{"id": "c339", "title": "Double-E ladder under the two named Shem bindings", "filename": "490d_C339_Double_E_Ladder_Shem_Binding_Control_20260924.md", "role": "Research action", "step": 339, "sha256": "4bc200d2bd0400ead5de95942d45ee842cf892636210a78c8b713357e4de58c1", "bytes": 3269, "origins": ["C01–C479/packets/C339/490d_C339_Double_E_Ladder_Shem_Binding_Control_20260924.md"], "download": "/sources/4bc200d2bd0400ead5de95942d45ee842cf892636210a78c8b713357e4de58c1.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Mirror / 529"], "excerpt": "490d C339 — Double-E ladder under the two named Shem bindings\n\n24 September2026. Step6 of10 in the C334–C343 authorization.\n\nThe endpoint relation1058×E²=1250 survives both valid Shem bindings. The complete intermediate1150 requires the Noah-bound500/100 choice in the named G2 control. This qualifies the ladder without", "content": "# 490d C339 — Double-E ladder under the two named Shem bindings\n\n24 September2026. Step6 of10 in the C334–C343 authorization.\n\n**The endpoint relation1058×E²=1250 survives both valid Shem bindings. The complete intermediate1150 requires the Noah-bound500/100 choice in the named G2 control.** This qualifies the ladder without treating502/98 as an error.\n\nHold SP G2 primary Noah N=3708, restored MT G3 Flood start R=2650, and base MT G2 Flood start F=2458. Use the two Shem bindings already retained in C329, retrieving their base-MT counterparts from C286.\n\n| Valid binding | Base MT Shem H | Base MT Noah→H / H→F | N−R, N−H, N−F | R−H / H−F |\n|---|---:|---|---|---|\n| Noah-bound, Shem G2 | 2558 | 500 /100 | **1058,1150,1250** | **92 /100** |\n| Arphaxad-bound, Shem G1 | 2556 | 502 /98 | **1058,1152,1250** | **94 /98** |\n\nBoth1058 and1250 are unchanged because their endpoints do not depend on the Shem binding. The first expansion predicts1150; the second expansion of the actual alternate1152 exceeds1250 by50/23. Thus the alternative remains a valid source comparison but is not a two-stage E ladder through its selected Shem endpoint.\n\nThis control uses the explicit G2/G1 comparison. It does not invent Gear0 to impose an analogous lower-Gear binding on G1. C338's companion limitation also remains: the one-year MT intermediate needed there is absent, regardless of the numerical appeal of1150.\n\nThe next useful step is to insert the already source-qualified SP Noah death. That will show how the newly matched92/100 increments relate to the retained950+108 dissection, and how the actual companion and alternate Shem records modify that interior reading.\n\n## Verification and limits\n\n13 main checks and 16 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\nTwo named valid bindings, not all conceivable modal combinations. Conditional fit does not prescribe or canonize the500/100 mode. No missing Gear or companion is created.\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**C340, proposed and unexecuted:** Insert the matched SP Noah death into the verified interval ladder. Resolve the primary, companion and named502/98 interior partitions from existing records, retaining the primary-death qualification and forbidding imported MT companions.\n\nC339 separates invariant outer endpoints from a modal interior. The previously established950+108 relation can now connect directly to the verified92/100 gains and reveal their dependence.\n"}