{"id": "c142", "title": "C142 — Exactly four of the thirty lie entirely on the 529 grid", "filename": "490d_C142_529_Mirror_Bounded_Step_20260923.md", "role": "Research action", "step": 142, "sha256": "896b2dfa26cf3be0e575e631a90ccbd6e9fc0af431bf525e26625256c25e86e0", "bytes": 2278, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_C136_C155_529_Square_Batch_20260923/C142/490d_C142_529_Mirror_Bounded_Step_20260923.md"], "download": "/sources/896b2dfa26cf3be0e575e631a90ccbd6e9fc0af431bf525e26625256c25e86e0.md", "format": "md", "derived": false, "tags": ["Regular", "Calendar Keys", "Mirror / 529"], "excerpt": "C142 — Exactly four of the thirty lie entirely on the 529 grid\n\n23 September 2026 · Bounded step 7 of 20 (C136–C155).\n\nThe exact 529 mask selects precisely N01–N04. All other 26 pairings keep their exact 17K diagonals but have off-diagonal residuals.\n\n\\\nS_d,j=\\beginpmatrix17K&19K+\\delta\\\\15K-\\delta&17K\\endpmatrix,\\qqua", "content": "# C142 — Exactly four of the thirty lie entirely on the 529 grid\n\n23 September 2026 · Bounded step 7 of 20 (C136–C155).\n\n**The exact 529 mask selects precisely N01–N04.** All other 26 pairings keep their exact 17K diagonals but have off-diagonal residuals.\n\n\\[\nS_{d,j}=\\begin{pmatrix}17K&19K+\\delta\\\\15K-\\delta&17K\\end{pmatrix},\\qquad \\delta=d+1-j.\n\\]\n\nIn 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.\n\n| Row | d | j | U | D | M | X |\n|---|---|---|---|---|---|---|\n| N01 | 1 | 2 | 6346 | 5288 | 2647 | 3705 |\n| N02 | 2 | 3 | 6345 | 5287 | 2648 | 3706 |\n| N03 | 3 | 4 | 6344 | 5286 | 2649 | 3707 |\n| N04 | 4 | 5 | 6343 | 5285 | 2650 | 3708 |\n\nThus 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.\n\nThe four exact results are inherited comparisons recovered by a unified formula. Their reappearance is a consistency check, not four new discoveries or independent evidence.\n\n## Source and scope control\n\nAll 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.\n\nRegular 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.\n\nEvidence: `CERTIFICATE.json`; **51 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.\n"}