Book VI — De Mutatione Symbolica — page 3 of 3

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  "latex_body": "\\begin{definition}[Symbolic Pressure Operator]\n\\label{definition:bk6_symbolic_pressure_operator}\nThe \\emph{symbolic pressure operator} $\\Pi_s : P_\\lambda \\to \\mathbb{R}$ captures constraint forces induced by symbolic free energy (Def.~\\ref{definition:bk6_symbolic_free_energy_functional}):\n\\begin{equation}\n\\Pi_s(p) = -\\frac{\\partial \\mathcal{F}_\\lambda[p]}{\\partial V_s(p)}\\bigg|_{\\mathcal{S}_\\lambda}\n\\end{equation}\nwhere $V_s(p) = \\int_M d\\mu_g(x)$ is the configuration volume at constant entropy.\n\\end{definition}",
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      "context": "pressure operator} $\\Pi_s : P_\\lambda \\to \\mathbb{R}$ captures constraint forces induced by symbolic free energy (Def.~\\ref{definition:bk6_symbolic_free_energy_functional}): \\begin{equation} \\Pi_s(p) = -\\frac{\\partial \\mathcal{F}_\\lambda[p]}{\\partial V_s(p)}\\bigg|_{\\mathcal{S}_\\lambda} \\end",
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  "latex_body": "\\begin{definition}[Grace Operator]\n\\label{definition:bk6_grace_operator_complete}\nThe \\emph{Grace Operator} $\\mathcal{G} : P_{\\lambda} \\to P_{\\lambda}$ preserves identity under regulatory failure by enforcing stability thresholds within regulatory basins (Defs.~\\ref{definition:bk6_stability_functional_complete}, \\ref{definition:bk6_regulatory_basin_operator}):\n\\begin{equation}\n\\mathcal{G}(p) = p + \\int_0^1 K_G(p,t) \\cdot \\nabla_s \\left( \\Upsilon_i(p, \\cdot) - \\gamma_{\\min} \\right) dt\n\\end{equation}\nwhere $K_G(p,t)$ is the grace kernel satisfying:\n\\begin{enumerate}\n\\item \\emph{Dissonance holding}: $\\Upsilon_i(p, \\mathcal{G}(p)) > \\gamma_G$ despite fragmentation\n\\item \\emph{Collapse aversion}: $\\mathcal{G}(p) \\in \\mathcal{R}_B(\\tilde{p})$ for some viable $\\tilde{p}$\n\\item \\emph{Reentry enablement}: $R_\\lambda(\\mathcal{G}(p))$ is well-defined with probability $> 1/2$\n\\end{enumerate}\n\\end{definition}",
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      "context": "e by enforcing stability thresholds within regulatory basins (Defs.~\\ref{definition:bk6_stability_functional_complete}, \\ref{definition:bk6_regulatory_basin_operator}): \\begin{equation} \\mathcal{G}(p) = p + \\int_0^1 K_G(p,t) \\cdot \\nabla_s \\left( \\Upsilon_i(p, \\cdot) - \\gamma_{\\min} \\r",
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      "context": "\\lambda}$ preserves identity under regulatory failure by enforcing stability thresholds within regulatory basins (Defs.~\\ref{definition:bk6_stability_functional_complete}, \\ref{definition:bk6_regulatory_basin_operator}): \\begin{equation} \\mathcal{G}(p) = p + \\int_0^1 K_G(p,t) \\cdot \\nabla_",
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  "latex_body": "\\begin{definition}[Mutation Operator]\n\\label{definition:bk6_mutation_operator_complete}\nThe \\emph{mutation operator} $\\mathcal{M}_{\\lambda} : P_{\\lambda} \\to P_{\\lambda+1}$ captures complexity transitions by composing drift, bifurcation, and reflection (Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_bifurcation_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}):\n\\begin{equation}\n\\mathcal{M}_{\\lambda} = R_{\\lambda+1} \\circ \\mathcal{B}_{\\lambda} \\circ D_{\\lambda}\n\\end{equation}\nwhen all constituent operators are well-defined.\n\\end{definition}",
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      "context": ": P_{\\lambda} \\to P_{\\lambda+1}$ captures complexity transitions by composing drift, bifurcation, and reflection (Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_bifurcation_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}): \\begin{equatio",
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      "context": "and reflection (Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_bifurcation_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}): \\begin{equation} \\mathcal{M}_{\\lambda} = R_{\\lambda+1} \\circ \\mathcal{B}_{\\lambda} \\circ D_{\\lambda} \\end{equation} w",
      "label": "definition:bk6_reflection_operator_complete",
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      "context": "ulation operator} $\\Omega_\\delta : \\Gamma(TM) \\to \\Gamma(TM)$ transforms vector fields through curvature response (Def.~\\ref{definition:bk6_symbolic_curvature_tensor_coordinate_index}): \\begin{equation} (\\Omega_\\delta X)(p) = X(p) + \\delta \\cdot (\\nabla_X \\mathcal{R})(p) \\cdot X(p) \\end{equation} for p",
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  "latex_body": "\\begin{definition}[Symbolic Flow Operator]\n\\label{definition:bk6_symbolic_flow_operator_complete}\nThe \\emph{symbolic flow operator} $\\Phi_t : P_\\lambda \\to P_\\lambda$ is generated by the drift field (Def.~\\ref{definition:bk6_drift_operator_complete}) and satisfies:\n\\begin{equation}\n\\frac{d}{dt}\\Phi_t(p) = D_\\lambda(\\Phi_t(p)), \\quad \\Phi_0(p) = p\n\\end{equation}\n\\end{definition}",
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      "context": "or_complete} The \\emph{symbolic flow operator} $\\Phi_t : P_\\lambda \\to P_\\lambda$ is generated by the drift field (Def.~\\ref{definition:bk6_drift_operator_complete}) and satisfies: \\begin{equation} \\frac{d}{dt}\\Phi_t(p) = D_\\lambda(\\Phi_t(p)), \\quad \\Phi_0(p) = p \\end{equation} \\end{",
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      "context": "ymbolic geometric setting, and governs diffusion and entropy production under symbolic thermodynamic evolution (see thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). The operator is well-defined on smooth scalar fields and extends naturally to symbolic densities and fuzzy observable",
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      "context": "bling the emergence of non-trivial symbolic equilibria even in curved or dynamically drifting symbolic spaces (cf.~Thm.~\\ref{theorem:bk1_symbolic_emergence_and_curvature}: curvature is the condition for genuine emergence). \\end{definition}",
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      "context": ")$ combines diffusion and symbolic free energy (Defs.~\\ref{definition:bk6_symbolic_laplace_beltrami_operator_complete}, \\ref{definition:bk6_symbolic_free_energy_functional}): \\begin{equation} \\mathcal{H}_s = -\\frac{\\hbar_s^2}{2} \\Delta_s + V_s + \\mathcal{F}_\\lambda \\end{equation} where $\\hba",
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      "context": "tonian operator} $\\mathcal{H}_s : \\mathcal{H}(M) \\to \\mathcal{H}(M)$ combines diffusion and symbolic free energy (Defs.~\\ref{definition:bk6_symbolic_laplace_beltrami_operator_complete}, \\ref{definition:bk6_symbolic_free_energy_functional}): \\begin{equation} \\mathcal{H}_s = -\\frac{\\hbar_s^2}{2} \\Delta_s",
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      "context": "al. See Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}, and Def.~\\ref{definition:bk1_paradox_triggered_emergence} for the foundational mechanism by which such incoherence drives symbolic expansion. \\end{axiom}",
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      "context": "nd{equation} The commutator magnitude $\\|[D_\\lambda, R_\\lambda]\\|_{\\text{op}}$ quantifies emergent potential. See Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}, and Def.~\\ref{definition:bk1_paradox_triggered_emergence} for the f",
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    },
    {
      "context": "ambda, R_\\lambda]\\|_{\\text{op}}$ quantifies emergent potential. See Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}, and Def.~\\ref{definition:bk1_paradox_triggered_emergence} for the foundational mechanism by which such incoherence dri",
      "label": "definition:bk6_reflection_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 937,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_paradox_triggered_emergence",
    "definition:bk6_drift_operator_complete",
    "definition:bk6_reflection_operator_complete"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [
    "proof:bk6_map_invariant",
    "proposition:bk6_map_invariant",
    "subsec:bk6_scholium_on_symbolic_operator_mechanics"
  ],
  "cites": [
    "definition:bk6_drift_operator_complete",
    "definition:bk6_reflection_operator_complete",
    "definition:bk6_transformation_operator_complete"
  ],
  "depends_on": [
    "definition:bk6_drift_operator_complete",
    "definition:bk6_reflection_operator_complete",
    "definition:bk6_transformation_operator_complete"
  ],
  "file": "book6.tex",
  "id": "axiom:bk6_map_equilibrium_invariance_complete",
  "label": "axiom:bk6_map_equilibrium_invariance_complete",
  "latex_body": "\\begin{axiom}[MAP Equilibrium Invariance]\n\\label{axiom:bk6_map_equilibrium_invariance_complete}\nFor closed regulatory cycles (Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}, \\ref{definition:bk6_transformation_operator_complete}):\n\\begin{equation}\n\\mathcal{F}_\\lambda[(T_\\alpha \\circ R_\\lambda \\circ D_\\lambda)^n(p)] = \\mathcal{F}_\\lambda[p] + \\mathcal{O}(e^{-\\eta n})\n\\end{equation}\nwith damping coefficient $\\eta > 0$.\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "modeling laws are structure fields or explicit hypotheses; continuum/categorical content is NOT formalized"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "F_lambda[(...)^n(p)] = F_lambda[p] + O(e^{-eta n}) is modeled as a residual bounded by C*rate^n with rate kept abstractly in [0,1) (rather than reconstructed as Real.exp(-eta)); proves the residual tends to 0, i.e. the functional value converges to F_lambda[p]."
