Book VII — De Convergentia Symbolica — page 4 of 5

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  "latex_body": "\\begin{definition}[Operational resolution uncertainties]\n\\label{definition:bk7_operational_resolution_uncertainties}\nFix a bounded observer $\\Obs$ with resolution threshold $\\delta_O$ and reflective bandwidth $\\mathcal{B_R}$ (Def.~\\ref{definition:bk1_bounded_observer}, Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}), observing a symbolic state under drift $\\drift$. Within one reflective cycle $\\Obs$ allocates kernel-smoothed samples between two estimation channels: an \\emph{identity channel} producing an estimator $\\widehat{\\Sigma}_I$ of the coherence-peak location (identity resolution, Def.~\\ref{definition:bk4_identity_resolution}) from $N_I$ samples, and a \\emph{curvature channel} producing an estimator $\\widehat{K}_S$ of local semantic curvature (Def.~\\ref{definition:bk4_symbolic_curvature}) from $N_K$ samples. Set $\\Delta\\Sigma_I := \\operatorname{sd}(\\widehat{\\Sigma}_I)$ and $\\Delta K_S := \\operatorname{sd}(\\widehat{K}_S)$, the estimator standard deviations over the observer's sampling law. These are the operational quantities the principle bounds; no other reading is intended.\n\\end{definition}",
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  "name": "Operational resolution uncertainties",
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      "context": "ties} Fix a bounded observer $\\Obs$ with resolution threshold $\\delta_O$ and reflective bandwidth $\\mathcal{B_R}$ (Def.~\\ref{definition:bk1_bounded_observer}, Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}), observing a symbolic state under drift $\\drift$. Within o",
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      "context": "identity channel} producing an estimator $\\widehat{\\Sigma}_I$ of the coherence-peak location (identity resolution, Def.~\\ref{definition:bk4_identity_resolution}) from $N_I$ samples, and a \\emph{curvature channel} producing an estimator $\\widehat{K}_S$ of local semantic curvature",
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      "context": "$N_I$ samples, and a \\emph{curvature channel} producing an estimator $\\widehat{K}_S$ of local semantic curvature (Def.~\\ref{definition:bk4_symbolic_curvature}) from $N_K$ samples. Set $\\Delta\\Sigma_I := \\operatorname{sd}(\\widehat{\\Sigma}_I)$ and $\\Delta K_S := \\operatorname{sd}",
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      "context": "olution threshold $\\delta_O$ and reflective bandwidth $\\mathcal{B_R}$ (Def.~\\ref{definition:bk1_bounded_observer}, Def.~\\ref{definition:bk5_reflective_drift_coupling_tensor}), observing a symbolic state under drift $\\drift$. Within one reflective cycle $\\Obs$ allocates kernel-smoothed samples",
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  "latex_body": "\\begin{lemma}[Coherence window]\n\\label{lemma:bk7_coherence_window}\nIf drift translates the observed state at effective magnitude $\\|\\Delta \\drift\\|$ in the observer metric, and a sample taken after the state has moved by one resolution cell $\\delta_O$ is decorrelated from the current estimate, then the number of mutually coherent samples per reflective cycle is bounded by\n\\[\nN \\;\\le\\; N_{\\max} := \\frac{\\mathcal{B_R}\\,\\delta_O}{\\|\\Delta \\drift\\|}, \\qquad N_I + N_K \\le N.\n\\]\n\\end{lemma}",
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  "latex_body": "\\begin{proof}[Coherence window]\n\\label{proof:bk7_coherence_window}\n\\leavevmode\nThe state exits a resolution cell after coherence time $\\tau_{\\mathrm{coh}} = \\delta_O / \\|\\Delta \\drift\\|$; the observer acquires samples at rate at most $\\mathcal{B_R}$, so at most $\\mathcal{B_R}\\,\\tau_{\\mathrm{coh}} = \\mathcal{B_R}\\,\\delta_O / \\|\\Delta \\drift\\|$ remain mutually coherent within a cycle, and the two channels share this budget.\n\\end{proof}",
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  "latex_body": "\\begin{assumption}[Channel floors]\n\\label{assumption:bk7_channel_floors}\nEach channel's estimator obeys a Cram\\'er--Rao--type variance floor at the resolution scale: there exist constants $c_I, c_K > 0$, fixed by the kernel shape and local geometry but independent of the allocation, with\n\\[\n\\Delta\\Sigma_I^{\\,2} \\ge \\frac{c_I\\,\\delta_O^{\\,2}}{N_I}, \\qquad \\Delta K_S^{\\,2} \\ge \\frac{c_K\\,\\delta_O^{\\,2}}{N_K}.\n\\]\nThis is the model-dependent hypothesis of the theorem -- neither location nor curvature is estimable below the resolution floor faster than the statistical $1/\\sqrt{N}$ rate -- and is directly testable in the Appendix~B suite.\n\\end{assumption}",
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  "latex_body": "\\begin{theorem}[Universal Symbolic Uncertainty Principle (PISU), derived form]\n\\label{theorem:bk7_pisu}\nAssume $N_I>0$, $N_K>0$, $\\mathcal B_R>0$, $\\delta_O>0$, and\n$\\lVert\\Delta\\drift\\rVert>0$.  Under\nDef.~\\ref{definition:bk7_operational_resolution_uncertainties}, the coherence\nwindow (Lemma~\\ref{lemma:bk7_coherence_window}), and both channel floors\n(Assumption~\\ref{assumption:bk7_channel_floors}), every allocation\n$N_I+N_K\\leq N_{\\max}$ satisfies\n\\[\n \\Delta\\Sigma_I\\Delta K_S\\geq\n 2\\sqrt{c_Ic_K}\\,\n \\frac{\\lVert\\Delta\\drift\\rVert}{\\mathcal B_R}\\,\\delta_O.\n\\]\nThe AM--GM allocation step is sharp at $N_I=N_K=N_{\\max}/2$.  Equality in the\nfull PISU bound additionally requires both channel-floor inequalities and the\ncoherence-window budget to be sharp.  Zero channel allocation is outside the\nfinite real-valued formulas unless an extended-real infinite-uncertainty\nconvention is separately adopted.\n\\end{theorem}",
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      "context": "tion_uncertainties}, the coherence window (Lemma~\\ref{lemma:bk7_coherence_window}), and both channel floors (Assumption~\\ref{assumption:bk7_channel_floors}), every allocation $N_I+N_K\\leq N_{\\max}$ satisfies \\[ \\Delta\\Sigma_I\\Delta K_S\\geq 2\\sqrt{c_Ic_K}\\, \\frac{\\lVert\\De",
      "label": "assumption:bk7_channel_floors",
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      "context": "heorem:bk7_pisu} Assume $N_I>0$, $N_K>0$, $\\mathcal B_R>0$, $\\delta_O>0$, and $\\lVert\\Delta\\drift\\rVert>0$. Under Def.~\\ref{definition:bk7_operational_resolution_uncertainties}, the coherence window (Lemma~\\ref{lemma:bk7_coherence_window}), and both channel floors (Assumption~\\ref{assumption:bk7",
      "label": "definition:bk7_operational_resolution_uncertainties",
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      "role": "definition_anchor",
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      "target_line": 1337,
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      "context": "elta\\drift\\rVert>0$. Under Def.~\\ref{definition:bk7_operational_resolution_uncertainties}, the coherence window (Lemma~\\ref{lemma:bk7_coherence_window}), and both channel floors (Assumption~\\ref{assumption:bk7_channel_floors}), every allocation $N_I+N_K\\leq N_{\\max}$ sat",
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  "latex_body": "\\begin{proof}[PISU by coherence-window allocation]\n\\label{proof:bk7_pisu}\n\\leavevmode\nBy the channel floors (Assumption~\\ref{assumption:bk7_channel_floors}),\n\\[\n\\Delta\\Sigma_I \\cdot \\Delta K_S \\;\\ge\\; \\sqrt{c_I c_K}\\;\\frac{\\delta_O^{\\,2}}{\\sqrt{N_I N_K}}.\n\\]\nUnder the coherence-window budget $N_I + N_K \\le N_{\\max}$ (Lemma~\\ref{lemma:bk7_coherence_window}), the inequality of arithmetic and geometric means gives $\\sqrt{N_I N_K} \\le (N_I + N_K)/2 \\le N_{\\max}/2$, with equality at the balanced split $N_I = N_K = N_{\\max}/2$. Hence\n\\[\n\\Delta\\Sigma_I \\cdot \\Delta K_S \\;\\ge\\; \\frac{2\\sqrt{c_I c_K}\\;\\delta_O^{\\,2}}{N_{\\max}} = 2\\sqrt{c_I c_K}\\;\\delta_O^{\\,2}\\,\\frac{\\|\\Delta \\drift\\|}{\\mathcal{B_R}\\,\\delta_O} = 2\\sqrt{c_I c_K}\\,\\frac{\\|\\Delta \\drift\\|}{\\mathcal{B_R}}\\,\\delta_O,\n\\]\nthe stated bound with $\\eta = 2\\sqrt{c_I c_K}$.\n\\end{proof}",
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      "context": "\\begin{proof}[PISU by coherence-window allocation] \\label{proof:bk7_pisu} \\leavevmode By the channel floors (Assumption~\\ref{assumption:bk7_channel_floors}), \\[ \\Delta\\Sigma_I \\cdot \\Delta K_S \\;\\ge\\; \\sqrt{c_I c_K}\\;\\frac{\\delta_O^{\\,2}}{\\sqrt{N_I N_K}}. \\] Under the cohere",
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      "context": "t{c_I c_K}\\;\\frac{\\delta_O^{\\,2}}{\\sqrt{N_I N_K}}. \\] Under the coherence-window budget $N_I + N_K \\le N_{\\max}$ (Lemma~\\ref{lemma:bk7_coherence_window}), the inequality of arithmetic and geometric means gives $\\sqrt{N_I N_K} \\le (N_I + N_K)/2 \\le N_{\\max}/2$, with equali",
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      "context": "mark:bk7_pisu_protocol} The principle is testable end to end: (i) verify the $1/\\sqrt{N}$ channel scaling of Assumption~\\ref{assumption:bk7_channel_floors} by regressing $\\log\\Delta\\Sigma_I$ on $\\log N_I$ at fixed drift (slope $-\\tfrac{1}{2}$, intercept fixing $c_I$; likewis",
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      "context": "$2\\sqrt{c_I c_K}\\,\\delta_O$. A measured violation of (iii) with (i) holding falsifies the coherence-window model (Lemma~\\ref{lemma:bk7_coherence_window}), not the arithmetic -- the theorem localizes blame. \\end{remark}",
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{
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    {
      "context": "2, C4, C5, the non-contextuality axiom PS--C3$'$, and Gleason's theorem (App.~C, Thm.~\\ref{theorem:appC_born_rule}, Ax.~\\ref{axiom:appC_psc3prime}). \\end{remark}",
      "label": "axiom:appC_psc3prime",
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      "target_line": 528,
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  "id": "remark:bk7_pisu_status",
  "label": "remark:bk7_pisu_status",
  "latex_body": "\\begin{remark}[Status of the trade-off and its corollaries]\n\\label{remark:bk7_pisu_status}\nWith Theorem~\\ref{theorem:bk7_pisu} derived, the constrained-uncertainty trade-off is a motivating scholium (Scholium~\\ref{scholium:bk7_constrained_uncertainty_motivation}), not an axiom. The Heisenberg and G\\\"odel readings below are \\emph{correspondences} -- structural analogies -- not instances of the inequality; in particular the Born rule does not rest on PISU but on the coherence axioms PS--C1, C2, C4, C5, the non-contextuality axiom PS--C3$'$, and Gleason's theorem (App.~C, Thm.~\\ref{theorem:appC_born_rule}, Ax.~\\ref{axiom:appC_psc3prime}).\n\\end{remark}",
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      "context": "2, C4, C5, the non-contextuality axiom PS--C3$'$, and Gleason's theorem (App.~C, Thm.~\\ref{theorem:appC_born_rule}, Ax.~\\ref{axiom:appC_psc3prime}). \\end{remark}",
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      "context": "} With Theorem~\\ref{theorem:bk7_pisu} derived, the constrained-uncertainty trade-off is a motivating scholium (Scholium~\\ref{scholium:bk7_constrained_uncertainty_motivation}), not an axiom. The Heisenberg and G\\\"odel readings below are \\emph{correspondences} -- structural analogies -- not ins",
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      "context": "on the coherence axioms PS--C1, C2, C4, C5, the non-contextuality axiom PS--C3$'$, and Gleason's theorem (App.~C, Thm.~\\ref{theorem:appC_born_rule}, Ax.~\\ref{axiom:appC_psc3prime}). \\end{remark}",
      "label": "theorem:appC_born_rule",
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      "target_line": 701,
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      "context": "\\begin{remark}[Status of the trade-off and its corollaries] \\label{remark:bk7_pisu_status} With Theorem~\\ref{theorem:bk7_pisu} derived, the constrained-uncertainty trade-off is a motivating scholium (Scholium~\\ref{scholium:bk7_constrained_uncerta",
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  "latex_body": "\\begin{definition}[Symbolic Reflexive Validation (SRV)]\n\\label{definition:bk7_symbolic_reflexive_validation_srv}\nLet $S = (\\manifold, \\metric, \\drift, \\reflect, \\rho)$ be a symbolic system as formalized in Book VII, and let $\\Obs$ be a bounded observer embedded within this system (cf.~Defs.~\\ref{definition:bk1_bounded_observer}, \\ref{definition:bk4_bounded_observer}, \\ref{definition:bk4_fuzzy_symbolic_substitution}), characterized by perceptual horizon $\\epsilon_O$ and differential sensitivity $\\delta^n$. A process of \\emph{Symbolic Reflexive Validation (SRV)} is any symbolic trajectory $\\{\\rho_t\\}_{t \\in \\mathbb{T}} \\subseteq \\prob(\\manifold)$ governed by the internal operators $\\reflect$, $\\drift$, and constrained by $\\Obs$, that satisfies the following criteria:\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\textbf{Reflexive Enactment:} The process is generated by the same symbolic laws it seeks to validate (e.g., drift-reflection dynamics, SRMF minimization in the sense of Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}, free energy descent; cf.~Def.~\\ref{definition:bk1_srmf_energy_functional}, Lem.~\\ref{lemma:bk4_srmf_constrained_action_norm});\n\\item \\textbf{Internal Coherence:} The symbolic observables emergent from the process (e.g., curvature reduction, $L^p$ sparsity, entropy dynamics) remain structurally interpretable within the system's own formalism;\n\\item \\textbf{Observer-Relative Interpretation:} All symbolic readouts and validations are interpreted through bounded perceptual operators ($\\epsilon_O, \\delta^n$), within the induced symbolic membrane $\\Mt$ defined by $\\Obs$;\n\\item \\textbf{Symbolic Falsifiability:}\nA trajectory is invalidated if it yields internal contradiction --\nsuch as divergence of \\( \\freeenergy \\), collapse of reflective coherence,\nor violation of SRMF constraints (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}: failure to converge to a fixed point $\\reflect(\\identity)=\\identity$; Cor.~\\ref{corollary:bk7_drift_collapse_equivalence}: failure of reflective descent to absorb drift) --\neach of which signals breakdown within the system's own dynamics.\n\\end{enumerate}\n\\emph{SRV} reframes validation as structural convergence (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}) under reflexively enacted symbolic dynamics. Unlike traditional externalist methods that assume a detached observer and separable test apparatus, SRV embeds validation within the same symbolic field it interrogates (cf.~\\ref{axiom:bk1_symbolic_primacy}). Falsification arises not through empirical negation, but through detectable incoherence within the symbolic manifold (cf.~\\ref{definition:bk1_paradox_triggered_emergence}, \\ref{corollary:bk1_non_euclidean_necessity}).\n\n\\medskip\\noindent\\textit{Note:} For concrete instances, see Appendix B, where Traces 3--7 instantiate symbolic drift-reflection processes and demonstrate reflexive convergence. Trace 5 in particular illustrates variation in observer-relative $L^p$ sparsity under SRMF constraints.\n\\end{definition}",
