The Fifth Coordinate — the Worldsheet-Flux Law for Θ

Derivation note, R&D Division, November 8, 2026. Successor to 'The Fifth Coordinate — the Θ/θ Dictionary' (November 8, 2026), which identified Θ as the epoch difference of the axial holonomy and left the angle's debt sharpened to a single named object: a law for the worldsheet flux of dS between epochs. This note derives that law. The transport law is DERIVED in closed form: the flux between two epoch hypersurfaces equals (𝒯_A/s_A) times the difference of the axial spin current's circulation about the lineage axis at the two endpoint epochs, modulo 2π. The worldsheet flux has no bulk term: the Nieh–Yan phase sector contributes only the winding integer, which is identical at both epochs and cancels, so that sector is structurally incapable of moving the fractional holonomy; the exact Cartan tie converts the algebraic component's circulation to the source's circulation outright. Evaluated on the matter contents the corpus owns — the static cord of the winding ansatz and the flagship's incoherent Weyssenhoff fluid, whose mean axial current is zero — the law returns β_frame ≡ 0, upgraded here from corpus report to DERIVED in the owned matter model. The value Θ(recombination) remains NOT DERIVED, and the underdetermination is proved by exhibition: the freedom sits in exactly one place, the post-founding generation of a mean axial spin current, so the debt shrinks from a functional law to one real datum per epoch pair, ΔCirc_σ, with its address named. Against the targets of Θ9, which remain targets and never inputs, that datum would have to satisfy ΔCirc_σ = (s_A/𝒯_A)·Θ_target with Θ_target in [4.40×10⁻³, 7.54×10⁻³] rad; no numerical value of s_A as a circulation scale is priced in the record, and none is printed. What closes the value is named: one constitutive law pricing the mean axial spin current of the cosmological register. The identification of Θ remains CONSTRAINED; the fifth coordinate χ remains CONSTRAINED (charge (ii) untouched). Four falsifiers are printed (F-F1–F-F4), including the inherited empirical one: a measured rotation that accumulates along the trip would kill the boundary-difference form, the dictionary, and this note with it.