The Embedding Equation of Motion of Absolute Frame Theory: The Tension as a Nambu–Goto Term and a Regge–Teitelboim Equation with a Routing Source

Version of July 27th, 2026, following the development of Absolute Frame Theory. The foundational paper of the Absolute Frame Theory (AFT) states that it does not derive the equation of motion of the embedding, and defers that derivation to separate work. This paper derives it. Under the exact reading of Axiom I — the metric of the observable manifold is the pullback of the immersion, and the immersion is the sole fundamental field (Branch A) — the Dirichlet scalar of the tension term is identically four, the tension term is the Nambu–Goto four-volume, and the Euler–Lagrange equation of the immersion is a Regge–Teitelboim equation with a source: the normal gradient of the pilot wave, entering at the same order in the gravitational coupling as the geometric term. The contribution is fourfold. First, the equation itself, together with two corrections to the interaction Lagrangian of AFT: the Howe–Tucker constant is restored — selected by the declared requirements of the framework, not by its axioms — and with it the relation among the multipliers previously recorded as open, which would have demanded a cancellation of order 10¹²⁰, is removed, the tension acting in the equation of motion as the mean-curvature force of a brane and not as a term of the Einstein equation. Second, a negative result, stated as such: the routing source decoheres over lengths of order microns on a galactic background — of order tens of metres even at the lowest spectral ceiling the framework permits, a gap of thirty-eight orders of magnitude — and its galactic mean vanishes; the routing supplies no coherent first-order galactic source, and what it does supply, through its variance, is a smooth pressureless component. Third, the closure of the inherited solution degeneracy on the framework's own background: the expanding cosmological background — a four-sphere in the Euclidean substratum, whose extrinsic curvature carries the temporal component that static treatments set to zero — removes the static spherically symmetric extra sector identically, reduces the radial freedom of the background embedding to a gauge choice, and closes the homogeneous branch conditionally on one declared hypothesis — confined by the channel derivation of the companion to the matter sector — the branch reopening as an onset magnitude if the hypothesis fails. Fourth, the relation between the two readings of the theory: Branch A, adopted here, and Branch B, the auxiliary-variable reading of the foundational paper and of the cosmological companion, in which the metric is varied independently of the pullback. The two readings are reconciled rather than opposed: for the tension sector the equation of motion of the auxiliary metric returns the Branch-A identification, every solution of the Branch-B field equations solves the Branch-A equation, and what fails to transfer between them is the evaluation of specific objects, not the solutions. The scope is delimited: the native basis is the interaction Lagrangian under Axioms I–IV, of which the Regge–Teitelboim structure is the consequence; the Einstein–Hilbert term is inherited from the foundational action as a postulate, its derivation from the ℳ–𝒜 channel supplied in separate work; and the composite reading of the gauge sector, an imported assumption the framework does not adopt, is confined to the open problems as an annex.