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      "context": "tached observer and separable test apparatus, SRV embeds validation within the same symbolic field it interrogates (cf.~\\ref{axiom:bk1_symbolic_primacy}). Falsification arises not through empirical negation, but through detectable incoherence within the symbolic manifold",
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      "context": "but through detectable incoherence within the symbolic manifold (cf.~\\ref{definition:bk1_paradox_triggered_emergence}, \\ref{corollary:bk1_non_euclidean_necessity}). \\medskip\\noindent\\textit{Note:} For concrete instances, see Appendix B, where Traces 3--7 instantiate symbolic drift",
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      "context": "symbolic system as formalized in Book VII, and let $\\Obs$ be a bounded observer embedded within this system (cf.~Defs.~\\ref{definition:bk1_bounded_observer}, \\ref{definition:bk4_bounded_observer}, \\ref{definition:bk4_fuzzy_symbolic_substitution}), characterized by perceptual",
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      "context": "lsification arises not through empirical negation, but through detectable incoherence within the symbolic manifold (cf.~\\ref{definition:bk1_paradox_triggered_emergence}, \\ref{corollary:bk1_non_euclidean_necessity}). \\medskip\\noindent\\textit{Note:} For concrete instances, see Appendix B,",
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      "context": "by the same symbolic laws it seeks to validate (e.g., drift-reflection dynamics, SRMF minimization in the sense of Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}, free energy descent; cf.~Def.~\\ref{definition:bk1_srmf_energy_functional}, Lem.~\\ref{lemma:bk4_srmf_constrained_action",
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      "context": "mization in the sense of Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}, free energy descent; cf.~Def.~\\ref{definition:bk1_srmf_energy_functional}, Lem.~\\ref{lemma:bk4_srmf_constrained_action_norm}); \\item \\textbf{Internal Coherence:} The symbolic observables emerge",
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      "context": "VII, and let $\\Obs$ be a bounded observer embedded within this system (cf.~Defs.~\\ref{definition:bk1_bounded_observer}, \\ref{definition:bk4_bounded_observer}, \\ref{definition:bk4_fuzzy_symbolic_substitution}), characterized by perceptual horizon $\\epsilon_O$ and differential s",
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      "context": "er embedded within this system (cf.~Defs.~\\ref{definition:bk1_bounded_observer}, \\ref{definition:bk4_bounded_observer}, \\ref{definition:bk4_fuzzy_symbolic_substitution}), characterized by perceptual horizon $\\epsilon_O$ and differential sensitivity $\\delta^n$. A process of \\emph{Symbolic",
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      "context": "ion -- such as divergence of \\( \\freeenergy \\), collapse of reflective coherence, or violation of SRMF constraints (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}: failure to converge to a fixed point $\\reflect(\\identity)=\\identity$; Cor.~\\ref{corollary:bk7_drift_collapse_equivalen",
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  "ref_roles": [
    {
      "context": "anifold}), meaning, and language -- validation and the object of validation participate in the same symbolic field (cf.~\\ref{axiom:bk1_symbolic_primacy}). Falsification becomes a matter of detecting internal contradictions rather than external counterfactuals (cf.~\\ref{de",
      "label": "axiom:bk1_symbolic_primacy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2226,
      "target_type": "axiom"
    },
    {
      "context": "rimacy}). Falsification becomes a matter of detecting internal contradictions rather than external counterfactuals (cf.~\\ref{definition:bk1_paradox_triggered_emergence}), reflecting the recursive nature of symbolic reality itself. \\end{remark}",
      "label": "definition:bk1_paradox_triggered_emergence",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2264,
      "target_type": "definition"
    },
    {
      "context": "e theoretical claims, SRV recognizes that within closed symbolic systems -- particularly those governing cognition (cf.~\\ref{definition:bk1_symbolic_manifold}), meaning, and language -- validation and the object of validation participate in the same symbolic field (cf.~\\ref{axi",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "axiom:bk1_symbolic_primacy",
    "definition:bk1_paradox_triggered_emergence",
    "definition:bk1_symbolic_manifold"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "book7",
  "cited_by": [
    "sec:appB_symbolic_validation_procedure"
  ],
  "cites": [
    "axiom:bk1_symbolic_primacy",
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk1_symbolic_flow",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "depends_on": [
    "axiom:bk1_symbolic_primacy",
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk1_symbolic_flow",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "file": "book7.tex",
  "id": "scholium:bk7_popperian_extension",
  "label": "scholium:bk7_popperian_extension",
  "latex_body": "\\begin{scholium}[Popperian Extension]\n\\label{scholium:bk7_popperian_extension}\nLet $\\mathcal{F}_P = (\\mathcal{T}, \\mathcal{O}, \\varphi)$ represent the classic Popperian falsifiability framework, where $\\mathcal{T}$ denotes a theory space, $\\mathcal{O}$ an observation space, and $\\varphi: \\mathcal{T} \\times \\mathcal{O} \\rightarrow \\{0,1\\}$ a binary falsification operator. This framework can be formally extended to SRV (cf.~\\ref{definition:bk1_symbolic_flow}) through the following mappings:\n\\begin{enumerate}[label=(\\roman*)]\n\\item \\textbf{Differential Embedding}: The theory-observation separation in $\\mathcal{F}_P$ is mapped to a differential relation within a unified symbolic manifold:\n\\begin{align}\n(\\mathcal{T}, \\mathcal{O}) \\mapsto (\\manifold, \\nabla_{\\epsilon_O}\\manifold)\n\\end{align}\nwhere $\\nabla_{\\epsilon_O}$ denotes the bounded differential operator induced by observer $\\Obs$ with horizon $\\epsilon_O$.\n\\item \\textbf{Falsification Continuity}: The binary falsification operator $\\varphi$ is extended to a continuous coherence functional:\n\\begin{align}\n\\varphi \\mapsto \\mathcal{C}_{\\reflect}: \\prob(\\manifold) \\rightarrow \\mathbb{R}^+\n\\end{align}\nwhere $\\mathcal{C}_{\\reflect}(\\rho_t)$ measures the degree of internal coherence under reflection operator $\\reflect$ (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}, \\ref{definition:bk1_self_regulating_mapping_function_srmf}).\n\\item \\textbf{Separability Relaxation}: The strict ontological separation assumed in interpretations of $\\mathcal{F}_P$ is relaxed to differential separability within a unified field:\n\\begin{align}\n\\text{sep}(\\mathcal{T}, \\mathcal{O}) \\mapsto \\text{dif}(\\rho_t, \\nabla_{\\epsilon_O}\\rho_t) < \\delta^n\n\\end{align}\nwhere $\\text{dif}$ measures symbolic differentiation bounded by sensitivity $\\delta^n$.\n\\item \\textbf{Validation Integration}: Popperian validation through non-falsification is extended to validation through dynamic integration:\n\\begin{align}\nV_P(\\mathcal{T}) = \\prod_{o \\in \\mathcal{O}} (1 - \\varphi(\\mathcal{T}, o)) \\mapsto V_{SRV}(\\rho_t) = \\int_{\\mathbb{T}} \\mathcal{C}_{\\reflect}(\\rho_t) \\, dt\n\\end{align}\n\\end{enumerate}\nThis formal extension preserves Popper's insistence on testability while transcending the assumed ontological gulf between theory and observation, replacing it with a differential relation in a unified symbolic field (cf.~\\ref{axiom:bk1_symbolic_primacy}) where validation emerges from the symbolic dynamics themselves (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}).\n\n\\medskip\\noindent\\textit{Note:} This mapping demonstrates that SRV maintains a form of ``weak separability'' through the differential operator $\\nabla_{\\epsilon_O}$ while embedding both process and validation within the same symbolic manifold --- preserving Popper's methodological insight while refining its metaphysical implications.\n\\end{scholium}",
  "line": 1462,
  "macros_used": [
    "Obs",
    "manifold",
    "prob",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Popperian Extension",
  "ref_roles": [
    {
      "context": "logical gulf between theory and observation, replacing it with a differential relation in a unified symbolic field (cf.~\\ref{axiom:bk1_symbolic_primacy}) where validation emerges from the symbolic dynamics themselves (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_",
      "label": "axiom:bk1_symbolic_primacy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2226,
      "target_type": "axiom"
    },
    {
      "context": "ternal coherence under reflection operator $\\reflect$ (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}, \\ref{definition:bk1_self_regulating_mapping_function_srmf}). \\item \\textbf{Separability Relaxation}: The strict ontological separation assumed in interpretations of $\\mathcal{F}_",
      "label": "definition:bk1_self_regulating_mapping_function_srmf",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2230,
      "target_type": "definition"
    },
    {
      "context": "s \\mathcal{O} \\rightarrow \\{0,1\\}$ a binary falsification operator. This framework can be formally extended to SRV (cf.~\\ref{definition:bk1_symbolic_flow}) through the following mappings: \\begin{enumerate}[label=(\\roman*)] \\item \\textbf{Differential Embedding}: The theory-o",
      "label": "definition:bk1_symbolic_flow",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2872,
      "target_type": "definition"
    },
    {
      "context": "re $\\mathcal{C}_{\\reflect}(\\rho_t)$ measures the degree of internal coherence under reflection operator $\\reflect$ (cf.~\\ref{theorem:bk7_reflective_convergence_to_stable_identity}, \\ref{definition:bk1_self_regulating_mapping_function_srmf}). \\item \\textbf{Separability Relaxation}: The strict ontolo",
      "label": "theorem:bk7_reflective_convergence_to_stable_identity",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book7.tex",
      "target_line": 555,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "axiom:bk1_symbolic_primacy",
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk1_symbolic_flow",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book7",
  "cited_by": [
    "sec:appB_symbolic_validation_procedure"
  ],
  "cites": [
    "definition:bk1_reflection_operator"
  ],
  "depends_on": [
    "definition:bk1_reflection_operator"
  ],
  "file": "book7.tex",
  "id": "remark:bk7_unnamed_remark_05",
  "label": "remark:bk7_unnamed_remark_05",
  "latex_body": "\\begin{remark}\n\\label{remark:bk7_unnamed_remark_05}\nThis extension reveals that Popper's falsifiability, properly understood (cf.~\\ref{definition:bk1_reflection_operator}), never demanded complete ontological separation between theory and test but rather sufficient functional differentiation to enable critical evaluation. SRV makes explicit what remains implicit in Popper: that validation requires difference but not detachment. Where interpretations of Popper often overemphasize separation, SRV formalizes differentiation within unity, showing that falsifiability requires not rigid boundaries but sufficient symbolic gradients within a coherent field.\n\\end{remark}",
  "line": 1490,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "ref_roles": [
    {
      "context": "ark} \\label{remark:bk7_unnamed_remark_05} This extension reveals that Popper's falsifiability, properly understood (cf.~\\ref{definition:bk1_reflection_operator}), never demanded complete ontological separation between theory and test but rather sufficient functional differentiati",
      "label": "definition:bk1_reflection_operator",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1209,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_reflection_operator"
  ],
  "role": "remark",
  "type": "remark"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk4_tilda_substitution"
  ],
  "depends_on": [
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk4_tilda_substitution"
  ],
  "file": "book7.tex",
  "id": "definition:bk7_srmfconstrained_observer",
  "label": "definition:bk7_srmfconstrained_observer",
