{"id": "c123", "title": "C123 — Minimal equations and the local complement", "filename": "490d_C123_Local_Complement_Grammar_20260922.md", "role": "Research action", "step": 123, "sha256": "496f7495afc02c0742f4a7ae21ba82fce1318ebca8664d50a773b215be831156", "bytes": 1663, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_C119_C128_529_Conquest_Batch_20260922/C123/490d_C123_Local_Complement_Grammar_20260922.md"], "download": "/sources/496f7495afc02c0742f4a7ae21ba82fce1318ebca8664d50a773b215be831156.md", "format": "md", "derived": false, "tags": ["Mirror / 529"], "excerpt": "C123 — Minimal equations and the local complement\n\nThree equations determine the relative four-endpoint construction. With positive BC bases c,z and AD pair bases q,f:\n\n\\\nc+q=11K,\\qquad z+q=8K,\\qquad f-q=2K.\n\\\n\nThey imply:\n\n\\\n\\boxedc-z=3K,\\quad z+f=10K,\\quad c+f=13K.\n\\\n\nTherefore the same admitted paths have refinement", "content": "# C123 — Minimal equations and the local complement\n\n**Three equations determine the relative four-endpoint construction.** With positive BC bases c,z and AD pair bases q,f:\n\n\\[\nc+q=11K,\\qquad z+q=8K,\\qquad f-q=2K.\n\\]\n\nThey imply:\n\n\\[\n\\boxed{c-z=3K,\\quad z+f=10K,\\quad c+f=13K.}\n\\]\n\nTherefore 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.\n\nThe 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.\n\nThe local symmetry has an exact operator description. Keep the established global Mirror **R(x)=−x**, and let **T₈K(x)=x+8K**. Their composition is:\n\n\\[\nS(x)=T_{8K}(R(x))=8K-x,\\qquad S^2(x)=x.\n\\]\n\nS 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.\n\nThe43 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.\n"}