Complete structured record
{
"book": "book7",
"cited_by": [
"axiom:bk7_caristi_descent_for_reflection",
"corollary:bk7_fixed_point_tracking_within_evolving_reciprocity",
"corollary:bk7_geometric_convergence_rate",
"corollary:bk7_observer_converges",
"corollary:bk7_recursive_convergence_principle",
"corollary:bk7_stability_innovation_equilibrium",
"definition:bk7_symbolic_reflexive_validation_srv",
"definition:bk8_identitystability",
"definition:bk9_recursive_liberation",
"demonstratio:bk7_banach_convergence_reflection",
"demonstratio:bk7_convergence_within_reflective_basin",
"lemma:bk9_mutual_convergence_criterion",
"proof:bk4_maximal_freedom_autonomous_constraints",
"proof:bk7_drift_collapse_equivalence",
"proof:bk7_geometric_convergence_rate",
"proof:bk7_observer_converges",
"proof:bk7_recursive_convergence_principle",
"proof:bk7_stability_innovation_equilibrium",
"proof:bk7_stabilization_as_orbit_limit",
"proof:bk9_good_as_lyapunov_basin",
"proof:bk9_mutual_convergence_criterion",
"proof:bk9_symbolic_viability",
"proposition:bk7_stabilization_as_orbit_limit",
"remark:bk4_ttpr_descent_route",
"remark:bk9_recursive_seeking",
"scholium:bk7_popperian_extension",
"subsec:bk7_formalizing_reflective_selection_confidence_loss_and_symbolic_"
],
"cites": [
"axiom:bk7_reflective_stabilization",
"corollary:bk1_fixed_point",
"definition:bk1_reflection_operator",
"definition:bk1_self_regulating_mapping_function_srmf",
"definition:bk1_symbolic_manifold",
"definition:bk2_symbolic_entropy",
"definition:bk2_symbolic_free_energy",
"definition:bk2_symbolic_temperature",
"definition:bk2_symbolic_wasserstein_met",
"definition:bk6_reflection_operator_complete",
"definition:bk7_reflective_operator"
],
"depends_on": [
"axiom:bk7_reflective_stabilization",
"corollary:bk1_fixed_point",
"definition:bk1_reflection_operator",
"definition:bk1_self_regulating_mapping_function_srmf",
"definition:bk1_symbolic_manifold",
"definition:bk2_symbolic_energy",
"definition:bk2_symbolic_entropy",
"definition:bk2_symbolic_free_energy",
"definition:bk2_symbolic_temperature",
"definition:bk2_symbolic_wasserstein_met",
"definition:bk6_reflection_operator_complete",
"definition:bk7_reflective_operator"
],
"file": "book7.tex",
"id": "theorem:bk7_reflective_convergence_to_stable_identity",
"label": "theorem:bk7_reflective_convergence_to_stable_identity",
"latex_body": "\\begin{theorem}[Reflective Convergence to Stable Identity]\n\\label{theorem:bk7_reflective_convergence_to_stable_identity}\nLet $S = (\\manifold, \\metric, \\drift, \\reflect, \\rho)$ be a symbolic system (cf.~\\ref{definition:bk1_symbolic_manifold}). Let $(\\prob(\\manifold), \\wass)$ be the space of probability densities on $\\manifold$ equipped with the Wasserstein-2 metric (cf.~\\ref{definition:bk2_symbolic_wasserstein_met}), forming a complete metric space. Let $\\reflect : \\prob(\\manifold) \\to \\prob(\\manifold)$ be the reflective stabilization operator (cf.~Def.~\\ref{definition:bk7_reflective_operator}, Ax.~\\ref{axiom:bk7_reflective_stabilization}). If $\\reflect$ satisfies:\n\\begin{itemize}\n \\item[(i)] \\textbf{Free-energy descent (Caristi inequality):} The symbolic free energy $\\freeenergy[\\rho] = E[\\rho] - T_S S[\\rho]$ (cf.~\\ref{definition:bk2_symbolic_free_energy}, \\ref{definition:bk2_symbolic_entropy}, \\ref{definition:bk2_symbolic_temperature}) is bounded below and lower semicontinuous on the closed basin $B(\\identity) \\subseteq \\prob(\\manifold)$, is continuous along $W_2$-convergent reflective orbits in that basin, and every reflective update pays for its displacement in free energy:\n \\[\n \\wass(\\rho, \\reflect(\\rho)) \\;\\leq\\; \\freeenergy[\\rho] - \\freeenergy[\\reflect(\\rho)]\n \\qquad \\text{for all } \\rho \\in B(\\identity).\n \\]\n This is the inequality the symbolic free energy must satisfy, not mere monotonicity (cf.~\\ref{definition:bk1_reflection_operator}, \\ref{definition:bk6_reflection_operator_complete}): displacement is bounded by the potential actually spent.