Canonical Spectral Response from Twisted Shell Geometry: Projective Reduction, Common-Support Invariance, and Modulus Closure

Abstract

This working preprint develops a canonical continuum bridge between the finite twisted-shell construction introduced in Twisted Shell Topology and Spectral Closure and the primitive stability response derived in The Chronos Stability Coordinate x. The construction begins with four phase channels and removes their uniform mode, producing a canonical three-dimensional active space. This rank-three structure is then represented using projective geometry, with a real projective three-space description and an oriented double cover. The resulting topology supplies a continuum representation of the two-step return structure inherited from the finite twisted-shell model, while the local tangent geometry generates the tetrahedral tight frame and associated projector curvature. For the primitive product mode, the angular and return contributions act on the same rank-three projector support. Because they share this common support, their response ratio is invariant under basis choice, normalized trace, operator norm, state amplitude, and more generally any positive homogeneous invariant response functional. This establishes a canonical shell-to-spectrum response ratio without directly identifying shell amplitude with Hamiltonian energy. When this canonical spectral response is identified with the previously derived primitive shell response, the formerly free dimensionless modulus becomes fixed. Under the same bridge, the bounded spectral share returns the previously derived Chronos stability coordinate. The result therefore closes the observable-level amplitude-versus-energy ambiguity that remained in the earlier twisted-shell construction. The paper carefully separates exact mathematical results from conditional physical interpretation. The projective geometry, rank-three reduction, return topology, projector curvature, common-support invariance, and primitive spectral response are established within the stated construction. The physical identification of the continuum realization, uniqueness of a microscopic action, absolute physical scale, and empirical universality remain open questions.