\\Delta=n(2-\\sqrt2)$ for the diagonal transition. Thus $\\drift$ measures precisely the cost of forcing geometric optimality through symbolic constraints.\n\\end{proof}",
  "line": 2448,
  "macros_used": [
    "drift"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "corollary:bk5_symbolic_decoherence_theory",
  "ref_roles": [
    {
      "context": "avevmode The symbolic decoherence is well-defined and nonnegative. By the shortest-path representability analysis (Prf.~\\ref{proof:bk5_shortest_path_representability}) the symbolically representable paths between two configurations are a proper subset of all geometric paths: a symbolic",
      "label": "proof:bk5_shortest_path_representability",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2414,
      "target_type": "proof"
    }
  ],
  "refs": [
    "proof:bk5_shortest_path_representability"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "remark:bk5_lattice_field_theory_analogy",
  "label": "remark:bk5_lattice_field_theory_analogy",
  "latex_body": "\\begin{remark}[Lattice Field Theory Analogy]\n\\label{remark:bk5_lattice_field_theory_analogy}\nThe transition from $\\ell_1$ to $\\ell_2$ geometry mirrors the \\textbf{continuum limit} in lattice field theory (cf.~Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}):\n- \\textbf{Discrete lattice} ($\\ell_1$): perfect symbolic integrability, but geometric distortion.\n- \\textbf{Continuum limit} ($\\ell_2$): geometric accuracy, but elementary diagonal fracture.\n- \\textbf{Renormalization}: the ratio $\\sqrt{2}$ measures the first unit-square cost of replacing axis traversal by Euclidean diagonal traversal.\n\\end{remark}",
  "line": 2458,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Lattice Field Theory Analogy",
  "ref_roles": [
    {
      "context": "he transition from $\\ell_1$ to $\\ell_2$ geometry mirrors the \\textbf{continuum limit} in lattice field theory (cf.~Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}): - \\textbf{Discrete lattice} ($\\ell_1$): perfect symbolic integrability, but geometric distortion. - \\textbf{Continuum",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proposition:bk5_fractal_dimension_connection",
  "label": "proposition:bk5_fractal_dimension_connection",
  "latex_body": "\\begin{proposition}[Effective Support-Dimension Diagnostic]\n\\label{proposition:bk5_fractal_dimension_connection}\nFor an equal-magnitude transition $\\vec{v}$ with support size\n$s=s(\\vec{v})\\geq2$, define its effective support dimension at norm exponent\n$p$ by\n\\[\nD_{\\mathrm{eff}}(p,s)\n=1+\\frac{\\log\\left(\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p\\right)}{\\log s}.\n\\]\nThen\n\\[\nD_{\\mathrm{eff}}(p,s)=2-\\frac{1}{p},\\qquad\nD_{\\mathrm{eff}}(1,s)=1,\\qquad\nD_{\\mathrm{eff}}(2,s)=\\frac32,\\qquad\nD_{\\mathrm{eff}}(\\infty,s)=2.\n\\]\nFor support size $s=1$, the transition is axis-integrable and the effective\ndimension is defined to be $D_{\\mathrm{eff}}=1$.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "finite support represented by Fin n",
      "named nonempty finite support",
      "nonzero common magnitude for equality witness",
      "nonzero coordinate magnitude and positive support for the ratio",
      "positive finite real p",
      "positive supremum cost for division-form ratio bounds",
      "support size at least two for logarithmic dimension",
      "support size at least two for logarithmic effective dimension"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The logarithmic diagnostic is 2-1/p for finite p on support at least two, with the support-one convention and p=1,2,infinity values certified."
    ],
    "record_ids": [
      "MAP-BOOK5-040"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5.effectiveSupportDimensionInfinity_eq_two",
      "Book5.effectiveSupportDimensionInfinity_log_ratio",
      "Book5.effectiveSupportDimension_eq",
      "Book5.effectiveSupportDimension_one",
      "Book5.effectiveSupportDimension_two"
    ]
  },
  "line": 2466,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Effective Support-Dimension Diagnostic",
  "proof_labels": [
    "proof:bk5_fractal_dimension_connection"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_fractal_dimension_connection",
  "label": "proof:bk5_fractal_dimension_connection",
  "latex_body": "\\begin{proof}[Effective support dimension]\n\\label{proof:bk5_fractal_dimension_connection}\n\\leavevmode\n\nFor equal-magnitude support size $s$, equality holds in\nThm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, so\n\\[\n\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p}=s^{1-1/p}.\n\\]\nSubstitution into the definition gives\n\\[\nD_{\\mathrm{eff}}(p,s)\n=1+\\frac{\\log(s^{1-1/p})}{\\log s}\n=1+\\left(1-\\frac1p\\right)\n=2-\\frac1p.\n\\]\nThe displayed special cases follow by evaluating at $p=1$, $p=2$, and\n$p=\\infty$.  When $s=1$, $\\log s=0$, so the formula is not used; the transition\nis a single-axis transition with no support expansion, and the diagnostic is\ndefined as $1$.\n\\end{proof}",
  "line": 2486,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Effective support dimension",
  "proves": "proposition:bk5_fractal_dimension_connection",
  "ref_roles": [
    {
      "context": "label{proof:bk5_fractal_dimension_connection} \\leavevmode For equal-magnitude support size $s$, equality holds in Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, so \\[ \\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p}=s^{1-1/p}. \\] Substitution into the definition gives \\[ D_{\\mathrm{eff}}(p,s",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk5_symbolic_regime_detection"
  ],
  "cites": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_compression_experiment",
  "label": "definition:bk5_symbolic_compression_experiment",
  "latex_body": "\\begin{definition}[Symbolic Compression Experiment]\n\\label{definition:bk5_symbolic_compression_experiment}\nTo test the $\\ell_p$-fracture theory, design an experiment measuring \\textbf{symbolic compression ratio} (cf.~Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}):\n$$R_p = \\frac{\\text{Length of symbolic encoding}}{\\text{Length of geometric path}}$$\nfor various $p$ values. The theory predicts:\n- $R_1 = 1$ (perfect compression)\n- $R_2 = \\sqrt{2}$ (minimal fracture)\n- $R_\\infty = s(\\vec{v})$ for equal-magnitude transitions with support size $s(\\vec{v})$\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-BOOK5-092"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5Residue.compression_ratio_R1",
      "Book5Residue.compression_ratio_R2",
      "Book5Residue.l1_eq_card_mul_c"
    ]
  },