\n \\item[(ii)] \\textbf{Recursive stability:} $\\reflect$ maps the basin into itself, $\\reflect(B(\\identity)) \\subseteq B(\\identity)$, so recursive stabilization remains within the domain of convergent identity formation (cf.~\\ref{corollary:bk1_fixed_point}).\n\\end{itemize}\nthen for any initial symbolic state density $\\rho_0 \\in B(\\identity)$ the orbit $\\rho_{n+1} = \\reflect(\\rho_n)$ has summable increments,\n\\[\n\\sum_{n=0}^{\\infty}\\wass(\\rho_n,\\rho_{n+1}) \\;\\le\\; \\freeenergy[\\rho_0] - \\inf_{B(\\identity)}\\freeenergy \\;<\\; \\infty,\n\\]\nis $W_2$-Cauchy, and converges to a stable symbolic identity $\\identity \\in B(\\identity)$ with $\\freeenergy[\\rho_n] \\downarrow \\freeenergy[\\identity]$. If moreover $\\reflect$ has closed graph in $B(\\identity)\\times B(\\identity)$ -- in particular if $\\reflect$ is $W_2$-continuous -- then $\\identity$ is the fixed point $\\reflect(\\identity) = \\identity$ (cf.~\\ref{corollary:bk1_fixed_point}). If the stall set $\\mathcal{S} := \\{\\rho \\in B(\\identity) : \\freeenergy[\\reflect(\\rho)] = \\freeenergy[\\rho]\\}$ is the singleton $\\{\\identity\\}$, the limit is independent of $\\rho_0$ and $\\identity$ is the thermodynamically optimal coherent state minimizing $\\freeenergy$ within $B(\\identity)$ (cf.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}). A $\\kappa$-contraction with $\\freeenergy$ comparable to $\\wass(\\cdot,\\identity)$ is the special case in which descent holds automatically.\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 summable-increments / bounded-total-displacement consequence of hypothesis (i) is proved, by telescoping. The W_2-Cauchy convergence to an actual limit, hypothesis (ii)'s self-map clause, and the closed-graph/fixed-point conclusion all require completeness of (prob(M), W_2) and are not modeled."
],
"record_ids": [
"MAP-BOOK7-008"
],
"statuses": [
"open_bridge"
],
"witnesses": [
"Book7.caristiDescent_sum_le_energy_drop",
"Book7.caristiDescent_total_displacement_bound"
]
},
"line": 555,
"macros_used": [
"drift",
"freeenergy",
"identity",
"manifold",
"metric",
"prob",
"reflect",
"wass"
],
"matter_region": "mainmatter",
"matter_role": "canonical_book",
"name": "Reflective Convergence to Stable Identity",
"proof_labels": [
"proof:bk7_reflective_convergence_to_stable_identity"
],
"proof_status": "proven",
"ref_roles": [
{
"context": "\\to \\prob(\\manifold)$ be the reflective stabilization operator (cf.~Def.~\\ref{definition:bk7_reflective_operator}, Ax.~\\ref{axiom:bk7_reflective_stabilization}). If $\\reflect$ satisfies: \\begin{itemize} \\item[(i)] \\textbf{Free-energy descent (Caristi inequality):} The symbol",
"label": "axiom:bk7_reflective_stabilization",
"logical_support": true,
"role": "cf_near_match",
"target_file": "book7.tex",
"target_line": 403,
"target_type": "axiom"
},
{
"context": "y)) \\subseteq B(\\identity)$, so recursive stabilization remains within the domain of convergent identity formation (cf.~\\ref{corollary:bk1_fixed_point}). \\end{itemize} then for any initial symbolic state density $\\rho_0 \\in B(\\identity)$ the orbit $\\rho_{n+1} = \\reflect(",
"label": "corollary:bk1_fixed_point",
"logical_support": true,
"role": "cf_near_match",
"target_file": "scholium_symbolicum.tex",
"target_line": 3003,
"target_type": "corollary"
},
{
"context": "o \\in B(\\identity). \\] This is the inequality the symbolic free energy must satisfy, not mere monotonicity (cf.~\\ref{definition:bk1_reflection_operator}, \\ref{definition:bk6_reflection_operator_complete}): displacement is bounded by the potential actually spent. \\item",
"label": "definition:bk1_reflection_operator",
"logical_support": true,
"role": "cf_near_match",
"target_file": "scholium_symbolicum.tex",
"target_line": 1209,
"target_type": "definition"
},
{
"context": "_0$ and $\\identity$ is the thermodynamically optimal coherent state minimizing $\\freeenergy$ within $B(\\identity)$ (cf.~\\ref{definition:bk1_self_regulating_mapping_function_srmf}). A $\\kappa$-contraction with $\\freeenergy$ comparable to $\\wass(\\cdot,\\identity)$ is the special case in which descent",