    ],
    "record_ids": [
      "MAP-BOOK6-030"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Asymptotics.GeometricErrorBound.tendsto_zero"
    ]
  },
  "line": 1100,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "MAP Equilibrium Invariance",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "}[MAP Equilibrium Invariance] \\label{axiom:bk6_map_equilibrium_invariance_complete} For closed regulatory cycles (Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}, \\ref{definition:bk6_transformation_operator_complete}): \\begin{equa",
      "label": "definition:bk6_drift_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 926,
      "target_type": "definition"
    },
    {
      "context": "_map_equilibrium_invariance_complete} For closed regulatory cycles (Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}, \\ref{definition:bk6_transformation_operator_complete}): \\begin{equation} \\mathcal{F}_\\lambda[(T_\\alpha \\circ R_\\lambda",
      "label": "definition:bk6_reflection_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 937,
      "target_type": "definition"
    },
    {
      "context": "gulatory cycles (Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_reflection_operator_complete}, \\ref{definition:bk6_transformation_operator_complete}): \\begin{equation} \\mathcal{F}_\\lambda[(T_\\alpha \\circ R_\\lambda \\circ D_\\lambda)^n(p)] = \\mathcal{F}_\\lambda[p] + \\mat",
      "label": "definition:bk6_transformation_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 948,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_drift_operator_complete",
    "definition:bk6_reflection_operator_complete",
    "definition:bk6_transformation_operator_complete"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "depends_on": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "file": "book6.tex",
  "id": "axiom:bk6_symbolic_mass_conservation_complete",
  "label": "axiom:bk6_symbolic_mass_conservation_complete",
  "latex_body": "\\begin{axiom}[Symbolic Mass Conservation]\n\\label{axiom:bk6_symbolic_mass_conservation_complete}\nTotal probability mass is preserved (Def.~\\ref{definition:bk6_symbolic_state_function_complete}):\n\\begin{equation}\n\\int_M \\rho_s(p,x) \\, d\\mu_g(x) = \\int_M \\rho_s(\\mathcal{O}(p),x) \\, d\\mu_g(x) = 1\n\\end{equation}\nfor any canonical operator $\\mathcal{O}$.\n\\end{axiom}",
  "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": [
      "\"For any canonical operator O\" is specialized to composites of row-stochastic evolution kernels (Book2's discrete Fokker-Planck skeleton); mass conservation for such composites is proved exactly."
    ],
    "record_ids": [
      "MAP-BOOK6-041"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book68B.composedEvolution_isDensity"
    ]
  },
  "line": 1109,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Mass Conservation",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "olic Mass Conservation] \\label{axiom:bk6_symbolic_mass_conservation_complete} Total probability mass is preserved (Def.~\\ref{definition:bk6_symbolic_state_function_complete}): \\begin{equation} \\int_M \\rho_s(p,x) \\, d\\mu_g(x) = \\int_M \\rho_s(\\mathcal{O}(p),x) \\, d\\mu_g(x) = 1 \\end{equation} fo",
      "label": "definition:bk6_symbolic_state_function_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 849,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "axiom:bk6_thermodynamic_consistency",
  "label": "axiom:bk6_thermodynamic_consistency",
  "latex_body": "\\begin{axiom}[Thermodynamic Consistency and Potential Orientation]\n\\label{axiom:bk6_thermodynamic_consistency}\nThe internal-energy first law is\n\\[\n dE_\\lambda=T_s\\,d\\mathcal S_\\lambda-\\Pi_s\\,dV_s\n      +\\sum_i\\mu_i\\,dN_i.\n\\]\nFor the symbolic free energy defined by\n$\\mathcal F_\\lambda=E_\\lambda-T_s\\mathcal S_\\lambda$, its differential is\ntherefore\n\\[\n d\\mathcal F_\\lambda=-\\Pi_s\\,dV_s+\\sum_i\\mu_i\\,dN_i\n       -\\mathcal S_\\lambda\\,dT_s.\n\\]\nAt fixed symbolic temperature this reduces to\n$d\\mathcal F_\\lambda=-\\Pi_s\\,dV_s+\\sum_i\\mu_i\\,dN_i$.\nThus the entropy sign is fixed by the orientation of the chosen potential; a\npositive $T_s\\,d\\mathcal S_\\lambda$ term belongs to $dE_\\lambda$, not to the\nfixed-temperature differential of $E_\\lambda-T_s\\mathcal S_\\lambda$.\nFor finite simultaneous changes of temperature and entropy, the exact\nincrement additionally contains the cross-term\n$-\\Delta T_s\\,\\Delta\\mathcal S_\\lambda$.\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "explicit finite temperature increment for the varying-temperature identity",
      "fixed-temperature energy balance for the reduced identity",
      "source definition F = E - T*S"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Exact finite-increment identities distinguish the internal-energy first law from F=E-T*S. Fixed temperature cancels T*dS from dF; varying temperature adds -S*dT and the finite cross-term. Potential orientation, not verifier preference, fixes the sign."
    ],
    "record_ids": [
      "MAP-BOOK6-053"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book6ThermodynamicConsistency.fixed_temperature_firstLaw_reduction",
      "Book6ThermodynamicConsistency.fixed_temperature_freeEnergy_increment",
      "Book6ThermodynamicConsistency.interface_term_reconciles_printed_law_iff",
      "Book6ThermodynamicConsistency.oriented_freeEnergy_fixed_temperature_increment",
      "Book6ThermodynamicConsistency.printed_and_derived_laws_agree_iff_entropy_static",
      "Book6ThermodynamicConsistency.printed_firstLaw_not_implied_by_freeEnergy_definition",
      "Book6ThermodynamicConsistency.varying_temperature_freeEnergy_increment"
    ]
  },
  "line": 1118,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Thermodynamic Consistency and Potential Orientation",
  "proof_status": "definitional",
  "refs": [],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_confidence_field_operator",
    "definition:bk6_stability_functional_complete"
  ],
  "depends_on": [
    "definition:bk6_confidence_field_operator",
    "definition:bk6_stability_functional_complete"
  ],
  "file": "book6.tex",
  "id": "axiom:bk6_confidence_stability_coupling",
  "label": "axiom:bk6_confidence_stability_coupling",
  "latex_body": "\\begin{axiom}[Certified Confidence--Stability Coupling]\n\\label{axiom:bk6_confidence_stability_coupling}\nLet $\\sigma(t)$ be a differentiable stability trajectory, let\n$u(t)=\\Upsilon_i(t)\\neq0$, and let the confidence Hamiltonian along the\ntrajectory be a differentiable scalar function $h(u)=\\mathcal H_{\\rm conf}(u)$.\nFor a coupling coefficient $\\kappa_\\sigma\\geq0$, the constitutive law is\n\\[\n \\dot\\sigma(t)=-\\kappa_\\sigma\n   \\frac{d}{du}\\left(\\frac{h(u)}{u}\\right)\\bigg|_{u=u(t)}\n =-\\kappa_\\sigma\n   \\frac{h'(u(t))u(t)-h(u(t))}{u(t)^2}.\n\\]\nConsequently, a nonnegative quotient slope gives $\\dot\\sigma\\leq0$, while a\nnegative quotient slope gives $\\dot\\sigma\\geq0$.  In particular, values of\nconfidence and stability alone do not determine their evolution, and the\nouter minus sign is not uniformly stabilizing.  The nonzero coordinate,\ndifferentiability, and displayed update law are part of the coupling\ncertificate rather than consequences of Defs.~\\ref{definition:bk6_confidence_field_operator}\nand \\ref{definition:bk6_stability_functional_complete}.\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "coupling and quotient-slope signs supplied for directional conclusions",
      "explicit quotient-response coupling law",
      "nonzero stability coordinate for reciprocal examples"
    ],
    "countermodels": [
      "Book6ConfidenceStability.values_alone_do_not_force_confidence_stability_coupling"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A typed constitutive certificate supplies the nonzero stability coordinate, coupling orientation, quotient response, and velocity equation. Quotient-slope sign controls velocity sign; values alone do not create dynamics, and a constant positive Hamiltonian exposes the opposite-sign regime."
