Exact LaTeX body
\begin{theorem}[Two-Way Street Convergence]
\label{theorem:bk7_two_way_street_convergence}
Let \( \mathbf{P} \) be an interactive pair (Def.~\ref{definition:bk7_interactive_drift_reflection_pair}).
Assume the reflective interaction operators
\[
\reflect_{\mathcal{A}} : \manifold_{\mathcal{B}} \to \manifold_{\mathcal{A}},
\qquad
\reflect_{\mathcal{B}} : \manifold_{\mathcal{A}} \to \manifold_{\mathcal{B}}
\]
(as in Def.~\ref{definition:bk7_reflective_interaction_operator_}) are contractions with constants \( \kappa_A \) and \( \kappa_B \), respectively, such that:
\[
d_{\mathcal{A}}(\reflect_{\mathcal{A}}(y_B), \reflect_{\mathcal{A}}(y'_B))
\le \kappa_A\, d_{\mathcal{B}}(y_B, y'_B),
\]
\[
d_{\mathcal{B}}(\reflect_{\mathcal{B}}(x_A), \reflect_{\mathcal{B}}(x'_A))
\le \kappa_B\, d_{\mathcal{A}}(x_A, x'_A).
\]
Define the joint reflective interaction operator:
\[
\Phi(x_A, y_B) :=
\big( \reflect_{\mathcal{A}}(y_B),\, \reflect_{\mathcal{B}}(x_A) \big).
\]
If \( \kappa' := \max\{ \kappa_A, \kappa_B \} < 1 \), then \( \Phi \) is a contraction
on the product space \( \manifold_{\mathcal{A}} \times \manifold_{\mathcal{B}} \),
with metric \( d_P \), and contraction constant \( \kappa' \).
Consequently, if \( \manifold_{\mathcal{A}} \) and \( \manifold_{\mathcal{B}} \) are complete metric spaces,
then \( \Phi \) admits a unique fixed point \( (x^{\ast}, y^{\ast}) \in \manifold_{\mathcal{A}} \times \manifold_{\mathcal{B}} \), satisfying:
\[
x^{\ast} = \reflect_{\mathcal{A}}(y^{\ast}),
\qquad
y^{\ast} = \reflect_{\mathcal{B}}(x^{\ast}).
\]
Furthermore, for any initial pair \( (x_0, y_0) \),
the joint iteration
\[
(x_{n+1}, y_{n+1}) = \Phi(x_n, y_n)
\]
converges to \( (x^{\ast}, y^{\ast}) \) as \( n \to \infty \).
If the reciprocity domain \( \recipdomain \) (Def.~\ref{definition:bk7_reciprocity_domain})
is non-empty and contains \( (x^{\ast}, y^{\ast}) \),
this represents convergence to mutual symbolic alignment.
\end{theorem}
Complete structured record
{
"book": "book7",
"cited_by": [
"corollary:bk7_fixed_point_tracking_within_evolving_reciprocity",
"corollary:bk7_stability_near_reciprocity",
"demonstratio:bk7_meta_drift_reflective_tracking",
"proof:bk7_map_compatible_reciprocity",
"proof:bk7_stability_near_reciprocity",
"proof:bk8_resonant_cognition",
"proposition:bk7_map_compatible_reciprocity",
"remark:bk7_empathy_as_dynamical_invariant",
"scholium:bk7_on_symbolic_reciprocity",
"scholium:bk7_srmf_coupled_agents"
],
"cites": [
"definition:bk7_interactive_drift_reflection_pair",
"definition:bk7_reciprocity_domain",
"definition:bk7_reflective_interaction_operator_"
],
"depends_on": [
"definition:bk7_interactive_drift_reflection_pair",
"definition:bk7_reciprocity_domain",
"definition:bk7_reflective_interaction_operator_"
],
"file": "book7.tex",
"id": "theorem:bk7_two_way_street_convergence",
"label": "theorem:bk7_two_way_street_convergence",
"latex_body": "\\begin{theorem}[Two-Way Street Convergence]\n\\label{theorem:bk7_two_way_street_convergence}\nLet \\( \\mathbf{P} \\) be an interactive pair (Def.~\\ref{definition:bk7_interactive_drift_reflection_pair}). \nAssume the reflective interaction operators\n\\[\n\\reflect_{\\mathcal{A}} : \\manifold_{\\mathcal{B}} \\to \\manifold_{\\mathcal{A}}, \n\\qquad\n\\reflect_{\\mathcal{B}} : \\manifold_{\\mathcal{A}} \\to \\manifold_{\\mathcal{B}}\n\\]\n(as in Def.~\\ref{definition:bk7_reflective_interaction_operator_}) are contractions with constants \\( \\kappa_A \\) and \\( \\kappa_B \\), respectively, such that:\n\\[\nd_{\\mathcal{A}}(\\reflect_{\\mathcal{A}}(y_B), \\reflect_{\\mathcal{A}}(y'_B)) \n\\le \\kappa_A\\, d_{\\mathcal{B}}(y_B, y'_B),\n\\]\n\\[\nd_{\\mathcal{B}}(\\reflect_{\\mathcal{B}}(x_A), \\reflect_{\\mathcal{B}}(x'_A)) \n\\le \\kappa_B\\, d_{\\mathcal{A}}(x_A, x'_A).