"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": "ective_convergence_to_stable_identity} Let $S = (\\manifold, \\metric, \\drift, \\reflect, \\rho)$ be a symbolic system (cf.~\\ref{definition:bk1_symbolic_manifold}). Let $(\\prob(\\manifold), \\wass)$ be the space of probability densities on $\\manifold$ equipped with the Wasserstein-2",
"label": "definition:bk1_symbolic_manifold",
"logical_support": true,
"role": "cf_near_match",
"target_file": "scholium_symbolicum.tex",
"target_line": 1188,
"target_type": "definition"
},
{
"context": ":} The symbolic free energy $\\freeenergy[\\rho] = E[\\rho] - T_S S[\\rho]$ (cf.~\\ref{definition:bk2_symbolic_free_energy}, \\ref{definition:bk2_symbolic_entropy}, \\ref{definition:bk2_symbolic_temperature}) is bounded below and lower semicontinuous on the closed basin $B(\\identity)",
"label": "definition:bk2_symbolic_entropy",
"logical_support": true,
"role": "cf_near_match",
"target_file": "book2.tex",
"target_line": 114,
"target_type": "definition"
},
{
"context": "bf{Free-energy descent (Caristi inequality):} The symbolic free energy $\\freeenergy[\\rho] = E[\\rho] - T_S S[\\rho]$ (cf.~\\ref{definition:bk2_symbolic_free_energy}, \\ref{definition:bk2_symbolic_entropy}, \\ref{definition:bk2_symbolic_temperature}) is bounded below and lower semiconti",
"label": "definition:bk2_symbolic_free_energy",
"logical_support": true,
"role": "cf_near_match",
"target_file": "book2.tex",
"target_line": 135,
"target_type": "definition"
},
{
"context": "y[\\rho] = E[\\rho] - T_S S[\\rho]$ (cf.~\\ref{definition:bk2_symbolic_free_energy}, \\ref{definition:bk2_symbolic_entropy}, \\ref{definition:bk2_symbolic_temperature}) is bounded below and lower semicontinuous on the closed basin $B(\\identity) \\subseteq \\prob(\\manifold)$, is continuous",
"label": "definition:bk2_symbolic_temperature",
"logical_support": true,
"role": "cf_near_match",
"target_file": "book2.tex",
"target_line": 148,
"target_type": "definition"
},
{
"context": "ob(\\manifold), \\wass)$ be the space of probability densities on $\\manifold$ equipped with the Wasserstein-2 metric (cf.~\\ref{definition:bk2_symbolic_wasserstein_met}), forming a complete metric space. Let $\\reflect : \\prob(\\manifold) \\to \\prob(\\manifold)$ be the reflective stabilizati",
"label": "definition:bk2_symbolic_wasserstein_met",
"logical_support": true,
"role": "cf_near_match",
"target_file": "book2.tex",
"target_line": 302,
"target_type": "definition"
},
{
"context": "inequality the symbolic free energy must satisfy, not mere monotonicity (cf.~\\ref{definition:bk1_reflection_operator}, \\ref{definition:bk6_reflection_operator_complete}): displacement is bounded by the potential actually spent. \\item[(ii)] \\textbf{Recursive stability:} $\\reflect$ map",
"label": "definition:bk6_reflection_operator_complete",
"logical_support": true,
"role": "cf_near_match",
"target_file": "book6.tex",
"target_line": 937,
"target_type": "definition"
},
{
"context": "metric space. Let $\\reflect : \\prob(\\manifold) \\to \\prob(\\manifold)$ be the reflective stabilization operator (cf.~Def.~\\ref{definition:bk7_reflective_operator}, Ax.~\\ref{axiom:bk7_reflective_stabilization}). If $\\reflect$ satisfies: \\begin{itemize} \\item[(i)] \\textbf{Free-en",
"label": "definition:bk7_reflective_operator",
"logical_support": true,
"role": "cf_near_match",
"target_file": "book7.tex",
"target_line": 451,
"target_type": "definition"
}
],
"refs": [
"axiom:bk7_reflective_stabilization",
"corollary:bk1_fixed_point",
"definition:bk1_reflection_operator",
"definition:bk1_self_regulating_mapping_function_srmf",
"definition:bk1_symbolic_manifold",
"definition:bk2_symbolic_entropy",
"definition:bk2_symbolic_free_energy",
"definition:bk2_symbolic_temperature",
"definition:bk2_symbolic_wasserstein_met",
"definition:bk6_reflection_operator_complete",
"definition:bk7_reflective_operator"
],
"role": "theorem",
"type": "theorem"
}