{"id": "c407", "title": "500 and 502 source bindings and the finite Gear boundary", "filename": "490d_C407_500_and_502_Source_Bindings_and_Finite_Gear_Boundary_20260924.md", "role": "Research action", "step": 407, "sha256": "ba88c308ab17e763da3c36ffbe4c85946743d909e100ffd7b9a1a1b2d9c345df", "bytes": 3306, "origins": ["C01–C479/packets/C407/490d_C407_500_and_502_Source_Bindings_and_Finite_Gear_Boundary_20260924.md"], "download": "/sources/ba88c308ab17e763da3c36ffbe4c85946743d909e100ffd7b9a1a1b2d9c345df.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Mirror / 529"], "excerpt": "490d C407 — 500 and 502 source bindings and the finite Gear boundary\n\n24 September 2026. Step 4 of 10 in the C404–C413 authorization.\n\nBoth retained Noah-to-Shem bindings work on their available source domains. The502/98 correspondence is shorter because it lowers the target Gear by one.\n\n|Binding|Labelled loops admitt", "content": "# 490d C407 — 500 and 502 source bindings and the finite Gear boundary\n\n24 September 2026. Step 4 of 10 in the C404–C413 authorization.\n\n**Both retained Noah-to-Shem bindings work on their available source domains. The502/98 correspondence is shorter because it lowers the target Gear by one.**\n\n|Binding|Labelled loops admitted|Coordinate loops|Complete E/P/J graphs|Boundary|\n|---|---:|---:|---:|---|\n|500/100|12|6|2|Both originalG1–G2 andG2–G3 segments|\n|502/98|6|3|1|OriginalG2–G3 becomes targetG1–G2|\n\nAll24 binding/loop cases check every required endpoint and pivot against its role, tradition, chronology context, Gear and person rail. In the six unavailable502 cases, the lower and held Shem births would require Gear0. No such point is adopted.\n\nThe surviving complete502 graph is2531↔2533 with E pivot2556, J death1933 and P death2393. It is the same source configuration already reached by the500 correspondence from the preceding Noah segment. Thus502 contributes a valid alternative coupling, not a new coordinate witness or additional loop.\n\nSP companion loops keep their person rail and use the independent Flood-close source. Neither MT nor LXX receives an SP companion. The result does not invalidate502/98 or privilege500/100 as the sole biblical binding; it states why500 covers the full recorded three-Gear diagram and502 covers a proper subdiagram.\n\nWith these modal boundaries settled, the next useful comparison is with the earlier E-expansion ladders. Their endpoints belong to different event roles, so the successful return-loop translation must not be assumed to carry every intermediate source.\n\n## Verification and limits\n\n50 main checks and 17 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. These checks and the source interpretation were performed by the root agent; no new independent-agent review is claimed. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.\n\nA missingGear0 is a finite domain boundary. Both500/100 and502/98 remain valid; repeated image coordinates are not independent witnesses.\n\nMirror/529 focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction January AD2; no civil year zero. Regular Gears are Noah/Shem/Flood only. SP minus-one person companions remain local to SP Noah/Shem; Flood close is a separate boundary. No upstream/cumulative Gear, new endpoint or phase, fitted anchor, global axis, primer reopening, canonical or graph edit. Capture now; canonize later. Source identity, context, modal binding, generated coordinates and phase ancestry stay distinct.\n\n## Decision after this result\n\n**C408, proposed and unexecuted:** Apply the fixed−500 translation to the two primary C343 Noah-held1058→1150→1250 ladders and compare their source endpoints with the C363 Shem-held squared-E construction. Identify precisely which intermediate roles remain sourced, preserving the original two17K contexts and all recorded phase qualifications.\n\nThe new source correspondence is complete for the return-loop grammar, but the older double-E ladder has different intermediate roles. Testing it can connect the two existing constructions while exposing the exact source boundary.\n"}