Palatini–PT-Even Torsion: Uniqueness, Three-Chain Equivalence, and Exact Luminal Tensors

What is new. Within a first–order (Palatini) setting we impose a scalar PT projector on all observable scalar densities, restricting to at most one derivative per building block. Under this setting the framework implies: (i) a unique, pure–trace torsion aligned with ∂ϵ (C1); (ii) a bulk equivalence of three ostensibly different rank–one constructions with the same quadratic coefficient A⋆ = λ2/8 (C2); and (iii) a unique weight ratio between the rank–one determinant and closed–metric routes that removes TT–nonTT mixing and enforces the equal–coefficient identity K=G (C3), hence cT=1 at quadratic order. Setup. All observable scalars are mapped by a scalar PT projector ΠPT( · ) to a real, even sector; the internal phase ϵ(x) is a nondynamical spurion that enters only through ∂ϵ. With global assumptions A1–A6 (PT–invariant domain/measure, PT , ∗=0, projection–variation commutation, boundary/topology posture, Nieh–Yan as boundary counterterm, and a trace lock), we work at quadratic order with at most one derivative per building block. The trace lock is enforced algebraically so that Tμ = 3η ∂μϵ (no new canonical pairs under A4). Results. (C1) Palatini algebraicity plus the scalar projector force Sμ=0, qλμν=0, and TA BC = 2η δA B∂Cϵ, yielding the invariant IT = −6η2 ΠPT(∂ϵ)2. (C2) After C1 and projection, the three routes—(i) a rank–one determinant built from the canonical traceless matrix, (ii) a closed–metric rank–one deformation, and (iii) the PT–even CS/Nieh–Yan shadow—collapse to the same bulk quadratic piece A⋆ √−g (σϵ IT) up to improvements, with A⋆ = λ2/8. (Our “ROD” is a parity–even volume deformation, not Born–Infeld gravity.) (C3) On flat FRW (unitary gauge) the two independent TT–nonTT mixing entries are proportional and fix the locked ratio wROD : wCM = 2 : 3 . On generic admissible patches the two mixing equations are non–collinear, selecting the same unique ratio; with this locking we prove K(w)=G(w) (a total– divergence identity on the admissible variational domain), hence c2 T = 1 and a GR TT sector with no extra propagating degrees of freedom. Reproducibility and scope. All figures and coefficients are reproduced by the accompanying open–source notebooks and scripts; the 2:3 ratio is extracted by an automated mixing–matrix check on flat FRW and its generic uniqueness follows from the non–collinearity argument. Matter couplings respect the posture (axial channel vanishes; the trace coupling is removable up to a Nieh–Yan boundary shift). The leading PT–even NLO correction predicts δc2 T(k) = b k2/Λ2, offering band–limited constraints on Λ. Detailed proofs, improvement currents, and coefficient tables appear in Apps. A–E.