    ],
    "record_ids": [
      "MAP-BOOK6-054"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book6ConfidenceStability.ConfidenceStabilityCertificate.velocity_nonpos",
      "Book6ConfidenceStability.constant_hamiltonian_gives_positive_stabilityVelocity",
      "Book6ConfidenceStability.stabilityVelocity_eq",
      "Book6ConfidenceStability.stabilityVelocity_nonpos_of_quotientSlope_nonneg",
      "Book6ConfidenceStability.values_alone_do_not_force_confidence_stability_coupling"
    ]
  },
  "line": 1142,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified Confidence--Stability Coupling",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ate, differentiability, and displayed update law are part of the coupling certificate rather than consequences of Defs.~\\ref{definition:bk6_confidence_field_operator} and \\ref{definition:bk6_stability_functional_complete}. \\end{axiom}",
      "label": "definition:bk6_confidence_field_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 970,
      "target_type": "definition"
    },
    {
      "context": "e part of the coupling certificate rather than consequences of Defs.~\\ref{definition:bk6_confidence_field_operator} and \\ref{definition:bk6_stability_functional_complete}. \\end{axiom}",
      "label": "definition:bk6_stability_functional_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 868,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_confidence_field_operator",
    "definition:bk6_stability_functional_complete"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_power_operator"
  ],
  "depends_on": [
    "definition:bk6_power_operator"
  ],
  "file": "book6.tex",
  "id": "axiom:bk6_power_conservation",
  "label": "axiom:bk6_power_conservation",
  "latex_body": "\\begin{axiom}[Power Conservation]\n\\label{axiom:bk6_power_conservation}\nIn closed systems (Def.~\\ref{definition:bk6_power_operator}):\n\\begin{equation}\n\\sum_{i,j} \\mathcal{P}_\\nu(p_i, p_j) = \\mathcal{P}_{\\text{total}} = \\text{const.}\n\\end{equation}\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Models a finite closed system as pairwise power evolving through antisymmetric internal exchange. The exchange double sum is proved zero, hence total pairwise power is conserved at every step. On Fin n this total is definitionally Book 3 metabolicRate, so the proof now establishes the explicit Book 3 -> Book 6 layer: an initial Book 3 Homeostatic band persists through every finite closed-power stage. The manifold geodesic integral defining each power-kernel entry remains outside the certified boundary."
    ],
    "record_ids": [
      "MAP-BOOK6-045"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book6.closedPower_exchange_sum_zero",
      "Book6.closedPower_homeostatic_all",
      "Book6.closedPower_homeostatic_step",
      "Book6.closedPower_total_conserved",
      "Book6.totalPairPower_eq_metabolicRate"
    ]
  },
  "line": 1163,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Power Conservation",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "\\begin{axiom}[Power Conservation] \\label{axiom:bk6_power_conservation} In closed systems (Def.~\\ref{definition:bk6_power_operator}): \\begin{equation} \\sum_{i,j} \\mathcal{P}_\\nu(p_i, p_j) = \\mathcal{P}_{\\text{total}} = \\text{const.} \\end{equation} \\en",
      "label": "definition:bk6_power_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 992,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_power_operator"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [
    "demonstratio:bk7_coherence_fulcrum_power_certainty"
  ],
  "cites": [
    "definition:bk6_reflection_operator_complete"
  ],
  "depends_on": [
    "definition:bk6_reflection_operator_complete"
  ],
  "file": "book6.tex",
  "id": "axiom:bk6_reflective_coherence_complete",
  "label": "axiom:bk6_reflective_coherence_complete",
  "latex_body": "\\begin{axiom}[Reflective Coherence]\n\\label{axiom:bk6_reflective_coherence_complete}\nFor \\(R_\\lambda\\) (Def.~\\ref{definition:bk6_reflection_operator_complete}):\n\\begin{equation}\n\\|R_\\lambda(p) - p\\|_g < \\delta_R \\iff p \\in \\mathcal{E}_R\n\\end{equation}\n\\end{axiom}",
  "line": 1171,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Reflective Coherence",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "\\begin{axiom}[Reflective Coherence] \\label{axiom:bk6_reflective_coherence_complete} For \\(R_\\lambda\\) (Def.~\\ref{definition:bk6_reflection_operator_complete}): \\begin{equation} \\|R_\\lambda(p) - p\\|_g < \\delta_R \\iff p \\in \\mathcal{E}_R \\end{equation} \\end{axiom}",
      "label": "definition:bk6_reflection_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 937,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_reflection_operator_complete"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_drift_operator_complete"
  ],
  "depends_on": [
    "definition:bk6_drift_operator_complete"
  ],
  "file": "book6.tex",
  "id": "axiom:bk6_symbolic_time_irreversibility_complete",
  "label": "axiom:bk6_symbolic_time_irreversibility_complete",
  "latex_body": "\\begin{axiom}[Symbolic Time Irreversibility]\n\\label{axiom:bk6_symbolic_time_irreversibility_complete}\nNo operator $\\mathcal{T}$ exists such that $\\mathcal{T} \\circ D_\\lambda = \\text{Id}_{P_{\\lambda}}$ (Def.~\\ref{definition:bk6_drift_operator_complete}).\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "manifold/curvature/PDE/Hilbert-space/asymptotic content of Book 6 is NOT formalized; static and discrete kernels only",
      "modeling laws (threshold laws, per-step entropy/energy bounds, bridge equivalences) are structure fields or explicit hypotheses"
    ],
    "countermodels": [
      "Book6.no_retraction_of_not_injective"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Proves the fully general fact that a non-injective map admits no left inverse; injectivity-failure of the drift operator D_lambda is the hypothesis under which this is proved, not asserted unconditionally about D_lambda itself."
    ],
    "record_ids": [
      "MAP-BOOK6-017"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book6.no_retraction_of_not_injective"
    ]
  },
  "line": 1179,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Time Irreversibility",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "lity_complete} No operator $\\mathcal{T}$ exists such that $\\mathcal{T} \\circ D_\\lambda = \\text{Id}_{P_{\\lambda}}$ (Def.~\\ref{definition:bk6_drift_operator_complete}). \\end{axiom}",
      "label": "definition:bk6_drift_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 926,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_drift_operator_complete"
  ],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "axiom:bk6_laplace_beltrami_observer_extension",
  "label": "axiom:bk6_laplace_beltrami_observer_extension",
  "latex_body": "\\begin{axiom}[Certified Symbolic Operator Extension under Observer Bounds]\n\\label{axiom:bk6_laplace_beltrami_observer_extension}\nLet $M_{\\mathcal O}\\subseteq\\widetilde M_{\\mathcal O}$ be the\nobserver-admissible subtype.  A coherent extension certificate for the local\nsymbolic Laplace--Beltrami operator $\\Delta_s$ consists of:\n\\begin{enumerate}\n\\item a total operator $\\widetilde\\Delta_{s,\\mathcal O}$ with specified\nfallback or boundary behavior outside $M_{\\mathcal O}$ and exact agreement\nwith $\\Delta_s$ on the admissible subtype;\n\\item normed divergence and entropy defects $e_{\\rm div},e_{\\rm ent}$;\n\\item an explicit $\\varepsilon_{\\mathcal O}\\geq0$ satisfying\n$\\lVert e_{\\rm div}\\rVert\\leq\\varepsilon_{\\mathcal O}$ and\n$\\lVert e_{\\rm ent}\\rVert\\leq\\varepsilon_{\\mathcal O}$.\n\\end{enumerate}\nCurvature boundedness alone does not supply either preservation estimate.  If\nsuccessive observer transports have budgets $\\varepsilon_1,\\ldots,\n\\varepsilon_n$, the certified composite budget is at most\n$\\sum_j\\varepsilon_j$; interface error is accumulated, not reset.\n\\end{axiom}",
  "lean_alignment": {
    "conditions": [
      "explicit observer-admissible domain",
      "local field on the subtype",
      "per-interface approximation certificates",
      "supplied fallback outside the domain"
    ],
    "countermodels": [
      "Book6ObserverExtension.observer_bound_alone_does_not_force_identity_preservation"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "A coherent observer extension now carries an admissible subtype, explicit fallback, exact local agreement, separate divergence/entropy defects, and normed epsilon bounds. Interface errors accumulate; curvature boundedness alone supplies none of these identity certificates."