  "line": 2508,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Compression Experiment",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ent} To test the $\\ell_p$-fracture theory, design an experiment measuring \\textbf{symbolic compression ratio} (cf.~Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}): $$R_p = \\frac{\\text{Length of symbolic encoding}}{\\text{Length of geometric path}}$$ for various $p$ values. The theo",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk6_symbolic_bifurcation",
    "demonstratio:bk5_geometry_symbol_unity",
    "remark:bk5_open_questions",
    "theorem:bk5_grand_unified_symbolic_geometric"
  ],
  "cites": [
    "corollary:bk5_symbolic_decoherence_theory",
    "proposition:bk5_symbolic_integrability_classes",
    "theorem:bk5_fundamental_dichotomy",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "corollary:bk5_symbolic_decoherence_theory",
    "proposition:bk5_symbolic_integrability_classes",
    "theorem:bk5_fundamental_dichotomy",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_fundamental_norm_fracture",
  "label": "theorem:bk5_fundamental_norm_fracture",
  "latex_body": "\\begin{theorem}[Fundamental Theorem of Norm-Induced Symbolic Fracture]\n\\label{theorem:bk5_fundamental_norm_fracture}\nIn any fuzzy symbolic manifold $\\tilde{M}$ with axis-generated symbolic paths,\nnorm-induced symbolic fracture is governed by the ratio\n$\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p$ (cf.~Prop.~\\ref{proposition:bk5_symbolic_integrability_classes},\nCor.~\\ref{corollary:bk5_symbolic_decoherence_theory},\nThm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy},\nThm.~\\ref{theorem:bk5_fundamental_dichotomy}):\n\n1. \\textbf{Symbolic integrability} is exact under $\\ell_1$ geometry.\n2. \\textbf{Symbolic fracture} emerges under $\\ell_2$ geometry for every transition with support size at least $2$.\n3. \\textbf{Support collapse} appears under $\\ell_\\infty$ geometry, where length remembers only the largest coordinate.\n\nThe constant $\\sqrt{2}$ is the elementary two-coordinate fracture ratio, while\n$\\varphi$ remains the balanced recursive-memory resonance ratio from\nThm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "balanced two-step memory characteristic equation",
      "finite nonempty geometric and symbolic candidate sets",
      "finite nonempty support",
      "finite-support norm kernel from LPS-P20 through LPS-P24",
      "named nonempty finite support",
      "positive integer n for strict diagonal gap",
      "real exponent p > 1",
      "real-valued coordinates",
      "shared real-valued path cost",
      "symbolic candidates included in geometric candidates",
      "transition diagnostics are local classifications, not agent identities"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Finite-support typed kernel assembling all norm regimes, equality cases, support collapse, diagonal decoherence, and the distinct memory resonance channel."
    ],
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      "MAP-BOOK5-038"
    ],
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      "exact"
    ],
    "witnesses": [
      "Book5.diagonalDecoherence_pos",
      "Book5.fundamental_norm_fracture_kernel",
      "Book5.infinity_sharp_iff_equal_magnitudes",
      "Book5.sharp_norm_bound_iff_equal_magnitudes"
    ]
  },
  "line": 2518,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Fundamental Theorem of Norm-Induced Symbolic Fracture",
  "proof_labels": [
    "proof:bk5_fundamental_norm_fracture"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "verned by the ratio $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p$ (cf.~Prop.~\\ref{proposition:bk5_symbolic_integrability_classes}, Cor.~\\ref{corollary:bk5_symbolic_decoherence_theory}, Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, Thm.~\\ref{theorem:bk5_fundamental_dichotomy}): 1. \\textbf{Symbolic",
      "label": "corollary:bk5_symbolic_decoherence_theory",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2441,
      "target_type": "corollary"
    },
    {
      "context": "erated symbolic paths, norm-induced symbolic fracture is governed by the ratio $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p$ (cf.~Prop.~\\ref{proposition:bk5_symbolic_integrability_classes}, Cor.~\\ref{corollary:bk5_symbolic_decoherence_theory}, Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, Thm.~\\ref{the",
      "label": "proposition:bk5_symbolic_integrability_classes",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2341,
      "target_type": "proposition"
    },
    {
      "context": "classes}, Cor.~\\ref{corollary:bk5_symbolic_decoherence_theory}, Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, Thm.~\\ref{theorem:bk5_fundamental_dichotomy}): 1. \\textbf{Symbolic integrability} is exact under $\\ell_1$ geometry. 2. \\textbf{Symbolic fracture} emerges under $\\e",
      "label": "theorem:bk5_fundamental_dichotomy",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 2256,
      "target_type": "theorem"
    },
    {
      "context": "ementary two-coordinate fracture ratio, while $\\varphi$ remains the balanced recursive-memory resonance ratio from Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}. \\end{theorem}",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "~Prop.~\\ref{proposition:bk5_symbolic_integrability_classes}, Cor.~\\ref{corollary:bk5_symbolic_decoherence_theory}, Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}, Thm.~\\ref{theorem:bk5_fundamental_dichotomy}): 1. \\textbf{Symbolic integrability} is exact under $\\ell_1$ geometry. 2",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk5_symbolic_decoherence_theory",
    "proposition:bk5_symbolic_integrability_classes",
    "theorem:bk5_fundamental_dichotomy",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
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    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_fundamental_norm_fracture",
  "label": "proof:bk5_fundamental_norm_fracture",
  "latex_body": "\\begin{proof}[Norm-induced fracture theorem]\n\\label{proof:bk5_fundamental_norm_fracture}\n\\leavevmode\n\nTheorem~\\ref{theorem:bk5_lp_norm_fracture_hierarchy} proves that the ratio\n$\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p$ is the controlling quantity.  When $p=1$, this\nratio is identically $1$, so there is no fracture.  When $p=2$ and\n$s(\\vec{v})\\geq2$, the upper-bound case for an elementary diagonal gives the\nfirst nontrivial ratio $\\sqrt{2}$.  When $p=\\infty$, the denominator is\n$\\max_i |v_i|$, so all coordinate information below the maximum is invisible to\nthe geometric length; this is support collapse.  The final sentence follows by\ncombining the elementary fracture result with the balanced-memory spectral\nresult of Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}.\n\\end{proof}",
