C205 — Fifty-step synthesis: source conditions and interval-center geometry

23 September 2026 · Step 50 of 50, continuing C155.

All fifty bounded steps C156–C205 are complete. The strongest result is a source-sensitive explanation of the868/190 relation, followed by its exact interval-center geometry.

First,190=130+60 comes from retained MT switches, whereas868 selects h=e+3 between cumulative OFF and native Creation members. Their sum2K is exact on that alignment. The next-Flood-Gear rule was already explicit in C10; its recovery in C149 is verification.

Second, the singleδ model belongs to a defined source family. The general square needs diagonal residualsα/β, and its Creation route also depends on member/profile residualsη/γ/ρ. Concrete sourced counterexamples show why those conditions cannot be omitted.

Third, at the four exact common rows, the Creation/Flood interval's center divides the separation of the two17K diagonal centers as 868+190=2K. In the deformed family the gaps become 868+δ/2 and 190+δ/2. P6 therefore has868.5+190.5=1059 without moving a source endpoint.

The8K complement and expansion tests preserve their algebraic value while distinguishing generated images, annual-grid integrality and source admission. The aligned coordinates reduce to K,190 and two finite selectors; they do not establish a universal action on the chronologies.

Best next bounded step C206: compare this interval-center geometry along the two retained source-qualified6877→7475→8125 chains, asking whether the center offsets transform coherently under a single declared endpoint map. Use the existing C01 source certificates; do not rerun the earlier expansion campaign. C206 is proposed only; the authorized fifty-step batch ends here.

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. 69 assertions passed; see RESULTS.json and CERTIFICATE.json. Assertions, selected field combinations and inherited identities are not independent historical witness counts.