# C40 — One 24-fold path has five qualified two-part decompositions

22 September 2026 · Step 18 of the authorized 30-step Mirror continuation (C23–C52).

The N01 chain is fully endpointed:

| Point | Signed coordinate | Role |
| --- | --- | --- |
| H | -6350 | native ON cumulative Shem head |
| A | -5292 | MT cumulative ON Flood |
| X | -3705 | SP G1 companion Noah |
| Y | -2647 | alternate MT G2 Flood close |
| Z | -2 | comparison landing |
| D | 5288 | reflected LXX cumulative OFF Flood |
| U | 6346 | reflected native ON cumulative Shem |

Its consecutive lengths are 2K, 3K, 2K, 5K, 10K and 2K. The retained cuts yield:

| Cut | Decomposition |
| --- | --- |
| X | 5 + 19 = 24 |
| Y | 7 + 17 = 24 |
| Z | 12 + 12 = 24 |
| D | 22 + 2 = 24 |
| A | 2 + 22 = 24 |

The right-hand 19-fold interval X→U is the canonical reflection of u→s. The right-hand 17-fold interval Y→U reflects u→v. Thus 5+19 and 7+17 preserve explicit source roles rather than being arbitrary numerical partitions. The alternative 2+22, 12+12 and 22+2 cuts use the already qualified cumulative and landing endpoints.

These decompositions are incidence identities of one path. They provide a useful integrated display but do not constitute five extra confirmations of 529. Nor does the N01 chain acquire a valid MT Noah birth for the full Creation expansion merely because its longer outer interval closes.

**Validation.** 7 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.** Solve the exact source-field bounds for raw and joined 24-fold intervals.