{"id": "c117", "title": "490d C117 — Local 529 Mirror graph integration", "filename": "490d_C117_Local_529_Mirror_Graph_Integration_20260922.md", "role": "Research action", "step": 117, "sha256": "aab8d2a630931c2a970ab5681f4b5248ee303f0954e9417678f0ead202bbf7dd", "bytes": 7989, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_C117_local_529_graph_integration_20260922/490d_C117_Local_529_Mirror_Graph_Integration_20260922.md"], "download": "/sources/aab8d2a630931c2a970ab5681f4b5248ee303f0954e9417678f0ead202bbf7dd.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Calendar Keys", "Mirror / 529"], "excerpt": "490d C117 — Local 529 Mirror graph integration\n\nDate: 2026-09-22. One bounded step following C116. Completed.\n\nThe C113–C116 local 4/8×529 construction integrates consistently with the retained C112 10/20×529 graph. Relative to the chosen source-chronology relations, one local placement condition is sufficient. The rem", "content": "# 490d C117 — Local 529 Mirror graph integration\n\nDate: 2026-09-22. One bounded step following C116. **Completed.**\n\n**The C113–C116 local 4/8×529 construction integrates consistently with the retained C112 10/20×529 graph.** Relative to the chosen source-chronology relations, one local placement condition is sufficient. The remaining named paths and phase relations follow without a second placement condition. The Joseph land-purchase identification remains proposed at **1875 or perhaps 1874 BC**, following Dean’s latest clarification.\n\nThis is a new working graph, not an overwrite of the canonical graph or historical reports. The unit remains **K=529**, local radius **4K=2116**, local width **8K=4232**, outer radius **10K=5290**, outer width **20K=10580**. Regular Gears remain restricted to **Noah/Shem/Flood**; this step applies no new Gear.\n\n## Narrative clarification and the two candidate years\n\nGenesis 47:17–18 explicitly advances from the livestock-exchange year to the following year. That relative marker supports the sequence; assigning either BC year also requires its placement against the retained Entry chronology. Entry remains **1876 BC**.\n\n| Proposed transaction-year boundary | Sum with Peleg’s birth, 2357 BC | Relation to 8×529 | Status |\n|---|---:|---:|---|\n| 1875 BC | 4232 | exact | Generated complement; event identification proposed |\n| 1874 BC | 4231 | one year short | Alternative event year retained |\n\nFor 1875, the componentwise interval is **1876t/1875n BC → AD 2357t/2358n = 4232**, ending at Peleg’s ordinary reflected birth pair. For 1874, **1875t/1874n BC → AD 2357t/2358n = 4231**. The calendar-pair convention is unchanged; neither result comes from a civil year-zero shortcut.\n\nThe famine’s central year remains **1875n → 1874n BC**. Its opening and closing boundaries can frame the proposed transaction year without specifying the transaction’s season. In the retained 1878n–1871n famine, the exact midpoint is **1875t BC**, half a year after 1875n BC. Thus the central year, its opening boundary, and the exact midpoint remain distinct.\n\nGenesis 41:56–57 supplies the universal famine-and-gathering language. Genesis 47:13–15 extends the famine and market to Egypt and Canaan; 47:20 specifies Egyptian land purchase, with the priestly exception in 47:22. Israel’s separate provision and possession appear in 47:12,27. Dean’s interpretation connects this world-embracing narrative with **Peleg and the nations**, within his divine-authorship and integrated chiastic/fractal reading. The graph captures that interpretation without making it an extra numerical equation or assigning Canaan’s land to the Egyptian transaction.\n\nPrimary text reviewed: [Genesis 41:56–57, YLT](https://www.biblegateway.com/passage/?search=Genesis+41:56-57&version=YLT) and [Genesis 47, YLT](https://www.biblegateway.com/passage/?search=Genesis+47&version=YLT), accessed 2026-09-22. The text supplies relative sequence, not an explicit BC date.\n\n## What fixes the local placement\n\nSixteen local chronological records and their ordinary reflections are first linked by fifteen source-chronology intervals. They include the qualified Terah-firstborn reconstruction and the derived Moses age-40 boundary, with those statuses retained. These relative intervals leave one local position undetermined.\n\nThe held relation **Abram BC → reflected Isaac birth = 4232** fixes it. With their 100-year birth separation:\n\n\\[\nA+I=8K=4232,\\qquad A-I=100,\n\\]\n\\[\nA=4K+50=2166,\\qquad I=4K-50=2066.