490d C452 — Source and Affine Structure of the Unique Mixed Feedback Component
25 September 2026 UTC. Step 8 of 25, C445–C469.
The only enlarged strong component joins two former reflection components. Its unique simple E-bearing directed cycle is2731→3765→2735→2733→2731. The two E legs each expand a signed−23 pivot gap to−25, moving the Noah-death coordinate down one Gear. The full word fixes2731 with slope625/529 and is not an identity.
| Leg | Operator | Result |
|---|---|---|
| 2731→3765 | R3248 | Noah death to SP companion Noah birth |
| 3765→2735 | R3250 | Back to a Noah-death coordinate four years above2731 |
| 2735→2733 | E2758 | −23 becomes−25 |
| 2733→2731 | E2756 | −23 becomes−25 |
The E descent can consistently use SP primary or native LXX Noah death, C1/T1/F0, with G3→G2→G1. Coordinate2735 also has a separate SP companion Shem-death alias. The reflection detour uses SP companion Noah birth3765, C0/T1/F1/G1. Those identities remain distinct.
For the word starting2731, the full affine map is (625x−262176)/529, equivalently2731+(625/529)(x−2731). Its denominator529 comes from two25/23 operations. The closed path has zero net endpoint gap; this denominator is not a newly found529-year historical span.
The two descending legs have a compact source explanation: pivot states carry full430, while the Noah-death inputs carry Cainan+Terah and one higher Gear. The residual is215−130−60−2=23. Testing that mechanism across its stated finite source domain is more informative than treating the cycle as an unexplained coincidence.
Verification and scope
1 calculation checks and 20 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.
The affine fixed point is computed from a word already realized at2731; it is not added as a new global axis. The reflection detour still crosses source contexts.
Mirror/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.
Best next bounded step
C453, proposed and unexecuted: Derive and census the same-role source-state identity215−130−60−2=23 across the finite register, in both signs, and identify exactly which resulting one-Gear E arrows use the eight fixed pivots.
The mixed cycle reveals an exact23-year source residual. A same-role template census will distinguish the general source-state mechanism from the particular pivots admitted in this graph.