# C152 — Residual-preserving shifts are partial source correspondences

23 September 2026 · Bounded step 17 of 20 (C136–C155).

**Moving both finite selector indices together preserves δ and the square matrix, but only while both endpoints remain in their admitted domains.** This is a partial correspondence, not a global chronology operator.

For T(d,j)=(d+1,j+1), δ=d+1−j is invariant. The finite runs have these lengths:

| δ | Number of admitted pairs |
|---|---|
| -3 | 1 |
| -2 | 2 |
| -1 | 3 |
| 0 | 4 |
| 1 | 5 |
| 2 | 5 |
| 3 | 4 |
| 4 | 3 |
| 5 | 2 |
| 6 | 1 |

The two-index-step move (d,j)→(d+2,j+2) preserves Noah's primary/companion bit and Flood's start/close bit while increasing both regular Gears by one. That is consistent with their regular two-year Gear spacing where available. In contrast, a one-index-step move can switch event-state bits; it is not uniformly a Gear change.

Other coordinates also change along the comparison: A and X increase, D and native b decrease, and reflected q increases. The b/q changes select different fixed source-field members; they are not Creation or Conquest Gears. The regular Gear interpretation is restricted to the Noah/Flood selectors.

The exact δ=0 run terminates at d=4,j=5. Its next formal index pair d=5,j=6 is outside the allowed MT field. No wraparound, fourth Gear or missing event is admitted. Invariance of a matrix along a finite run cannot establish a source-preserving transformation beyond that run.

## 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`; **64 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.