\n\\]\n\nThe retained chronology then gives:\n\n\\[\nP_{death}=A-48=4K+2=2118,\n\\]\n\\[\nP_{birth}=P_{death}+239=4K+241=2357,\n\\]\n\\[\nJ=8K-P_{birth}=4K-241=1875.\n\\]\n\nThis shows precisely how Peleg’s 239-year life and his death’s two-year offset from the fulcrum generate the 1875 complement. It does not independently date the transaction. Assigning the proposed event to 1874 changes the event annotation and its interval to Peleg’s reflection; it does not change K or move the 1875 generated complement.\n\nThe outer opening is the existing **cumulative LXX derived Cainan-OFF Flood comparison pair**, 5291t/5290n BC and its ordinary reflection. No new regular Gear is applied to these local events.\n\n| Path within the retained 20K opening | Left flank | Inner interval | Right flank |\n|---|---:|---:|---:|\n| Generated 4K fulcrum and reflection | 3174 | 4232 | 3174 |\n| Abram birth → reflected Isaac birth | 3124 | 4232 | 3224 |\n| Isaac birth → reflected Abram birth | 3224 | 4232 | 3124 |\n| Peleg death → its reflection | 3172 | 4236 | 3172 |\n| Nahor death → its reflection | 3173 | 4234 | 3173 |\n\nEach row totals **10580=20K**. Alternatively, fixing the Abram left flank at 3124 fixes the same local placement; subsequently adding the 4232 Abram–Isaac relation increases rank by zero. These are equivalent placement conditions within this model, not separate independent confirmations.\n\n## Isaac and the 3.5-year components remain intact\n\nThe generated BC chain remains **2707t/2706n → 2347t/2346n → 2167t/2166n → 1447t/1446n**, with componentwise intervals **360+180+720=1260=(2+1+4)×180**. Its 1260 total remains the derivative **8K−(1526+1446)**.\n\nGenerated Isaac birth and sourced Abram birth occupy the same coordinate while remaining separate records. The analogous Abram/Isaac and Terah-firstborn/Jacob coincidences are explicit equalities, not merged event identities.\n\nThe adopted C114 template retains:\n\n| Track | Opening | Middle | End |\n|---|---|---|---|\n| Aaron-derived selected components | AD 183t | AD 1443t | AD 2703t |\n| Moses-derived selected components | AD 187n | AD 1447n | AD 2707n |\n\nEach horizontal segment is 1260. Each selected vertical separation is **3.5**, although the underlying pair-center separation is **3**. The crossed intervals remain **1263.5** and **1256.5**. Generated template roles retain their C114 conditional status; no lifespan or death boundary is relabelled.\n\n## Verification and recoverable checkpoint\n\n| Stage | Pair-record variables | Equation rank | Remaining freedoms |\n|---|---:|---:|---:|\n| Frozen C112 baseline | 104 | 101 | 3 |\n| Local chronology and ordinary reflections | 136 | 132 | 4 |\n| One local 8K placement condition | 136 | 133 | 3 |\n| Translations and their reflections | 168 | 165 | 3 |\n| Generated fulcrum | 170 | 167 | 3 |\n| Alternative 1874 boundary | 172 | 169 | 3 |\n| Adopted C114 template boundaries | 178 | 175 | 3 |\n| Remaining named paths, equalities and phase equations | 178 | 175 | 3 |\n\nThe working graph contains **178 nodes, 310 duration edges, 72 reflections, 8 role-preserving coordinate equalities, and 27 component equations referencing 20 component records**. Component records are expressions in existing pair variables, not additional free variables. The three remaining freedoms are inherited C5/C6/C7 placement freedoms; the existing diagnostic source pins resolve them, but are excluded from reported rank.\n\n**658 exact checks passed**, including recombination of **115 dependency certificates**. Eleven inputs were frozen with hashes. One isolated replay reproduced the four generated JSON outputs byte-for-byte. Baseline collections remain unchanged prefixes of the new graph, and source/canonical files remain unchanged. Prior campaigns were not rerun. Rank measures this encoded system’s constraints; it does not measure independent historical evidence or theological probability.\n\nThe evidence ZIP includes the full working graph, checker, input sources, certificates, dependency combinations, replay verification and manifests. Existing historical recovery status remains inherited; this step makes no new claim about interrupted Reports 11–25.\n\n**Best next bounded step:** isolate the minimum source relations behind **4K±241** and the **1875/1874 famine-year boundary**, distinguishing which Peleg–Joseph correspondences are forced by the 529 construction and which require the narrative assignment. **High sufficient.**\n"}