# C155 — A single residual connects the finite Mirror constructions

23 September 2026 · Bounded step 20 of 20 (C136–C155).

**C136–C155 establish one compact finite comparison model linking the 10K Noah construction, the 13K Creation family and the 17K equal-diagonal square.** The controlling residual is δ=X−M−2K, equivalently d+1−j.

| Construction | Finite formula | Exact condition |
|---|---|---|
| 17K square | [[17K, 19K+δ], [15K−δ, 17K]] | All entries on 529 grid iff δ=0 |
| Creation→ordinary Flood | 13K−δ | 13K iff δ=0 |
| Noah route with alternate Flood | 3K + (2K+δ) + (5K−δ) | 3+2+5 and 5+5 iff δ=0 |
| Creation→generated preimage→Conquest | 3K+8K=11K | All five Noah/native alignments |
| Sourced cumulative-to-Noah route | 868+11K+(2K−δ)+190+(2K+δ)=17K | All thirty admitted pairings |

The strongest new boundary result is P6's admitted 1059-sided square: **[[8993,10052],[7934,8993]]**. It retains 17K diagonals while its 13K prefix is 6876 and its alternate-Flood midpoint divides 10K into 2646+2644. The excluded 1058-sided square remains excluded.

The source interpretation is finite and typed. The four exact rows require the MT Flood Gear one above the SP Noah Gear, with matched primary/start or companion/close bits. P6's Noah is already G3 companion, so no exact continuation exists in the allowed three-Gear domain. Those matched bits are correspondences between different source states, not identical events.

The thirty squares are endpoint recombinations, and the exact conditions are algebraically dependent. The equal 3K comparisons still differ in provenance because their Creation-side preimages are generated. The rank, reflection and phase checks preserve those limitations.

**Next bounded question (C156):** recover the 868 and 190 offsets from their chronology inputs independently of the target 529 equations, then assess precisely what their 868+190=2K sum explains. This is a proposal only; the authorized twenty-step batch ends here. No interpretive expansion or new campaign is counted as completed.

## Source and scope control

All numerical bases in this packet stand for paired coordinates: BC base B is (B+1)t/Bn BC; reflected AD base A is AD At/(A+1)n. Thus a cross-Mirror same-season span is B+A, after the civil no-year-zero subtraction. AD stations are reflected source events, not historical AD occurrences. K=529 throughout.

Regular Gears remain restricted to Noah/Shem/Flood. Creation and Conquest are fixed source-field selections. C133–C135 corrections remain controlling. The C127 graph and canonical controls are frozen; source-field admission is distinguished from an existing graph node and from a generated comparison coordinate. No wider N01/N03 expansion or missing N03 Shem join is repaired. Original Reports 01–10 remain previously verified references; interrupted original Reports 11–25 remain unestablished. Check counts are computational assertions, not independent historical witnesses.

Evidence: `CERTIFICATE.json`; **40 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.