From Colour--Casimir Recoupling to Near-Threshold Scattering: Algebraic Diagnostics and a Pion-Mass Test for the Tcc⁺ Channel

The near-threshold Tcc⁺ is a clean testing ground for how constituent-level colour structure connects to hadronic scattering. We report three results and state what each does and does not establish. (i)~Colour algebra. In the two-dimensional colour-singlet basis of cc u d, a general pairwise colour operator has direct DD^* matrix element VMM=-43(v₁₃+v₂₄), containing only the two intra-meson coefficients, while meson--hidden-colour mixing is VMHᵒᶠᶠ=√2/3Δᵥ with Δᵥ=v₁₄+v₂₃-v₁₂-v₃₄; uniform edges give no mixing. The projection is exact within this basis and is checked numerically on 10\,000 random edge assignments. A same-Hamiltonian decomposition shows that it re-expresses the central colour operator ($116.46$ MeV off-diagonal) rather than adding an independent dynamical input. (ii)~Negative dynamical results. For Coulomb-type Gaussian edges the hidden-colour diagonal lies below the meson diagonal, so the attractive second-order shift -2Δᵥ²/(9ΔH) does not apply. In a Gaussian constituent Hamiltonian calibrated to MD and MD^*, the lowest compact level lies $148.6$ MeV above its model DD^* threshold (iii)~Pion-mass test. Couplings fixed at the m_π280 MeV lattice pole ($-5.2$ MeV) are carried to m_π391 MeV by changing only the reduced mass and compact-sector gap, both refitted to the lattice D and D^* masses. A one-channel separable Lippmann--Schwinger benchmark ($Λ=650$ MeV, $μ=967.7$ MeV, CG-36.15~GeV⁻²) is calibrated to the $5.2$ MeV pole and serves only as a regulator-sensitivity study. We do not reproduce the complex lattice pole or perform a full coupled-channel calculation.