From a Gravity-Calibrated Helix to a Matter–Spacetime Interface: Closure Obstruction and a Factorized Mass-Squared Ansatz

The Emergent Frame (TEF) investigates whether matter and open spacetime may be different global organizations of compatible local structure. In the working “track” picture developed here, the track is not a classical rail carrying a particle: it represents structural constraints on how energy may be organized, propagated, and rerouted. This paper studies a prerequisite for such a picture: whether the frozen native TEF helix can close directly into matter-like structures. For every oriented complete nonzero-pitch circular helix segment, the endpoint tangent and net displacement satisfy tᵢ · Δrᵢ > 0. It follows that a finite sequence of such complete segments cannot form a closed oriented curve by direct sequential C¹ joining alone. Some additional completion mechanism—such as rerouting, reconnection, partial-turn structure, or more general geometry—is therefore required. When one frozen native turn is retained as part of a smooth closed completion, Fenchel’s theorem further gives the necessary bound Kᵢₙₜ ≥ 2π[1 − 1/√(1 + q²)]. For the previously frozen TEF shape parameter q = 0.556324180045…, the corresponding dimensionless Fenchel deficit is 0.1261280215…. Motivated by this closure obstruction, the paper constructs a geometric interface map from prescribed post-rerouting closure data and studies the positive Gram operator Gₙ = Cₙ† Cₙ. For minimal Euclidean closures, its rank and kernel follow exactly from the specified geometry, including a single interface-null combination. The identification of this operator with a physical mass-squared contribution remains an ansatz. Particle masses, QCD dynamics, and a fundamental TEF energy law are not yet derived. The present result instead advances the program from geometric closure toward a quantitative matter–spacetime interface, with the longer-term goal of connecting topology, energy, and multiple independent physical observables without process-by-process fitting. This manuscript is released as a preprint for archival citation and open scientific discussion.