%
\label{thm:ontic-identity-invariance}
%   I.D90, I.D91, I.L02, I.T37, I.T39
In the coherence kernel ($\KAxiom{0}$--$\KAxiom{6}$),
every admissible construction preserves ontic identity:
\begin{enumerate}[\normalfont(i)]
    \item The normalization map to canonical form
          is unique and path-independent.
    \item No admissible construction can introduce
          alternative identity witnesses,
          partial identifications,
          or shadow identities
          (Definition~\ref{def:shadow-identity}, I.D91).
    \item Ontic identity is invariant
          under the full group of admissible symmetries.
\end{enumerate}
In particular, identity slippage
(Definition~\ref{def:identity-slippage}, I.D90)
is zero at every stage of every admissible construction.

[Proof sketch]
The argument proceeds in five steps.

\textbf{Step 1: Admissible constructions factor
through the program monoid.}
Every admissible construction in the coherence kernel
is a finite composition of generator applications
within the program monoid
(Definition~\ref{def:program-monoid}, I.D14,
Chapter~\ref{ch:program-monoid}).
An admissible construction is a sequence
of $\rho$-instructions and swap operations
that respects the constraints
of $\KAxiom{0}$--$\KAxiom{6}$.

\textbf{Step 2: NF-Confluence ensures unique output.}
By NF-Confluence
(Lemma~\ref{lem:nf-confluence}, I.L02),
any two reduction paths from a given input
converge to the same normal form.
This means that the output of normalization
--- the canonical representative of the result ---
is uniquely determined by the input,
independent of the reduction strategy.

\textbf{Step 3: K5 prevents contraction artifacts.}
By $\KAxiom{5}$
(Definition~\ref{def:diagonal-discipline}, I.D03),
no step in the construction can introduce
free contraction.
A $\rho$-instruction consumes its input channel
and produces its output channel;
no instruction duplicates a channel
without explicitly earning the copy.
In the terminology of
Chapter~\ref{ch:diagonal-resonance},
this blocks the (L) component:
no construction step can produce
two references to the same object
without an explicit morphism mediating between them.

\textbf{Step 4: Three-level equality prevents
implicit identification.}
By three-level equality
(Definition~\ref{def:three-equality}, I.D15),
syntactic identity (normal-form comparison),
denotational identity (program-monoid equivalence),
and categorical identity (isomorphism)
are stratified and non-collapsible.
No construction step can substitute
across levels:
an isomorphism between $A$ and $B$
does not make $A$ and $B$ syntactically identical.
This blocks the (E) component:
no implicit identification channel exists.

\textbf{Step 5: Star-autonomous structure prevents
primitive self-products.}
By the K5 Structural Exclusion Theorem
(Theorem~\ref{thm:k5-structural-exclusion}, I.T39),
$\tau$ lives on the star-autonomous side
of the CCC--linear dichotomy.
Self-products $A \otimes A$ are not primitive:
they must be earned by explicit construction.
No construction step can materialize
a self-product that was not already present
in the input.
This blocks the (P) component:
the ambient space for shadow identities
does not exist unless explicitly constructed.

\textbf{Combining.}
Steps 3--5 block the three components
(L), (E), (P) of diagonal resonance (I.D89).
Since diagonal resonance requires all three components
to produce identity slippage (I.D90),
and each component is independently blocked,
no admissible construction can produce
identity slippage.
By Step~2,
the normalization map produces
a unique canonical form.
By Steps~3--5,
no alternative identity witnesses or shadow identities
can be introduced.
Therefore, ontic identity is preserved:
the canonical normal form IS the ontic representative,
with no quotient, no equivalence class,
and no choice of representative.