  "line": 2536,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Norm-induced fracture theorem",
  "proves": "theorem:bk5_fundamental_norm_fracture",
  "ref_roles": [
    {
      "context": "The final sentence follows by combining the elementary fracture result with the balanced-memory spectral result of 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,
      "target_type": "theorem"
    },
    {
      "context": "\\begin{proof}[Norm-induced fracture theorem] \\label{proof:bk5_fundamental_norm_fracture} \\leavevmode Theorem~\\ref{theorem:bk5_lp_norm_fracture_hierarchy} proves that the ratio $\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p$ is the controlling quantity. When $p=1$, this ratio is identically",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
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    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "proof",
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}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_fundamental_norm_fracture"
  ],
  "depends_on": [
    "theorem:bk5_fundamental_norm_fracture"
  ],
  "file": "book5.tex",
  "id": "demonstratio:bk5_geometry_symbol_unity",
  "label": "demonstratio:bk5_geometry_symbol_unity",
  "latex_body": "\\begin{demonstratio}[The Deep Unity of Geometry and Symbol]\n\\label{demonstratio:bk5_geometry_symbol_unity}\nThis analysis reveals that the \\textbf{crisis of symbolic representation} is not merely a computational issue, but reflects a fundamental tension between \\textbf{discrete symbolic logic} and \\textbf{continuous geometric reality} (cf.~Thm.~\\ref{theorem:bk5_fundamental_norm_fracture}). \n\nThe parameter $p$ in $\\ell_p$ norms controls the \\textbf{degree of geometric realism} the symbolic system attempts to capture:\n- Low $p$: Symbolic purity, geometric distortion\n- High $p$: Geometric accuracy, symbolic chaos\n\nThe Euclidean point $p = 2$ is where the elementary unit-square diagonal first\nregisters as $\\sqrt{2}$ against the axis-symbolic length $2$.  This is why\n$\\sqrt{2}$ emerges as the \\textbf{minimal fracture constant}: it marks the first\nnon-axis-aligned cost of geometric realism in symbolic systems.\n\\end{demonstratio}",
  "line": 2551,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Deep Unity of Geometry and Symbol",
  "ref_roles": [
    {
      "context": "ects a fundamental tension between \\textbf{discrete symbolic logic} and \\textbf{continuous geometric reality} (cf.~Thm.~\\ref{theorem:bk5_fundamental_norm_fracture}). The parameter $p$ in $\\ell_p$ norms controls the \\textbf{degree of geometric realism} the symbolic system attempts",
      "label": "theorem:bk5_fundamental_norm_fracture",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2518,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_fundamental_norm_fracture"
  ],
  "role": "demonstration",
  "type": "demonstratio"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_fundamental_norm_fracture"
  ],
  "depends_on": [
    "theorem:bk5_fundamental_norm_fracture"
  ],
  "file": "book5.tex",
  "id": "remark:bk5_open_questions",
  "label": "remark:bk5_open_questions",
  "latex_body": "\\begin{remark}[Open Questions]\n\\label{remark:bk5_open_questions}\nThis framework raises several open questions\n(cf.~Thm.~\\ref{theorem:bk5_fundamental_norm_fracture}):\n\n1. \\textbf{Quantum Geometric Encoding}: Do quantum systems naturally operate in specific $\\ell_p$ regimes? Could quantum coherence correspond to symbolic integrability classes?\n\n2. \\textbf{Information Theoretic Bounds}: Can we establish fundamental limits on \\textbf{symbolic compression} based on the underlying geometric structure?\n\n3. \\textbf{Cognitive Symbolic Processing}: Do biological cognitive systems exhibit $\\ell_p$-dependent symbolic processing regimes? Is there a \\textbf{natural norm} for symbolic cognition?\n\n4. \\textbf{Computational Complexity}: How does the computational complexity of symbolic operations scale with the $\\ell_p$ parameter? Is there a \\textbf{complexity phase transition} at $p = 2$?\n\\end{remark}",
  "line": 2565,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Open Questions",
  "ref_roles": [
    {
      "context": "\\begin{remark}[Open Questions] \\label{remark:bk5_open_questions} This framework raises several open questions (cf.~Thm.~\\ref{theorem:bk5_fundamental_norm_fracture}): 1. \\textbf{Quantum Geometric Encoding}: Do quantum systems naturally operate in specific $\\ell_p$ regimes? Could qua",
      "label": "theorem:bk5_fundamental_norm_fracture",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2518,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_fundamental_norm_fracture"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "book5",
  "cited_by": [
    "definition:bk5_symbolic_curvature_operator_spectrum",
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "cites": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "theorem:bk5_symbolic_norm_spectrum",
  "label": "theorem:bk5_symbolic_norm_spectrum",
  "latex_body": "\\begin{theorem}[Symbolic Norm Spectrum]\n\\label{theorem:bk5_symbolic_norm_spectrum}\nThe symbolic behavior of fuzzy manifolds separates into two coupled spectra:\n\\[\n\\mathcal{S}_{\\mathrm{norm}}(p,\\vec{v})\n=\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p},\n\\qquad\n\\mathcal{S}_{\\mathrm{mem}}\n=\\rho\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}=\\varphi.\n\\]\nThe norm spectrum is governed by Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}:\n$\\mathcal{S}_{\\mathrm{norm}}=1$ at $p=1$,\n$\\mathcal{S}_{\\mathrm{norm}}=\\sqrt{2}$ for the elementary Euclidean diagonal,\nand $\\mathcal{S}_{\\mathrm{norm}}=s(\\vec{v})$ at $p=\\infty$ for equal-magnitude\nsupport size $s(\\vec{v})$.  The memory spectrum is governed by\nThm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}: $\\varphi$ is the Perron\nratio of balanced two-step symbolic memory.  Thus $\\varphi$ is not a special\n$\\ell_p$ norm exponent; it is the resonance eigenvalue of the recursive memory\nchannel coupled to the geometric fracture channel.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "balanced two-step memory characteristic equation",
      "finite-support norm kernel from LPS-P20 through LPS-P24",
      "transition diagnostics are local classifications, not agent identities"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The norm and memory spectra are explicit independent product coordinates; sqrt(2) and phi coexist without identification."
