C173 — The residual model detects a regular/cumulative Cainan substitution

23 September 2026 · Step 18 of 50, continuing C155.

Using 130 where cumulative removal requires 460 shifts the proposed OFF coordinate by 330 years and breaks source admission. This is an explicitly invalid diagnostic, not an alternate accepted chronology.

At P6, replacing the cumulative removal would change D5284 to D5614. The SP diagonal residual α and cumulative/native gap residual η would both become +330. They cancel in the formula for the independently held Creation→ordinary-Flood span, but neither the 17K diagonal nor the 868 source alignment survives.

This explains a practical limit of total-span checks: a held interval elsewhere can remain correct even while an upstream operator has been misapplied. The retained cumulative field rejects 5614 directly. Regular Cainan 130, cumulative Cainan 460 and the regular residual 650−190=460 must remain separately typed.

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.