{"id": "c115", "title": "C115 — Isaac and Peleg attach the local 8×529 construction to the 20×529 Mirror", "filename": "490d_C115_Isaac_Peleg_529_Mirror_Attachment_20260922.md", "role": "Research action", "step": 115, "sha256": "e2c1ea35904c52c1803cc33d5e4c0aaef7847c9bd3df824e5d2ddd414eeca008", "bytes": 9047, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_C115_Isaac_Peleg_529_20260922/490d_C115_Isaac_Peleg_529_Mirror_Attachment_20260922.md"], "download": "/sources/e2c1ea35904c52c1803cc33d5e4c0aaef7847c9bd3df824e5d2ddd414eeca008.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Rounded", "Mirror / 529"], "excerpt": "C115 — Isaac and Peleg attach the local 8×529 construction to the 20×529 Mirror\n\n22 September 2026 · One bounded continuation after C114.\n\nIsaac supplies the named 360+180+720 division, Peleg supplies a sourced ±2 bracket about 2116, and the local 8×529 interval has an exact endpoint attachment to the retained 20×529 o", "content": "# C115 — Isaac and Peleg attach the local 8×529 construction to the 20×529 Mirror\n\n22 September 2026 · One bounded continuation after C114.\n\n**Isaac supplies the named 360+180+720 division, Peleg supplies a sourced ±2 bracket about 2116, and the local 8×529 interval has an exact endpoint attachment to the retained 20×529 opening span.** The controlling unit remains K=529. Isaac’s 180-year life measures the subdivision of the derived 1260; it does not replace K.\n\nThis step follows Dean’s supplied Moses–Isaac observation and his instruction to give Peleg priority, with Nahor as a secondary comparison. It also completes C114’s proposed bounded comparison of local 4/8-fold widths with the established 10/20-fold opening structure.\n\n## 1. Source settings and the Moses–Isaac sequence\n\nFile_18 §§2.2–2.3 supplies the MT minimum rows. The retained +215/full-430 selection gives:\n\n| Person | Selected birth–death, BC | Lifespan | Source calculation |\n| --- | --- | ---: | --- |\n| Isaac | 2066–1886 | 180 | 1851–1671, each +215 |\n| Peleg | 2357–2118 | 239 | 2142–1903, each +215 |\n| Nahor, Terah’s father | 2265–2117 | 148 | 2050–1902, each +215 |\n\nAbram’s birth is 2166=1951+215. Moses’ birth 1526 and Exodus 1446 remain fixed by §2.4. These Peleg/Nahor records use the ordinary MT profile; no +60 Terah displacement or regular Gear is applied to them.\n\nFor W=8K=4232, retain the complementary pair-base rule C_W(b)=W−b. Its exact operation is a forward translation by W followed by reflection about the existing January AD 1 axis, with component order reversed. It generates the requested BC sequence:\n\n| BC boundary pair | Derivation or source | Forward interval |\n| --- | --- | ---: |\n| 2707t/2706n | Complement of Moses’ birth: 4232−1526 | 360 |\n| 2347t/2346n | Complement of Isaac’s death: 4232−1886 | 180 |\n| 2167t/2166n | Complement of Isaac’s birth: 4232−2066; also Abram’s sourced birth pair | 720 |\n| 1447t/1446n | Fixed Exodus pair | — |\n\nEvery interval holds on both seasonal components:\n\n\\[\n\\boxed{360+180+720=(2+1+4)\\times180=1260\n=8\\times529-(1526+1446).}\n\\]\n\nThis verifies the proposed **time + half a time + times** ordering within the 529 construction. The BC complement places Isaac’s derived death before his derived birth. Its birth landing exactly reuses the Abram source coordinate admitted in C113. Under the ordinary global reflection, the AD sequence reads forward as Exodus→Abram/derived Isaac birth→derived Isaac death→derived Moses birth, with intervals **720+180+360**.\n\nThe dependency is compact:\n\n| Segment | Source expression | What supplies it |\n| --- | --- | --- |\n| 360 | 1886−1526 = 2×180 | Isaac’s death to Moses’ birth; W cancels |\n| 180 | 2066−1886 | Isaac’s lifespan; W cancels |\n| 720 | W−2066−1446 = 2166−1446 = 4×180 | The eightfold Abram/Isaac complement and the fixed Exodus anchor |\n\nThe advance is therefore a named biographical partition of the already verified residual. The translated and reflected copies are dependent representations of those same source relations.\n\n## 2. Peleg gives the primary near-center source comparison\n\nThe local complementary center is C=4K=2116. Peleg’s sourced death lies two years before it in BC chronological order. His complementary boundary lies two years after it:\n\n\\[\n2118=C+2,\\qquad C_W(2118)=2114=C-2.\n\\]\n\nNahor supplies the adjacent one-year relation. The exact AD comparison is:\n\n| Death source, BC | Ordinary death mirror | Source advanced by W | Offsets around AD 2116t/2117n |\n| --- | --- | --- | --- |\n| Peleg: 2119t/2118n | AD 2118t/2119n | AD 2114t/2115n | +2 / −2 |\n| Nahor: 2118t/2117n | AD 2117t/2118n | AD 2115t/2116n | +1 / −1 |\n\nThus Peleg’s ordinary and translated death pairs are **four years apart**, bracketing the generated center at ±2. Nahor’s are two years apart, bracketing it at ±1. The BC complements are respectively 2115t/2114n and 2116t/2115n. The AD translations and BC complements retain their different directions and labels.