Corrections required before the gates

. Injectivity is not an ordering comparison. The inherited equations

and the slab witness ${{\mathcal{Q}}^{\mathcal{R}}}_{g,f}\ne 0$ imply

They do not imply ${\mathcal{Q}}_{f,g}\ne {{\mathcal{Q}}^{\mathcal{R}}}_{g,f}$. For example, two injective linear maps can be the same map. The difference of the two contractions must be computed independently.

. Contacts cannot repair inconsistent prescribed values. If both support orderings hold, symmetry and factorization require the new value to equal both fixed free values. If those values are unequal, no alteration of the new value can satisfy both equations. This argument applies even on the permitted contact set.

. Fixed old coefficients must agree exactly. A contact is normalization freedom only where normalization is unfixed. A contact discrepancy from a supplied, fixed old coefficient remains a failed overlap until it is removed by a permitted normalization preserving all other conditions.

. Double ordering means causal disjointness. The sentence that a two-way pair requires overlapping functional supports is incorrect. With the ordering used here, both orderings exclude every causal relation between the two supports, including coincidence and null relation. Proposition prop:geometry proves the precise statement.

. A time slab proves an ordering, not necessarily its uniqueness. A later slab proves $W_g\gtrsim B_f$. It does not, by itself, rule out $B_f\gtrsim W_g$: supports in different time slabs can still be spacelike separated. Uniqueness of the ordering needs an actual causal relation between actual functional supports.