# C111 — Ratio, scale and origin require distinct bindings

22 September 2026 · Step 9 of the authorized ten-step Mirror continuation (C103–C112).

The reduced junction model has four continuous parameters: apparent origin α, anchor difference d, base unit k and common ratio r. Its assumptions have separate jobs:

| Bindings admitted | Parameters still free | Exact local rank |
| --- | --- | --- |
| Both junction equations | k and α | 2 |
| Then the source interval 600 | α | 3 |
| Then the source Flood coordinate 2456 | none | 4 |

The rank calculation is the exact Jacobian rank at the reconstructed solution. The explicit factorization in C108 and calibration in C109 also give uniqueness throughout the declared positive expansion domain; local rank alone would not establish that global statement.

Either junction alone is insufficient. In a formal unit model, the Flood junction admits r=13/12 with a positive anchor difference while missing the Noah junction; the Noah junction admits r=12/11 while missing the Flood junction. These are diagnostic parameter examples, not alternate sourced chronologies. Other source constraints are intentionally not imported into this reduced-model deletion test.

Even after all four numeric parameters are fixed, geometry supplies a coordinate rather than a person's identity. Calling the first apparent target Joseph still requires the original Kohath +110 source rule and its admitted member. That requirement cannot be replaced by matrix rank.

The full fixed-width graph has a different count: 104 variables and rank 101. Its three remaining freedoms occur only in the isolated C5/C6/C7 Creation–Conquest pairs. One already recorded source coordinate per pair raises a diagnostic pinned system to rank 104 without inconsistency. The stored relation graph remains unchanged at rank 101; these explicit source pins are reported separately. The selected junction construction is not left underdetermined by those unrelated graph components.

**Validation.** 9 exact arithmetic/structural checks passed. The 20 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 and Appendix A (paired Mirror); File_18 §§2.1.3, 3.1 and 6A (regular fields and cumulative source members); File_61 §11.2 (original backward Joseph +110); File_62 collateral-scope guards; C05 admitted apparent field; C43 frozen graph; C95 and C100–C102 named carrier rows; preceding current certificates. The forthcoming 529 × 4 × 2 cycle has not been explored. The declared phase model does not infer an unstated birthday or event month from an annual source row.

**Best next bounded step — High sufficient.** Synthesize the graph extension and the conditional ratio/scale reconstruction, publish the completed ten-step checkpoint, then await the user’s 529 × 4 × 2 cycle.