{"id": "c439", "title": "Source Complement Rule for Noah Birth–Death Reflections", "filename": "490d_C439_Noah_Birth_Death_Reflection_Complement_Rule_20260925.md", "role": "Research action", "step": 439, "sha256": "da9431e882e012bf682540b064a696510517504f8e4b13881605bb39fd9a13e3", "bytes": 3844, "origins": ["C01–C479/packets/C439/490d_C439_Noah_Birth_Death_Reflection_Complement_Rule_20260925.md"], "download": "/sources/da9431e882e012bf682540b064a696510517504f8e4b13881605bb39fd9a13e3.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Calendar Keys", "Mirror / 529"], "excerpt": "490d C439 — Source Complement Rule for Noah Birth–Death Reflections\n\n25 September 2026 UTC. Step 5 of 10, C435–C444.\n\nThe entire Noah birth/death reflection class reduces to one complementary source rule. All36,864 role-pair comparisons at the four fixed centers agree with it; no hit falls outside the rule.\n\nLet the ce", "content": "# 490d C439 — Source Complement Rule for Noah Birth–Death Reflections\n\n25 September 2026 UTC. Step 5 of 10, C435–C444.\n\nThe entire Noah birth/death reflection class reduces to **one complementary source rule**. All36,864 role-pair comparisons at the four fixed centers agree with it; no hit falls outside the rule.\n\nLet the center be c=3188+2(g−1)+60T, where g is1 or2 and T is0 or1. For a Noah birth and Noah death to reflect about c:\n\n- Both belong to SP or the explicitly indexed LXX comparison field.\n- Their Cainan flags sum to1, and their sojourn flags sum to1.\n- Both retain the center's Terah setting T.\n- Their Gear numbers sum to2g.\n- Exactly one endpoint is the local SP companion.\n\nThe SP-normalized source base sum is3491+2541=6032. The complementary variants contribute130+215−1=344, producing6376=2×3188. Gear and common Terah shifts then supply the other three centers.\n\nExplicitly, the reflection condition is\n\n`130(Cb+Cd)+60(Tb+Td)+215(Fb+Fd)+2(Gb+Gd−2)−(εb+εd)=344+4(g−1)+120T`.\n\nThe finite flag ranges yield only the stated sum solution. The equation is not inferred merely from selected positive rows; all1,620 abstract bounded context-sum candidates were checked as well.\n\n| Center | Allowed endpoint Gear pairs | Coordinate pairs | Labelled pairs |\n|---|---|---:|---:|\n|3188|G1/G1|8|16|\n|3190|G1/G3, G2/G2, G3/G1|24|48|\n|3248|G1/G1|8|16|\n|3250|G1/G3, G2/G2, G3/G1|24|48|\n\nExactly one endpoint must be SP companion. Its primary partner may carry SP or the appropriate LXX label; LXX does not receive a companion rail. Native LXX Cainan remains ON, while OFF cases retain their theoretical status.\n\nMT cannot participate in this role class at these centers: any pair containing an MT Noah endpoint has maximum coordinate sum6200, below the smallest required6376. The MT Noah record serves as the center, not one of these reflected endpoints.\n\nThese64 coordinate-center pairs and128 labelled pairs account for the largest part of C438's inventory. They exchange complementary source contexts and do not constitute128 single-biography lifespan comparisons. Matched biographies continue to preserve950.\n\n## Verification and scope\n\n11 calculation checks and 18 source/predecessor authentication checks passed. Main result files replay byte for byte. These are root-agent checks, not an independent-agent review or independent historical witnesses.\n\nFinite source-context complement rule. No companion is copied to MT/LXX, no new biography is assembled, and no global reflection operation is inferred.\n\nMirror/529 focus: K=529, E=25/23, P=70/69; J=300/299 remains File02 gap conversion, not a calendar date operator or Residue Protocol. Regular Gears are Noah/Shem/Flood only. SP person companions are local to Noah/Shem; Flood close is an independent event boundary. Both500/100 and502/98 bindings remain valid; SP primary Noah death retains its inclusive/harmonic qualification. LXX native Cainan is ON; OFF is theoretical removal. Global paired Mirror January AD1; local junction January AD2; no civil year zero. No new source point, phase, fitted axis, upstream/cumulative Gear, primer reopening or canonical/graph edit. Capture now; canonize later. Coordinates, source labels, context aliases and derived comparisons remain distinct. Counts are not independent historical witnesses.\n\n## Best next bounded step\n\n**C440, proposed and unexecuted:** Test whether the Noah-birth/Shem-death reflection class and the Shem-birth/Flood-boundary class are exactly related by opposed500-year source shifts. Preserve primary/companion and Flood-start/close distinctions, all aliases, and any change in pair ordering.\n\nThe dominant class now has one source rule. The next two role classes appear related by existing500-year source correspondences; testing that relation could reduce the remaining reflection grammar.\n", "evidenceLinks": [{"id": "d-24203a4c3b282f9b", "title": "File 02"}]}