    ],
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      "MAP-BOOK5-024"
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      "exact"
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      "Book5.balancedSpectrum_memory",
      "Book5.elementary_spectrum_values",
      "Book5.euclidean_balanced_product_spectrum",
      "Book5.norm_and_memory_coordinates_independent"
    ]
  },
  "line": 2579,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Norm Spectrum",
  "proof_labels": [
    "proof:bk5_symbolic_norm_spectrum"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "orm}}=s(\\vec{v})$ at $p=\\infty$ for equal-magnitude support size $s(\\vec{v})$. The memory spectrum is governed by Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}: $\\varphi$ is the Perron ratio of balanced two-step symbolic memory. Thus $\\varphi$ is not a special $\\ell_p$ norm exp",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "\\mathcal{S}_{\\mathrm{mem}} =\\rho\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}=\\varphi. \\] The norm spectrum is governed by Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}: $\\mathcal{S}_{\\mathrm{norm}}=1$ at $p=1$, $\\mathcal{S}_{\\mathrm{norm}}=\\sqrt{2}$ for the elementary Euclidean diagonal",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
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    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_norm_spectrum",
  "label": "proof:bk5_symbolic_norm_spectrum",
  "latex_body": "\\begin{proof}[Product spectrum]\n\\label{proof:bk5_symbolic_norm_spectrum}\n\\leavevmode\n\nThe norm component is exactly the ratio proved in\nThm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}.  Its special values follow\nby substituting $p=1$, $(p,\\vec{v})=(2,e_i+e_j)$, and $p=\\infty$ for\nequal-magnitude support.  The memory component is exactly the spectral radius\ncomputed in Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}.  Since the\nfirst quantity is a geometric representability ratio and the second is a\nrecursive-memory eigenvalue, the two components are coupled but not identical.\n\\end{proof}",
  "line": 2600,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Product spectrum",
  "proves": "theorem:bk5_symbolic_norm_spectrum",
  "ref_roles": [
    {
      "context": "_j)$, and $p=\\infty$ for equal-magnitude support. The memory component is exactly the spectral radius computed in Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}. Since the first quantity is a geometric representability ratio and the second is a recursive-memory eigenvalue, the t",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "spectrum] \\label{proof:bk5_symbolic_norm_spectrum} \\leavevmode The norm component is exactly the ratio proved in Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}. Its special values follow by substituting $p=1$, $(p,\\vec{v})=(2,e_i+e_j)$, and $p=\\infty$ for equal-magnitude suppor",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_symbolic_curvature_control",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_symbolic_norm_spectrum"
  ],
  "depends_on": [
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    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_symbolic_norm_spectrum"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_curvature_operator_spectrum",
  "label": "definition:bk5_symbolic_curvature_operator_spectrum",
  "latex_body": "\\begin{definition}[Symbolic Curvature Operator Spectrum]\n\\label{definition:bk5_symbolic_curvature_operator_spectrum}\nFor a fuzzy symbolic manifold $\\tilde{M}$ with observer resolution\n$\\epsilon_\\mathcal{O}$, the \\textbf{Symbolic Curvature Operator}\n$\\hat{\\mathcal{K}}_p$ acts on symbolic transitions $\\vec{v}$ according to\n(cf.~Def.~\\ref{definition:bk5_symbolic_curvature_control},\nThm.~\\ref{theorem:bk5_symbolic_norm_spectrum}):\n\n\\[\n\\hat{\\mathcal{K}}_p[\\vec{v}]\n=\\left(\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p}-1\\right)\\vec{v}\n=\\kappa_p(\\vec{v})\\vec{v}.\n\\]\n\nFor balanced memory-coupled transitions, the separate memory multiplier is the\nPerron ratio $\\varphi$ from\nThm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}; it is not inserted as\na value of $p$ in the curvature operator.\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-BOOK5-093"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book5Residue.curvature_control_p2_nonneg"
    ]
  },
  "line": 2613,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Curvature Operator Spectrum",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "textbf{Symbolic Curvature Operator} $\\hat{\\mathcal{K}}_p$ acts on symbolic transitions $\\vec{v}$ according to (cf.~Def.~\\ref{definition:bk5_symbolic_curvature_control}, Thm.~\\ref{theorem:bk5_symbolic_norm_spectrum}): \\[ \\hat{\\mathcal{K}}_p[\\vec{v}] =\\left(\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}",
      "label": "definition:bk5_symbolic_curvature_control",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2287,
      "target_type": "definition"
    },
    {
      "context": "v}. \\] For balanced memory-coupled transitions, the separate memory multiplier is the Perron ratio $\\varphi$ from Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}; it is not inserted as a value of $p$ in the curvature operator. \\end{definition}",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 1886,
      "target_type": "theorem"
    },
    {
      "context": "_p$ acts on symbolic transitions $\\vec{v}$ according to (cf.~Def.~\\ref{definition:bk5_symbolic_curvature_control}, Thm.~\\ref{theorem:bk5_symbolic_norm_spectrum}): \\[ \\hat{\\mathcal{K}}_p[\\vec{v}] =\\left(\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p}-1\\right)\\vec{v} =\\kappa_p(\\vec{v})\\vec{v}",
      "label": "theorem:bk5_symbolic_norm_spectrum",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2579,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_symbolic_curvature_control",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_symbolic_norm_spectrum"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book5",
  "cited_by": [
    "corollary:bk5_phi_centrality_principle",
    "proof:bk5_complete_symbolic_regime_classification",
    "proof:bk5_phi_centrality_principle",
    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "cites": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_symbolic_norm_spectrum"
  ],
  "depends_on": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant",
    "theorem:bk5_symbolic_norm_spectrum"
  ],
  "file": "book5.tex",
  "id": "lemma:bk5_phi_critical_resonant_norm",
  "label": "lemma:bk5_phi_critical_resonant_norm",
  "latex_body": "\\begin{lemma}[$\\varphi$ as Balanced Memory Resonance]\n\\label{lemma:bk5_phi_critical_resonant_norm}\nThe Golden Ratio $\\varphi$ is the unique positive resonance ratio of balanced\ntwo-step symbolic memory.  It enters the symbolic norm spectrum only through\nthe memory channel $\\mathcal{S}_{\\mathrm{mem}}$, not as a critical value of the\nnorm exponent $p$ (cf.~Def.~\\ref{definition:bk5_balanced_two_step_memory_closure},\nThm.~\\ref{theorem:bk5_symbolic_norm_spectrum}).\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "balanced two-step memory characteristic equation",
      "finite-support norm kernel from LPS-P20 through LPS-P24",
      "transition diagnostics are local classifications, not agent identities"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Phi is the unique positive balanced-recursion root in the memory coordinate and is proved distinct from the Euclidean norm ratio."
