proofmainmatter
Mutation-Bifurcation Duality
proof:bk6_mutation_bifurcation_duality
Exact LaTeX body
\begin{proof}[Mutation-Bifurcation Duality]
\label{proof:bk6_mutation_bifurcation_duality}
\leavevmode
\textbf{(Mutation $\Rightarrow$ Bifurcation.)}
By Axiom~\ref{axiom:bk6_symbolic_mutation_as_curvature_transition},
mutation at time $t_0$ produces a curvature discontinuity
$\Delta\kappa(t_0) \neq 0$.
The curvature tensor $\kappa$ is built from the commutator structure
of the connection:
$R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z$
(Def.~\ref{definition:bk6_symbolic_curvature_tensor}).
A discontinuity in $\kappa$ is therefore a discontinuity in the
commutator structure of $D$ and $R$, which forces
$\|D \circ R - R \circ D\|_{\mathrm{op}} > \gamma$
(Def.~\ref{definition:bk6_symbolic_mutation}).
By Prop.~\ref{proposition:bk6_bifurcation_threshold}, this is
equivalent to the contradictory tension exceeding threshold:
$\tau(x) = \|D(x) \times R(D(x))\|_g > \tau_c$.
The bifurcation threshold condition implies
$\det(\mathcal{J}(t_0)) = 0$ where
$\mathcal{J} = \nabla D + \nabla R$ is the combined Jacobian
(Def.~\ref{definition:bk6_symbolic_bifurcation}), since the
singular Jacobian is the linearized expression of the same
drift-reflection misalignment that $\tau$ measures globally.
Hence mutation implies bifurcation.
\textbf{(Bifurcation $\Rightarrow$ Mutation.)}
Conversely, suppose bifurcation occurs at $t_0$:
$\det(\mathcal{J}(t_0)) = 0$. Then the flow $\Phi_t$ branches,
producing distinct evolution pathways $\{x_1, \ldots, x_n\}$
(Axiom~\ref{axiom:bk6_bifurcation_as_emergence_operator}).
The contradictory tension
$\tau(x) = \|D(x) \times R(D(x))\|_g$
(Prop.~\ref{proposition:bk6_bifurcation_threshold}) exceeds
$\tau_c$, which implies $\|D \circ R - R \circ D\|_{\mathrm{op}} > \gamma$
(Def.~\ref{definition:bk6_symbolic_mutation}). This is the trigger
condition for mutation. Hence bifurcation implies mutation.
\textbf{(Distributional form.)}
Since the mutation operator decomposes as
$\mathcal{M}_t = R_t \circ \mathcal{B}_t \circ D_t$
(Def.~\ref{definition:bk6_mutation_operator}), the operator inner
product $\langle \mathcal{M}_t, \mathcal{B}_t \rangle_\mathcal{H}$
is nonzero if and only if $\mathcal{B}_t$ has nontrivial action.
By the equivalence above, this occurs precisely at mutation times.
The bifurcation indicator $\chi_\text{bifurcation}$
(Def.~\ref{definition:bk6_mutation_rate}) has support on isolated
points $\{t_0\}$; in the distributional limit of a single event,
$\langle \mathcal{M}_t, \mathcal{B}_t \rangle_\mathcal{H} = \delta(t - t_0)$.
\end{proof}Depends on
Cites
Reference roles
| Target | Role | Logical support |
|---|---|---|
axiom:bk6_bifurcation_as_emergence_operator | definition_anchor | yes |
axiom:bk6_symbolic_mutation_as_curvature_transition | definition_anchor | yes |
definition:bk6_mutation_operator | definition_anchor | yes |
definition:bk6_mutation_rate | definition_anchor | yes |
definition:bk6_symbolic_bifurcation | definition_anchor | yes |
definition:bk6_symbolic_curvature_tensor | definition_anchor | yes |
definition:bk6_symbolic_mutation | definition_anchor | yes |
proposition:bk6_bifurcation_threshold | proof_support | yes |
Complete structured record
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"axiom:bk6_bifurcation_as_emergence_operator",
"axiom:bk6_symbolic_mutation_as_curvature_transition",
"definition:bk6_mutation_operator",
"definition:bk6_mutation_rate",
"definition:bk6_symbolic_bifurcation",
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"id": "proof:bk6_mutation_bifurcation_duality",
"label": "proof:bk6_mutation_bifurcation_duality",