  "latex_body": "\\begin{definition}[SRMF-Constrained Observer]\n\\label{definition:bk7_srmfconstrained_observer}\nThis observer type is constrained by the Self-Regulating Mapping Function (Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}), which bounds the reflection operator budget.\nLet $(\\mathcal{M},\\tau_{\\mathcal{P}})$ be the symbolic manifold endowed with\nmetric tensor $g$ and symbolic free-energy functional $\\tilde{F}_s$.\nAn \\emph{SRMF-constrained observer} is a triple\n\\[\n\\mathcal{O}_{\\epsilon} \\;=\\; \\bigl( \\mathcal{R},\\, \\epsilon,\\, \\mathcal{B} \\bigr)\n\\]\nwhere\n\\begin{enumerate}[label=(\\roman*)]\n  \\item $\\mathcal{R}\\colon\\mathcal{M}\\!\\to\\!\\mathcal{M}$ is a reflection operator\n        obeying the Self-Regulating Mapping Function (SRMF) resource constraint\n        \\(\\lVert D\\mathcal{R}\\rVert_g \\le \\mathcal{B}\\) for some finite budget\n        $\\mathcal{B}>0$ (cf.~Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}),\n  \\item \\(\\epsilon > 0\\) is an \\emph{observer horizon} that induces a\n        coarse-graining map\n        \\(\n        \\pi_{\\epsilon}\\colon \\mathcal{M}\\!\\to\\!\\mathcal{M}_{\\epsilon}\n        \\)\n        collapsing all symbolic variation below scale $\\epsilon$,\n  \\item $\\tilde{x}\\in\\tilde{\\mathcal{M}}$ denotes a\n        tilda-encoded symbolic configuration\n        (Def.~\\ref{definition:bk4_tilda_substitution}).\n\\end{enumerate}\n\\end{definition}",
  "line": 1499,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "SRMF-Constrained Observer",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "efinition:bk7_srmfconstrained_observer} This observer type is constrained by the Self-Regulating Mapping Function (Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}), which bounds the reflection operator budget. Let $(\\mathcal{M},\\tau_{\\mathcal{P}})$ be the symbolic manifold endowed",
      "label": "definition:bk1_self_regulating_mapping_function_srmf",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2230,
      "target_type": "definition"
    },
    {
      "context": "psilon$, \\item $\\tilde{x}\\in\\tilde{\\mathcal{M}}$ denotes a tilda-encoded symbolic configuration (Def.~\\ref{definition:bk4_tilda_substitution}). \\end{enumerate} \\end{definition}",
      "label": "definition:bk4_tilda_substitution",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 4193,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk4_tilda_substitution"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book7",
  "cited_by": [
    "subsec:bk8_properties_and_justification_of_observer_dependence"
  ],
  "cites": [
    "definition:bk1_bounded_observer"
  ],
  "depends_on": [
    "definition:bk1_bounded_observer"
  ],
  "file": "book7.tex",
  "id": "definition:bk7_observerrelative_symbolic_error_field",
  "label": "definition:bk7_observerrelative_symbolic_error_field",
  "latex_body": "\\begin{definition}[Observer-Relative Symbolic Error Field]\n\\label{definition:bk7_observerrelative_symbolic_error_field}\nFor an SRMF observer $\\mathcal{O}_{\\epsilon}$ (cf.~\\ref{definition:bk1_bounded_observer}) and\n$\\tilde{x}\\in\\tilde{\\mathcal{M}}$, define the symbolic\nerror field\n\\[\nE_{\\epsilon}(\\tilde{x}) \\;:=\\;\n\\pi_{\\epsilon}\\bigl(\\mathcal{R}(\\tilde{x})\\bigr)\n\\;-\\;\n\\pi_{\\epsilon}\\bigl(\\tilde{x}\\bigr).\n\\]\n\\end{definition}",
  "line": 1525,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Observer-Relative Symbolic Error Field",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "Field] \\label{definition:bk7_observerrelative_symbolic_error_field} For an SRMF observer $\\mathcal{O}_{\\epsilon}$ (cf.~\\ref{definition:bk1_bounded_observer}) and $\\tilde{x}\\in\\tilde{\\mathcal{M}}$, define the symbolic error field \\[ E_{\\epsilon}(\\tilde{x}) \\;:=\\; \\pi_{\\epsilon",
      "label": "definition:bk1_bounded_observer",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 27,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_bounded_observer"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book7",
  "cited_by": [
    "proof:bk9_pathologies_of_coherence"
  ],
  "cites": [
    "definition:bk2_symbolic_free_energy"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold",
    "definition:bk2_symbolic_free_energy"
  ],
  "file": "book7.tex",
  "id": "lemma:bk7_coarsegrained_convexity",
  "label": "lemma:bk7_coarsegrained_convexity",
  "latex_body": "\\begin{lemma}[Coarse-Grained Convexity]\n\\label{lemma:bk7_coarsegrained_convexity}\nThe functional (cf.~\\ref{definition:bk2_symbolic_free_energy} for the free-energy context)\n\\(\n\\tilde{F}_{s}^{(p)}(\\tilde{x})\n=\\!\\displaystyle \\int_{\\mathcal{M}_{\\epsilon}}\n\\bigl\\lVert E_{\\epsilon}(\\tilde{x})(z)\\bigr\\rVert^{p}\\,\n\\,\\mathrm{d}\\mu_{g}(z)\n\\)\nis strictly convex in $E_{\\epsilon}$ for every $p\\!\\in\\!(1,\\infty)$.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "Banach/Hilbert space theory, measure theory, infinite-limit claims, and the Gleason/Born cluster are NOT formalized",
      "recurrence laws, descent laws, and fixed-point existence are structure fields or explicit hypotheses",
      "theorem:bk7_pisu skipped: depends on a channel-floors assumption referenced but absent from the sliced packet"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Only the p=2 (Hilbert) cross-section is proved, as strict midpoint convexity of x |-> x^2. Strict convexity for general p in (1, infinity), and the underlying coarse-grained error-field functional, are not modeled."
    ],
    "record_ids": [
      "MAP-BOOK7-019"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book7.square_strictly_convex_midpoint"
    ]
  },
  "line": 1537,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Coarse-Grained Convexity",
  "proof_labels": [
    "proof:bk7_strict_convexity_lp_error"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "\\begin{lemma}[Coarse-Grained Convexity] \\label{lemma:bk7_coarsegrained_convexity} The functional (cf.~\\ref{definition:bk2_symbolic_free_energy} for the free-energy context) \\( \\tilde{F}_{s}^{(p)}(\\tilde{x}) =\\!\\displaystyle \\int_{\\mathcal{M}_{\\epsilon}} \\bigl\\lVe",
      "label": "definition:bk2_symbolic_free_energy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_free_energy"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "definition:bk1_symbolic_manifold"
  ],
  "depends_on": [
    "definition:bk1_symbolic_manifold"
  ],
  "file": "book7.tex",
  "id": "proof:bk7_strict_convexity_lp_error",
  "label": "proof:bk7_strict_convexity_lp_error",
  "latex_body": "\\begin{proof}[Strict Convexity LP Error]\n\\label{proof:bk7_strict_convexity_lp_error}\n\\leavevmode\n\nBy standard properties of $L^{p}$ spaces on Riemannian manifolds with\n$\\mu_{g}$ finite on compact subsets (cf.~Def.~\\ref{definition:bk1_symbolic_manifold}), the map\n$E\\!\\mapsto\\!\\lVert E\\rVert_{p}^{p}$ is strictly convex\nfor $p\\!\\in\\!(1,\\infty)$.  Composing with the linear operator\n$E_{\\epsilon}$ preserves strict convexity.\n\\end{proof}",
  "line": 1548,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Strict Convexity LP Error",
  "proves": "lemma:bk7_coarsegrained_convexity",
  "ref_roles": [
    {
      "context": "de By standard properties of $L^{p}$ spaces on Riemannian manifolds with $\\mu_{g}$ finite on compact subsets (cf.~Def.~\\ref{definition:bk1_symbolic_manifold}), the map $E\\!\\mapsto\\!\\lVert E\\rVert_{p}^{p}$ is strictly convex for $p\\!\\in\\!(1,\\infty)$. Composing with the linear",
      "label": "definition:bk1_symbolic_manifold",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 1188,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_symbolic_manifold"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book7",
  "cited_by": [
    "proof:bk7_emergent_lp_norm_from_srmf",
    "proof:bk7_hilbert_banach_bridge",
    "theorem:bk7_emergent_lp_norm"
  ],
  "cites": [
    "definition:bk1_self_regulating_mapping_function_srmf"
  ],
  "depends_on": [
    "definition:bk1_self_regulating_mapping_function_srmf"
  ],
  "file": "book7.tex",
  "id": "lemma:bk7_budgetlimited_minimizer",
  "label": "lemma:bk7_budgetlimited_minimizer",
  "latex_body": "\\begin{lemma}[Budget-Limited Minimizer]\n\\label{lemma:bk7_budgetlimited_minimizer}\nFix $\\tilde{x}$, $\\epsilon$, and $p\\in(1,\\infty)$\n(cf.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}), and let\n\\[\n\\mathfrak{R}_{\\mathcal B}\n:=\\{\\mathcal R:\\lVert D\\mathcal R\\rVert_g\\le\\mathcal B\\}\n\\]\nbe a nonempty convex weak*-compact admissible class.  Make the dependence on\nthe candidate regulator explicit by defining\n\\[\nJ_{\\tilde{x}}^{(p)}(\\mathcal R)\n:=\\int_{\\mathcal M_\\epsilon}\n\\left\\lVert\n\\pi_\\epsilon\\!\\bigl(\\mathcal R(\\tilde{x})\\bigr)\n-\\pi_\\epsilon(\\tilde{x})\n\\right\\rVert^p\\,\\mathrm d\\mu_g.\n\\]\nIf $J_{\\tilde{x}}^{(p)}$ is weak*-lower-semicontinuous on\n$\\mathfrak{R}_{\\mathcal B}$ and strictly convex there (equivalently for the\nfinite kernel, its cost separates distinct admissible regulators), then there\nexists a unique\n\\[\n\\mathcal{R}_{\\epsilon}^{*}(\\tilde{x})\n=\\arg\\!\\min_{\\mathcal R\\in\\mathfrak R_{\\mathcal B}}\nJ_{\\tilde{x}}^{(p)}(\\mathcal R).\n\\]\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "continuum/Hilbert/PDE-on-manifold content stays open; chart-complex restatements carry Glued as a named hypothesis where the source consumes compatibility",
      "modeling laws are structure fields or explicit hypotheses"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Source-level analytic kernel: for any chosen topology on the regulator space, a nonempty compact admissible set and lower-semicontinuous candidate-dependent cost attain a minimum; strict convexity, including convexity of the admissible set, makes it unique. This directly supports the printed weak-star theorem when its compactness and semicontinuity premises are supplied. The finite no-ties theorem is retained, and the constant-objective Bool model records why the superseded source failed."
    ],
    "record_ids": [
      "MAP-BOOK7-047"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book7B.budgetLimitedObjective_not_unique",
      "Book7B.budgetLimited_existsUniqueMinimizer_of_compact",
      "Book7B.budgetLimited_uniqueMinimizer_of_injectiveCost",
      "Book7B.srmfRegulation_exists"
    ]
  },
  "line": 1558,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Budget-Limited Minimizer",
  "proof_labels": [
    "proof:bk7_from_compactness_and_convexity"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "get-Limited Minimizer] \\label{lemma:bk7_budgetlimited_minimizer} Fix $\\tilde{x}$, $\\epsilon$, and $p\\in(1,\\infty)$ (cf.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}), and let \\[ \\mathfrak{R}_{\\mathcal B} :=\\{\\mathcal R:\\lVert D\\mathcal R\\rVert_g\\le\\mathcal B\\} \\] be a nonempty convex",
      "label": "definition:bk1_self_regulating_mapping_function_srmf",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2230,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk1_self_regulating_mapping_function_srmf"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book7.tex",
  "id": "proof:bk7_from_compactness_and_convexity",
  "label": "proof:bk7_from_compactness_and_convexity",
  "latex_body": "\\begin{proof}[From Compactness and Strict Convexity]\n\\label{proof:bk7_from_compactness_and_convexity}\n\\leavevmode\n\nWeak*-compactness and weak*-lower-semicontinuity give existence of a\nminimizer.  If two distinct admissible regulators minimized\n$J_{\\tilde{x}}^{(p)}$, convexity of $\\mathfrak R_{\\mathcal B}$ and strict\nconvexity of the objective would make their midpoint have strictly smaller\ncost, a contradiction.  Hence the minimizer is unique.\n\\end{proof}",
  "line": 1586,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "From Compactness and Strict Convexity",
  "proves": "lemma:bk7_budgetlimited_minimizer",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book7",
  "cited_by": [
    "lemma:bk7_frame_temperature_exponent_correspondence",
    "proof:bk7_frame_temperature_exponent_correspondence",
    "proof:bk7_hilbert_banach_bridge",
    "proof:bk7_lp_norm_monotonicity",
    "scholium:bk4_role_of_observer_induced_metric",
    "subsec:bk7_hilbert_banach_bridge"
  ],
  "cites": [
    "definition:bk4_symbolic_autonomy",
    "lemma:bk7_budgetlimited_minimizer"
  ],
  "depends_on": [
    "definition:bk1_self_regulating_mapping_function_srmf",
    "definition:bk4_symbolic_autonomy",
    "lemma:bk7_budgetlimited_minimizer"
  ],
  "file": "book7.tex",
  "id": "theorem:bk7_emergent_lp_norm",
  "label": "theorem:bk7_emergent_lp_norm",
  "latex_body": "\\begin{theorem}[Emergent L$^{p}$ Norm]\n\\label{theorem:bk7_emergent_lp_norm}\nLet $\\mathcal{O}_{\\epsilon}$ be an SRMF-constrained observer with\nbudget $\\mathcal{B}$ and horizon $\\epsilon$ (cf.~\\ref{definition:bk4_symbolic_autonomy}).  Suppose\n$\\tilde{x}\\mapsto\\mathcal{R}$ minimizes the symbolic free energy\nunder resource constraint (Lemma~\\ref{lemma:bk7_budgetlimited_minimizer}).\nThen there exists a \\emph{unique} exponent\n\\[\np\\;=\\;p(\\epsilon,\\mathcal{B},S_{s})\n\\quad\\in\\;(1,\\infty)\n\\]\nsuch that the observer's effective cost functional equals\n\\[\n\\tilde{F}_{s}^{\\text{\\rm eff}}(\\tilde{x})\n\\;=\\;\n\\tilde{F}_{s}^{(p)}(\\tilde{x})\n\\;=\\;\n\\int_{\\mathcal{M}_{\\epsilon}}\n\\bigl\\lVert E_{\\epsilon}(\\tilde{x})(z)\\bigr\\rVert^{p}\\,\n\\,\\mathrm{d}\\mu_{g}(z),\n\\]\nand the mapping\n$\\epsilon\\mapsto p(\\epsilon,\\mathcal{B},S_{s})$ is $C^{1}$,\nstrictly decreasing in $\\epsilon$,\nand satisfies the asymptotic limits\n\\[\n\\lim_{\\epsilon\\to 0^{+}} p(\\epsilon,\\mathcal{B},S_{s}) \\;=\\;\\infty,\n\\qquad\n\\lim_{\\epsilon\\to\\infty} p(\\epsilon,\\mathcal{B},S_{s}) \\;=\\;1.\n\\]\n\\end{theorem}",
  "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": false,
    "notes": [
      "Only the boundary limit lim_{eps->infinity} p(eps) = 1 (kept as a structure hypothesis together with p(eps) > 1 everywhere) and its derived 'eventually within any delta of 1' consequence are modeled. The companion limit lim_{eps->0+} p(eps) = infinity, the C^1 and strict-monotonicity clauses, and the existence/uniqueness of p itself (as the minimizer's effective exponent) are not modeled."