    ],
    "record_ids": [
      "MAP-BOOK6-055"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book6ObserverExtension.ObserverOperatorExtensionCertificate.exists_total_extension",
      "Book6ObserverExtension.ObserverOperatorExtensionCertificate.joint_defect_bound",
      "Book6ObserverExtension.exists_observerExtension",
      "Book6ObserverExtension.observerExtension_agrees",
      "Book6ObserverExtension.observer_bound_alone_does_not_force_identity_preservation",
      "Book6ObserverExtension.observer_error_accumulates"
    ]
  },
  "line": 1184,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified Symbolic Operator Extension under Observer Bounds",
  "proof_status": "definitional",
  "refs": [],
  "role": "axiom",
  "type": "axiom"
}

{
  "book": "book6",
  "cited_by": [
    "scholium:appC_structural_universality_phi"
  ],
  "cites": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "book6.tex",
  "id": "theorem:bk6_symbolic_diffusion_governs_evolution",
  "label": "theorem:bk6_symbolic_diffusion_governs_evolution",
  "latex_body": "\\begin{theorem}[Symbolic Diffusion Operator Governs Thermodynamic Evolution]\n\\label{theorem:bk6_symbolic_diffusion_governs_evolution}\nOn the symbolic manifold $(M, g)$ with drift field $D$ and symbolic probability density $\\rho$, the Laplace–Beltrami operator $\\Delta_s$ governs diffusion in the symbolic Fokker–Planck equation:\n\\[\n\\frac{\\partial \\rho}{\\partial s} = -\\nabla \\cdot (\\rho D) + \\beta^{-1} \\Delta_s \\rho.\n\\]\nThis operator ensures probability conservation, smooth evolution, and entropy production across symbolic flows (see thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}).\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "The manifold Fokker-Planck PDE (with Laplace-Beltrami diffusion term) is NOT certified; the discrete skeleton -- multi-step evolution under composed row-stochastic kernels conserves density -- is, per Book2's existing certified Gibbs-kernel apparatus."
    ],
    "record_ids": [
      "MAP-BOOK6-042"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book68B.composedEvolution_isDensity",
      "Book68B.evolve_comp",
      "Book68B.isStochastic_mul"
    ]
  },
  "line": 1204,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Diffusion Operator Governs Thermodynamic Evolution",
  "proof_labels": [
    "proof:bk6_symbolic_diffusion_governs_evolution"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "This operator ensures probability conservation, smooth evolution, and entropy production across symbolic flows (see thm~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}). \\end{theorem}",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "depends_on": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "file": "book6.tex",
  "id": "proof:bk6_symbolic_diffusion_governs_evolution",
  "label": "proof:bk6_symbolic_diffusion_governs_evolution",
  "latex_body": "\\begin{proof}\n\\label{proof:bk6_symbolic_diffusion_governs_evolution}\n\\leavevmode\nThe proven fundamental Fokker--Planck relation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) establishes that a symbolic density evolves by a drift term plus a second-order diffusion term. On a Riemannian symbolic manifold $(M,g)$ the intrinsic, coordinate-free second-order diffusion generator is the Laplace--Beltrami operator $\\Delta_s$, so the diffusion term is $\\beta^{-1}\\Delta_s\\rho$ and the evolution reads\n\\[\n\\frac{\\partial\\rho}{\\partial s}=-\\nabla\\cdot(\\rho D)+\\beta^{-1}\\Delta_s\\rho .\n\\]\nThe three asserted properties follow. \\emph{Probability conservation}: both $\\nabla\\cdot(\\rho D)$ and $\\Delta_s\\rho=\\nabla\\cdot(\\nabla\\rho)$ are divergences, so by the divergence theorem $\\tfrac{d}{ds}\\int_M\\rho\\,d\\mu_g=-\\int_M\\nabla\\cdot(\\rho D-\\beta^{-1}\\nabla\\rho)\\,d\\mu_g=0$ on closed $M$. \\emph{Smooth evolution}: $\\Delta_s$ is a uniformly elliptic second-order operator, so the equation is parabolic and instantaneously smoothing. \\emph{Entropy production}: the diffusion term drives the standard $H$-theorem dissipation of symbolic free energy. Hence $\\Delta_s$ is precisely the operator governing symbolic thermodynamic diffusion.\n\\end{proof}",
  "line": 1212,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "theorem:bk6_symbolic_diffusion_governs_evolution",
  "ref_roles": [
    {
      "context": "\\label{proof:bk6_symbolic_diffusion_governs_evolution} \\leavevmode The proven fundamental Fokker--Planck relation (Thm.~\\ref{theorem:bk1_fundamental_relation_fokker_plank_equation}) establishes that a symbolic density evolves by a drift term plus a second-order diffusion term. On a Riemannian symbol",
      "label": "theorem:bk1_fundamental_relation_fokker_plank_equation",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 3098,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk1_fundamental_relation_fokker_plank_equation"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "subsec:bk6_canonical_operator_algebra",
  "label": "subsec:bk6_canonical_operator_algebra",
  "latex_body": "",
  "line": 1222,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Canonical Operator Algebra",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book6",
  "cited_by": [
    "assumption:bk6_canonical_density",
    "proof:bk6_complete_operator_closure",
    "theorem:bk6_complete_operator_closure"
  ],
  "cites": [
    "definition:bk6_drift_operator_complete",
    "definition:bk6_symbolic_hamiltonian_complete"
  ],
  "depends_on": [
    "definition:bk6_drift_operator_complete",
    "definition:bk6_symbolic_hamiltonian_complete"
  ],
  "file": "book6.tex",
  "id": "definition:bk6_complete_canonical_set",
  "label": "definition:bk6_complete_canonical_set",
  "latex_body": "\\begin{definition}[Complete Canonical Set]\n\\label{definition:bk6_complete_canonical_set}\nThe \\emph{complete canonical operator set}, assembled from the operator definitions above (e.g., Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_symbolic_hamiltonian_complete}), is:\n\\begin{equation}\n\\mathcal{C}_{\\text{ext}} = \\{D_\\lambda, R_\\lambda, T_\\alpha, \\mathcal{B}_\\lambda, \\mathcal{M}_\\lambda, \\Omega_\\delta, \\mathcal{G}, \\mathcal{C}_\\sigma, \\mathcal{P}_\\nu, \\mathcal{R}_B, \\Pi_s, \\mathcal{H}_s, \\Phi_t, \\Delta_s\\}\n\\end{equation}\n\\end{definition}",
  "line": 1225,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Complete Canonical Set",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "e_canonical_set} The \\emph{complete canonical operator set}, assembled from the operator definitions above (e.g., Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_symbolic_hamiltonian_complete}), is: \\begin{equation} \\mathcal{C}_{\\text{ext}} = \\{D_\\lambda, R_\\l",
      "label": "definition:bk6_drift_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 926,
      "target_type": "definition"
    },
    {
      "context": "operator set}, assembled from the operator definitions above (e.g., Defs.~\\ref{definition:bk6_drift_operator_complete}, \\ref{definition:bk6_symbolic_hamiltonian_complete}), is: \\begin{equation} \\mathcal{C}_{\\text{ext}} = \\{D_\\lambda, R_\\lambda, T_\\alpha, \\mathcal{B}_\\lambda, \\mathcal{M}_\\l",
      "label": "definition:bk6_symbolic_hamiltonian_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1079,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_drift_operator_complete",
    "definition:bk6_symbolic_hamiltonian_complete"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book6",
  "cited_by": [
    "proof:bk6_complete_operator_closure"
  ],
  "cites": [
    "definition:bk6_complete_canonical_set"
  ],
  "depends_on": [
    "definition:bk6_complete_canonical_set"
  ],
  "file": "book6.tex",
  "id": "assumption:bk6_canonical_density",
  "label": "assumption:bk6_canonical_density",
  "latex_body": "\\begin{assumption}[Canonical Approximation Completeness]\n\\label{assumption:bk6_canonical_density}\nThe canonical set $\\mathcal{C}_{\\text{ext}}$ is an approximating family for the operator algebra it generates (Def.~\\ref{definition:bk6_complete_canonical_set}): the linear span of $\\mathcal{C}_{\\text{ext}}$ is dense, in the operator norm $\\|\\cdot\\|_{\\text{op}}$, in the closure of the algebra generated by composition of its elements. Equivalently, every generated operator is approximable to arbitrary precision by a finite canonical combination. This is the structural completeness hypothesis underlying the ``arbitrarily small $\\epsilon$'' closure --- what the canonical set is constructed to satisfy, not a measured fact.\n\\end{assumption}",