"latex_body": "\\begin{proof}[Mutation-Bifurcation Duality]\n\\label{proof:bk6_mutation_bifurcation_duality}\n\\leavevmode\n\n\\textbf{(Mutation $\\Rightarrow$ Bifurcation.)}\nBy Axiom~\\ref{axiom:bk6_symbolic_mutation_as_curvature_transition},\nmutation at time $t_0$ produces a curvature discontinuity\n$\\Delta\\kappa(t_0) \\neq 0$.\nThe curvature tensor $\\kappa$ is built from the commutator structure\nof the connection:\n$R(X,Y)Z = \\nabla_X\\nabla_Y Z - \\nabla_Y\\nabla_X Z - \\nabla_{[X,Y]}Z$\n(Def.~\\ref{definition:bk6_symbolic_curvature_tensor}).\nA discontinuity in $\\kappa$ is therefore a discontinuity in the\ncommutator structure of $D$ and $R$, which forces\n$\\|D \\circ R - R \\circ D\\|_{\\mathrm{op}} > \\gamma$\n(Def.~\\ref{definition:bk6_symbolic_mutation}).\nBy Prop.~\\ref{proposition:bk6_bifurcation_threshold}, this is\nequivalent to the contradictory tension exceeding threshold:\n$\\tau(x) = \\|D(x) \\times R(D(x))\\|_g > \\tau_c$.\nThe bifurcation threshold condition implies\n$\\det(\\mathcal{J}(t_0)) = 0$ where\n$\\mathcal{J} = \\nabla D + \\nabla R$ is the combined Jacobian\n(Def.~\\ref{definition:bk6_symbolic_bifurcation}), since the\nsingular Jacobian is the linearized expression of the same\ndrift-reflection misalignment that $\\tau$ measures globally.\nHence mutation implies bifurcation.\n\n\\textbf{(Bifurcation $\\Rightarrow$ Mutation.)}\nConversely, suppose bifurcation occurs at $t_0$:\n$\\det(\\mathcal{J}(t_0)) = 0$. Then the flow $\\Phi_t$ branches,\nproducing distinct evolution pathways $\\{x_1, \\ldots, x_n\\}$\n(Axiom~\\ref{axiom:bk6_bifurcation_as_emergence_operator}).\nThe contradictory tension\n$\\tau(x) = \\|D(x) \\times R(D(x))\\|_g$\n(Prop.~\\ref{proposition:bk6_bifurcation_threshold}) exceeds\n$\\tau_c$, which implies $\\|D \\circ R - R \\circ D\\|_{\\mathrm{op}} > \\gamma$\n(Def.~\\ref{definition:bk6_symbolic_mutation}). This is the trigger\ncondition for mutation. Hence bifurcation implies mutation.\n\n\\textbf{(Distributional form.)}\nSince the mutation operator decomposes as\n$\\mathcal{M}_t = R_t \\circ \\mathcal{B}_t \\circ D_t$\n(Def.~\\ref{definition:bk6_mutation_operator}), the operator inner\nproduct $\\langle \\mathcal{M}_t, \\mathcal{B}_t \\rangle_\\mathcal{H}$\nis nonzero if and only if $\\mathcal{B}_t$ has nontrivial action.\nBy the equivalence above, this occurs precisely at mutation times.\nThe bifurcation indicator $\\chi_\\text{bifurcation}$\n(Def.~\\ref{definition:bk6_mutation_rate}) has support on isolated\npoints $\\{t_0\\}$; in the distributional limit of a single event,\n$\\langle \\mathcal{M}_t, \\mathcal{B}_t \\rangle_\\mathcal{H} = \\delta(t - t_0)$.\n\\end{proof}",
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"context": "cal{J}(t_0)) = 0$. Then the flow $\\Phi_t$ branches, producing distinct evolution pathways $\\{x_1, \\ldots, x_n\\}$ (Axiom~\\ref{axiom:bk6_bifurcation_as_emergence_operator}). The contradictory tension $\\tau(x) = \\|D(x) \\times R(D(x))\\|_g$ (Prop.~\\ref{proposition:bk6_bifurcation_threshold}) e",
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"context": "ty] \\label{proof:bk6_mutation_bifurcation_duality} \\leavevmode \\textbf{(Mutation $\\Rightarrow$ Bifurcation.)} By Axiom~\\ref{axiom:bk6_symbolic_mutation_as_curvature_transition}, mutation at time $t_0$ produces a curvature discontinuity $\\Delta\\kappa(t_0) \\neq 0$. The curvature tensor $\\kappa$ is",
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"context": "tributional form.)} Since the mutation operator decomposes as $\\mathcal{M}_t = R_t \\circ \\mathcal{B}_t \\circ D_t$ (Def.~\\ref{definition:bk6_mutation_operator}), the operator inner product $\\langle \\mathcal{M}_t, \\mathcal{B}_t \\rangle_\\mathcal{H}$ is nonzero if and only if $\\mat",
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"context": "e equivalence above, this occurs precisely at mutation times. The bifurcation indicator $\\chi_\\text{bifurcation}$ (Def.~\\ref{definition:bk6_mutation_rate}) has support on isolated points $\\{t_0\\}$; in the distributional limit of a single event, $\\langle \\mathcal{M}_t, \\math",
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"context": "nuity in the commutator structure of $D$ and $R$, which forces $\\|D \\circ R - R \\circ D\\|_{\\mathrm{op}} > \\gamma$ (Def.~\\ref{definition:bk6_symbolic_mutation}). By Prop.~\\ref{proposition:bk6_bifurcation_threshold}, this is equivalent to the contradictory tension exceeding thres",
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