# C41 — Raw 24-fold pairs and the joined path have different domains

22 September 2026 · Step 19 of the authorized 30-step Mirror continuation (C23–C52).

Both native-ON Shem endpoints must lie in 6343–6350 BC. For H's magnitude 12K+j this gives −5≤j≤2; for u's magnitude 12K−j it gives −2≤j≤5. Their intersection is **−2≤j≤2**.

| j | BC Shem numeral | Reflected Shem numeral | OFF member also admitted | Full joined base |
| --- | --- | --- | --- | --- |
| -2 | 6346 | 6350 | no | no |
| -1 | 6347 | 6349 | no | no |
| 0 | 6348 | 6348 | yes | no |
| 1 | 6349 | 6347 | yes | no |
| 2 | 6350 | 6346 | yes | yes |

These five ordered pairs all have width 12696. Reflection swaps j and −j, giving two opposite-orientation pairs and one self-Mirror case at j=0 (6348 BC reflected to its own paired AD image). Thus there are three interval classes up to canonical reflection within this specifically defined source-only family, not five independent witnesses.

Requiring the original OFF Flood field restricts j to 0…7 and leaves 0,1,2. Requiring the retained SP Noah/MT cumulative base further restricts j to 2…6, leaving **j=2 only**, the complete N01 chain.

The negative j rows are admitted only as raw ON-Shem endpoint pairs. Their putative OFF member lies outside its field, so they do not enlarge the joined chronology. This is a closed-inequality derivation on a named field, not a broad search across historical reports or arbitrary event pairs.

**Validation.** 11 exact arithmetic/structural checks passed. The ten frozen dependencies passed hash verification; all three generated outputs replayed byte for byte in isolation, and the original dependencies remained unchanged. This step reads completed certificates and does not execute their campaigns.

**Scope and provenance.** Controlling File_18 and File_09 remain unchanged. Regular Gears apply only to Noah/Shem/Flood; SP person companions only to Noah/Shem. Creation and Conquest fields remain fixed. Native LXX ON, explicitly derived OFF, independent Flood start/close, and paired suffix inversion at January AD 1 remain in force. No civil year zero, upstream Gear transport, primers, Rounded campaign, or localized −33 is introduced. These are new continuation reports, not recovered historical Reports 11–25. Evidence is conditional on the supplied project model, not a proof of literary intention or independent historical dating.

**Source locators.** File_09 §3.1 (paired Mirror); File_18 §§2.1.3, 3.1–3.2, 6B and 6D (qualified regular and cumulative fields); frozen C15 typed graph, C16 domain certificate, and C22 reduced scale/constraint certificates.

**Best next bounded step — High sufficient.** Check exactly where the new families coexist with the full 6877 expansion.