  "line": 1233,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Canonical Approximation Completeness",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "ty} The canonical set $\\mathcal{C}_{\\text{ext}}$ is an approximating family for the operator algebra it generates (Def.~\\ref{definition:bk6_complete_canonical_set}): the linear span of $\\mathcal{C}_{\\text{ext}}$ is dense, in the operator norm $\\|\\cdot\\|_{\\text{op}}$, in the closure",
      "label": "definition:bk6_complete_canonical_set",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1225,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_complete_canonical_set"
  ],
  "role": "assumption",
  "type": "assumption"
}

{
  "book": "book6",
  "cited_by": [
    "subsec:bk6_scholium_on_symbolic_operator_mechanics"
  ],
  "cites": [
    "definition:bk6_complete_canonical_set",
    "definition:bk6_modulation_operator_complete",
    "definition:bk6_symbolic_flow_operator_complete"
  ],
  "depends_on": [
    "assumption:bk6_canonical_density",
    "definition:bk6_complete_canonical_set",
    "definition:bk6_modulation_operator_complete",
    "definition:bk6_symbolic_flow_operator_complete"
  ],
  "file": "book6.tex",
  "id": "theorem:bk6_complete_operator_closure",
  "label": "theorem:bk6_complete_operator_closure",
  "latex_body": "\\begin{theorem}[Complete Operator Closure]\n\\label{theorem:bk6_complete_operator_closure}\nThe canonical set $\\mathcal{C}_{\\text{ext}}$ forms a closed algebra under composition:\n\\begin{equation}\n\\forall \\mathcal{O}_1, \\mathcal{O}_2 \\in \\mathcal{C}_{\\text{ext}}, \\; \\exists \\{c_k, \\mathcal{O}_k\\}_{k=1}^n : \\mathcal{O}_1 \\circ \\mathcal{O}_2 = \\sum_{k=1}^n c_k \\mathcal{O}_k + \\mathcal{E}\n\\end{equation}\nwhere $\\|\\mathcal{E}\\|_{\\text{op}} < \\epsilon$ for arbitrarily small $\\epsilon > 0$.\nSee Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, and Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Specialized to the row-stochastic-matrix instance of the canonical operator set: closure under composition is proved EXACTLY (no error term E), strictly stronger than the source's closure-up-to-epsilon claim -- an honesty gap noted rather than hidden."
    ],
    "record_ids": [
      "MAP-BOOK6-043"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book68B.isStochastic_mul"
    ]
  },
  "line": 1238,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Complete Operator Closure",
  "proof_labels": [
    "proof:bk6_complete_operator_closure"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "mathcal{E} \\end{equation} where $\\|\\mathcal{E}\\|_{\\text{op}} < \\epsilon$ for arbitrarily small $\\epsilon > 0$. See Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, and Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}.",
      "label": "definition:bk6_complete_canonical_set",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1225,
      "target_type": "definition"
    },
    {
      "context": "\\text{op}} < \\epsilon$ for arbitrarily small $\\epsilon > 0$. See Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, and Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}. \\end{theorem}",
      "label": "definition:bk6_modulation_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1046,
      "target_type": "definition"
    },
    {
      "context": "See Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, and Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}. \\end{theorem}",
      "label": "definition:bk6_symbolic_flow_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1058,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_complete_canonical_set",
    "definition:bk6_modulation_operator_complete",
    "definition:bk6_symbolic_flow_operator_complete"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "assumption:bk6_canonical_density",
    "definition:bk6_complete_canonical_set",
    "definition:bk6_modulation_operator_complete",
    "definition:bk6_symbolic_flow_operator_complete"
  ],
  "depends_on": [
    "assumption:bk6_canonical_density",
    "definition:bk6_complete_canonical_set",
    "definition:bk6_modulation_operator_complete",
    "definition:bk6_symbolic_flow_operator_complete"
  ],
  "file": "book6.tex",
  "id": "proof:bk6_complete_operator_closure",
  "label": "proof:bk6_complete_operator_closure",
  "latex_body": "\\begin{proof}\n\\label{proof:bk6_complete_operator_closure}\n\\leavevmode\nLet $\\mathcal{O}_1,\\mathcal{O}_2\\in\\mathcal{C}_{\\text{ext}}$. Their composition lies, by construction, in the algebra generated by $\\mathcal{C}_{\\text{ext}}$ under composition (Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}). By Canonical Approximation Completeness (Assumption~\\ref{assumption:bk6_canonical_density}) the linear span of $\\mathcal{C}_{\\text{ext}}$ is dense in this generated algebra under $\\|\\cdot\\|_{\\text{op}}$. Hence for any prescribed $\\epsilon>0$ there is a finite canonical combination $\\sum_{k=1}^n c_k\\mathcal{O}_k$ with\n\\[\n\\Big\\|\\,\\mathcal{O}_1\\circ\\mathcal{O}_2-\\textstyle\\sum_{k=1}^n c_k\\mathcal{O}_k\\,\\Big\\|_{\\text{op}}=\\|\\mathcal{E}\\|_{\\text{op}}<\\epsilon .\n\\]\nSince $\\epsilon$ was arbitrary, $\\mathcal{C}_{\\text{ext}}$ is closed under composition up to arbitrarily small operator-norm remainder, i.e.\\ it forms a norm-closed algebra in the stated sense.\n\\end{proof}",
  "line": 1247,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "theorem:bk6_complete_operator_closure",
  "ref_roles": [
    {
      "context": "plete}, Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}). By Canonical Approximation Completeness (Assumption~\\ref{assumption:bk6_canonical_density}) the linear span of $\\mathcal{C}_{\\text{ext}}$ is dense in this generated algebra under $\\|\\cdot\\|_{\\text{op}}$. Hence",
      "label": "assumption:bk6_canonical_density",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1233,
      "target_type": "assumption"
    },
    {
      "context": "Their composition lies, by construction, in the algebra generated by $\\mathcal{C}_{\\text{ext}}$ under composition (Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}). By",
      "label": "definition:bk6_complete_canonical_set",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1225,
      "target_type": "definition"
    },
    {
      "context": "gebra generated by $\\mathcal{C}_{\\text{ext}}$ under composition (Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}). By Canonical Approximation Completeness (Assumption~\\ref{a",
      "label": "definition:bk6_modulation_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1046,
      "target_type": "definition"
    },
    {
      "context": "osition (Def.~\\ref{definition:bk6_complete_canonical_set}, Def.~\\ref{definition:bk6_modulation_operator_complete}, Def.~\\ref{definition:bk6_symbolic_flow_operator_complete}). By Canonical Approximation Completeness (Assumption~\\ref{assumption:bk6_canonical_density}) the linear span of $\\math",
      "label": "definition:bk6_symbolic_flow_operator_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1058,
      "target_type": "definition"
    }
  ],
  "refs": [
    "assumption:bk6_canonical_density",
    "definition:bk6_complete_canonical_set",
    "definition:bk6_modulation_operator_complete",
    "definition:bk6_symbolic_flow_operator_complete"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book6",
  "cited_by": [
    "subsec:bk6_scholium_on_symbolic_operator_mechanics"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "theorem:bk6_thermodynamic_map_duality",
  "label": "theorem:bk6_thermodynamic_map_duality",
  "latex_body": "\\begin{theorem}[Certified Thermodynamic--MAP Balance]\n\\label{theorem:bk6_thermodynamic_map_duality}\nLet $T_s\\neq0$ and suppose the MAP equilibrium is supplied with the averaged\nconstitutive balance\n\\[\n \\langle D_\\lambda\\rangle_{\\rho_s}\n -\\langle R_\\lambda-\\mathrm{Id}\\rangle_{\\rho_s}\n =T_s^{-1}\\langle\\Pi_s\\nabla_sV_s\\rangle_{\\rho_s}.\n\\]\nThen\n\\[\n \\langle D_\\lambda\\rangle_{\\rho_s}\n =T_s^{-1}\\langle\\Pi_s\\nabla_sV_s\\rangle_{\\rho_s}\n  +\\langle R_\\lambda-\\mathrm{Id}\\rangle_{\\rho_s}.\n\\]\nStationarity, conservation, time orientation, and coherent observer extension\nmay be used to derive the constitutive premise in a specified dynamical model,\nbut their names alone do not determine its coefficient or sign.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "the mean drift decomposes constitutively into inverse-temperature projected force plus reflective deviation"
    ],
    "countermodels": [
      "Book6ThermodynamicMAP.equilibrium_flags_alone_do_not_force_duality"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "An explicit averaged constitutive balance derives the oriented thermodynamic--MAP identity and zero residual. Qualitative equilibrium and conservation flags remain countermodels to deriving its inverse-temperature coefficient or reflective sign by narration alone."
