# C142 — Exactly four of the thirty lie entirely on the 529 grid

23 September 2026 · Bounded step 7 of 20 (C136–C155).

**The exact 529 mask selects precisely N01–N04.** All other 26 pairings keep their exact 17K diagonals but have off-diagonal residuals.

\[
S_{d,j}=\begin{pmatrix}17K&19K+\delta\\15K-\delta&17K\end{pmatrix},\qquad \delta=d+1-j.
\]

In this finite domain −3≤δ≤6, divisibility of all four spans by K requires δ=0. There is no nonzero multiple of 529 in that residual range.

| Row | d | j | U | D | M | X |
|---|---|---|---|---|---|---|
| N01 | 1 | 2 | 6346 | 5288 | 2647 | 3705 |
| N02 | 2 | 3 | 6345 | 5287 | 2648 | 3706 |
| N03 | 3 | 4 | 6344 | 5286 | 2649 | 3707 |
| N04 | 4 | 5 | 6343 | 5285 | 2650 | 3708 |

Thus the expanded source comparison family leaves C10's exact four-square result intact. Negative residuals are admitted in some cross-row pairings; P6 alone has only positive residuals 1…6. This asymmetry follows from the finite upper MT Flood boundary and is not a change of Mirror axis or calendar phase.

The four exact results are inherited comparisons recovered by a unified formula. Their reappearance is a consistency check, not four new discoveries or independent evidence.

## 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`; **51 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.