C165 — Source audit: the 2K bridge is exact and conditional
23 September 2026 · Step 10 of 50, continuing C155.
C156–C165 locate the 2K bridge in three explicit ingredients: a selected cumulative/native member relation (868), a retained regular MT profile difference (190), and a shared numerical 460 arising by different mechanisms.
The source arithmetic is
The final −3 in the first formula is the relation between selected field offsets, not an independently established chronology-wide shift. The two 460s agree numerically but are not the same operator. The 598 next-Gear result is inherited from C08/C10.
The finite intersections explain why related constructions have different counts: seven 868 pairs, six shared 868/11K/17K rows, five with the cumulative Noah 3K start, and four with the exact 2K Flood/square attachment. No additional historical node or Gear was needed.
The next ten steps test how the equations change if the selected diagonal, native-member or profile conditions are relaxed. This is necessary to distinguish an exact conditional grammar from an unrestricted equivalence among the chronologies.
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.