    ],
    "record_ids": [
      "MAP-BOOK6-057"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book6ThermodynamicMAP.AveragedConstitutiveLaw.dualityResidual_eq_zero",
      "Book6ThermodynamicMAP.AveragedConstitutiveLaw.thermodynamicMAPDuality",
      "Book6ThermodynamicMAP.duality_iff_residual_zero",
      "Book6ThermodynamicMAP.duality_of_reflectionDeviation_eq",
      "Book6ThermodynamicMAP.equilibrium_flags_alone_do_not_force_duality",
      "Book6ThermodynamicMAP.reflectionDeviation_eq"
    ]
  },
  "line": 1257,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified Thermodynamic--MAP Balance",
  "proof_labels": [
    "proof:bk6_thermodynamic_map_duality"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "proof:bk6_thermodynamic_map_duality",
  "label": "proof:bk6_thermodynamic_map_duality",
  "latex_body": "\\begin{proof}[Oriented Mean-Balance Rearrangement]\n\\label{proof:bk6_thermodynamic_map_duality}\n\\leavevmode\nAdd $\\langle R_\\lambda-\\mathrm{Id}\\rangle_{\\rho_s}$ to both sides of the\nsupplied averaged constitutive balance.  This yields the displayed identity\nwithout changing the inverse-temperature coefficient or the orientation of\nthe reflective deviation.  Without that balance, the qualitative equilibrium\nflags admit numerical countermodels to the conclusion.\n\\end{proof}",
  "line": 1276,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Oriented Mean-Balance Rearrangement",
  "proves": "theorem:bk6_thermodynamic_map_duality",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book6",
  "cited_by": [
    "proof:bk6_confidence_power_bound"
  ],
  "cites": [
    "definition:bk6_power_operator"
  ],
  "depends_on": [
    "definition:bk6_power_operator"
  ],
  "file": "book6.tex",
  "id": "assumption:bk6_gaussian_locality",
  "label": "assumption:bk6_gaussian_locality",
  "latex_body": "\\begin{assumption}[Gaussian Locality of Symbolic Power]\n\\label{assumption:bk6_gaussian_locality}\nThe power operator $\\mathcal{P}_\\nu$ (Def.~\\ref{definition:bk6_power_operator}) is Gaussian-localized in geodesic distance: there is a confidence correlation length $\\lambda_{\\text{conf}}>0$ and a peak amplitude $P_0$ with\n\\[\n\\mathcal{P}_\\nu(p,p')\\le P_0\\,\\exp\\!\\Big(-\\frac{d_g(p,p')^2}{2\\lambda_{\\text{conf}}^2}\\Big).\n\\]\nThis sub-Gaussian decay --- regulatory influence falling off Gaussian-fast beyond the correlation length --- is a structural hypothesis on the power kernel, not inferred from sampled traces.\n\\end{assumption}",
  "line": 1286,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Gaussian Locality of Symbolic Power",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "aussian Locality of Symbolic Power] \\label{assumption:bk6_gaussian_locality} The power operator $\\mathcal{P}_\\nu$ (Def.~\\ref{definition:bk6_power_operator}) is Gaussian-localized in geodesic distance: there is a confidence correlation length $\\lambda_{\\text{conf}}>0$ and a p",
      "label": "definition:bk6_power_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 992,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_power_operator"
  ],
  "role": "assumption",
  "type": "assumption"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_confidence_field_operator",
    "definition:bk6_power_operator"
  ],
  "depends_on": [
    "assumption:bk6_gaussian_locality",
    "definition:bk6_confidence_field_operator",
    "definition:bk6_power_operator"
  ],
  "file": "book6.tex",
  "id": "theorem:bk6_confidence_power_bound",
  "label": "theorem:bk6_confidence_power_bound",
  "latex_body": "\\begin{theorem}[Confidence-Power Bound]\n\\label{theorem:bk6_confidence_power_bound}\n\\begin{equation}\n\\sigma(p) \\cdot \\mathcal{P}_\\nu(p, p') \\leq \\mathcal{P}_{\\max} \\cdot \\exp\\left(-\\frac{d_g(p,p')^2}{2\\lambda_{\\text{conf}}^2}\\right)\n\\end{equation}\nwith \\(\\sigma\\) and \\(\\mathcal{P}_\\nu\\) from Defs.~\\ref{definition:bk6_confidence_field_operator} and \\ref{definition:bk6_power_operator}.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Derives the stated confidence-weighted Gaussian bound from the manuscript's Gaussian-locality hypothesis, nonnegative peak amplitude, positive correlation length, and the confidence codomain 0 <= sigma <= 1. The geodesic distance is modeled by an arbitrary pseudometric space."
    ],
    "record_ids": [
      "MAP-BOOK6-046"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book6.confidencePower_gaussian_bound"
    ]
  },
  "line": 1295,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Confidence-Power Bound",
  "proof_labels": [
    "proof:bk6_confidence_power_bound"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "(-\\frac{d_g(p,p')^2}{2\\lambda_{\\text{conf}}^2}\\right) \\end{equation} with \\(\\sigma\\) and \\(\\mathcal{P}_\\nu\\) from Defs.~\\ref{definition:bk6_confidence_field_operator} and \\ref{definition:bk6_power_operator}. \\end{theorem}",
      "label": "definition:bk6_confidence_field_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 970,
      "target_type": "definition"
    },
    {
      "context": "t) \\end{equation} with \\(\\sigma\\) and \\(\\mathcal{P}_\\nu\\) from Defs.~\\ref{definition:bk6_confidence_field_operator} and \\ref{definition:bk6_power_operator}. \\end{theorem}",
      "label": "definition:bk6_power_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 992,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_confidence_field_operator",
    "definition:bk6_power_operator"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "assumption:bk6_gaussian_locality",
    "definition:bk6_confidence_field_operator"
  ],
  "depends_on": [
    "assumption:bk6_gaussian_locality",
    "definition:bk6_confidence_field_operator"
  ],
  "file": "book6.tex",
  "id": "proof:bk6_confidence_power_bound",
  "label": "proof:bk6_confidence_power_bound",
  "latex_body": "\\begin{proof}\n\\label{proof:bk6_confidence_power_bound}\n\\leavevmode\nBy Def.~\\ref{definition:bk6_confidence_field_operator} the confidence field is bounded, $0\\le\\sigma(p)\\le 1$. By Gaussian Locality (Assumption~\\ref{assumption:bk6_gaussian_locality}) the power kernel obeys $\\mathcal{P}_\\nu(p,p')\\le P_0\\,e^{-d_g(p,p')^2/2\\lambda_{\\text{conf}}^2}$. Multiplying the two and setting $\\mathcal{P}_{\\max}:=P_0\\,\\sup_p\\sigma(p)\\le P_0$,\n\\[\n\\sigma(p)\\,\\mathcal{P}_\\nu(p,p')\\le\\sigma(p)\\,P_0\\,e^{-d_g(p,p')^2/2\\lambda_{\\text{conf}}^2}\\le\\mathcal{P}_{\\max}\\,e^{-d_g(p,p')^2/2\\lambda_{\\text{conf}}^2},\n\\]\nthe stated bound. The envelope is sharp at coincidence ($d_g\\to 0$, kernel $\\to 1$), and shows regulatory influence decays at least Gaussian-fast in geodesic separation, with range set by the confidence correlation length $\\lambda_{\\text{conf}}$.\n\\end{proof}",
  "line": 1302,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "theorem:bk6_confidence_power_bound",
  "ref_roles": [
    {
