# C25 — A compact endpoint form for the retained Mirror branches

22 September 2026 · Step 3 of the authorized 30-step Mirror continuation (C23–C52).

Let K=529, ℓ=598, κ=460=2K−ℓ, p=190 and let j identify the cumulative OFF member 5290−j. In the signed paired coordinate convention, the following formulas reproduce every registered point on N01–N04 and the available P6 base points.

| Point and role | Signed coordinate |
| --- | --- |
| c: cumulative OFF Flood | −10K+j |
| u: native ON cumulative Shem member | −12K+j |
| cm: derived OFF Shem member | −10K−ℓ+j |
| b: fixed-field SP Creation | −8K−p+j |
| q: reflected Conquest | 3K−p+j |
| f: reflected ordinary MT Flood | 5K−p+j |
| v: reflected alternate MT Flood | 5K+j |
| s: reflected SP Noah | 7K+j |
| r: reflected SP Flood | 7K−ℓ+j |
| mtN: reflected MT Noah | 5K+ℓ−p+j |
| spDeath: reflected SP companion death | 5K+ℓ/2−p+j |
| A, X, Y, Z, D | −10K−j, −7K−j, −5K−j, −j, 10K−j |
| H: the optional left Shem head | −12K−j |

The admitted indices are N01=2, N02=3, N03=4, N04=5 and P6=6. A formula is not an admission: H exists as the retained head only on N01, while mtN and spDeath complete the full expansion only on N02/N04. The evidence compares only points actually present in the completed typed graph.

The formulas expose two opposite field-index signs. Direct branch points carry +j; their reflections and the BC partition points carry −j. They also separate the profile offset p from the common 529 unit. The Creation formula selects among fixed source members; it does not move Creation through the regular Gears.

This is an exact compression of the existing geometry, not a replacement for the role, Gear, boundary and companion tags preserved in the frozen graph.

**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.** Determine which data identify the field index and which widths leave it free.