490d C395 — E/J endpoint loops, affine fixed points and biography limits

24 September 2026. Step 2 of 10 in the C394–C403 authorization.

The two E/J source loops return their selected endpoints, but their affine compositions are nonidentity dilations. They neither make E and J global inverses nor preserve a950-year Noah biography.

Let x be the higher-Gear Noah input to E, y=x−2 the lower-Gear Noah output, N the held E Noah, and A the existing return J Flood. The source distances are N−x23, N−y25, y−A598 and x−A600. Thus N−A623=23+600=25+598.

Writing F_c^r(t)=c+r(t−c), the exact composition laws are

J_A ∘ E_N = F_x^(7500/6877) and E_N ∘ J_A = F_y^(7500/6877).

Each composition fixes a different existing source point. Its multiplier is1+623/6877, not1. The order defect, J after E minus E after J, is the constant−1246/6877. Repeating a source loop keeps its selected endpoint fixed while multiplying any displacement from it; this is not a translation of the whole chronology.

All32 assignments of the inherited±1/4 offsets to the four points/pivots were tested. Four retain both source edges: the two common-offset assignments per loop. A single fixed composite stretches half-year separation by623/13754. The matched-component comparisons therefore require their corresponding source-component pivots.

The SP and LXX source Noah biographies both retain950-year birth/death spans. E's death image misses the corresponding next source death by−1900/23; J's by−950/299. A complete J-after-E circuit misses its starting death by−591850/6877. The actual adjacent-Gear birth/death pairs still translate together by2; that ordinary source comparison is distinct from these dilations. SP primary death retains its inclusive/harmonic qualification.

The positive result is an exact source endpoint loop with fully specified affine behavior. It warrants testing the same motif throughout the fixed regular family menu, without assuming that only the originally selected SP/LXX centers can support it.

Verification and limits

20 main checks and 14 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.

The loops concern selected source endpoints and exact affine comparisons. No independent-witness claim, inverse chronology protocol, preserved biography or new phase is inferred.

Mirror/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.

Decision after this result

C396, proposed and unexecuted: Enumerate the same adjacent-Gear E/J return-loop motif across all coherent regular Noah families and both existing Flood boundaries. Then classify whether each surviving loop also has an existing P return pivot, without prefiltering by tradition or529.

The fixed-point result makes the source-loop motif precise. A full finite family census can determine its actual domain and whether the missing P return is specific to the two initial cases.