C175 — The one-residual model is a precise restricted grammar
23 September 2026 · Step 20 of 50, continuing C155.
C166–C175 establish the conditions under which the compact δ model applies. They also provide admitted counterexamples outside that alignment.
| Residual | Measurement | What zero fixes |
|---|---|---|
| α | D+X−17K | SP diagonal |
| β | U+M−17K | MT diagonal |
| η | D−b−868 | Cumulative/native member pairing |
| γ | b+q−11K | Creation/Conquest pairing |
| ρ | M−f−190 | MT profile and event-state pairing |
| δ | X−M−2K | Regular side width |
On α=β=η=γ=ρ=0, the source-qualified family reduces to the earlier square and 13K−δ/10K partition formulas. Away from it, the general equations identify which assumption changed. Exact totals can coexist with altered internal subdivisions.
The gain is a smaller, explicit conditional grammar rather than a claim of unrestricted equivalence. The next ten steps carry these distinctions through the 8K complement so the post-Flood/Moses framework remains connected to the 529 analysis without generated coordinates acquiring unwarranted event identities.
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. 20 assertions passed; see RESULTS.json and CERTIFICATE.json. Assertions, selected field combinations and inherited identities are not independent historical witness counts.