Finite Blaschke Packet Model Spaces

This paper is the first installment of Volume VI of the HoTT/Yoneda Riemann hypothesis programme. Volume V proved a conditional theorem $\texttt{RH\_classical\_of\_no\_phantom\_language\_breakthrough}:\textsf{NoPhantomLanguageBreakthrough}\Longrightarrow\textsf{RH\_classical}$, whose two payload fields are $\textsf{OffCriticalZeroDefectKernel}$ (the existence of a nonzero vector in the Burnol/Blaschke model space whenever an off-critical zeta zero exists) and $\textsf{YonedaBlaschkeDetectorCalculus}$ (the RKHS detector / rational-dilation externalization / quotient orthogonality / admissibility package). The present paper supplies the very first non-fake step toward the first payload: the construction of finite Blaschke packet model spaces and the proof that any nonempty finite packet has a nonempty model-space carrier.

We define the structure $\mathsf{FiniteBlaschkePacket}$ (a finite-index family of points $\alpha_i \in \mathbb{C}$ together with off-critical flags), construct the finite Blaschke product $B_P(z) = \prod_{i\in I}(z-\alpha_i)/(1-\overline{\alpha_i}\,z)$, specify the associated finite-dimensional model space $K_{B_P} = H^2 \ominus B_P H^2$, exhibit a reproducing-kernel vector at any chosen packet zero, and prove that $K_{B_P}$ is nonempty. The Lean module $\texttt{Vol6.FiniteBlaschkePacket}$ formalizes all of this and proves the principal theorem $\texttt{finite\_packet\_model\_space\_nonempty}$ with no $\texttt{sorry}$, $\texttt{admit}$, new $\texttt{axiom}$, or new $\texttt{opaque}$. An accompanying runnable Haskell program verifies the construction numerically on a 3-zero packet.