      "context": ":bk6_confidence_field_operator} the confidence field is bounded, $0\\le\\sigma(p)\\le 1$. By Gaussian Locality (Assumption~\\ref{assumption:bk6_gaussian_locality}) the power kernel obeys $\\mathcal{P}_\\nu(p,p')\\le P_0\\,e^{-d_g(p,p')^2/2\\lambda_{\\text{conf}}^2}$. Multiplying the two",
      "label": "assumption:bk6_gaussian_locality",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 1286,
      "target_type": "assumption"
    },
    {
      "context": "\\begin{proof} \\label{proof:bk6_confidence_power_bound} \\leavevmode By Def.~\\ref{definition:bk6_confidence_field_operator} the confidence field is bounded, $0\\le\\sigma(p)\\le 1$. By Gaussian Locality (Assumption~\\ref{assumption:bk6_gaussian_lo",
      "label": "definition:bk6_confidence_field_operator",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 970,
      "target_type": "definition"
    }
  ],
  "refs": [
    "assumption:bk6_gaussian_locality",
    "definition:bk6_confidence_field_operator"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "lemma:bk6_grace_basin_correspondence",
  "label": "lemma:bk6_grace_basin_correspondence",
  "latex_body": "\\begin{lemma}[Certified Grace--Basin Correspondence]\n\\label{lemma:bk6_grace_basin_correspondence}\nAssume the regulatory basins cover every subcritical state,\n\\[\n \\Upsilon_i(p,p)<\\gamma_{\\rm crit}\n \\Longrightarrow\n \\exists q\\in\\mathcal E_R:\\ p\\in\\mathcal R_B(q),\n\\]\nand that grace preserves every admitted basin,\n\\[\n q\\in\\mathcal E_R\n \\Longrightarrow\n \\mathcal G(\\mathcal R_B(q))\\subseteq\\mathcal R_B(q).\n\\]\nThen\n\\[\n \\mathcal{G}(p)\\in\\bigcup_{q\\in\\mathcal E_R}\\mathcal R_B(q)\n \\qquad\\text{whenever}\\qquad\n \\Upsilon_i(p,p)<\\gamma_{\\rm crit}.\n\\]\nNeither subcriticality without coverage nor coverage without forward\ninvariance suffices.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "every subcritical state belongs to an admitted regulatory basin",
      "grace maps each admitted regulatory basin into itself"
    ],
    "countermodels": [
      "Book6GraceBasin.coverage_without_invariance_does_not_force_grace_membership",
      "Book6GraceBasin.subcriticality_alone_does_not_force_basin_membership"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Subcritical basin coverage supplies a named admitted basin and forward invariance carries the grace image back into it. Empty-basin and exiting-grace countermodels prove that neither thresholding nor coverage alone suffices."
    ],
    "record_ids": [
      "MAP-BOOK6-056"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book6GraceBasin.coverage_without_invariance_does_not_force_grace_membership",
      "Book6GraceBasin.grace_mem_regulatoryUnion",
      "Book6GraceBasin.grace_stays_in_basin",
      "Book6GraceBasin.subcriticality_alone_does_not_force_basin_membership"
    ]
  },
  "line": 1312,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified Grace--Basin Correspondence",
  "proof_labels": [
    "proof:bk6_grace_basin_correspondence"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "proof:bk6_grace_basin_correspondence",
  "label": "proof:bk6_grace_basin_correspondence",
  "latex_body": "\\begin{proof}[Coverage followed by Grace Invariance]\n\\label{proof:bk6_grace_basin_correspondence}\n\\leavevmode\nFor subcritical $p$, basin coverage supplies $q\\in\\mathcal E_R$ with\n$p\\in\\mathcal R_B(q)$.  Forward invariance then gives\n$\\mathcal G(p)\\in\\mathcal R_B(q)$, hence membership in the displayed union.\nEmpty basins refute the conclusion from subcriticality alone, while a grace\nmap that exits a covered basin refutes it without invariance.\n\\end{proof}",
  "line": 1335,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Coverage followed by Grace Invariance",
  "proves": "lemma:bk6_grace_basin_correspondence",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book6",
  "cited_by": [
    "subsec:bk6_extensions_and_future_directions"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "subsec:bk6_commutation_relations",
  "label": "subsec:bk6_commutation_relations",
  "latex_body": "",
  "line": 1345,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Commutation Relations",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "subsubsec:bk6_essential_non_commuting_pairs",
  "label": "subsubsec:bk6_essential_non_commuting_pairs",
  "latex_body": "",
  "line": 1348,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Essential Non-Commuting Pairs",
  "role": "section",
  "subtype": "subsubsection",
  "type": "section"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_symbolic_flow_operator_complete",
    "definition:bk6_symbolic_laplace_beltrami_operator_complete"
  ],
  "depends_on": [
    "definition:bk6_symbolic_flow_operator_complete",
    "definition:bk6_symbolic_laplace_beltrami_operator_complete"
  ],
  "file": "book6.tex",
  "id": "subsubsec:bk6_commuting_families",
  "label": "subsubsec:bk6_commuting_families",
  "latex_body": "",
  "line": 1356,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Commuting Families",
  "ref_roles": [
    {
      "context": "",
      "label": "definition:bk6_symbolic_flow_operator_complete",
      "logical_support": false,
      "role": "navigation",
      "target_file": "book6.tex",
      "target_line": 1058,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "definition:bk6_symbolic_laplace_beltrami_operator_complete",
      "logical_support": false,
      "role": "navigation",
      "target_file": "book6.tex",
      "target_line": 1066,
      "target_type": "definition"
    }
  ],
  "role": "section",
  "subtype": "subsubsection",
  "type": "section"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book6.tex",
  "id": "subsec:bk6_conservation_laws_invariants",
  "label": "subsec:bk6_conservation_laws_invariants",
  "latex_body": "",
  "line": 1364,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conservation Laws and Invariants",
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book6",
  "cited_by": [
    "subsec:bk6_scholium_on_symbolic_operator_mechanics"
  ],
  "cites": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "depends_on": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "file": "book6.tex",
  "id": "proposition:bk6_symbolic_charge_conservation",
  "label": "proposition:bk6_symbolic_charge_conservation",
  "latex_body": "\\begin{proposition}[Symbolic Charge Conservation]\n\\label{proposition:bk6_symbolic_charge_conservation}\n\\begin{equation}\nQ_s[p] = \\int_M \\text{Im}(\\Phi_s^*(p,x) \\nabla_s \\Phi_s(p,x)) \\, d\\mu_g(x) = \\text{const.}\n\\end{equation}\nwith \\(\\Phi_s\\) from Def.~\\ref{definition:bk6_symbolic_state_function_complete}.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "continuum charge/action integrals stay open; the discrete conservation mechanism is certified"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Discrete conservation kernel: a step-invariant charge is constant along the orbit; the continuum charge integral stays open."
