C171 — Six residual measurements leave one common placement freedom
23 September 2026 · Step 16 of 50, continuing C155.
On the seven coordinates D,X,U,M,b,q,f, the six residual measurements have rank six. Their definitions are independent linear forms, leaving one formal placement direction when their values are held.
The direction is
In signed coordinates, where BC bases enter with a minus sign, this is a common translation of the whole configuration. It preserves spans, not membership in finite source fields. It is not a Gear applied to Creation or Conquest.
Holding α,β,η,γ,ρ at zero gives rank five and leaves two numerical freedoms, consistent with C147 after the dependent A/Z/z variables are restored. Adding δ=0 leaves one formal placement; a source coordinate fixes it. This local rank statement explains the reduced grammar without treating linear independence as independent historical attestation.
Evidence and scope
K=529. Positive BC base B denotes (B+1)t/Bn BC; reflected AD base A denotes AD At/(A+1)n. Their same-season civil span is B+A, after the no-year-zero correction. AD source projections are not historical AD occurrences. Generated coordinates and interval midpoints do not acquire event identity from numerical coincidence.
Regular Gears remain confined to Noah/Shem/Flood. Creation and Conquest use fixed source-field members; no fourth Gear is admitted. File_18/File_09 and the C127 graph are unchanged. C133–C155 source corrections remain controlling, with the C10 ancestry of the next-Flood-Gear rule now explicit. No historical campaign is rerun, no quarantined primer is reopened, and no N01/N03 expansion or missing Shem join is repaired. Original Reports 01–10 remain previously verified; interrupted original Reports 11–25 remain unestablished.
Sources: controlling File_18 §§2–3, 6A–6D; File_09 paired Mirror rules; selectively retained C05, C08, C10, C123, C128 and C133–C155 evidence. Specific dependencies are named in the body/certificate. 23 assertions passed; see RESULTS.json and CERTIFICATE.json. Assertions, selected field combinations and inherited identities are not independent historical witness counts.