{"id": "c70", "title": "C70 — Exact family closure and reflection do not determine half-year phase", "filename": "490d_C70_Mirror_Report_20260922.md", "role": "Research action", "step": 70, "sha256": "3c1a03fed03207fa65470b7b1fda66b5ba25413ed6a996cce8026f82c6278106", "bytes": 3439, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_10_step_phase_lift_20260922/C70/490d_C70_Mirror_Report_20260922.md"], "download": "/sources/3c1a03fed03207fa65470b7b1fda66b5ba25413ed6a996cce8026f82c6278106.md", "format": "md", "derived": false, "tags": ["Regular", "Rounded", "Calendar Keys", "Mirror / 529"], "excerpt": "C70 — Exact family closure and reflection do not determine half-year phase\n\n22 September 2026 · Step 8 of the authorized ten-step Mirror continuation (C63–C72).\n\nA coherent countermodel makes the phase requirement explicit. Replace every pair P(x)=(x−¼,x+¼) by Pₕ(x)=(x−h,x+h), using the same h throughout a branch. For ", "content": "# C70 — Exact family closure and reflection do not determine half-year phase\n\n22 September 2026 · Step 8 of the authorized ten-step Mirror continuation (C63–C72).\n\nA coherent countermodel makes the phase requirement explicit. Replace every pair P(x)=(x−¼,x+¼) by Pₕ(x)=(x−h,x+h), using the same h throughout a branch. For both h=¼ and h=¾:\n\n- every 6877/299/299/650 same-component interval is unchanged;\n- the source-qualified midpoint's coordinate remains the numerical midpoint;\n- reflection still satisfies R(Pₕ(x))=Pₕ(−x);\n- the correspondingly defined component-wise affine expansions reach the same pair centres.\n\n| Branch | Half-width h | Internal pair width | Geometric identities | File_09 phase admitted |\n| --- | --- | --- | --- | --- |\n| N02 | 1/4 | 1/2 | pass | yes |\n| N02 | 3/4 | 3/2 | pass | no |\n| N04 | 1/4 | 1/2 | pass | yes |\n| N04 | 3/4 | 3/2 | pass | no |\n\nThe h=¾ model has 1.5-year pairs. It fails File_09's seasonal phase convention even though all listed geometric identities survive. Assigning the old civil suffix labels to these altered component times would be incorrect. This is a formal countermodel, not an alternative chronology or a proposed calendar.\n\nThe reason is structural: equal phase offsets cancel in same-component differences, while symmetric offsets survive reflection. Width equations and reflection determine neither the internal width of every pair nor the calendar meaning of its labels. The half-year phase comes from the controlling seasonal model. Its agreement with the 529 geometry is a compatibility result, not another independent numerical determination of 529.\n\nC61's one-sided suffix error could be detected by unequal component spans. This stronger control changes every pair coherently, so those span checks alone no longer detect it. A complete certificate must include an explicit source phase condition.\n\n**Validation.** 17 exact arithmetic/structural checks passed. The 14 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.\n\n**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.\n\n**Source locators.** File_09 §3 and Appendix A (paired Mirror and exact phase); File_18 §§2.1.3, 2.4, 3.1 and 3.2 (regular source bindings and seasonal limits); frozen C15/C16/C22/C43/C50/C61/C62 certificates (retained graph and phase); C01/C04/C05 reports (completed expansions and their source limits). The declared phase model does not infer an unstated birthday or event month from an annual source row.\n\n**Best next bounded step — High sufficient.** Derive the smallest sufficient phase certificate for the retained five-node chain, keeping source membership as a separate requirement.\n", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-45851004c5864229", "title": "File 09"}]}