%
\label{def:diagonal-resonance}
A foundation exhibits \textbf{diagonal resonance}
if it simultaneously provides the following
three structural capabilities at the kernel level:
\begin{enumerate}
    \item[\textbf{(L)}]
          \textbf{Meta-level contraction / free token reuse.}
          Variables can be freely reused:
          if a term $t$ appears once,
          it may appear again without cost.
          In sequent-calculus terms,
          the contraction rule
          \[
              \frac{\Gamma, A, A \vdash B}{\Gamma, A \vdash B}
          \]
          is admissible.
          Identity-of-reference ---
          the meta-level assertion
          ``this is the same token'' ---
          is free in syntax.
          This is standard in classical logic,
          in intuitionistic logic,
          and in the Calculus of Inductive Constructions.
    \item[\textbf{(E)}]
          \textbf{Equality-as-congruence with full substitution.}
          The system admits an equality relation
          $= \,\subseteq X \times X$
          such that if $a = b$ and $P(a)$ holds,
          then $P(b)$ holds.
          Substitution makes equality behave
          like identification:
          the truth of a proposition about $a$
          transfers to $b$ whenever $a = b$.
          This is Leibniz's law,
          the indiscernibility of identicals,
          and it is standard in every major foundation.

          The system thereby admits two distinct notions
          of ``sameness'':
          \emph{identity-of-reference}
          (meta-level: ``this is the same token'')
          and \emph{equality-as-relation}
          (object-level: $a = b$ as a proposition).
          Full substitution forces these two notions
          to interact:
          object-level equality inherits
          the operational behavior
          of meta-level identity.
    \item[\textbf{(P)}]
          \textbf{Ontic self-products with diagonal materialization.}
          For any type or set $A$,
          the self-product $A \times A$ exists,
          and the diagonal subset
          $\Delta_A = \{(a, a) \mid a \in A\}$
          can be carved out by comprehension
          or constructed as a morphism
          $\Delta_A : A \to A \times A$
          via $a \mapsto (a, a)$.
          The theory can materialize
          two ports to the same type
          and identify elements across those ports.
          This is standard in ZFC
          (Cartesian products and separation),
          in CIC (sigma-types and pattern matching),
          and in every cartesian closed category.
\end{enumerate}
No single component is pathological.
Each is individually reasonable, well-motivated,
and practically indispensable.
The pathology arises from their \emph{joint presence}:
when all three are simultaneously active
at the kernel level,
diagonal resonance becomes unavoidable
once the system has sufficient expressivity
to encode self-referential constructions.