# C163 — The cumulative 2K interval selects a two-member displacement

23 September 2026 · Step 8 of 50, continuing C155.

**The two cumulative fields yield six 2K source-field comparisons before regular endpoints are attached.** Use U=6350−u in cumulative ON and D=5290−v in OFF, with u,v=0,…,7.

Removing cumulative Cainan from U gives W=5890−u. Hence

\[
U-D=1060-u+v=460+(600-u+v).
\]

Exact 1058 requires u−v=2, leaving W−D=598. The six admitted pairs are:

| U ON | W derived OFF | D OFF | W−D |
|---|---|---|---|
| 6348 | 5888 | 5290 | 598 |
| 6347 | 5887 | 5289 | 598 |
| 6346 | 5886 | 5288 | 598 |
| 6345 | 5885 | 5287 | 598 |
| 6344 | 5884 | 5286 | 598 |
| 6343 | 5883 | 5285 | 598 |

This displacement selects cumulative field members; it is not a regular Gear applied to cumulative or Creation nodes. Cumulative support alone does not admit the required regular Noah/Flood endpoints of every joint square. That source intersection is tested next.

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