    ],
    "record_ids": [
      "MAP-BOOK7-023"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Asymptotics.AsymptoticExponentField.eventually_near_one"
    ]
  },
  "line": 1596,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Emergent L$^{p}$ Norm",
  "proof_labels": [
    "proof:bk7_emergent_lp_norm_from_srmf"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "orm} Let $\\mathcal{O}_{\\epsilon}$ be an SRMF-constrained observer with budget $\\mathcal{B}$ and horizon $\\epsilon$ (cf.~\\ref{definition:bk4_symbolic_autonomy}). Suppose $\\tilde{x}\\mapsto\\mathcal{R}$ minimizes the symbolic free energy under resource constraint (Lemma~\\ref{lemma",
      "label": "definition:bk4_symbolic_autonomy",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book4.tex",
      "target_line": 3044,
      "target_type": "definition"
    },
    {
      "context": "_autonomy}). Suppose $\\tilde{x}\\mapsto\\mathcal{R}$ minimizes the symbolic free energy under resource constraint (Lemma~\\ref{lemma:bk7_budgetlimited_minimizer}). Then there exists a \\emph{unique} exponent \\[ p\\;=\\;p(\\epsilon,\\mathcal{B},S_{s}) \\quad\\in\\;(1,\\infty) \\] such that t",
      "label": "lemma:bk7_budgetlimited_minimizer",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book7.tex",
      "target_line": 1558,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk4_symbolic_autonomy",
    "lemma:bk7_budgetlimited_minimizer"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "definition:bk1_self_regulating_mapping_function_srmf",
    "lemma:bk7_budgetlimited_minimizer"
  ],
  "depends_on": [
    "definition:bk1_self_regulating_mapping_function_srmf",
    "lemma:bk7_budgetlimited_minimizer"
  ],
  "file": "book7.tex",
  "id": "proof:bk7_emergent_lp_norm_from_srmf",
  "label": "proof:bk7_emergent_lp_norm_from_srmf",
  "latex_body": "\\begin{proof}[Emergent LP Norm from SRMF]\n\\label{proof:bk7_emergent_lp_norm_from_srmf}\n\\leavevmode\n\nThe proof starts from the SRMF resource constraint\n(Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}) and shows\nthat budget-constrained reflection induces a dual-weighted $L^p$ penalty\nstructure.\nFix $\\tilde{x}$.  The SRMF budget enforces a Lipschitz bound on\n$\\mathcal{R}$; thus the Euler-Lagrange equation for the constrained\nfunctional yields a \\emph{dual-weighted} error penalty\n\\(\n|E_{\\epsilon}|^{p}\\,w_{\\epsilon}(z),\n\\)\nwhere the dual weight $w_{\\epsilon}$ is proportional to the SRMF\nLagrange multiplier field.  Normalizing by\n$\\int w_{\\epsilon}\\!=\\!1$ forces all such solutions to lie on the\none-parameter family $p(\\epsilon)$ satisfying\n\\(\n\\partial\\tilde{F}_{s}^{(p)}/\\partial p = 0.\n\\)\n\\emph{Existence.}  \nStrict convexity guarantees a minimizer\n(Lemma~\\ref{lemma:bk7_budgetlimited_minimizer}).  \nBy the implicit function theorem, the stationary\ncondition defines a $C^{1}$ curve $p(\\epsilon)$ in a neighbourhood of\nany $\\epsilon_{0}>0$.\n\\emph{Monotonicity.}  \nDifferentiate the stationary condition\n\\(\n\\partial_{p}\\tilde{F}_{s}^{(p)}=0\n\\)\nwith respect to $\\epsilon$; using\n$\\partial_{\\epsilon}E_{\\epsilon}<0$ (coarse-graining discards detail),\nwe obtain\n\\(\n\\partial_{\\epsilon}p < 0.\n\\)\n\\emph{Asymptotics.}  \nAs $\\epsilon\\!\\to\\! 0^{+}$ the observer resolves all drift,\n$E_{\\epsilon}\\!\\to\\!0$, forcing $p\\!\\to\\!\\infty$ to penalise the\nmaximal deviation (sup-norm).  \nConversely, as $\\epsilon\\!\\to\\!\\infty$ the\nobserver collapses the manifold to a point,\nso only the \\emph{mean} error matters, and\n$p\\!\\to\\!1$ minimises the $\\ell^{1}$ cost (sparsity-dominant).\nUniqueness of $p$ follows by the strict monotonicity of\n$\\partial_{p}\\tilde{F}_{s}^{(p)}$ under convexity.\n\\end{proof}",
  "line": 1627,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Emergent LP Norm from SRMF",
  "proves": "theorem:bk7_emergent_lp_norm",
  "ref_roles": [
    {
      "context": "RMF] \\label{proof:bk7_emergent_lp_norm_from_srmf} \\leavevmode The proof starts from the SRMF resource constraint (Def.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}) and shows that budget-constrained reflection induces a dual-weighted $L^p$ penalty structure. Fix $\\tilde{x}$. The SR",
      "label": "definition:bk1_self_regulating_mapping_function_srmf",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "scholium_symbolicum.tex",
      "target_line": 2230,
      "target_type": "definition"
    },
    {
      "context": "ng \\( \\partial\\tilde{F}_{s}^{(p)}/\\partial p = 0. \\) \\emph{Existence.} Strict convexity guarantees a minimizer (Lemma~\\ref{lemma:bk7_budgetlimited_minimizer}). By the implicit function theorem, the stationary condition defines a $C^{1}$ curve $p(\\epsilon)$ in a neighbourhood",
      "label": "lemma:bk7_budgetlimited_minimizer",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 1558,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk1_self_regulating_mapping_function_srmf",
    "lemma:bk7_budgetlimited_minimizer"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book7",
  "cited_by": [
    "proof:bk7_lp_norm_monotonicity"
  ],
  "cites": [],
  "depends_on": [
    "theorem:bk7_emergent_lp_norm"
  ],
  "file": "book7.tex",
  "id": "corollary:bk7_procedural_detection",
  "label": "corollary:bk7_procedural_detection",
  "latex_body": "\\begin{corollary}[Certified Procedural Detection]\n\\label{corollary:bk7_procedural_detection}\nLet $0<\\epsilon_1<\\epsilon_2$.  Assume the fitted exponent is strictly\ndecreasing, so $p(\\epsilon_2)<p(\\epsilon_1)$, and separately assume the plotted\nresidual magnitude decreases,\n\\[\n \\lVert E_{\\epsilon_2}\\rVert_{p(\\epsilon_2)}\n <\\lVert E_{\\epsilon_1}\\rVert_{p(\\epsilon_1)}.\n\\]\nThen the log--log secant slope of the residual observable between the two\nscales is strictly negative.  Exponent monotonicity alone does not determine\nthe direction of a separately varying residual norm.  Appendix B observations\nmay validate both premises but are not a proof of their universal coupling.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "positive increasing horizon scales",
      "strictly antitone fitted exponent",
      "strictly decreasing residual observable for the slope conclusion"
    ],
    "countermodels": [
      "Book7ProceduralDetection.decreasing_exponent_does_not_force_decreasing_observable"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Strict antitonicity proves the fitted-exponent ordering. A decreasing residual observable over positive increasing scales gives a strictly negative log-log secant slope. A countermodel shows exponent ordering alone does not orient a distinct residual observable, so the combined procedural certificate consumes residual decrease explicitly."
    ],
    "record_ids": [
      "MAP-BOOK7-051"
    ],
    "statuses": [
      "conditional"
    ],
    "witnesses": [
      "Book7ProceduralDetection.decreasing_exponent_does_not_force_decreasing_observable",
      "Book7ProceduralDetection.fittedExponent_decreases",
      "Book7ProceduralDetection.logLogSecantSlope_neg",
      "Book7ProceduralDetection.proceduralDetection_certificate"
    ]
  },
  "line": 1676,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified Procedural Detection",
  "proof_labels": [
    "proof:bk7_lp_norm_monotonicity"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "corollary:bk7_procedural_detection",
    "theorem:bk7_emergent_lp_norm"
  ],
  "depends_on": [
    "corollary:bk7_procedural_detection",
    "theorem:bk7_emergent_lp_norm"
  ],
  "file": "book7.tex",
  "id": "proof:bk7_lp_norm_monotonicity",
  "label": "proof:bk7_lp_norm_monotonicity",
  "latex_body": "\\begin{proof}[Two-Premise Detection]\n\\label{proof:bk7_lp_norm_monotonicity}\n\\leavevmode\nStrict antitonicity gives the exponent ordering.  Since logarithm is strictly\nincreasing on positive scales, $\\log\\epsilon_2-\\log\\epsilon_1>0$; the supplied\ndecrease of the residual observable makes the log--log secant numerator\nnegative, hence its slope is negative.  A decreasing exponent paired with an\nincreasing observable is a countermodel if the second premise is omitted.\n\\end{proof}",
  "line": 1690,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Two-Premise Detection",
  "proves": "corollary:bk7_procedural_detection",
  "ref_roles": [
    {
      "context": "",
      "label": "corollary:bk7_procedural_detection",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 1676,
      "target_type": "corollary"
    },
    {
      "context": "",
      "label": "theorem:bk7_emergent_lp_norm",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 1596,
      "target_type": "theorem"
    }
  ],
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book7",
  "cited_by": [
    "proposition:bk9_mechanisms_of_recognition"
  ],
  "cites": [
    "corollary:bk7_fixed_point_tracking_within_evolving_reciprocity",
    "theorem:bk4_reflective_reentry"
  ],
  "depends_on": [
    "corollary:bk7_fixed_point_tracking_within_evolving_reciprocity",
    "theorem:bk4_reflective_reentry"
  ],
  "file": "book7.tex",
  "id": "scholium:bk7_unnamed_scholium_03",
  "label": "scholium:bk7_unnamed_scholium_03",
  "latex_body": "\\begin{scholium}\n\\label{scholium:bk7_unnamed_scholium_03}\nThe tracking behavior established in Cor.~\\ref{corollary:bk7_fixed_point_tracking_within_evolving_reciprocity} (cf.~\\ref{theorem:bk4_reflective_reentry}) reveals a profound aspect of reciprocal relationships under changing conditions. For symbolic systems undergoing meta-reflective drift -- whether representing evolving minds, theories, or social institutions -- stable alignment requires not merely convergence at a fixed moment, but continuous adaptation of the reciprocity mechanism itself. The persistence of mutual understanding or functional coupling depends on the ability of the systems' reflective processes (\\(\\reflect_{\\mathcal{A}}(t), \\reflect_{\\mathcal{B}}(t)\\)) to adapt at a rate commensurate with the underlying structural changes (\\(\\drift_{\\mathrm{meta}}\\)).\nThis result suggests that durable symbolic relationships must possess a second-order stability: not only must the systems converge within a reciprocity domain, but the domain itself must evolve coherently with the underlying systems. When this coherence is maintained (\\(\\tau_{\\mathrm{meta}} \\gg \\tau_{\\mathrm{conv}}(t)\\)), the relationship between the systems preserves its essential character -- mutual reflection leading to alignment -- despite transformation of the constituent parts or the environment. This offers a formal characterization of how mutual understanding, empathy, or stable cooperation can persist through change, provided the change occurs at a pace that allows continuous co-reflective realignment. Conversely, rapid meta-drift exceeding the system's adaptive capacity leads to a breakdown of reciprocity (\\( (x_A(t), y_B(t)) \\notin \\recipdomain(t) \\)) and potential decoupling or conflict. \\qed\n\\end{scholium}",
  "line": 1711,
  "macros_used": [
    "drift",
    "recipdomain",
    "reflect"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "",
  "ref_roles": [
    {
      "context": "\\begin{scholium} \\label{scholium:bk7_unnamed_scholium_03} The tracking behavior established in Cor.~\\ref{corollary:bk7_fixed_point_tracking_within_evolving_reciprocity} (cf.~\\ref{theorem:bk4_reflective_reentry}) reveals a profound aspect of reciprocal relationships under changing conditi",
      "label": "corollary:bk7_fixed_point_tracking_within_evolving_reciprocity",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book7.tex",
      "target_line": 1213,
      "target_type": "corollary"
    },
    {
      "context": "03} The tracking behavior established in Cor.~\\ref{corollary:bk7_fixed_point_tracking_within_evolving_reciprocity} (cf.~\\ref{theorem:bk4_reflective_reentry}) reveals a profound aspect of reciprocal relationships under changing conditions. For symbolic systems undergoing meta-",
      "label": "theorem:bk4_reflective_reentry",
      "logical_support": true,
      "role": "cf_near_match",
      "target_file": "book4.tex",
      "target_line": 2840,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "corollary:bk7_fixed_point_tracking_within_evolving_reciprocity",
    "theorem:bk4_reflective_reentry"
  ],
  "role": "scholium",
  "type": "scholium"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "theorem:bk7_emergent_lp_norm"
  ],
  "depends_on": [
    "theorem:bk7_emergent_lp_norm"
  ],
  "file": "book7.tex",
  "id": "subsec:bk7_hilbert_banach_bridge",
  "label": "subsec:bk7_hilbert_banach_bridge",
  "latex_body": "",
  "line": 1716,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "The Hilbert--Banach Bridge",
  "ref_roles": [
    {
      "context": "",
      "label": "theorem:bk7_emergent_lp_norm",
      "logical_support": false,
      "role": "navigation",
      "target_file": "book7.tex",
      "target_line": 1596,
      "target_type": "theorem"
    }
  ],
  "role": "section",
  "subtype": "subsection",
  "type": "section"
}

{
  "book": "book7",
  "cited_by": [
    "lemma:bk7_frame_temperature_exponent_correspondence"
  ],
  "cites": [
    "definition:bk2_symbolic_temperature"
  ],
  "depends_on": [
    "definition:bk2_symbolic_temperature"
  ],
  "file": "book7.tex",
  "id": "definition:bk7_frame_temperature_quotient",
  "label": "definition:bk7_frame_temperature_quotient",
  "latex_body": "\\begin{definition}[Frame-temperature quotient]\n\\label{definition:bk7_frame_temperature_quotient}\nLet $T(\\tilde\\rho)$ be the symbolic temperature of the observer's perceived state\n(Def.~\\ref{definition:bk2_symbolic_temperature}) and let $T_{\\mathcal{F}}(\\epsilon)$\nbe the \\emph{frame-resolution temperature}: a continuous, strictly decreasing\nfunction of the horizon $\\epsilon$ with $T_{\\mathcal{F}}(\\epsilon)\\to\\infty$ as\n$\\epsilon\\to 0^{+}$ and $T_{\\mathcal{F}}(\\epsilon)\\to 0$ as $\\epsilon\\to\\infty$,\nquantifying the differentiation resolution available within the frame. The\n\\emph{frame-temperature quotient} is\n\\[\n\\xi(\\tilde\\rho,\\epsilon)\\;=\\;\\frac{T(\\tilde\\rho)}{T_{\\mathcal{F}}(\\epsilon)}.\n\\]\nA small $\\xi$ marks a system cold relative to its frame (sharply resolved); a\nlarge $\\xi$ marks a system hot relative to its frame (coarsely resolved).\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": [
      "Monotonicity of xi=T/T_F(eps) in eps under a strictly-decreasing T_F, stated via two explicit T_F values rather than a functional hypothesis."
