C51 — Family closure alone is not a sufficient source certificate

22 September 2026 · Step 29 of the authorized 30-step Mirror continuation (C23–C52).

The reduced continuation needs the source scale, the profile offset, the regular/cumulative match and the endpoint domain in addition to exact family arithmetic. Four bounded countermodels show why:

Changing p from 190 to 191 translates both Creation and ordinary Flood by the same amount in the compact formulas, preserving 6877. It also leaves 19 and 22 unchanged. Yet it violates the retained 130+60 and 650−460 profile binding and makes the required MT Noah 3055, absent from the ordinary field.

Using K=530 can preserve the formal family coefficients, but with ℓ=598 it requires κ=462 instead of source Cainan 460. Choosing j=6 preserves the algebraic widths but requires ON Shem 6342 below its field. Releasing the selected-length match by one changes the new 19/22 residuals while leaving the Creation 13-fold residual zero.

These examples are formal controls, not proposed chronology variants. They establish that a list of exact multiples is insufficient. C22's four reduced scalar conditions plus the source scale remain necessary to its calibrated model; the typed profile and endpoint bindings must then be retained. This compact guard is scoped to the present continuation and does not replace the full controlling files.

No broad negative-case campaign or statistical inference is made. The controls address specific remaining risks: rescaling, role-offset changes, out-of-field endpoints and a mismatched cumulative member.

Validation. 5 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. Publish the C23–C52 synthesis and an exact restart locator for the completed 30-step batch.