# C26 — Widths do not determine the branch

22 September 2026 · Step 4 of the authorized 30-step Mirror continuation (C23–C52).

The family widths leave j free. A single *typed* coordinate can recover it once K and the opening attachment are fixed:

\[
j=c+10K=-Z=s-7K=10K-D.
\]

| Branch | j | c | s | Z |
| --- | --- | --- | --- | --- |
| N01 | 2 | -5288 | 3705 | -2 |
| N02 | 3 | -5287 | 3706 | -3 |
| N03 | 4 | -5286 | 3707 | -4 |
| N04 | 5 | -5285 | 3708 | -5 |
| P6 | 6 | -5284 | 3709 | -6 |

All five rows have B=17K, A→D=20K and A→Z=10K. Measuring those lengths alone cannot distinguish a primary Noah row from a companion row, cannot select 3 BC rather than 5 BC as a comparison landing, and cannot authorize the missing MT Noah on N01/N03.

The agreement of four reconstructions is a consistency check on retained assignments, not four independent determinations. A formal increment of j moves c and s forward by one and their reflected coordinates backward by one. Source admission must then be checked separately: arbitrary real or integer j is not an admitted chronology.

This distinction will matter for any longer Mirror path. Adding a multiple of 529 may preserve its length for every j while its required endpoint occurs in the source field only on a restricted set of rows.

**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.** Verify the signed formulas against paired civil counting before using longer reflected paths.