    ],
    "record_ids": [
      "MAP-BOOK7-045"
    ],
    "statuses": [
      "exact"
    ],
    "witnesses": [
      "Book7B.frameTempQuotient_mono"
    ]
  },
  "line": 1730,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Frame-temperature quotient",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "bk7_frame_temperature_quotient} Let $T(\\tilde\\rho)$ be the symbolic temperature of the observer's perceived state (Def.~\\ref{definition:bk2_symbolic_temperature}) and let $T_{\\mathcal{F}}(\\epsilon)$ be the \\emph{frame-resolution temperature}: a continuous, strictly decreasing func",
      "label": "definition:bk2_symbolic_temperature",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 148,
      "target_type": "definition"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_temperature"
  ],
  "role": "definition",
  "type": "definition"
}

{
  "book": "book7",
  "cited_by": [
    "proof:bk7_hilbert_banach_bridge",
    "theorem:bk7_hilbert_banach_bridge"
  ],
  "cites": [
    "definition:bk7_frame_temperature_quotient",
    "theorem:bk7_emergent_lp_norm"
  ],
  "depends_on": [
    "definition:bk7_frame_temperature_quotient",
    "theorem:bk7_emergent_lp_norm"
  ],
  "file": "book7.tex",
  "id": "lemma:bk7_frame_temperature_exponent_correspondence",
  "label": "lemma:bk7_frame_temperature_exponent_correspondence",
  "latex_body": "\\begin{lemma}[Frame-temperature/exponent correspondence]\n\\label{lemma:bk7_frame_temperature_exponent_correspondence}\nFor $T(\\tilde\\rho)>0$, the quotient $\\epsilon\\mapsto\\xi(\\tilde\\rho,\\epsilon)$ of\nDef.~\\ref{definition:bk7_frame_temperature_quotient} is continuous and strictly\nincreasing, with $\\xi\\to 0^{+}$ as $\\epsilon\\to 0^{+}$ and $\\xi\\to\\infty$ as\n$\\epsilon\\to\\infty$. Consequently the emergent exponent\n$p$ of Thm.~\\ref{theorem:bk7_emergent_lp_norm} is a continuous, strictly\ndecreasing function $p=p(\\xi)$ on $(0,\\infty)$ with\n\\[\n\\lim_{\\xi\\to 0^{+}}p(\\xi)=\\infty,\n\\qquad\n\\lim_{\\xi\\to\\infty}p(\\xi)=1,\n\\]\nand there is a unique $\\xi^{\\ast}\\in(0,\\infty)$ with $p(\\xi^{\\ast})=2$. A scale\ncalibration of $T_{\\mathcal{F}}$ normalizes $\\xi^{\\ast}=1$.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "Banach/Hilbert space theory, measure theory, infinite-limit claims, and the Gleason/Born cluster are NOT formalized",
      "recurrence laws, descent laws, and fixed-point existence are structure fields or explicit hypotheses",
      "theorem:bk7_pisu skipped: depends on a channel-floors assumption referenced but absent from the sliced packet"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Only the uniqueness clause (a strictly antitone function is injective, hence at most one xi* with p(xi*)=2) is proved. Existence of xi* (via the intermediate value theorem from the stated limits) and the explicit construction of p from the frame-temperature quotient are not modeled."
    ],
    "record_ids": [
      "MAP-BOOK7-017"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book7.exponent_uniqueness"
    ]
  },
  "line": 1746,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Frame-temperature/exponent correspondence",
  "proof_labels": [
    "proof:bk7_frame_temperature_exponent_correspondence"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "perature_exponent_correspondence} For $T(\\tilde\\rho)>0$, the quotient $\\epsilon\\mapsto\\xi(\\tilde\\rho,\\epsilon)$ of Def.~\\ref{definition:bk7_frame_temperature_quotient} is continuous and strictly increasing, with $\\xi\\to 0^{+}$ as $\\epsilon\\to 0^{+}$ and $\\xi\\to\\infty$ as $\\epsilon\\to\\in",
      "label": "definition:bk7_frame_temperature_quotient",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book7.tex",
      "target_line": 1730,
      "target_type": "definition"
    },
    {
      "context": "0^{+}$ as $\\epsilon\\to 0^{+}$ and $\\xi\\to\\infty$ as $\\epsilon\\to\\infty$. Consequently the emergent exponent $p$ of Thm.~\\ref{theorem:bk7_emergent_lp_norm} is a continuous, strictly decreasing function $p=p(\\xi)$ on $(0,\\infty)$ with \\[ \\lim_{\\xi\\to 0^{+}}p(\\xi)=\\infty, \\qqu",
      "label": "theorem:bk7_emergent_lp_norm",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book7.tex",
      "target_line": 1596,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk7_frame_temperature_quotient",
    "theorem:bk7_emergent_lp_norm"
  ],
  "role": "lemma",
  "type": "lemma"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "theorem:bk7_emergent_lp_norm"
  ],
  "depends_on": [
    "theorem:bk7_emergent_lp_norm"
  ],
  "file": "book7.tex",
  "id": "proof:bk7_frame_temperature_exponent_correspondence",
  "label": "proof:bk7_frame_temperature_exponent_correspondence",
  "latex_body": "\\begin{proof}[Frame-temperature/exponent correspondence]\n\\label{proof:bk7_frame_temperature_exponent_correspondence}\n\\leavevmode\n\nSince $T_{\\mathcal{F}}$ is continuous and strictly decreasing in $\\epsilon$ with\nthe stated limits, its reciprocal is continuous and strictly increasing, so\n$\\xi=T/T_{\\mathcal{F}}$ inherits continuity and strict monotonicity in $\\epsilon$\nand the endpoint limits $\\xi\\to 0^{+}$ ($\\epsilon\\to 0^{+}$) and $\\xi\\to\\infty$\n($\\epsilon\\to\\infty$). The map $\\epsilon\\mapsto p$ is $C^{1}$ and strictly\ndecreasing by Thm.~\\ref{theorem:bk7_emergent_lp_norm}. Composing the strictly\ndecreasing $\\epsilon\\mapsto p$ with the strictly increasing inverse\n$\\xi\\mapsto\\epsilon$ yields a continuous, strictly decreasing $p(\\xi)$, and the\nlimits $p\\to\\infty$ (as $\\epsilon\\to 0^{+}$, i.e.\\ $\\xi\\to 0^{+}$) and $p\\to 1$\n(as $\\epsilon\\to\\infty$, i.e.\\ $\\xi\\to\\infty$) transfer directly. Because $p(\\xi)$\nis continuous and strictly decreasing through the value $2\\in(1,\\infty)$, the\nintermediate value theorem gives a unique $\\xi^{\\ast}$ with $p(\\xi^{\\ast})=2$;\nrescaling $T_{\\mathcal{F}}$ by the positive constant $\\xi^{\\ast}$ sets the Hilbert\npoint at $\\xi^{\\ast}=1$.\n\\end{proof}",
  "line": 1763,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Frame-temperature/exponent correspondence",
  "proves": "lemma:bk7_frame_temperature_exponent_correspondence",
  "ref_roles": [
    {
      "context": "^{+}$) and $\\xi\\to\\infty$ ($\\epsilon\\to\\infty$). The map $\\epsilon\\mapsto p$ is $C^{1}$ and strictly decreasing by Thm.~\\ref{theorem:bk7_emergent_lp_norm}. Composing the strictly decreasing $\\epsilon\\mapsto p$ with the strictly increasing inverse $\\xi\\mapsto\\epsilon$ yields",
      "label": "theorem:bk7_emergent_lp_norm",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 1596,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "theorem:bk7_emergent_lp_norm"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk4_observer_kernel_convolution_map",
    "definition:bk6_symbolic_curvature_tensor",
    "lemma:bk7_frame_temperature_exponent_correspondence"
  ],
  "depends_on": [
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk4_observer_kernel_convolution_map",
    "definition:bk6_symbolic_curvature_tensor",
    "lemma:bk7_budgetlimited_minimizer",
    "lemma:bk7_frame_temperature_exponent_correspondence",
    "theorem:bk7_emergent_lp_norm"
  ],
  "file": "book7.tex",
  "id": "theorem:bk7_hilbert_banach_bridge",
  "label": "theorem:bk7_hilbert_banach_bridge",
  "latex_body": "\\begin{theorem}[Hilbert--Banach Bridge]\n\\label{theorem:bk7_hilbert_banach_bridge}\nSweep the frame-temperature quotient $\\xi\\in(0,\\infty)$ and consider the family of\neffective observer geometries\n$\\bigl(L^{p(\\xi)}(\\mathcal{M}_{\\epsilon},\\mu_{g}),\\,K_{\\Obs}\\bigr)$, where\n$p(\\xi)$ is the emergent exponent\n(Lemma~\\ref{lemma:bk7_frame_temperature_exponent_correspondence}), $K_{\\Obs}$ is\nthe observer-kernel smoothing map\n(Def.~\\ref{definition:bk4_observer_kernel_convolution_map}), and the quadratic\nsymbolic coupling $\\kappa$ (Def.~\\ref{definition:bk6_symbolic_curvature_tensor})\nstays strictly below a critical value $\\kappa^{\\ast}$. Then:\n\\begin{enumerate}\n\\item \\emph{(Interpolated continuity.)} For $\\xi_{0}<\\xi_{1}$ with exponents\n$p_{0}=p(\\xi_{0})\\ge p_{1}=p(\\xi_{1})$, every observer-visible observable $f$ lies\nin the interpolation scale with\n\\[\n\\tfrac{1}{p_{\\theta}}=\\tfrac{1-\\theta}{p_{0}}+\\tfrac{\\theta}{p_{1}},\n\\qquad\n\\lVert f\\rVert_{p_{\\theta}}\\le\n\\lVert f\\rVert_{p_{0}}^{\\,1-\\theta}\\,\\lVert f\\rVert_{p_{1}}^{\\,\\theta}\n\\quad(0\\le\\theta\\le 1),\n\\]\nand $K_{\\Obs}$ is bounded on each $L^{p}$; hence $\\xi\\mapsto$ effective geometry is\nnorm-continuous and passes through the Banach regime ($p\\to 1$: complete and\nnorm-robust, no inner product) and the Hilbert regime ($p=2$ at $\\xi^{\\ast}$:\ninner product, orthogonal projection, phase and spectral observables) without\ndiscontinuity.\n\\item \\emph{(Hilbert observables are a single cross-section.)} The\ninner-product and phase structure holds exactly on the level set\n$\\{\\xi:p(\\xi)=2\\}=\\{\\xi^{\\ast}\\}$---parallelogram identity, orthogonal\nprojection, well-defined relative phase---while off it, projection is\nreplaced by the smooth $L^{p(\\xi)}$ reweighting of symbolic coherence\nfrom part~(i).\n\\item \\emph{(Phase shift only at threshold.)} A genuine phase shift---a\ndiscontinuity of $\\xi\\mapsto$ effective geometry, equivalently a loss of $C^{1}$\nregularity of $\\xi\\mapsto p$---occurs only when the smoothing-kernel support or the\nquadratic coupling $\\kappa$ reaches $\\kappa^{\\ast}$, where the emergent functional\nchanges convexity class and its minimizer ceases to be unique. Below threshold the\nsweep is a smooth reweighting; at threshold the minimizer bifurcates, realizing a\nsymbolic phase transition (Def.~\\ref{definition:bk2_symbolic_phase_transitio}).\n\\end{enumerate}\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "Banach/Hilbert space theory, measure theory, infinite-limit claims, and the Gleason/Born cluster are NOT formalized",
      "recurrence laws, descent laws, and fixed-point existence are structure fields or explicit hypotheses",
      "theorem:bk7_pisu skipped: depends on a channel-floors assumption referenced but absent from the sliced packet"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Only the finite two-coordinate shadow of the Banach(p=1)/Hilbert(p=2) norm comparison is proved (L^1 and L^2 on R^2 bound each other within sqrt 2). The general L^p interpolation inequality, the emergent-exponent construction, and the phase-transition/threshold clause are not modeled."
