{"id": "c112", "title": "C112 — The named Mirror grid now has a graph and a compact conditional reconstruction", "filename": "490d_C112_Mirror_Report_20260922.md", "role": "Research action", "step": 112, "sha256": "ece9ec1653a24f60080f4ae62aaf753a7bad3c4fbd899c77fc01ff38b7fd3ed1", "bytes": 6606, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_10_step_graph_confluence_20260922/C112/490d_C112_Mirror_Report_20260922.md"], "download": "/sources/ece9ec1653a24f60080f4ae62aaf753a7bad3c4fbd899c77fc01ff38b7fd3ed1.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Rounded", "Mirror / 529"], "excerpt": "C112 — The named Mirror grid now has a graph and a compact conditional reconstruction\n\n22 September 2026 · Step 10 of the authorized ten-step Mirror continuation (C103–C112).\n\nC103–C112 are complete. The central advance is a reduction of the named 11/12/13 construction to a small set of distinct premises. Two shared so", "content": "# C112 — The named Mirror grid now has a graph and a compact conditional reconstruction\n\n22 September 2026 · Step 10 of the authorized ten-step Mirror continuation (C103–C112).\n\n**C103–C112 are complete.** The central advance is a reduction of the named 11/12/13 construction to a small set of distinct premises. Two shared source junctions force the carrier ratio; the same-Gear Noah/Flood interval fixes its scale; one source coordinate fixes its origin. The source names remain separately controlled.\n\n| Step | Focus | Completed finding |\n| --- | --- | --- |\n| 103 | Source records | Four new records; seven width edges; the opening G1 Flood joins the new row. |\n| 104 | Relation rank | Rank 77→82 for four new variables: one existing freedom is removed. |\n| 105 | Joseph source bindings | Kohath 1794/1796/1798 plus 110 agrees with Joseph 1904/1906/1908. |\n| 106 | Mirror completion | Sixteen counterparts and 18 reflected durations; copied durations add no rank. |\n| 107 | Source Gear links | Both junction paths give 25 for N02 and 27 for N04. |\n| 108 | Ratio reconstruction | Shared 11/12/13 junctions imply kr(23r−25)=0. |\n| 109 | Scale reconstruction | The same-Gear 600-year source interval fixes k=529. |\n| 110 | N04 residuals | Equal source residuals 2 preserve r; d−e=25 on both branches. |\n| 111 | Assumption diagnostic | Two junctions fix shape; 600 fixes scale; 2456 fixes origin; identity stays sourced. |\n| 112 | Completed checkpoint | The extended graph and all ten reports are ready for the next user-supplied cycle. |\n\n## The structural result\n\nWrite k>0 for the initially unknown unit, r>1 for the common ratio, and d=b−α for the ordinary/apparent Creation difference. The G1 Flood and Noah junctions impose\n\n\\[\nd+k(11r^2-12r)=0,\\qquad d+k(13r-12r^2)=0.\n\\]\n\nTheir difference is **kr(23r−25)=0**, so r=25/23 and d/k=25/529. Here 23=11+12 and 25=12+13. This is conditional on the given family labels, common-ratio form and actual junction identities.\n\nThe sourced same-Gear Noah/Flood span then gives\n\n\\[\n600=12kr(r-1)=\\frac{600}{529}k,\n\\qquad k=529.\n\\]\n\nWith G1 Flood 2456, α=−4444 and b=−4419. The reconstructed apparent row is Joseph 1904 → Flood 2456 → Noah 3056, giving 6348→6900→7500. Joseph's identity is supplied by the original Kohath closing member 1794 plus 110. Numeric reconstruction alone does not identify a source event.\n\nN04 retains its different ordinary anchor −4417. Both ordinary junctions are 2 later than the apparent G1 records. Equal residuals preserve the ratio and give d−e=27−2=25. This comparison does not merge G1/G2 events or apply a Gear to Creation, Conquest or Joseph. The paired Mirror reverses the oriented differences while preserving the same relation.\n\n## The integrated graph\n\nThe graph is explicitly an extension of frozen C43. Its new records and edges include the original Joseph/Kohath members, the apparent row, exact reuse of existing ordinary source records, their paired counterparts, and the two permitted regular Gear comparisons.\n\n| State | Nodes | Durations | Reflections | Fixed-width rank |\n| --- | --- | --- | --- | --- |\n| C43 baseline | 81 | 161 | 19 | 77 |\n| Completed C112 extension | 104 | 193 | 35 | 101 |\n\nThe first seven new width relations add five ranks for four new variables: they connect the earlier opening Flood construction to the selected N02 network. Reflected duration copies and explicit Gear comparison copies add no duplicate numerical rank once their parent relations are present.\n\nThe final three graph freedoms belong only to the isolated C5/C6/C7 Creation–Conquest pairs. Their known source coordinates can pin them consistently; those pins are reported separately from the stored relation graph. The reduced four-parameter junction model, the fixed-width graph rank and independent historical evidence are distinct notions.\n\n## Checkpoint and next input\n\nAll ten reports have individual frozen-input evidence and isolated replay checks. The complete packet includes the final typed graph in this certificate, the unchanged controlling sources, selected prior dependencies, the report index and a restart checkpoint. No completed historical campaign was executed. The original graph and canonical files remain unchanged.\n\nRegular Gears remain restricted to Noah/Shem/Flood. Joseph remains original cumulative collateral; apparent Creation uses the existing fixed +30 field. Both primers, Rounded, localized −33 and H7/T14 remain inactive. No upstream Gear transport, civil year zero or unstated historical event season has been introduced. Existing family labels, full 6877 branches N02/N04 and joined 24 on N01 are preserved.\n\nThe authorized batch ends at C112. The **529 × 4 × 2** after-Flood-to-Moses Mirror cycle is reserved for the user's introduction. No endpoint selection or interpretation for that cycle has been attempted.\n\n**Validation.** 12 exact arithmetic/structural checks passed. The 21 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); File_18 §§2.1.3, 3.1 and 6A (regular fields and cumulative source members); File_61 §11.2 (original backward Joseph +110); File_62 collateral-scope guards; C05 admitted apparent field; C43 frozen graph; C95 and C100–C102 named carrier rows; preceding current certificates. The forthcoming 529 × 4 × 2 cycle has not been explored. 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.** Receive the user’s 529 × 4 × 2 cycle and its source endpoints, then compare its stated after-Flood-to-Moses path with the completed typed Mirror graph. Await that introduction before C113; High is sufficient initially.\n", "evidenceLinks": [{"id": "d-be42daca1d82dac9", "title": "File 62"}, {"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-d580afd48d6e12d6", "title": "File 61"}, {"id": "d-45851004c5864229", "title": "File 09"}]}