%
\label{prop:four-ray-rigidity}
The four ABCD rays $(A, B, C, D)$
provide sufficient structure
for holomorphic rigidity on $\tau^3$
without importing quaternionic algebra.
Specifically:
\begin{enumerate}
    \item \textbf{Completeness.}
          The ABCD chart is a complete coordinate system:
          every object of $\tau$ receives a unique
          ABCD quadruple
          (by hyperfactorization, I.T04).
    \item \textbf{Bipolar compatibility.}
          The fiber coordinates $(B, C)$
          carry the bipolar sector assignment
          (Proposition~\ref{prop:sector-inheritance}),
          which is compatible with the
          idempotent decomposition of $H_\tau$.
    \item \textbf{Coherence constraint.}
          The fibered product structure
          $\tau^3 = \tau^1 \times_f T^2$
          interlocks the four rays
          through the peel-order coupling,
          providing the rigidity
          that determines holomorphic functions
          up to the Identity Theorem.
    \item \textbf{No imports.}
          The four-ray structure is earned
          entirely from the kernel axioms
          $\KAxiom{0}$--$\KAxiom{6}$,
          the five generators
          $\{\alpha, \pi, \gamma, \eta, \omega\}$,
          and the progression operator $\rho$.
          No external algebraic structure
          (quaternions, Clifford algebras,
          division algebras)
          is imported.
\end{enumerate}

Part~(1) is the Hyperfactorization Theorem (I.T04).
Part~(2) is Proposition~\ref{prop:sector-inheritance}
(Chapter~\ref{ch:bipolar-interior}).
Part~(3): the fibered product coupling
means that a holomorphic function on $\tau^3$
must satisfy coherence conditions
on both the base projection $\mathrm{pr} \colon \tau^3 \to \tau^1$
and the fiber coordinates,
with the coupling ensuring
that base and fiber conditions are not independent.
The Identity Theorem for $\mathbb{L}$
(Book~I, Part~XII)
propagates: a holomorphic function on $\tau^3$
that vanishes on a sufficiently large
sub-tower of the primorial system
must vanish identically.
Part~(4): the ABCD chart, prime polarity,
split-complex algebra, and fibered product
are all constructed in Book~I
from the kernel axioms and generators,
with no external imports.