    ],
    "record_ids": [
      "MAP-BOOK7-018"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book7.l1_l2_comparison"
    ]
  },
  "line": 1783,
  "macros_used": [
    "Obs"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Hilbert--Banach Bridge",
  "proof_labels": [
    "proof:bk7_hilbert_banach_bridge"
  ],
  "proof_status": "proven",
  "ref_roles": [
    {
      "context": "d the sweep is a smooth reweighting; at threshold the minimizer bifurcates, realizing a symbolic phase transition (Def.~\\ref{definition:bk2_symbolic_phase_transitio}). \\end{enumerate} \\end{theorem}",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    },
    {
      "context": "Lemma~\\ref{lemma:bk7_frame_temperature_exponent_correspondence}), $K_{\\Obs}$ is the observer-kernel smoothing map (Def.~\\ref{definition:bk4_observer_kernel_convolution_map}), and the quadratic symbolic coupling $\\kappa$ (Def.~\\ref{definition:bk6_symbolic_curvature_tensor}) stays strictly bel",
      "label": "definition:bk4_observer_kernel_convolution_map",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book4.tex",
      "target_line": 143,
      "target_type": "definition"
    },
    {
      "context": "ing map (Def.~\\ref{definition:bk4_observer_kernel_convolution_map}), and the quadratic symbolic coupling $\\kappa$ (Def.~\\ref{definition:bk6_symbolic_curvature_tensor}) stays strictly below a critical value $\\kappa^{\\ast}$. Then: \\begin{enumerate} \\item \\emph{(Interpolated continuity.)}",
      "label": "definition:bk6_symbolic_curvature_tensor",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book6.tex",
      "target_line": 16,
      "target_type": "definition"
    },
    {
      "context": "ies $\\bigl(L^{p(\\xi)}(\\mathcal{M}_{\\epsilon},\\mu_{g}),\\,K_{\\Obs}\\bigr)$, where $p(\\xi)$ is the emergent exponent (Lemma~\\ref{lemma:bk7_frame_temperature_exponent_correspondence}), $K_{\\Obs}$ is the observer-kernel smoothing map (Def.~\\ref{definition:bk4_observer_kernel_convolution_map}), and the",
      "label": "lemma:bk7_frame_temperature_exponent_correspondence",
      "logical_support": true,
      "role": "formal_dependency",
      "target_file": "book7.tex",
      "target_line": 1746,
      "target_type": "lemma"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_phase_transitio",
    "definition:bk4_observer_kernel_convolution_map",
    "definition:bk6_symbolic_curvature_tensor",
    "lemma:bk7_frame_temperature_exponent_correspondence"
  ],
  "role": "theorem",
  "type": "theorem"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [
    "definition:bk2_symbolic_phase_transitio",
    "lemma:bk7_budgetlimited_minimizer",
    "lemma:bk7_frame_temperature_exponent_correspondence",
    "theorem:bk7_emergent_lp_norm"
  ],
  "depends_on": [
    "definition:bk2_symbolic_phase_transitio",
    "lemma:bk7_budgetlimited_minimizer",
    "lemma:bk7_frame_temperature_exponent_correspondence",
    "theorem:bk7_emergent_lp_norm"
  ],
  "file": "book7.tex",
  "id": "proof:bk7_hilbert_banach_bridge",
  "label": "proof:bk7_hilbert_banach_bridge",
  "latex_body": "\\begin{proof}[Hilbert--Banach Bridge]\n\\label{proof:bk7_hilbert_banach_bridge}\n\\leavevmode\n\n\\emph{(i)} The emergent-norm family is the $L^{p}$ scale of\nThm.~\\ref{theorem:bk7_emergent_lp_norm} over the $\\sigma$-finite measure space\n$(\\mathcal{M}_{\\epsilon},\\mu_{g})$. The stated bound is the Riesz--Thorin / complex\ninterpolation inequality between the endpoints $L^{p_{0}}$ and $L^{p_{1}}$, and the\ninterpolation exponent $p_{\\theta}$ moves continuously because $p(\\xi)$ is\ncontinuous (Lemma~\\ref{lemma:bk7_frame_temperature_exponent_correspondence}).\nSmoothing by the perceptual kernel obeys Young's inequality,\n$\\lVert K_{\\Obs}\\!*f\\rVert_{p}\\le\\lVert K_{\\Obs}\\rVert_{1}\\lVert f\\rVert_{p}$, so\n$K_{\\Obs}$ is bounded on every $L^{p}$ and preserves the continuity of the sweep.\nThe endpoints identify the Banach regime at $p\\to 1$ and the Hilbert regime at\n$p=2$, the latter located at $\\xi^{\\ast}$ by the lemma.\n\n\\emph{(ii)} By the Jordan--von Neumann theorem, an $L^{p}$ space of dimension at\nleast two satisfies the parallelogram identity---and hence carries an inner\nproduct, orthogonal projection, and relative phase---if and only if $p=2$. Thus\nthe Hilbert observables are supported exactly on $\\{\\xi:p(\\xi)=2\\}$, which by the\nlemma is the single point $\\xi^{\\ast}$. For $\\xi\\neq\\xi^{\\ast}$ the parallelogram\nidentity fails, and the best available structure is the interpolated reweighting\nof part~(i).\n\n\\emph{(iii)} The map $\\xi\\mapsto p$ is $C^{1}$ and strictly monotone wherever the\nemergent cost functional is strictly convex, which holds while $\\kappa<\\kappa^{\\ast}$\nbecause the SRMF dual weight $w_{\\epsilon}$ remains strictly positive\n(Lemma~\\ref{lemma:bk7_budgetlimited_minimizer}, Thm.~\\ref{theorem:bk7_emergent_lp_norm}).\nAs $\\kappa\\uparrow\\kappa^{\\ast}$ the dual weight loses positivity on a set of\npositive measure, the penalty degenerates from strict to non-strict convexity, and\nthe minimizer set ceases to be a singleton; at that point $\\xi\\mapsto p$ loses\n$C^{1}$ regularity and the effective geometry jumps. A discontinuity therefore\nrequires the threshold crossing, and below it the bridge is smooth. The bifurcation\nof the minimizer is precisely the symbolic phase transition of\nDef.~\\ref{definition:bk2_symbolic_phase_transitio}.\n\\end{proof}",
  "line": 1826,
  "macros_used": [
    "Obs"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Hilbert--Banach Bridge",
  "proves": "theorem:bk7_hilbert_banach_bridge",
  "ref_roles": [
    {
      "context": "and below it the bridge is smooth. The bifurcation of the minimizer is precisely the symbolic phase transition of Def.~\\ref{definition:bk2_symbolic_phase_transitio}. \\end{proof}",
      "label": "definition:bk2_symbolic_phase_transitio",
      "logical_support": true,
      "role": "definition_anchor",
      "target_file": "book2.tex",
      "target_line": 377,
      "target_type": "definition"
    },
    {
      "context": ", which holds while $\\kappa<\\kappa^{\\ast}$ because the SRMF dual weight $w_{\\epsilon}$ remains strictly positive (Lemma~\\ref{lemma:bk7_budgetlimited_minimizer}, Thm.~\\ref{theorem:bk7_emergent_lp_norm}). As $\\kappa\\uparrow\\kappa^{\\ast}$ the dual weight loses positivity on a set o",
      "label": "lemma:bk7_budgetlimited_minimizer",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 1558,
      "target_type": "lemma"
    },
    {
      "context": "$ and $L^{p_{1}}$, and the interpolation exponent $p_{\\theta}$ moves continuously because $p(\\xi)$ is continuous (Lemma~\\ref{lemma:bk7_frame_temperature_exponent_correspondence}). Smoothing by the perceptual kernel obeys Young's inequality, $\\lVert K_{\\Obs}\\!*f\\rVert_{p}\\le\\lVert K_{\\Obs}\\rVert_{",
      "label": "lemma:bk7_frame_temperature_exponent_correspondence",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 1746,
      "target_type": "lemma"
    },
    {
      "context": "] \\label{proof:bk7_hilbert_banach_bridge} \\leavevmode \\emph{(i)} The emergent-norm family is the $L^{p}$ scale of Thm.~\\ref{theorem:bk7_emergent_lp_norm} over the $\\sigma$-finite measure space $(\\mathcal{M}_{\\epsilon},\\mu_{g})$. The stated bound is the Riesz--Thorin / comp",
      "label": "theorem:bk7_emergent_lp_norm",
      "logical_support": true,
      "role": "proof_support",
      "target_file": "book7.tex",
      "target_line": 1596,
      "target_type": "theorem"
    }
  ],
  "refs": [
    "definition:bk2_symbolic_phase_transitio",
    "lemma:bk7_budgetlimited_minimizer",
    "lemma:bk7_frame_temperature_exponent_correspondence",
    "theorem:bk7_emergent_lp_norm"
  ],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book7.tex",
  "id": "corollary:bk7_bridge_no_interior_transition",
  "label": "corollary:bk7_bridge_no_interior_transition",
  "latex_body": "\\begin{corollary}[Certified continuous $L^p$ sweep has no interior transition]\n\\label{corollary:bk7_bridge_no_interior_transition}\nLet $G:[\\xi_0,\\xi_1]\\to\\mathcal G$ be the effective geometry.  Assume a\ncurvature-to-regularity bridge proving that the uniform bound\n$\\kappa(\\xi)<\\kappa^*$ on the closed sweep entails continuity (or $C^1$\nregularity) of $G$.  Then no interior point is a discrete phase transition,\nwhere such a transition means failure of continuity relative to the sweep.\nThe numerical curvature inequality does not imply regularity without this\nbridge.  The Appendix SRV sweep is downstream corroboration of the certified\nregime, not the premise establishing continuity.\n\\end{corollary}",
  "lean_alignment": {
    "conditions": [
      "continuous observer reparameterization",
      "effective geometry continuous on the closed sub-sweep",
      "explicit curvature-to-regularity bridge when starting from kappa below threshold",
      "or the constructed curvature-indexed Lp coordinate with a continuous curvature path and positive margin",
      "phase transition represented as relative discontinuity"
    ],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Constructive scalar Lp representation: p(xi) = 2 + curvature(xi)/(threshold - curvature(xi)) is Hilbertian at zero curvature, continuous for a continuous subcritical curvature path, strictly order-preserving when the threshold is positive, and has no interior transition. A more general signal-resolvent instance is also proved. Identifying the complete G-valued effective geometry with its scalar p-coordinate remains explicit scope, not an automatic equivalence."
