Signed Longitudinal Displacement Occupancy in the Axis Parent: Exact-Source Closure, a Constructive Charged-Cell Parent, Continuum Radial Localization, and the Three-Axis Composition Interface

This paper constructs and validates a localized charged cell within an explicit Axis parent theory. The cell has one signed longitudinal internal direction and a complementary transverse plane. Its signed source is carried by the existing massive U(1)GU(1)_GU(1)G response field, so the construction introduces no additional long-range gauge force. The parent action is specified analytically, its charged sector is isolated from the neutral heavy sector, and its reduction yields the exact scalar, charged-field, and massive-vector model used in the localization analysis. The mathematical program proceeds in two stages. Interval arithmetic first certifies that the finite radial collocation system has a unique solution inside a verified box and that the unit-charge sector contains configurations below the free same-charge threshold. A subsequent function-space proof establishes an exact continuum solution on the half-line with unit charge, finite energy, exponential tails, and local uniqueness in the certified Banach neighborhood. The total energy remains strictly below the lightest free state carrying the same parent charge. The paper also develops the parent composition interface. Under the ordinary one-cell-per-frame-line representation, three equal-sign cells define the first nonzero isotropic frame-scalar sector with the traditional Axis ledger of three longitudinal and six transverse units. This sector provides the canonical internal composition target for downstream multi-cell constructions, while spacetime spin remains an independent spinor factor. The resulting parent theory establishes a constructive localized charged-cell sector together with its exact representation, coefficient ledger, and continuum existence theorem. It provides the certified one-cell foundation on which the downstream composition, fermion, and Standard Model interface papers are built. This revision clarifies the paper’s scope by separating the certified classical one-cell theorem from the conditional postparent internal-carrier construction and the unresolved physical multi-cell particle.