    ],
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      "MAP-BOOK5-043"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5.balancedSpectrum_memory",
      "Book5.balanced_memory_resonance_unique",
      "Book5.euclidean_balanced_product_spectrum"
    ]
  },
  "line": 2633,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "$\\varphi$ as Balanced Memory Resonance",
  "proof_labels": [
    "proof:bk5_phi_critical_resonant_norm"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "nly through the memory channel $\\mathcal{S}_{\\mathrm{mem}}$, not as a critical value of the norm exponent $p$ (cf.~Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, Thm.~\\ref{theorem:bk5_symbolic_norm_spectrum}). \\end{lemma}",
      "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": "not as a critical value of the norm exponent $p$ (cf.~Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, Thm.~\\ref{theorem:bk5_symbolic_norm_spectrum}). \\end{lemma}",
      "label": "theorem:bk5_symbolic_norm_spectrum",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2579,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_symbolic_norm_spectrum"
  ],
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  "book": "book5",
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  "cites": [
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    "theorem:bk5_golden_ratio_spectral_invariant"
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    "theorem:bk5_golden_ratio_spectral_invariant"
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  "id": "proof:bk5_phi_critical_resonant_norm",
  "label": "proof:bk5_phi_critical_resonant_norm",
  "latex_body": "\\begin{proof}[$\\varphi$ as balanced memory resonance]\n\\label{proof:bk5_phi_critical_resonant_norm}\n\\leavevmode\n\nBy Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, balanced memory\nevolves by the matrix\n\\[\nA=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}.\n\\]\nTheorem~\\ref{theorem:bk5_golden_ratio_spectral_invariant} computes\n$\\rho(A)=\\varphi$ and proves projective convergence to the positive Perron\neigendirection.  The norm exponent $p$ does not appear in this computation;\ntherefore $\\varphi$ is a memory-resonance invariant.  When a transition has both\ngeometric norm-fracture and balanced memory, the observed interface carries both\n$\\mathcal{S}_{\\mathrm{norm}}(p,\\vec{v})$ and $\\varphi$ as separate factors.\n\\end{proof}",
  "line": 2642,
  "macros_used": [],
  "matter_region": "mainmatter",
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  "name": "$\\varphi$ as balanced memory resonance",
  "proves": "lemma:bk5_phi_critical_resonant_norm",
  "ref_roles": [
    {
      "context": "\\begin{proof}[$\\varphi$ as balanced memory resonance] \\label{proof:bk5_phi_critical_resonant_norm} \\leavevmode By Def.~\\ref{definition:bk5_balanced_two_step_memory_closure}, balanced memory evolves by the matrix \\[ A=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}. \\] Theorem~\\ref{theorem:bk5_golden_ra",
      "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": "d_two_step_memory_closure}, balanced memory evolves by the matrix \\[ A=\\begin{pmatrix}1&1\\\\1&0\\end{pmatrix}. \\] Theorem~\\ref{theorem:bk5_golden_ratio_spectral_invariant} computes $\\rho(A)=\\varphi$ and proves projective convergence to the positive Perron eigendirection. The norm exponent",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 1886,
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    }
  ],
  "refs": [
    "definition:bk5_balanced_two_step_memory_closure",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "corollary:bk5_phi_centrality_principle",
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    "theorem:bk5_symbolic_norm_spectrum"
  ],
  "depends_on": [
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  ],
  "file": "book5.tex",
  "id": "proposition:bk5_complete_symbolic_regime_classification",
  "label": "proposition:bk5_complete_symbolic_regime_classification",
  "latex_body": "\\begin{proposition}[Complete Symbolic Regime Classification]\n\\label{proposition:bk5_complete_symbolic_regime_classification}\nEvery fuzzy symbolic manifold with axis-generated paths decomposes along two\nindependent diagnostic axes (cf.~Thm.~\\ref{theorem:bk5_symbolic_norm_spectrum},\nLem.~\\ref{lemma:bk5_phi_critical_resonant_norm}):\n\n\\begin{enumerate}\n\\item \\textbf{Norm-fracture regime}: $p=1$ gives exact axis integrability;\n$p=2$ gives elementary Euclidean diagonal fracture $\\sqrt{2}$; $p=\\infty$\ngives support collapse.\n\\item \\textbf{Memory-resonance regime}: balanced two-step recursive memory gives\nthe Perron ratio $\\varphi$.\n\\end{enumerate}\n\nThe four older names---atomic order, resonant coherence, fracture emergence, and\nsupport collapse---are therefore regime labels for combinations of these two\naxes, not mutually exclusive universal states of a manifold.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "balanced two-step memory characteristic equation",
      "finite-support norm kernel from LPS-P20 through LPS-P24",
      "transition diagnostics are local classifications, not agent identities"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Transition-level diagnostic classification with independently varying norm and memory axes; not an essential identity of a manifold or agent."
    ],
    "record_ids": [
      "MAP-BOOK5-025"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5.changing_norm_preserves_balanced_memory",
      "Book5.elementary_spectrum_values",
      "Book5.norm_and_memory_coordinates_independent"
    ]
  },
  "line": 2659,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Complete Symbolic Regime Classification",
  "proof_labels": [
    "proof:bk5_complete_symbolic_regime_classification"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "nerated paths decomposes along two independent diagnostic axes (cf.~Thm.~\\ref{theorem:bk5_symbolic_norm_spectrum}, Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}): \\begin{enumerate} \\item \\textbf{Norm-fracture regime}: $p=1$ gives exact axis integrability; $p=2$ gives elementary",
      "label": "lemma:bk5_phi_critical_resonant_norm",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2633,
      "target_type": "lemma"
    },
    {
      "context": "ion} Every fuzzy symbolic manifold with axis-generated paths decomposes along two independent diagnostic axes (cf.~Thm.~\\ref{theorem:bk5_symbolic_norm_spectrum}, Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}): \\begin{enumerate} \\item \\textbf{Norm-fracture regime}: $p=1$ gives",
      "label": "theorem:bk5_symbolic_norm_spectrum",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2579,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "lemma:bk5_phi_critical_resonant_norm",