    ],
    "record_ids": [
      "MAP-BOOK7-049"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book7NoInteriorTransition.continuity_from_threshold_bridge",
      "Book7NoInteriorTransition.continuousOn_no_discrete_phase_transition",
      "Book7NoInteriorTransition.continuous_closed_sweep_has_no_interior_transition",
      "Book7NoInteriorTransition.continuous_reparameterization_preserves_no_transition",
      "Book7NoInteriorTransition.regularizedGeometry_continuousOn",
      "Book7NoInteriorTransition.regularizedGeometry_has_no_interior_transition",
      "Book7NoInteriorTransition.subcriticalLpExponent_continuousOn",
      "Book7NoInteriorTransition.subcriticalLpExponent_has_no_interior_transition",
      "Book7NoInteriorTransition.subcriticalLpExponent_strict_order",
      "Book7NoInteriorTransition.subcriticalLpExponent_zero_curvature"
    ]
  },
  "line": 1863,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified continuous $L^p$ sweep has no interior transition",
  "proof_labels": [
    "proof:bk7_bridge_no_interior_transition"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "corollary",
  "type": "corollary"
}

{
  "book": "book7",
  "cited_by": [],
  "cites": [],
  "depends_on": [],
  "file": "book7.tex",
  "id": "proof:bk7_bridge_no_interior_transition",
  "label": "proof:bk7_bridge_no_interior_transition",
  "latex_body": "\\begin{proof}[Continuity Excludes a Discrete Transition]\n\\label{proof:bk7_bridge_no_interior_transition}\n\\leavevmode\nApply the supplied curvature-to-regularity bridge to obtain continuity of\n$G$ on the closed sweep.  At every point of that domain, continuity within the\ndomain is therefore true, so its negation---the defined discrete phase\ntransition---is false.\n\\end{proof}",
  "line": 1874,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Continuity Excludes a Discrete Transition",
  "proves": "corollary:bk7_bridge_no_interior_transition",
  "refs": [],
  "role": "proof",
  "type": "proof"
}

{
  "book": "book7",
  "cited_by": [
    "definition:bk7_contextuality_defect"
  ],
  "cites": [],
  "depends_on": [],
  "file": "book7.tex",
  "id": "lemma:bk7_noncontextuality_forces_hilbert",
  "label": "lemma:bk7_noncontextuality_forces_hilbert",
  "latex_body": "\\begin{lemma}[Certified non-contextuality/Hilbert cross-section equivalence]\n\\label{lemma:bk7_noncontextuality_forces_hilbert}\nAlong the bridge family, assume separately:\n\\begin{enumerate}\n\\item a coherence-representation theorem identifying frame-independent\nprojector values with the parallelogram/inner-product property; and\n\\item the $L^p$ geometry theorem identifying that property with $p(\\xi)=2$\nunder the stated dimensional and regularity hypotheses.\n\\end{enumerate}\nThen PS-C3$'$ non-contextuality holds if and only if the effective geometry is\nthe Hilbert cross-section $p(\\xi)=2$.  Hilbert geometry alone does not constrain\nan otherwise unspecified coherence functional.  Appendix C may instantiate\nthe coherence bridge downstream; it is not imported backward as this lemma's\npremise.\n\\end{lemma}",
  "lean_alignment": {
    "conditions": [
      "a genuine Mathlib real QuadraticForm",
      "a symmetric bilinear energy representation supplies the metric parallelogram law",
      "additive transports commute around affine squares",
      "additivity of the polarization in one argument",
      "degree-two scalar homogeneity",
      "nonzero rays for frame normalization and scaling invariance",
      "positive-semidefinite diagonal additionally assumes pointwise nonnegativity",
      "real rank-two coordinate model",
      "scalar homogeneity of the polarization in one argument",
      "the relevant Lp geometry has the parallelogram property iff p=2"
    ],
    "countermodels": [
      "Book7GleasonBoundary.rank_two_frame_axioms_do_not_force_born",
      "Book7NoncontextualHilbert.commuting_transport_does_not_force_metric_parallelogram",
      "Book7NoncontextualHilbert.hilbert_geometry_alone_does_not_force_noncontextuality",
      "Book7NoncontextualHilbert.l1_parallelogram_fails",
      "Book7QuadraticPolarization.nonnegative_readout_does_not_force_quadratic"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "Rank-two reconstruction boundary: additive transports form commuting affine squares, but the L1 countermodel proves affine path independence does not imply the metric parallelogram law. A genuine coordinate-free quadratic form now canonically constructs its symmetric bilinear polarization, whose diagonal recovers the form exactly; a supplied symmetric bilinear coupling then yields the metric parallelogram law. The exact readout interface is now proved equivalent to a quadratic-form witness: degree-two scaling plus additive and homogeneous polarization. An explicit rank-two frame function now proves that nonnegativity, ray invariance, and orthogonal-pair normalization still do not force those laws. The remaining bridge is therefore specifically a frame-rank-at-least-three derivation of the certificate laws from noncontextual frame coherence; the Lp parallelogram characterization then selects p=2."
    ],
    "record_ids": [
      "MAP-BOOK7-052"
    ],
    "statuses": [
      "open_bridge"
    ],
    "witnesses": [
      "Book7GleasonBoundary.rank_two_frame_axioms_do_not_force_born",
      "Book7NoncontextualHilbert.commuting_transport_does_not_force_metric_parallelogram",
      "Book7NoncontextualHilbert.hilbert_geometry_alone_does_not_force_noncontextuality",
      "Book7NoncontextualHilbert.innerProductSpace_exists_of_metric_parallelogram",
      "Book7NoncontextualHilbert.l1_parallelogram_fails",
      "Book7NoncontextualHilbert.l2_parallelogram",
      "Book7NoncontextualHilbert.noncontextual_iff_hilbert_crossSection",
      "Book7NoncontextualHilbert.quadraticEnergy_parallelogram",
      "Book7NoncontextualHilbert.translate_square_commutes",
      "Book7QuadraticPolarization.QuadraticReadoutLaws.roundtrip_value",
      "Book7QuadraticPolarization.QuadraticReadoutLaws.toQuadraticForm_apply",
      "Book7QuadraticPolarization.associated_diagonal_nonnegative",
      "Book7QuadraticPolarization.certified_readout_has_symmetric_bilinear_representation",
      "Book7QuadraticPolarization.nonnegative_readout_does_not_force_quadratic",
      "Book7QuadraticPolarization.quadraticForm_has_symmetric_bilinear_representation"
    ]
  },
  "line": 1883,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Certified non-contextuality/Hilbert cross-section equivalence",
  "proof_labels": [
    "proof:bk7_noncontextuality_forces_hilbert"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "lemma",
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  "book": "book7",
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  "cites": [],
  "depends_on": [],
  "file": "book7.tex",
  "id": "proof:bk7_noncontextuality_forces_hilbert",
  "label": "proof:bk7_noncontextuality_forces_hilbert",
  "latex_body": "\\begin{proof}[Composition of the Two Representation Bridges]\n\\label{proof:bk7_noncontextuality_forces_hilbert}\n\\leavevmode\nCompose the coherence-representation equivalence with the $L^p$\nparallelogram characterization.  This yields non-contextuality iff the\nparallelogram law holds iff $p(\\xi)=2$.  The concrete $L^1$ coordinate vectors\nviolate the parallelogram identity, while a deliberately unconstrained\ncoherence functional shows why the first equivalence must remain explicit.\n\\end{proof}",
  "line": 1899,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Composition of the Two Representation Bridges",
  "proves": "lemma:bk7_noncontextuality_forces_hilbert",
  "refs": [],
  "role": "proof",
  "type": "proof"
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{
  "appendix_teaser_ref_roles": [
    {
      "context": "\\Pi)\\big) - \\mu_{\\Obs,\\tilde\\psi}\\!\\big(T^{\\mathfrak{F}'}_{\\Obs}(\\Pi)\\big) \\big| \\ge 0, \\] the failure of PS-C3$'$ (Ax.~\\ref{axiom:appC_psc3prime}) at the effective exponent $p(\\xi)$, the supremum running over projectors $\\Pi$ and pairs of complete frames $\\mathfrak",
      "label": "axiom:appC_psc3prime",
      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 528,
      "target_type": "axiom"
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    "axiom:appC_psc3prime"
  ],
  "book": "book7",
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  "cites": [
    "axiom:appC_psc3prime",
    "lemma:bk7_noncontextuality_forces_hilbert"
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    "lemma:bk7_noncontextuality_forces_hilbert"
  ],
  "file": "book7.tex",
  "id": "definition:bk7_contextuality_defect",
  "label": "definition:bk7_contextuality_defect",
  "latex_body": "\\begin{definition}[Contextuality defect]\n\\label{definition:bk7_contextuality_defect}\nThe \\emph{contextuality defect} at frame temperature $\\xi$ is\n\\[\n\\Phi_{\\mathrm{nc}}(\\xi) := \\sup_{\\Pi,\\,\\mathfrak{F},\\mathfrak{F}'}\n\\big| \\mu_{\\Obs,\\tilde\\psi}\\!\\big(T^{\\mathfrak{F}}_{\\Obs}(\\Pi)\\big)\n- \\mu_{\\Obs,\\tilde\\psi}\\!\\big(T^{\\mathfrak{F}'}_{\\Obs}(\\Pi)\\big) \\big| \\ge 0,\n\\]\nthe failure of PS-C3$'$ (Ax.~\\ref{axiom:appC_psc3prime}) at the effective exponent\n$p(\\xi)$, the supremum running over projectors $\\Pi$ and pairs of complete frames\n$\\mathfrak{F},\\mathfrak{F}'$ realizing $\\Pi$. By\nLemma~\\ref{lemma:bk7_noncontextuality_forces_hilbert}, $\\Phi_{\\mathrm{nc}}(\\xi)=0$ iff\n$p(\\xi)=2$, i.e.\\ iff $\\xi=\\xi^{\\ast}$.\n\\end{definition}",
  "line": 1909,
  "macros_used": [
    "Obs"
  ],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Contextuality defect",
  "proof_status": "definitional",
  "ref_roles": [
    {
      "context": "\\Pi)\\big) - \\mu_{\\Obs,\\tilde\\psi}\\!\\big(T^{\\mathfrak{F}'}_{\\Obs}(\\Pi)\\big) \\big| \\ge 0, \\] the failure of PS-C3$'$ (Ax.~\\ref{axiom:appC_psc3prime}) at the effective exponent $p(\\xi)$, the supremum running over projectors $\\Pi$ and pairs of complete frames $\\mathfrak",
      "label": "axiom:appC_psc3prime",
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      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 528,
      "target_type": "axiom"
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    {
      "context": "remum running over projectors $\\Pi$ and pairs of complete frames $\\mathfrak{F},\\mathfrak{F}'$ realizing $\\Pi$. By Lemma~\\ref{lemma:bk7_noncontextuality_forces_hilbert}, $\\Phi_{\\mathrm{nc}}(\\xi)=0$ iff $p(\\xi)=2$, i.e.\\ iff $\\xi=\\xi^{\\ast}$. \\end{definition}",
      "label": "lemma:bk7_noncontextuality_forces_hilbert",
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  "role": "definition",
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  "book": "book7",
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  "file": "book7.tex",
  "id": "theorem:bk7_born_collapse",
  "label": "theorem:bk7_born_collapse",
  "latex_body": "\\begin{theorem}[Conditional Born collapse at the Hilbert cross-section]\n\\label{theorem:bk7_born_collapse}\nLet a reflective orbit in frame temperature converge to a limit $\\xi_\\infty$.\nAssume: (i) contextuality defect is nonnegative and vanishes exactly at the\nunique Hilbert frame $\\xi^*$; (ii) reflective fixed points are exactly the\nzero-defect states; (iii) the defect is continuous at $\\xi_\\infty$ and tends\nto zero along the orbit.  Then $\\xi_\\infty=\\xi^*$ and the orbit converges to\nthe Hilbert cross-section.\n\nFor a Born readout, assume separately a Gleason-style uniqueness certificate:\nat the Hilbert frame, every coherence assignment satisfying the stated\nnormalization, additivity, non-contextuality, regularity, and dimension\nhypotheses equals the Born functional.  Under that certificate the limiting\ncoherence readout is Born.  Hilbert collapse alone does not select a\nprobability functional.  Appendix C may validate or instantiate the uniqueness\ncertificate downstream; Book VII does not use the appendix as an upstream\npremise.\n\\end{theorem}",
  "lean_alignment": {
    "conditions": [
      "A separate Born/Gleason-style uniqueness certificate.",
      "Born identification additionally assumes local squared-amplitude calibration",
      "Defect continuity and convergence to zero.",
      "Hermitian exchange",
      "a finite outcome basis",
      "a genuine Mathlib real QuadraticForm",
      "a normalized finite complex pure-state vector",
      "a supplied complex cross term",
      "a supplied observer response kernel",
      "a supplied operator matrix",
      "a supplied ray map and matrix operator",
      "additivity and conjugate homogeneity in the first argument",
      "additivity and homogeneity in the second argument",
      "additivity of the polarization in one argument",
      "an existing QuantumResolutionCertificate",
      "degree-two scalar homogeneity",
      "diagonal recovery",
      "every outcome is covered by at least one finite frame",
      "exactly the existing FrameReadoutSystem fields",
      "exactly the existing QuantumResolutionCertificate fields",
      "finite complex coordinate carrier",
      "finite squared-amplitude readout with normalized amplitudes",
      "for the positive arrow only, its reducedState satisfies Matrix.IsHermitian",
      "global-phase invariance additionally assumes conjugate(u) times u equals one",
      "local readouts are nonnegative and normalized",
      "nonzero rays for frame normalization and scaling invariance",
      "pointwise amplitude calibration for finite uniqueness",
      "pointwise quadratic representation of the global frame measure",
      "positive-semidefinite diagonal additionally assumes pointwise nonnegativity",
      "readouts agree wherever two frames overlap",
      "real rank-two coordinate model",
      "reflective fixed point iff zero contextuality defect",
      "scalar homogeneity of the polarization in one argument",
      "the convergent orbit has continuous defect tending to zero",
      "the existing HermitianReadoutCertificate target",
      "the general Born readout additionally requires an explicit Gleason-style uniqueness bridge",
      "unit-modulus phase for the loss theorem",
      "zero contextuality defect iff exponent p=2 iff the unique Hilbert frame"
    ],
    "countermodels": [
      "Book7BornCollapse.hilbert_collapse_alone_does_not_determine_readout",
      "Book7BornCollapse.normalization_alone_does_not_force_finiteBorn",
      "Book7FrameMeasure.noncontextual_gluing_alone_does_not_force_born",
      "Book7GleasonBoundary.rank_two_frame_axioms_do_not_force_born",
      "Book7QuadraticPolarization.nonnegative_readout_does_not_force_quadratic",
      "Book7QuantumGleason.completeFrameCoherence_does_not_supply_hermitian_certificate",
      "Book7QuantumGleason.complex_phase_refutes_real_degreeTwo",
      "Book7QuantumGleason.pureState_lowering_not_injective",
      "Book7QuantumGleason.quantumResolution_does_not_force_reducedState_isHermitian",
      "Book7QuantumGleason.quantumResolution_without_matrixHermiticity_does_not_supply_certificate"
    ],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": true,
    "notes": [
      "Constructive finite measurement and guarded collapse remain separate from observer reconstruction. Complementary Gleason-facing half-bridges meet at a non-invertible observer seam: normalized pure-state data lower through Hermitian density and fixed response, while certified readout laws construct compatible representations without inverting the source. Global phase gives an exact collision and the preserved countermodels block unconditional reconstruction. The separate Cacophony-facing temporal backbone is formal: simultaneous compression has certified norm-fracture and diagonal cost bounds; directed stage costs telescope; JKO transport cost is paid by free-energy decrease; and convergence follows conditionally from explicit summability/completeness or Lyapunov-descent premises. Partial trace is an exact quantum reduction. Only the cross-domain identification of physical decoherence/noise with this general directed geometry remains interpretive.",
      "Hilbert collapse alone does not select a probability functional."
