# C147 — A local affine model has two selector freedoms

23 September 2026 · Bounded step 12 of 20 (C136–C155).

**Eight independent affine equations organize the ten local coordinates; two numerical freedoms remain before source restrictions.** They can be represented by X and M, or equivalently d and j within the admitted finite product.

| Retained equation | Role |
|---|---|
| A-X=3K | Noah first leg |
| A-Z=10K | Noah total |
| D+X=17K | SP 17K diagonal |
| D-b=868 | Cumulative/native separation |
| b+q=11K | Creation–Conquest span |
| z+q=8K | Generated 8K preimage |
| M-f=190 | MT profile difference |
| U+M=17K | MT 17K diagonal |

Exact rational elimination gives coefficient rank 8 and augmented rank 8. Imposing X−M=2K increases the rank to 9. Fixing P6's X=3709 as well gives rank 10 and forces M=2651, outside the retained alternate MT field.

The model explains why one additional alignment equation can simultaneously enforce several exact partitions. **Field membership and endpoint type are extra conditions**, not consequences of matrix rank. Its variables also include generated z, so rank does not confer historical status on that point.

These ranks concern only this ten-variable local comparison system. The frozen 208-node historical graph and its inherited rank are unchanged. The eight equations summarize retained relationships; their linear independence does not mean eight independent historical attestations.

## Source and scope control

All numerical bases in this packet stand for paired coordinates: BC base B is (B+1)t/Bn BC; reflected AD base A is AD At/(A+1)n. Thus a cross-Mirror same-season span is B+A, after the civil no-year-zero subtraction. AD stations are reflected source events, not historical AD occurrences. K=529 throughout.

Regular Gears remain restricted to Noah/Shem/Flood. Creation and Conquest are fixed source-field selections. C133–C135 corrections remain controlling. The C127 graph and canonical controls are frozen; source-field admission is distinguished from an existing graph node and from a generated comparison coordinate. No wider N01/N03 expansion or missing N03 Shem join is repaired. Original Reports 01–10 remain previously verified references; interrupted original Reports 11–25 remain unestablished. Check counts are computational assertions, not independent historical witnesses.

Evidence: `CERTIFICATE.json`; **53 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.