    "theorem:bk5_symbolic_norm_spectrum"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_symbolic_integrability_classes",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "depends_on": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_symbolic_integrability_classes",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_complete_symbolic_regime_classification",
  "label": "proof:bk5_complete_symbolic_regime_classification",
  "latex_body": "\\begin{proof}[Regime classification]\n\\label{proof:bk5_complete_symbolic_regime_classification}\n\\leavevmode\n\nThe first axis follows from Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy}\nand Prop.~\\ref{proposition:bk5_symbolic_integrability_classes}.  The second\naxis follows from Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}.  Since a\nsingle transition can simultaneously have an $\\ell_p$ geometry and a balanced\nmemory update, the axes classify different structure maps and cannot be\nexclusive alternatives.  Their combinations yield the named regimes.\n\\end{proof}",
  "line": 2678,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Regime classification",
  "proves": "proposition:bk5_complete_symbolic_regime_classification",
  "ref_roles": [
    {
      "context": "_fracture_hierarchy} and Prop.~\\ref{proposition:bk5_symbolic_integrability_classes}. The second axis follows from Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}. Since a single transition can simultaneously have an $\\ell_p$ geometry and a balanced memory update, the axes classif",
      "label": "lemma:bk5_phi_critical_resonant_norm",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2633,
      "target_type": "lemma"
    },
    {
      "context": "me_classification} \\leavevmode The first axis follows from Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy} and Prop.~\\ref{proposition:bk5_symbolic_integrability_classes}. The second axis follows from Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}. Since a single transition can simultan",
      "label": "proposition:bk5_symbolic_integrability_classes",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2341,
      "target_type": "proposition"
    },
    {
      "context": "classification] \\label{proof:bk5_complete_symbolic_regime_classification} \\leavevmode The first axis follows from Thm.~\\ref{theorem:bk5_lp_norm_fracture_hierarchy} and Prop.~\\ref{proposition:bk5_symbolic_integrability_classes}. The second axis follows from Lem.~\\ref{lemma:bk5_phi_c",
      "label": "theorem:bk5_lp_norm_fracture_hierarchy",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2301,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_symbolic_integrability_classes",
    "theorem:bk5_lp_norm_fracture_hierarchy"
  ],
  "role": "proof",
  "type": "proof"
}

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  "book": "book5",
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    "definition:bk5_symbolic_regime_detection"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "theorem:bk5_symbolic_manifold_spectral_decomposition",
  "label": "theorem:bk5_symbolic_manifold_spectral_decomposition",
  "latex_body": "\\begin{theorem}[Symbolic Manifold Spectral Decomposition]\n\\label{theorem:bk5_symbolic_manifold_spectral_decomposition}\nSuppose a fuzzy symbolic manifold $\\tilde{M}$ admits an observer-resolved\ndirect-sum decomposition of its transition space into invariant components\n$\\mathcal{M}_1,\\mathcal{M}_2,\\mathcal{M}_\\infty,\\mathcal{M}_{\\varphi}$ for\naxis-integrable, Euclidean-fractured, support-collapsed, and balanced-memory\ndirections, respectively.  Then every transition in the span of these components\nhas a unique coordinate decomposition\n\n\\[\nX=\\alpha_1 X_1+\\alpha_2 X_2+\\alpha_\\infty X_\\infty+\\alpha_\\varphi X_\\varphi,\n\\]\n\nwith $X_i\\in\\mathcal{M}_i$.  The coefficients\n$\\{\\alpha_1,\\alpha_2,\\alpha_\\infty,\\alpha_\\varphi\\}$ determine the\n\\textbf{symbolic character} of the transition relative to this observer\ndecomposition.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Given the stated observer-resolved linear equivalence and diagonal evolution law, four-channel reconstruction, uniqueness, and invariance are proved. Existence of such a decomposition is not asserted universally."
    ],
    "record_ids": [
      "MAP-BOOK5-045"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5.ObserverSpectralDecomposition.component_is_invariant",
      "Book5.ObserverSpectralDecomposition.coordinate_representation_unique",
      "Book5.ObserverSpectralDecomposition.spectral_decomposition_kernel",
      "Book5.ObserverSpectralDecomposition.sum_components"
    ]
  },
  "line": 2690,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Manifold Spectral Decomposition",
  "proof_labels": [
    "proof:bk5_symbolic_manifold_spectral_decomposition"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "proof:bk5_symbolic_manifold_spectral_decomposition",
  "label": "proof:bk5_symbolic_manifold_spectral_decomposition",
  "latex_body": "\\begin{proof}[Direct-sum decomposition]\n\\label{proof:bk5_symbolic_manifold_spectral_decomposition}\n\\leavevmode\n\nThe statement is the standard uniqueness property of a direct-sum\ndecomposition.  By hypothesis, the listed components are invariant and their\nspan contains $X$ with pairwise-zero intersections.  Therefore each $X$ has a\nunique sum of components, and the scalar coordinates are its observer-resolved\nregime coefficients.\n\\end{proof}",
  "line": 2709,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Direct-sum decomposition",
  "proves": "theorem:bk5_symbolic_manifold_spectral_decomposition",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [
    "scholium:bk5_golden_rule_covenant"
  ],
  "cites": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "depends_on": [
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    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "file": "book5.tex",
  "id": "corollary:bk5_phi_centrality_principle",
  "label": "corollary:bk5_phi_centrality_principle",
  "latex_body": "\\begin{corollary}[The $\\varphi$-Centrality Principle]\n\\label{corollary:bk5_phi_centrality_principle}\nThe Golden Ratio $\\varphi$ occupies a central position in the memory component\nof the symbolic spectrum because it is the positive Perron ratio of balanced\ntwo-step recursion\n(cf.~Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm},\nProp.~\\ref{proposition:bk5_complete_symbolic_regime_classification}).  Its\ncentrality is recursive rather than metric: it governs stable memory\nproportions, while $\\sqrt{2}$ governs the elementary Euclidean fracture ratio.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "balanced two-step memory characteristic equation",
      "finite-support norm kernel from LPS-P20 through LPS-P24",
      "transition diagnostics are local classifications, not agent identities"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Balanced memory ratios converge to the unique phi coordinate, while sqrt(2) remains a distinct metric-fracture coordinate."