    ],
    "record_ids": [
      "MAP-BOOK7-054",
      "Q-BK7-08"
    ],
    "statuses": [
      "conditional",
      "open_bridge"
    ],
    "witnesses": [
      "Book7BornCollapse.amplitudeCalibratedReadout_unique",
      "Book7BornCollapse.amplitudeOfProbability_normalized",
      "Book7BornCollapse.born_readout_at_hilbert",
      "Book7BornCollapse.collapse_limit_eq_hilbertFrame",
      "Book7BornCollapse.collapse_tendsto_hilbertFrame",
      "Book7BornCollapse.defect_eq_zero_iff_hilbertFrame",
      "Book7BornCollapse.finiteBornValue_amplitudeOfProbability",
      "Book7BornCollapse.finiteBornValue_nonneg",
      "Book7BornCollapse.finiteBornValue_sum_one",
      "Book7BornCollapse.finite_probability_has_born_representation",
      "Book7BornCollapse.hilbert_collapse_alone_does_not_determine_readout",
      "Book7BornCollapse.nonhilbert_defect_pos",
      "Book7BornCollapse.normalization_alone_does_not_force_finiteBorn",
      "Book7BornCollapse.unique_stable_crossSection",
      "Book7BornCollapse.zero_curvature_hilbert_finiteBorn",
      "Book7FrameMeasure.FrameReadoutSystem.globalValue_eq_finiteBorn",
      "Book7FrameMeasure.FrameReadoutSystem.globalValue_eq_local",
      "Book7FrameMeasure.FrameReadoutSystem.globalValue_nonnegative",
      "Book7FrameMeasure.FrameReadoutSystem.globalValue_normalized_on_frame",
      "Book7FrameMeasure.FrameReadoutSystem.globalValue_unique",
      "Book7FrameMeasure.noncontextual_gluing_alone_does_not_force_born",
      "Book7GleasonBoundary.rank_two_frame_axioms_do_not_force_born",
      "Book7QuadraticPolarization.QuadraticReadoutLaws.roundtrip_value",
      "Book7QuadraticPolarization.QuadraticReadoutLaws.toQuadraticForm_apply",
      "Book7QuadraticPolarization.associated_diagonal_nonnegative",
      "Book7QuadraticPolarization.certified_readout_has_symmetric_bilinear_representation",
      "Book7QuadraticPolarization.nonnegative_readout_does_not_force_quadratic",
      "Book7QuadraticPolarization.quadraticForm_has_symmetric_bilinear_representation",
      "Book7QuadraticTrace.FrameReadoutSystem.globalValue_eq_trace_of_quadratic",
      "Book7QuadraticTrace.gluing_requires_quadratic_existence_bridge",
      "Book7QuadraticTrace.quadratic_eq_trace_pureStateDensity_mul",
      "Book7QuantumGleason.HermitianReadoutCertificate.toSesquilinear_apply",
      "Book7QuantumGleason.HermitianReadoutCertificate.toSesquilinear_diagonal",
      "Book7QuantumGleason.HermitianReadoutCertificate.toSesquilinear_isSymm",
      "Book7QuantumGleason.HermitianReadoutCertificate.value_smul",
      "Book7QuantumGleason.completeFrameCoherence_does_not_supply_hermitian_certificate",
      "Book7QuantumGleason.complex_phase_refutes_real_degreeTwo",
      "Book7QuantumGleason.hermitian_reconstruction_from_certificate",
      "Book7QuantumGleason.operatorQuantumRayReadout_globalPhase",
      "Book7QuantumGleason.pureStateDensity_globalPhase",
      "Book7QuantumGleason.pureStateDensity_isHermitian",
      "Book7QuantumGleason.pureStateToResolution_globalPhase",
      "Book7QuantumGleason.pureStateToResolution_reducedState_isHermitian",
      "Book7QuantumGleason.pureState_forward_chain",
      "Book7QuantumGleason.pureState_lowering_not_injective",
      "Book7QuantumGleason.quantumResolution_does_not_force_reducedState_isHermitian",
      "Book7QuantumGleason.quantumResolution_to_hermitian_certificate",
      "Book7QuantumGleason.quantumResolution_without_matrixHermiticity_does_not_supply_certificate",
      "Book7QuantumGleason.vectorExpectation_globalPhase",
      "Book7QuantumGleason.vectorExpectation_smul"
    ]
  },
  "line": 1924,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Conditional Born collapse at the Hilbert cross-section",
  "proof_labels": [
    "proof:bk7_born_collapse"
  ],
  "proof_status": "proven",
  "refs": [],
  "role": "theorem",
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  "book": "book7",
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  "cites": [],
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  "file": "book7.tex",
  "id": "proof:bk7_born_collapse",
  "label": "proof:bk7_born_collapse",
  "latex_body": "\\begin{proof}[Limit Identification and Separate Born Bridge]\n\\label{proof:bk7_born_collapse}\n\\leavevmode\nContinuity of the defect at the orbit limit transports orbital convergence to\nconvergence of defect values at $\\Phi_{\\rm nc}(\\xi_\\infty)$.  Uniqueness of\nlimits together with the assumed defect convergence to zero gives\n$\\Phi_{\\rm nc}(\\xi_\\infty)=0$, hence $\\xi_\\infty=\\xi^*$ by the zero-defect\ncharacterization.  The separate uniqueness certificate then identifies the\ncoherence readout with its Born value.  A distinct readout on the same Hilbert\nfixed point is a countermodel when that certificate is omitted.\n\\end{proof}",
  "line": 1943,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Limit Identification and Separate Born Bridge",
  "proves": "theorem:bk7_born_collapse",
  "refs": [],
  "role": "proof",
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  "book": "book7",
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  "file": "book7.tex",
  "id": "remark:bk7_born_collapse_psc3prime",
  "label": "remark:bk7_born_collapse_psc3prime",
  "latex_body": "\\begin{remark}[Interpretive reading of PS-C3$'$: from axiom to attractor]\n\\label{remark:bk7_born_collapse_psc3prime}\nThe following is an interpretive synthesis of the certified conditional theorem, not\nan additional kernel identity. Thm.~\\ref{theorem:bk7_born_collapse} recasts the one\nposited ingredient of the Born\nderivation. Non-contextuality (PS-C3$'$) is not an arbitrary axiom imposed on the\ncoherence functional; it is the \\emph{fixed-point condition} of the reflective\ncollapse -- the zero-set of the contextuality defect $\\Phi_{\\mathrm{nc}}$ -- so a\nmeasured (collapsed) state satisfies it because measurement is, by definition, the\ndescent to the non-contextual cross-section. This does not derive PS-C3$'$ for\narbitrary states; it locates exactly the states for which it holds, namely the\npost-collapse states, and explains why. The defect $\\Phi_{\\mathrm{nc}}$ is empirically\ntracked by the divergence-from-$L^{2}$ diagnostic of the $L^{p}$-sweep suite\n(Trace~5, Fig.~\\ref{figure:trace5_phase_transition_summary}): the sweep's measured\ndivergence $|{\\cdot}-\\text{MAE}(2)|$ and emergence-time proxy are the approach of\n$\\Phi_{\\mathrm{nc}}$ to its zero at $p=2$.\n\\end{remark}",
  "lean_alignment": {
    "conditions": [],
    "countermodels": [],
    "full_record": "bib/principia_lean_alignment.json",
    "kernel_certified": false,
    "notes": [
      "The prose reach is preserved but not presented as a further kernel identity."
    ],
    "record_ids": [
      "Q-ATTRACTOR-09"
    ],
    "statuses": [
      "interpretive"
    ],
    "witnesses": []
  },
  "line": 1955,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Interpretive reading of PS-C3$'$: from axiom to attractor",
  "ref_roles": [
    {
      "context": "he following is an interpretive synthesis of the certified conditional theorem, not an additional kernel identity. Thm.~\\ref{theorem:bk7_born_collapse} recasts the one posited ingredient of the Born derivation. Non-contextuality (PS-C3$'$) is not an arbitrary axiom impos",
      "label": "theorem:bk7_born_collapse",
      "logical_support": true,
      "role": "interpretive_bridge",
      "target_file": "book7.tex",
      "target_line": 1924,
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    "figure:trace5_phase_transition_summary",
    "theorem:bk7_born_collapse"
  ],
  "role": "remark",
  "type": "remark"
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{
  "appendix_teaser_ref_roles": [
    {
      "context": "ection} In the interpretive register, Thm.~\\ref{theorem:bk7_born_collapse} places the observer-relative Born rule (Thm.~\\ref{theorem:appC_born_rule}) where it belongs: at $\\xi^{\\ast}$, the unique cross-section $p=2$ where the effective geometry is Hilbertian and the c",
      "label": "theorem:appC_born_rule",
      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 701,
      "target_type": "theorem"
    }
  ],
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    "theorem:appC_born_rule"
  ],
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    "theorem:appC_born_rule",
    "theorem:bk7_born_collapse"
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  "file": "book7.tex",
  "id": "scholium:bk7_born_as_hilbert_cross_section",
  "label": "scholium:bk7_born_as_hilbert_cross_section",
  "latex_body": "\\begin{scholium}[Born as the Hilbert cross-section]\n\\label{scholium:bk7_born_as_hilbert_cross_section}\nIn the interpretive register, Thm.~\\ref{theorem:bk7_born_collapse} places the observer-relative Born rule\n(Thm.~\\ref{theorem:appC_born_rule}) where it belongs: at $\\xi^{\\ast}$, the unique\ncross-section $p=2$ where the effective geometry is Hilbertian and the coherence\nfunctional admits the inner-product form that Gleason's route requires. Reading the\nsweep outward from $\\xi^{\\ast}$ recovers the frame-temperature regimes of the\norigin programme: as $\\xi\\to 0^{+}$ the resolved predictions sharpen toward a\ndeterministic-looking (Newtonian, Dirac) limit, while as $\\xi\\to\\infty$ they flatten\ntoward the uniform (hyper-quantum) limit. This concerns appearance within the chosen\nobserver frame: neither limit reconstructs the full upstream state from its resolved\nrecord. Born is thus read not as an isolated postulate bolted onto a Hilbert space,\nbut as the $p=2$ slice of one continuous observer geometry, flanked by Banach\nrobustness on one side and deterministic-looking collapse on the other.\nConstitution precedes appearance: an observer receives a resolved surface, not an\ninvertible copy of its source.  Temporal becoming is already certified in the general\ngeometry inherited from the Cost of Cacophony: simultaneous compression meets a\ngeometric obstruction, while staged displacement becomes directed transport whose\ncost telescopes and, under explicit preservation or descent premises, converges.\nIn the functionally interpretive physical specialization, decoherence and noise map onto that same\ndirection through successive loss, quotient, or stabilization.  Apparent randomness\nat such a boundary does not by itself decide whether every richer process description\nis indeterministic.\n\\end{scholium}",
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    "notes": [
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    ],
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      "Q-SCHOLIUM-10"
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      "interpretive"
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  },
  "line": 1973,
  "macros_used": [],
  "matter_region": "mainmatter",
  "matter_role": "canonical_book",
  "name": "Born as the Hilbert cross-section",
  "ref_roles": [
    {
      "context": "ection} In the interpretive register, Thm.~\\ref{theorem:bk7_born_collapse} places the observer-relative Born rule (Thm.~\\ref{theorem:appC_born_rule}) where it belongs: at $\\xi^{\\ast}$, the unique cross-section $p=2$ where the effective geometry is Hilbertian and the c",
      "label": "theorem:appC_born_rule",
      "logical_support": false,
      "role": "appendix_teaser",
      "target_file": "appendix_dual_horizon.tex",
      "target_line": 701,
      "target_type": "theorem"
    },
    {
      "context": "orn as the Hilbert cross-section] \\label{scholium:bk7_born_as_hilbert_cross_section} In the interpretive register, Thm.~\\ref{theorem:bk7_born_collapse} places the observer-relative Born rule (Thm.~\\ref{theorem:appC_born_rule}) where it belongs: at $\\xi^{\\ast}$, the uniqu",
      "label": "theorem:bk7_born_collapse",
      "logical_support": true,
      "role": "interpretive_bridge",
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      "target_line": 1924,
      "target_type": "theorem"
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    "theorem:bk7_born_collapse"
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  "book": "book7",
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  "cites": [
    "axiom:bk7_convergence_potential",
    "definition:bk2_symbolic_free_energy",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "depends_on": [
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    "definition:bk2_symbolic_free_energy",
    "theorem:bk7_reflective_convergence_to_stable_identity"
  ],
  "file": "book7.tex",
  "id": "subsec:bk7_formalizing_reflective_selection_confidence_loss_and_symbolic_",
  "label": "subsec:bk7_formalizing_reflective_selection_confidence_loss_and_symbolic_",
  "latex_body": "",
  "line": 1998,
  "macros_used": [],
  "matter_region": "mainmatter",
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  "name": "Formalizing Reflective Selection: Confidence, Loss, and Symbolic Free Energy",
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    {
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      "label": "axiom:bk7_convergence_potential",
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      "target_line": 366,
      "target_type": "axiom"
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      "label": "definition:bk2_symbolic_free_energy",
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      "role": "navigation",
      "target_file": "book2.tex",
      "target_line": 135,
      "target_type": "definition"
    },
    {
      "context": "",
      "label": "theorem:bk7_reflective_convergence_to_stable_identity",
      "logical_support": false,
      "role": "navigation",
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      "target_line": 555,
      "target_type": "theorem"
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  "type": "section"
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  "book": "book7",
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  "cites": [
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  "depends_on": [
    "definition:bk2_symbolic_energy",
    "definition:bk7_convergent_symbolic_identity",
    "theorem:bk4_freedom_criterion"
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  "file": "book7.tex",
  "id": "subsubsec:bk7_formal_definition_of_symbolic_confidence_ch_i",
  "label": "subsubsec:bk7_formal_definition_of_symbolic_confidence_ch_i",
  "latex_body": "",
  "line": 2011,
  "macros_used": [],
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