From a Gravity-Calibrated Helix to a Strong-Interaction Confinement Correspondence

We formulate a conditional closure-energy framework for a restricted connected static-source channel in The Emergent Frame (TEF). The construction joins the inherited open-space helix of Paper I to the matter-closure question of Paper IV without specifying a microscopic quark topology. Closure-compatible states are organized into integer rollout sectors, and a closure-preserving map advances a discrete branch by a fixed separation increment P while increasing the sector count by one. Assuming that this connected branch admits no lower-count shortcut, that the static quadratic form reduces across number sectors, and that the residual energy remains subextensive after optimization over all higher-count sectors, we show that the averaged discrete static-energy slope tends to σ = ΔE/P. A separate finite-window bound states the accuracy required before string breaking, while any identification with a continuous-r constrained energy requires additional within-cell control. The TEF specialization sets P = 2πnqR_G; neither n nor the positive increment ΔE is derived here. A future gauge completion could relate the linear term to rectangular Wilson-loop behavior only under additional spectral and limit assumptions. The result is therefore a conditional statement about connected-channel closure energetics, not a microscopic derivation of QCD confinement. String breaking, flux-tube fluctuations, SU(3)c dynamics, the numerical origin of ΔE, and ultraviolet running remain unresolved. Context note: This preprint is Paper V in The Emergent Frame (TEF) research sequence. It connects two previously separated branches of the program. Paper I introduced the gravity-calibrated circular-helix ansatz for the open spatial sector and fixed the geometric parameters inherited here. Paper IV introduced the complementary closed-matter/open-space distinction, treated quark-like sectors as locally persistent but globally incomplete matter sub-sectors, and identified the energetics of maintaining closure under separation as a future strong-sector pressure test. The present work develops that pressure test into a matter-shape-agnostic mathematical interface. Its main result is a conditional spectral-slope statement on a restricted connected branch: fixed-increment closure extension together with a subextensive optimized residual energy yields an asymptotically linear averaged static-energy slope. The paper does not claim a microscopic derivation of QCD confinement, the QCD string tension, SU(3)c gauge dynamics, Wilson operators, or the strong coupling constant. Direct antecedents: Paper I: Xiaodan Wu, “A Helical Spacetime Ansatz Linking the Planck Scale and Electroweak Mixing,” Version 5.1. Zenodo. DOI: 10.5281/zenodo.22101000. Paper IV: Xiaodan Wu, “Closed Matter and Open Space: A Relative-Framing Ansatz for Incomplete Quark Sectors and the Matter–Space Interface,” Version 4.6. Zenodo. DOI: 10.5281/zenodo.22256714.