    ],
    "record_ids": [
      "MAP-BOOK5-044"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book5.balanced_memory_converges_to_spectrum_memory",
      "Book5.balanced_memory_resonance_unique",
      "Book5.euclidean_balanced_product_spectrum"
    ]
  },
  "line": 2720,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The $\\varphi$-Centrality Principle",
  "proof_labels": [
    "proof:bk5_phi_centrality_principle"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "ory component of the symbolic spectrum because it is the positive Perron ratio of balanced two-step recursion (cf.~Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}, Prop.~\\ref{proposition:bk5_complete_symbolic_regime_classification}). Its centrality is recursive rather than metric:",
      "label": "lemma:bk5_phi_critical_resonant_norm",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2633,
      "target_type": "lemma"
    },
    {
      "context": "is the positive Perron ratio of balanced two-step recursion (cf.~Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}, Prop.~\\ref{proposition:bk5_complete_symbolic_regime_classification}). Its centrality is recursive rather than metric: it governs stable memory proportions, while $\\sqrt{2}$ governs the e",
      "label": "proposition:bk5_complete_symbolic_regime_classification",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2659,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "depends_on": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "file": "book5.tex",
  "id": "proof:bk5_phi_centrality_principle",
  "label": "proof:bk5_phi_centrality_principle",
  "latex_body": "\\begin{proof}\n\\label{proof:bk5_phi_centrality_principle}\n\\leavevmode\nBy the complete symbolic regime classification (Prop.~\\ref{proposition:bk5_complete_symbolic_regime_classification}) the symbolic spectrum splits into a recursive memory component and a metric (geometric) component. Within the memory component, the balanced two-step recursion has companion matrix $A=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$ whose unique positive Perron root is $\\varphi$ (Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}); by Perron--Frobenius this dominant eigenvalue is the spectral center toward which the normalized memory ratios $a_{n+1}/a_n$ converge. Hence $\\varphi$ occupies the central position of the memory component: its centrality is recursive --- it fixes the stable proportion of retained reflective memory --- not metric. The metric component is governed instead by the elementary Euclidean fracture ratio $\\sqrt2$ (the first diagonal representability cost), spectrally distinct from $\\varphi$. Thus $\\varphi$ and $\\sqrt2$ are the characteristic ratios of the two components, with $\\varphi$ central to memory and $\\sqrt2$ to metric fracture.\n\\end{proof}",
  "line": 2730,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "corollary:bk5_phi_centrality_principle",
  "ref_roles": [
    {
      "context": "matrix $A=\\big(\\begin{smallmatrix}1&1\\\\1&0\\end{smallmatrix}\\big)$ whose unique positive Perron root is $\\varphi$ (Lem.~\\ref{lemma:bk5_phi_critical_resonant_norm}); by Perron--Frobenius this dominant eigenvalue is the spectral center toward which the normalized memory ratios $a_{n+",
      "label": "lemma:bk5_phi_critical_resonant_norm",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2633,
      "target_type": "lemma"
    },
    {
      "context": "gin{proof} \\label{proof:bk5_phi_centrality_principle} \\leavevmode By the complete symbolic regime classification (Prop.~\\ref{proposition:bk5_complete_symbolic_regime_classification}) the symbolic spectrum splits into a recursive memory component and a metric (geometric) component. Within the memory c",
      "label": "proposition:bk5_complete_symbolic_regime_classification",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book5.tex",
      "target_line": 2659,
      "target_type": "proposition"
    }
  ],
  "refs": [
    "lemma:bk5_phi_critical_resonant_norm",
    "proposition:bk5_complete_symbolic_regime_classification"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "definition:bk5_symbolic_compression_experiment",
    "theorem:bk5_symbolic_manifold_spectral_decomposition"
  ],
  "depends_on": [
    "definition:bk5_symbolic_compression_experiment",
    "theorem:bk5_symbolic_manifold_spectral_decomposition"
  ],
  "file": "book5.tex",
  "id": "definition:bk5_symbolic_regime_detection",
  "label": "definition:bk5_symbolic_regime_detection",
  "latex_body": "\\begin{definition}[Symbolic Regime Detection Experiment]\n\\label{definition:bk5_symbolic_regime_detection}\nTo empirically validate the product spectrum, measure the \\textbf{symbolic\ncompression ratio} and the \\textbf{balanced-memory ratio}\n(cf.~Def.~\\ref{definition:bk5_symbolic_compression_experiment},\nThm.~\\ref{theorem:bk5_symbolic_manifold_spectral_decomposition}):\n$$R_p = \\frac{\\text{Symbolic encoding length}}{\\text{Geometric path length}}$$\nand\n\\[\nM_n=\\frac{a_{n+1}}{a_n}.\n\\]\n\nThe theory predicts:\n- $R_1 = 1.000$ (perfect compression)\n- $R_2 = \\sqrt{2}$ for elementary Euclidean diagonals\n- $R_\\infty = s(\\vec{v})$ for equal-magnitude support size $s(\\vec{v})$\n- $M_n\\to\\varphi$ for balanced two-step memory\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": [
      "Balanced Fibonacci memory ratio prediction."
    ],
    "record_ids": [
      "MAP-BOOK5-019"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "balanced_memory_tendsto_gold"
    ]
  },
  "line": 2736,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Regime Detection Experiment",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "the product spectrum, measure the \\textbf{symbolic compression ratio} and the \\textbf{balanced-memory ratio} (cf.~Def.~\\ref{definition:bk5_symbolic_compression_experiment}, Thm.~\\ref{theorem:bk5_symbolic_manifold_spectral_decomposition}): $$R_p = \\frac{\\text{Symbolic encoding length}}{\\text",
      "label": "definition:bk5_symbolic_compression_experiment",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book5.tex",
      "target_line": 2508,
      "target_type": "definition"
    },
    {
      "context": "sion ratio} and the \\textbf{balanced-memory ratio} (cf.~Def.~\\ref{definition:bk5_symbolic_compression_experiment}, Thm.~\\ref{theorem:bk5_symbolic_manifold_spectral_decomposition}): $$R_p = \\frac{\\text{Symbolic encoding length}}{\\text{Geometric path length}}$$ and \\[ M_n=\\frac{a_{n+1}}{a_n}. \\] Th",
      "label": "theorem:bk5_symbolic_manifold_spectral_decomposition",
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    "theorem:bk5_symbolic_manifold_spectral_decomposition"
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{
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    "demonstratio:bk5_deep_unity_math_meaning",
    "remark:bk5_open_frontiers",
    "scholium:bk5_golden_rule_covenant"
  ],
  "cites": [
    "theorem:bk5_fundamental_norm_fracture",
    "theorem:bk5_golden_ratio_spectral_invariant"
  ],
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    "theorem:bk5_golden_ratio_spectral_invariant"
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  "id": "theorem:bk5_grand_unified_symbolic_geometric",
  "label": "theorem:bk5_grand_unified_symbolic_geometric",
  "latex_body": "\\begin{theorem}[Product Theorem of the Symbolic-Geometric Interface]\n\\label{theorem:bk5_grand_unified_symbolic_geometric}\nFor an observer-resolved symbolic system with axis-generated geometric paths and\nbalanced two-step memory, the symbolic-geometric interface has product\ninvariants\n\\[\n\\left(\\frac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p},\\ \\varphi\\right).\n\\]\nThe first invariant is determined by norm-induced fracture\n(Thm.~\\ref{theorem:bk5_fundamental_norm_fracture}); the second is determined by\nbalanced memory resonance\n(Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}).  In particular,\n$1$ marks exact axis integrability, $\\sqrt{2}$ marks the elementary Euclidean\ndiagonal fracture, $s(\\vec{v})$ marks equal-magnitude support collapse at\n$p=\\infty$, and $\\varphi$ marks balanced recursive memory.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "nonzero diagonal amplitude",
      "only the arithmetic, matrix, lattice, and limit content of the cited Book 5 statements",
      "two-coordinate special case"
    ],
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      "Book5.same_norm_does_not_fix_memory"
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    "kernel_certified": false,
    "notes": [
      "Typed product coordinates and their independence, not a universal manifold semantics."
