The Canonical Inner Product on Pre-Coherent Possibility Space: How the Tension Functional Determines the Analytical Structure of Mₛ

The Quantum Blueprint Formalism (QBF) derives the architecture of quantum field theory from the Mother Equation on the infinite-dimensional pre-coherent possibility space Ms (Schmieke, 2026t). This derivation requires an inner product on the tangent spaces of Ms — needed for gradients, Hessians, the spectral theorem, and the mode decomposition that defines the particle content. The Erratum (Schmieke, 2026z) identified this inner product as a hidden structural commitment within Assumption 2.1 (infinite-dimensional Hilbert manifold). This paper resolves the resulting open problem. We prove that projection stability — the requirement that the analytical structure be recognizable under all admissible projections πΘ — selects a unique equivalence class of inner products: the class generated by the Hessian of Φ at the vacuum pointer state σ0. Two inner products in this class are equivalent if and only if they yield the same spectral decomposition of Φ. Consequently, the mode decomposition, the mass spectrum, and the particle content of the universe are determined by Φ, not freely chosen. The inner product on Ms is emergent, not postulated. Assumption 2.1 reduces to a single genuine assumption: infinite dimensionality. The smooth Hilbert manifold structure, including the inner product, follows from Φ and projection stability.