{"id": "c140", "title": "C140 — P6 has a bounded residual family δ=1…6", "filename": "490d_C140_529_Mirror_Bounded_Step_20260923.md", "role": "Research action", "step": 140, "sha256": "14c8e121b3b1fd6a78e3f061eee7a6cc9910f5be1aad1962301894cfac024796", "bytes": 2237, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_C136_C155_529_Square_Batch_20260923/C140/490d_C140_529_Mirror_Bounded_Step_20260923.md"], "download": "/sources/14c8e121b3b1fd6a78e3f061eee7a6cc9910f5be1aad1962301894cfac024796.md", "format": "md", "derived": false, "tags": ["Regular", "Mirror / 529"], "excerpt": "C140 — P6 has a bounded residual family δ=1…6\n\n23 September 2026 · Bounded step 5 of 20 (C136–C155).\n\nP6's six squares obey one finite residual formula, with δ=1,…,6. The exact 529 grid is missed by at least one year at this boundary.\n\n\\\nS_\\delta=\\beginpmatrix17K&19K+\\delta\\\\15K-\\delta&17K\\endpmatrix,\\qquad\nU-D=X-M=2K+", "content": "# C140 — P6 has a bounded residual family δ=1…6\n\n23 September 2026 · Bounded step 5 of 20 (C136–C155).\n\n**P6's six squares obey one finite residual formula, with δ=1,…,6.** The exact 529 grid is missed by at least one year at this boundary.\n\n\\[\nS_\\delta=\\begin{pmatrix}17K&19K+\\delta\\\\15K-\\delta&17K\\end{pmatrix},\\qquad\nU-D=X-M=2K+\\delta.\n\\]\n\n| δ | Side | U+X | D+M |\n|---|---|---|---|\n| 6 | 1064 | 10057 | 7929 |\n| 5 | 1063 | 10056 | 7930 |\n| 4 | 1062 | 10055 | 7931 |\n| 3 | 1061 | 10054 | 7932 |\n| 2 | 1060 | 10053 | 7933 |\n| 1 | 1059 | 10052 | 7934 |\n\nThe diagonal sum and the off-diagonal sum are both 34K=17986. This conservation follows from recombining the same four endpoints; it is not an additional independent numerical witness. The residual shifts from one off-diagonal span to the other.\n\nThe P6 upper source boundary excludes δ=0: that would require M=2651 and U=6342. The observed minimum δ=1 uses the admitted G3 start. C136–C140 establish a wider finite family of equal-diagonal comparisons while preserving the original four exact C10 squares. The next five steps test whether the same residual controls the associated Creation-to-Flood and Noah partitions.\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`; **31 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.\n"}