\n\\]\nDefine the joint reflective interaction operator:\n\\[\n\\Phi(x_A, y_B) := \n\\big( \\reflect_{\\mathcal{A}}(y_B),\\, \\reflect_{\\mathcal{B}}(x_A) \\big).\n\\]\nIf \\( \\kappa' := \\max\\{ \\kappa_A, \\kappa_B \\} < 1 \\), then \\( \\Phi \\) is a contraction \non the product space \\( \\manifold_{\\mathcal{A}} \\times \\manifold_{\\mathcal{B}} \\), \nwith metric \\( d_P \\), and contraction constant \\( \\kappa' \\).\nConsequently, if \\( \\manifold_{\\mathcal{A}} \\) and \\( \\manifold_{\\mathcal{B}} \\) are complete metric spaces, \nthen \\( \\Phi \\) admits a unique fixed point \\( (x^{\\ast}, y^{\\ast}) \\in \\manifold_{\\mathcal{A}} \\times \\manifold_{\\mathcal{B}} \\), satisfying:\n\\[\nx^{\\ast} = \\reflect_{\\mathcal{A}}(y^{\\ast}), \n\\qquad \ny^{\\ast} = \\reflect_{\\mathcal{B}}(x^{\\ast}).\n\\]\nFurthermore, for any initial pair \\( (x_0, y_0) \\), \nthe joint iteration \n\\[\n(x_{n+1}, y_{n+1}) = \\Phi(x_n, y_n)\n\\]\nconverges to \\( (x^{\\ast}, y^{\\ast}) \\) as \\( n \\to \\infty \\).\nIf the reciprocity domain \\( \\recipdomain \\) (Def.~\\ref{definition:bk7_reciprocity_domain}) \nis non-empty and contains \\( (x^{\\ast}, y^{\\ast}) \\), \nthis represents convergence to mutual symbolic alignment.\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": true,
"notes": [
"For nonempty complete metric factors and cross-Lipschitz constants whose positive maximum is below one, the product map is packaged as a Book 4 contraction refinement. Its iterates converge from every initial pair to a canonical reciprocal limit, and that limit is the unique fixed pair."
],
"record_ids": [
"MAP-BOOK7-014"
],
"statuses": [
"conditional"
],
"witnesses": [
"Book7.mutualLimit_fixed",
"Book7.product_contraction",
"Book7.reciprocalPair_unique",
"Book7.tendsto_mutualRefinement"
]
},
"line": 1028,
"macros_used": [
"manifold",
"recipdomain",
"reflect"
],
"matter_region": "mainmatter",
"matter_role": "canonical_book",
"name": "Two-Way Street Convergence",
"proof_labels": [
"proof:bk7_two_way_street_convergence"
],
"proof_status": "proven",
"ref_roles": [
{
"context": "ay Street Convergence] \\label{theorem:bk7_two_way_street_convergence} Let \\( \\mathbf{P} \\) be an interactive pair (Def.~\\ref{definition:bk7_interactive_drift_reflection_pair}). Assume the reflective interaction operators \\[ \\reflect_{\\mathcal{A}} : \\manifold_{\\mathcal{B}} \\to \\manifold_{\\math",
"label": "definition:bk7_interactive_drift_reflection_pair",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book7.tex",
"target_line": 947,
"target_type": "definition"
},
{
"context": "n) \\] converges to \\( (x^{\\ast}, y^{\\ast}) \\) as \\( n \\to \\infty \\). If the reciprocity domain \\( \\recipdomain \\) (Def.~\\ref{definition:bk7_reciprocity_domain}) is non-empty and contains \\( (x^{\\ast}, y^{\\ast}) \\), this represents convergence to mutual symbolic alignment. \\end",
"label": "definition:bk7_reciprocity_domain",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book7.tex",
"target_line": 980,
"target_type": "definition"
},
{
"context": "fold_{\\mathcal{A}}, \\qquad \\reflect_{\\mathcal{B}} : \\manifold_{\\mathcal{A}} \\to \\manifold_{\\mathcal{B}} \\] (as in Def.~\\ref{definition:bk7_reflective_interaction_operator_}) are contractions with constants \\( \\kappa_A \\) and \\( \\kappa_B \\), respectively, such that: \\[ d_{\\mathcal{A}}(\\reflec",
"label": "definition:bk7_reflective_interaction_operator_",
"logical_support": true,
"role": "definition_anchor",
"target_file": "book7.tex",
"target_line": 972,
"target_type": "definition"
}
],
"refs": [
"definition:bk7_interactive_drift_reflection_pair",
"definition:bk7_reciprocity_domain",
"definition:bk7_reflective_interaction_operator_"
],
"role": "theorem",
"type": "theorem"
}