Proposition: Thermal-pair state and moment certificate

These preparations satisfy

For $L$ independent pairs with individual cutoffs $m_{\ell }$, the complete pure-state distance is

Appending these fixed preparations to the same logical isometry gives the same complete-channel allowance, independently of the code dimension.

Proof. The squared overlap of the complete pair preparations is $1-{\lambda }^{m+1}$, giving the pure-state distance. The discarded geometric tail has weight ${\lambda }^{m+1}$; the normalized retained distribution gives the marginal distance. Differentiating the finite geometric sum gives

Product overlaps multiply, and $1-{\prod }_{\ell }(1-t_{\ell })\le {\sum }_{\ell }{t}_{\ell }$. For any referenced logical input, the preparation difference is a tensor product with a trace-one logical output, so its trace norm is precisely the preparation trace norm.