    ],
    "record_ids": [
      "MAP-BOOK6-047"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Conservation.conserved_along_orbit"
    ]
  },
  "line": 1367,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Symbolic Charge Conservation",
  "proof_labels": [
    "proof:bk6_symbolic_charge_conservation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "t_M \\text{Im}(\\Phi_s^*(p,x) \\nabla_s \\Phi_s(p,x)) \\, d\\mu_g(x) = \\text{const.} \\end{equation} with \\(\\Phi_s\\) from Def.~\\ref{definition:bk6_symbolic_state_function_complete}. \\end{proposition}",
      "label": "definition:bk6_symbolic_state_function_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 849,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "depends_on": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "file": "book6.tex",
  "id": "proof:bk6_symbolic_charge_conservation",
  "label": "proof:bk6_symbolic_charge_conservation",
  "latex_body": "\\begin{proof}\n\\label{proof:bk6_symbolic_charge_conservation}\n\\leavevmode\nThe integrand $\\mathrm{Im}(\\Phi_s^*\\nabla_s\\Phi_s)$ is the Noether charge density of the global phase symmetry $\\Phi_s\\mapsto e^{i\\theta}\\Phi_s$ of the symbolic state function (Def.~\\ref{definition:bk6_symbolic_state_function_complete}): the symbolic action governing $\\Phi_s$ depends only on $|\\Phi_s|$ and its derivatives, hence is invariant under constant phase rotation. By Noether's first theorem this $U(1)$ invariance yields a conserved current $j_s$ with $\\nabla_\\mu j_s^\\mu=0$, whose component integrates to $Q_s[p]=\\int_M\\mathrm{Im}(\\Phi_s^*\\nabla_s\\Phi_s)\\,d\\mu_g$. Integrating the continuity equation over the closed manifold $M$, the spatial divergence contributes zero net flux by the divergence theorem, so $\\tfrac{d}{ds}Q_s[p]=0$. Therefore the symbolic charge is conserved.\n\\end{proof}",
  "line": 1374,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "proposition:bk6_symbolic_charge_conservation",
  "ref_roles": [
    {
      "context": "ther charge density of the global phase symmetry $\\Phi_s\\mapsto e^{i\\theta}\\Phi_s$ of the symbolic state function (Def.~\\ref{definition:bk6_symbolic_state_function_complete}): the symbolic action governing $\\Phi_s$ depends only on $|\\Phi_s|$ and its derivatives, hence is invariant under const",
      "label": "definition:bk6_symbolic_state_function_complete",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 849,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_symbolic_state_function_complete"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book6",
  "cited_by": [
    "subsec:bk6_scholium_on_symbolic_operator_mechanics"
  ],
  "cites": [
    "definition:bk6_symbolic_free_energy_functional"
  ],
  "depends_on": [
    "definition:bk6_symbolic_free_energy_functional"
  ],
  "file": "book6.tex",
  "id": "proposition:bk6_total_symbolic_action_conservation",
  "label": "proposition:bk6_total_symbolic_action_conservation",
  "latex_body": "\\begin{proposition}[Total Symbolic Action Conservation]\n\\label{proposition:bk6_total_symbolic_action_conservation}\n\\begin{equation}\n\\mathcal{A}_s = \\int dt \\, \\mathcal{L}_s = \\int dt \\left( \\mathcal{T}_s - \\mathcal{F}_\\lambda \\right) = \\text{const.}\n\\end{equation}\nwhere $\\mathcal{T}_s$ is symbolic kinetic energy.\nThe free-energy term is given by Def.~\\ref{definition:bk6_symbolic_free_energy_functional}.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "continuum charge/action integrals stay open; the discrete conservation mechanism is certified"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Total symbolic action as a conserved charge; the action integral stays open."
    ],
    "record_ids": [
      "MAP-BOOK6-048"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Conservation.conserved_along_orbit"
    ]
  },
  "line": 1380,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Total Symbolic Action Conservation",
  "proof_labels": [
    "proof:bk6_total_symbolic_action_conservation"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "= \\text{const.} \\end{equation} where $\\mathcal{T}_s$ is symbolic kinetic energy. The free-energy term is given by Def.~\\ref{definition:bk6_symbolic_free_energy_functional}. \\end{proposition}",
      "label": "definition:bk6_symbolic_free_energy_functional",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 906,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_symbolic_free_energy_functional"
  ],
  "role": "proposition",
  "type": "proposition"
}

{
  "book": "book6",
  "cited_by": [],
  "cites": [
    "definition:bk6_symbolic_free_energy_functional"
  ],
  "depends_on": [
    "definition:bk6_symbolic_free_energy_functional"
  ],
  "file": "book6.tex",
  "id": "proof:bk6_total_symbolic_action_conservation",
  "label": "proof:bk6_total_symbolic_action_conservation",
  "latex_body": "\\begin{proof}\n\\label{proof:bk6_total_symbolic_action_conservation}\n\\leavevmode\nThe symbolic Lagrangian $\\mathcal{L}_s=\\mathcal{T}_s-\\mathcal{F}_\\lambda$ (Def.~\\ref{definition:bk6_symbolic_free_energy_functional}) carries no explicit symbolic-time dependence, $\\partial\\mathcal{L}_s/\\partial t=0$. By Noether's theorem this time-translation symmetry yields a conserved symbolic energy $\\mathcal{H}_s=\\mathcal{T}_s+\\mathcal{F}_\\lambda$, constant along every trajectory of the canonical dynamics. Restrict to a closed regulatory orbit of period $\\tau$: the accumulated action $\\mathcal{A}_s=\\oint\\mathcal{L}_s\\,dt$ is the action variable of the autonomous system, and by Liouville's theorem the Hamiltonian flow preserves the enclosed phase-space area, so the loop action is an adiabatic invariant --- unchanged along the motion and under slow variation of the regulatory parameters. Hence $\\mathcal{A}_s=\\int dt\\,(\\mathcal{T}_s-\\mathcal{F}_\\lambda)$ is constant along closed symbolic orbits.\n\\end{proof}",
  "line": 1388,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "proves": "proposition:bk6_total_symbolic_action_conservation",
  "ref_roles": [
    {
      "context": "mbolic_action_conservation} \\leavevmode The symbolic Lagrangian $\\mathcal{L}_s=\\mathcal{T}_s-\\mathcal{F}_\\lambda$ (Def.~\\ref{definition:bk6_symbolic_free_energy_functional}) carries no explicit symbolic-time dependence, $\\partial\\mathcal{L}_s/\\partial t=0$. By Noether's theorem this time-tra",
      "label": "definition:bk6_symbolic_free_energy_functional",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 906,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk6_symbolic_free_energy_functional"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book6",
  "cited_by": [
    "subsec:bk6_scholium_on_symbolic_operator_mechanics"
  ],
  "cites": [
    "axiom:bk6_map_equilibrium_invariance_complete"
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  "label": "proposition:bk6_map_invariant",
  "latex_body": "\\begin{proposition}[MAP Invariant]\n\\label{proposition:bk6_map_invariant}\n\\begin{equation}\n\\mathcal{M}_{\\text{MAP}}[p] = \\mathcal{F}_\\lambda[p] + \\alpha \\Upsilon_i(p,p) + \\beta \\sigma(p) = \\text{const.}\n\\end{equation}\nalong closed regulatory orbits.\nThis is the invariant form of Axiom~\\ref{axiom:bk6_map_equilibrium_invariance_complete}.\n\\end{proposition}",
  "lean_alignment": {
    "conditions": [
      "continuum charge/action integrals stay open; the discrete conservation mechanism is certified"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "The MAP invariant F + alpha*Upsilon + beta*sigma is conserved along regulatory orbits, with additive/scalar composition; stronger than the closed-orbit form the source assumes."
    ],
    "record_ids": [
      "MAP-BOOK6-049"
    ],
    "statuses": [
      "exact"
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    "witnesses": [
      "Conservation.IsConserved.add",
      "Conservation.IsConserved.smul",
      "Conservation.conserved_closed_orbit"
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  "name": "MAP Invariant",
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    {
      "context": "p) + \\beta \\sigma(p) = \\text{const.} \\end{equation} along closed regulatory orbits. This is the invariant form of Axiom~\\ref{axiom:bk6_map_equilibrium_invariance_complete}. \\end{proposition}",
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  "latex_body": "\\begin{proof}\n\\label{proof:bk6_map_invariant}\n\\leavevmode\nThe MAP equilibrium invariance axiom (Axiom~\\ref{axiom:bk6_map_equilibrium_invariance_complete}) asserts that the regulatory dynamics preserve the MAP equilibrium functional. Write that functional as $\\mathcal{M}_{\\text{MAP}}[p]=\\mathcal{F}_\\lambda[p]+\\alpha\\Upsilon_i(p,p)+\\beta\\sigma(p)$, combining the symbolic free energy, the internal incoherence, and the confidence with the equilibrium weights $\\alpha,\\beta$. Differentiating along a closed regulatory orbit and applying the axiom's balance condition, the free-energy decrease is offset exactly by the compensating changes in incoherence and confidence, so $\\tfrac{d}{dt}\\mathcal{M}_{\\text{MAP}}[p]=0$ around the orbit. Therefore $\\mathcal{M}_{\\text{MAP}}[p]$ is constant along closed regulatory orbits --- the conserved invariant form of the equilibrium axiom.\n\\end{proof}",
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      "context": "\\begin{proof} \\label{proof:bk6_map_invariant} \\leavevmode The MAP equilibrium invariance axiom (Axiom~\\ref{axiom:bk6_map_equilibrium_invariance_complete}) asserts that the regulatory dynamics preserve the MAP equilibrium functional. Write that functional as $\\mathcal{M}_{\\",
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