    ],
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      "MAP-BOOK5-026"
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      "open_bridge"
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    "witnesses": [
      "Book5.ObserverSpectralDecomposition.spectral_decomposition_kernel",
      "Book5.same_memory_does_not_fix_norm",
      "Book5.same_norm_does_not_fix_memory",
      "closureMatrix_eigen_gold"
    ]
  },
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  "name": "Product Theorem of the Symbolic-Geometric Interface",
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    "proof:bk5_grand_unified_symbolic_geometric"
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  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "rac{\\|\\vec{v}\\|_1}{\\|\\vec{v}\\|_p},\\ \\varphi\\right). \\] The first invariant is determined by norm-induced fracture (Thm.~\\ref{theorem:bk5_fundamental_norm_fracture}); the second is determined by balanced memory resonance (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). In p",
      "label": "theorem:bk5_fundamental_norm_fracture",
      "logical_support": true,
      "role": "application",
      "target_file": "book5.tex",
      "target_line": 2518,
      "target_type": "theorem"
    },
    {
      "context": "racture (Thm.~\\ref{theorem:bk5_fundamental_norm_fracture}); the second is determined by balanced memory resonance (Thm.~\\ref{theorem:bk5_golden_ratio_spectral_invariant}). In particular, $1$ marks exact axis integrability, $\\sqrt{2}$ marks the elementary Euclidean diagonal fracture, $s(\\",
      "label": "theorem:bk5_golden_ratio_spectral_invariant",
      "logical_support": true,
      "role": "application",
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      "target_line": 1886,
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    "theorem:bk5_golden_ratio_spectral_invariant"
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  "id": "proof:bk5_grand_unified_symbolic_geometric",
  "label": "proof:bk5_grand_unified_symbolic_geometric",
  "latex_body": "\\begin{proof}[Product interface invariants]\n\\label{proof:bk5_grand_unified_symbolic_geometric}\n\\leavevmode\n\nThe geometric component follows from the norm-ratio theorem:\n$\\|\\vec{v}\\|_1/\\|\\vec{v}\\|_p$ is the complete axis/geometric representability\nratio under the stated hypotheses.  The memory component follows from the\nbalanced-memory theorem: the update matrix has Perron radius $\\varphi$.\nBecause these components act on different coordinates of the observer-resolved\nstate--geometric transition length and recursive memory amplitude--the interface\ninvariant is their ordered product.  The listed constants are the corresponding\nspecial cases already proved in the cited theorems.\n\\end{proof}",
  "line": 2772,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Product interface invariants",
  "proves": "theorem:bk5_grand_unified_symbolic_geometric",
  "refs": [],
  "role": "proof",
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{
  "book": "book5",
  "cited_by": [],
  "cites": [
    "theorem:bk5_grand_unified_symbolic_geometric"
  ],
  "depends_on": [
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  ],
  "file": "book5.tex",
  "id": "demonstratio:bk5_deep_unity_math_meaning",
  "label": "demonstratio:bk5_deep_unity_math_meaning",
  "latex_body": "\\begin{demonstratio}[The Deep Unity of Mathematics and Meaning]\n\\label{demonstratio:bk5_deep_unity_math_meaning}\nThis spectrum reveals that the constants used here are \\textbf{interface\ninvariants}: different ways that discrete symbolic logic can interface with\ncontinuous geometric reality and recursive memory\n(cf.~Thm.~\\ref{theorem:bk5_grand_unified_symbolic_geometric}).\n\n- \\textbf{$1$} represents \\textbf{axis integrability} (no representability gap).\n- \\textbf{$\\varphi$} represents \\textbf{balanced recursive memory}.\n- \\textbf{$\\sqrt{2}$} represents \\textbf{elementary Euclidean diagonal fracture}.\n- \\textbf{$s(\\vec{v})$} represents \\textbf{support collapse} at $p=\\infty$ for equal-magnitude transitions.\n\nThe placement of $\\varphi$ in the memory coordinate explains why it appears in\nsystems that balance current persistence against retained history.  The\nplacement of $\\sqrt{2}$ in the geometric coordinate explains why it appears\nwhenever a symbolic lattice first admits a direct orthogonal diagonal.\n\\end{demonstratio}",
  "line": 2786,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Deep Unity of Mathematics and Meaning",
  "ref_roles": [
    {
      "context": "ferent ways that discrete symbolic logic can interface with continuous geometric reality and recursive memory (cf.~Thm.~\\ref{theorem:bk5_grand_unified_symbolic_geometric}). - \\textbf{$1$} represents \\textbf{axis integrability} (no representability gap). - \\textbf{$\\varphi$} represents \\te",
      "label": "theorem:bk5_grand_unified_symbolic_geometric",
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  "cites": [
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  "depends_on": [
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  ],
  "file": "book5.tex",
  "id": "remark:bk5_open_frontiers",
  "label": "remark:bk5_open_frontiers",
  "latex_body": "\\begin{remark}[Open Frontiers]\n\\label{remark:bk5_open_frontiers}\nThis product framework opens concrete research directions\n(cf.~Thm.~\\ref{theorem:bk5_grand_unified_symbolic_geometric}):\n\n1. \\textbf{Biological Symbolic Processing}: Do biological systems exhibit balanced two-step memory ratios near $\\varphi$?\n\n2. \\textbf{Quantum Symbolic Mechanics}: Can observer-resolved state spaces separate geometric fracture ratios from memory-resonance ratios?\n\n3. \\textbf{Cognitive Symbolic Architecture}: Does human cognition switch between axis-integrable, fractured, and support-collapsed geometric encodings?\n\n4. \\textbf{Computational Symbolic Optimization}: Can algorithms tune $p$ for representability cost while separately tuning memory recurrence toward $\\varphi$?\n\n5. \\textbf{Physical Symbolic Fields}: Which physical constants are geometric representability ratios, and which are dynamical memory eigenvalues?\n\\end{remark}",
  "line": 2804,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Open Frontiers",
  "ref_roles": [
    {
      "context": "}[Open Frontiers] \\label{remark:bk5_open_frontiers} This product framework opens concrete research directions (cf.~Thm.~\\ref{theorem:bk5_grand_unified_symbolic_geometric}): 1. \\textbf{Biological Symbolic Processing}: Do biological systems exhibit balanced two-step memory ratios near $\\var",
      "label": "theorem:bk5_grand_unified_symbolic_geometric",
      "logical_support": true,
      "role": "cf_near_match",
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{
  "book": "book5",
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  "cites": [],
  "depends_on": [],
  "file": "book5.tex",
  "id": "subsec:bk5_golden_rule_ethics",
  "label": "subsec:bk5_golden_rule_ethics",
  "latex_body": "",
  "line": 2820,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Golden Rule as Recursive Ethics",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}