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\label{thm:structural-instability}
Diagonal-resonant foundations
cannot host identity-faithful reception of $\tau$.
More precisely:
if $\mathsf{S}$ is a foundation
exhibiting diagonal resonance (I.D89),
then no functor
$P : \mathcal{C}_\tau \to \mathcal{C}_{\mathsf{S}}$
satisfies all three conditions
of identity-faithful reception (I.D92).
The three components (L, E, P)
jointly create identity slack
that prevents any global projection
from preserving the distinctness
guaranteed by ontic identity invariance (I.T46).

[Proof sketch]
The argument proceeds by contradiction
in seven steps.

\textbf{Step 1: Assume identity-faithfulness.}
Suppose $P : \mathcal{C}_\tau \to \mathcal{C}_{\mathsf{S}}$
is identity-faithful ---
satisfying conditions (i), (ii), and (iii)
of Definition~\ref{def:identity-faithful-reception}.

\textbf{Step 2: In $\tau$, distinctness is absolute.}
The ontic identity invariance theorem (I.T46)
guarantees that distinct normal forms
in the program monoid $\mathfrak{P}_\tau$
correspond to distinct objects in $\mathcal{C}_\tau$.
If $w \neq w'$ are distinct normal forms
(irreducible words in the generators
$\alpha, \pi, \gamma, \eta, \omega$),
then the objects $[w]$ and $[w']$
are distinct in $\mathcal{C}_\tau$.
The no-identity-decoherence corollary (I.C03)
ensures that no partial identification is possible:
$[w] = [w']$ or $[w] \neq [w']$,
with no intermediate state.

\textbf{Step 3: In $\mathsf{S}$, diagonal resonance creates shadow identities.}
Since $\mathsf{S}$ exhibits diagonal resonance (I.D89),
it hosts shadow identities (I.D91):
implicit identification channels
that $\mathsf{S}$ cannot distinguish
from genuine identities.
A shadow identity between $\mathsf{S}$-objects $A$ and $B$
is a metamathematical artifact ---
a chain of extensional equalities,
forcing equivalences, or model-theoretic identifications ---
that makes $A$ and $B$
``the same'' in some models of $\mathsf{S}$
but not in others.

\textbf{Step 4: Shadow identities act on the image of $P$.}
The objects $P([w])$ and $P([w'])$
live in $\mathcal{C}_{\mathsf{S}}$.
Shadow identities in $\mathsf{S}$
may identify $P([w])$ and $P([w'])$
even when $[w] \neq [w']$ in $\mathcal{C}_\tau$.
The mechanism is the three-component resonance:
\begin{itemize}
    \item[\textbf{(L)}] Unlimited lambda abstraction
          permits free token reuse in $\mathsf{S}$.
          The names assigned to $P([w])$ and $P([w'])$
          may share meta-level references ---
          a bound variable in one definition
          may be $\alpha$-equivalent
          to a bound variable in another,
          creating a naming coincidence
          that $\mathsf{S}$ cannot distinguish
          from a structural identification.
    \item[\textbf{(E)}] Extensional, substitution-based equality in $\mathsf{S}$
          may force $P([w]) = P([w'])$
          at the congruence level.
          If $\mathsf{S}$ derives
          that two terms are extensionally equal
          (they produce the same outputs
          on all inputs in $\mathsf{S}$),
          then $P([w])$ and $P([w'])$
          become identified in $\mathsf{S}$'s identity type ---
          even if the corresponding $\tau$-objects
          are structurally distinct.
    \item[\textbf{(P)}] The diagonal $\Delta$ in $\mathsf{S}$
          may materialize a witness of ``sameness''
          between $P([w])$ and $P([w'])$
          that has no $\tau$-preimage.
          The diagonal self-application
          $P([w]) \mapsto (P([w]), P([w]))$
          enables the construction of isomorphisms
          in $\mathsf{S}$ that do not lift to $\tau$.
\end{itemize}

\textbf{Step 5: $P$ fails to reflect isomorphism.}
By Step~4, there exist $\tau$-objects
$[w] \neq [w']$ such that
$P([w]) \cong P([w'])$ in $\mathcal{C}_{\mathsf{S}}$.
The isomorphism is constructed
using the L+E+P resonance in $\mathsf{S}$
--- specifically, using the diagonal map $\Delta$
that $\tau$'s $\KAxiom{5}$ discipline refuses.
Since $[w] \not\cong [w']$ in $\mathcal{C}_\tau$
(by ontic identity invariance, I.T46),
the functor $P$ fails condition~(iii)
of identity-faithful reception:
it does not reflect isomorphism.

\textbf{Step 6: Alternatively, $P$ fails object distinctness.}
If the shadow identities in $\mathsf{S}$
are strong enough to force
$P([w]) = P([w'])$ (equality, not merely isomorphism),
then $P$ fails condition~(i) directly:
distinct $\tau$-objects
are sent to the same $\mathsf{S}$-object.

\textbf{Step 7: Contradiction.}
In either case, $P$ is not identity-faithful,
contradicting Step~1.
The contradiction establishes
that no identity-faithful reception functor exists
when $\mathsf{S}$ exhibits diagonal resonance.