C123 — Minimal equations and the local complement

Three equations determine the relative four-endpoint construction. With positive BC bases c,z and AD pair bases q,f:

c+q=11K,\qquad z+q=8K,\qquad f-q=2K.

They imply:

\boxed{c-z=3K,\quad z+f=10K,\quad c+f=13K.}

Therefore the same admitted paths have refinements 3+8+2, 3+10, and 11+2. These are dependent descriptions. In particular, generated z→reflected Flood has width10K; it is not a new independent calibration of the tenfold family.

The three relations have rank3 on four variables. Their one formal free direction is (Δc,Δz,Δq,Δf)=(−δ,−δ,+δ,+δ). A retained source endpoint fixes δ. The evidence provides exact consequence certificates and a deletion witness for each defining relation. The formal free direction does not authorize historical movement of Creation or Conquest.

The local symmetry has an exact operator description. Keep the established global Mirror R(x)=−x, and let T₈K(x)=x+8K. Their composition is:

S(x)=T_{8K}(R(x))=8K-x,\qquad S^2(x)=x.

S fixes the proof coordinate4K=2116; it reverses intervals and swaps paired component order. Also S∘R=T₈K and R∘S=T₋₈K. This is a derived affine complement, not a replacement global Mirror axis or an automatically source-preserving action on all chronologies. Only named endpoints retain their established source identities; other images remain generated.

The43 checks include the retained paired coordinates and operator identities. Next: apply this complement to the fixed Moses-birth, Exodus and Moses-death/Conquest boundaries. High sufficient.