\n\nPeleg also supplies a named division of the Abram-to-Isaac birth interval:\n\n\\[\n\\boxed{2166\\xrightarrow{48}2118\\xrightarrow{52}2066\\ \\mathrm{BC}.}\n\\]\n\nIncluding his generated complementary boundary exposes its symmetry:\n\n\\[\n2166\\xrightarrow{48}2118\\xrightarrow{4}2114\\xrightarrow{48}2066.\n\\]\n\nNahor gives **49+51**, or **49+2+49** when his complement is included. These are componentwise pair relations. They do not shift either source death. The center itself remains 50 years after Abram’s birth and 50 years before Isaac’s birth.\n\nThis gives Peleg interpretive priority within Dean’s Arphaxad–Abram division/nations reading: his actual death boundary sits next to the center of the sourced Abram/Isaac pairing. The endpoint remains labelled as his death. The ±2 geometry follows from 2118−2116; the Noah/Shem Gear operation is not applied here.\n\n## 3. The local construction attaches to the retained 10/20-fold opening\n\nThe existing C112 graph preserves the LXX cumulative Cainan-OFF opening pair **5291t/5290n BC** and its ordinary reflection **AD 5290t/5291n**, joined by `opening_self20`. File_18 §6D reproduces its radius from the actual native opening: **5750−460=5290=10K**. Its self-Mirror width is **10580=20K**.\n\nThe generated local fulcrum has radius 2116=4K. The radius difference is therefore 3174=6K, giving the symmetric containment:\n\n\\[\n20K=6K+8K+6K.\n\\]\n\nThe named Abram and Isaac endpoints supply the following source attachment:\n\n| Source path, in chronological order | Exact width |\n| --- | ---: |\n| LXX OFF opening 5291t/5290n BC → Abram birth 2167t/2166n BC | 3124 = 6K−50 |\n| Abram birth 2167t/2166n BC → Isaac birth mirror AD 2066t/2067n | 4232 = 8K |\n| Isaac birth mirror AD 2066t/2067n → LXX OFF opening mirror AD 5290t/5291n | 3224 = 6K+50 |\n| **Complete opening-to-reflection path** | **10580 = 20K** |\n\nBoth components verify, including the middle civil-span equations `2167+2066−1=4232` and `2166+2067−1=4232`. Reversing the Abram/Isaac selection exchanges the two flanks: **3224+4232+3124**.\n\nThis path attaches to the existing opening node identities. Its endpoint types remain cumulative LXX OFF at the outside and regular MT at the inside. The 5290 radius is distinguished from the separately registered tenfold elapsed intervals in earlier campaigns.\n\n## 4. Peleg and Nahor inside the same outer frame\n\nPeleg’s source death pair and its own ordinary reflection delimit **4236=8K+4**. Each outer flank is **5290−2118=3172=6K−2**. Nahor supplies the neighboring case:\n\n| Inner pair construction | Left flank | Inner width | Right flank | Total |\n| --- | ---: | ---: | ---: | ---: |\n| Generated 2116 fulcrum and reflection | 3174 = 6K | 4232 = 8K | 3174 = 6K | 10580 |\n| Peleg death and reflection | 3172 = 6K−2 | 4236 = 8K+4 | 3172 = 6K−2 | 10580 |\n| Nahor death and reflection | 3173 = 6K−1 | 4234 = 8K+2 | 3173 = 6K−1 | 10580 |\n\nThe Peleg route has actual sourced endpoints at both ends of each same-side flank: cumulative LXX opening to regular MT death, and their reflections. Peleg’s inner self-Mirror width is four years larger than 8K; his source-to-complement translation remains exactly 8K. Those two comparisons explain the ±2 bracket without identifying the 2116 generated center as Peleg’s death.\n\nAll three outer partitions are dependent on the retained opening and their selected inner coordinates. They attach the new local source comparisons to the known 20K interval; they do not create three independent calibrations of K.\n\n## 5. Checkpoint and next bounded question\n\n**52 exact source, phase and attachment checks passed.** A single isolated replay reproduced the certificate, results and report body byte for byte. Frozen inputs and archived file hashes were verified. C114’s complete 3.5-year witness construction remains admitted; C115 adds the Isaac subdivision and Peleg/Nahor source attachments.\n\nThe C112 integration graph is retained at **104 nodes, 193 durations, 35 reflections, rank 101**. The new finite paths are recorded in the C115 certificate against its existing opening node identities; they have not yet been merged into that graph. Regular Gears remain confined to **Noah/Shem/Flood**. No completed campaign, quarantined primer, Rounded analysis or localized −33 branch was reopened.\n\nSources: Dean’s current supplied sequence and Peleg instruction; updated branch handoff §§3–4 and 8; File_18 §§2.2–2.4 and 6D; File_09 Appendix A.3; the original Mirror handoff §7; C114 certificate/checkpoint; and the frozen C112 graph/certificate. These are calculations within the selected project chronology. Dean’s integrated chiastic/fractal interpretive framework is retained.\n\n**Best next bounded step — High sufficient:** examine the other end of Peleg’s biography, **2357+1875=4232**, against the fixed Entry and famine boundaries. Determine the source status of the proposed 1875 Joseph boundary while preserving the 8×529 relation. The present step stops at C115.\n", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-45851004c5864229", "title": "File 09"}]}