490d C453 — The 23-Year Source-State Residual and One-Gear E Arrows
25 September 2026 UTC. Step 9 of 25, C445–C469.
The residual215−130−60−2=23 generates88 same-role source-label E templates, representing64 distinct coordinate arrows. Five label templates realize three of the four one-Gear arrows at the fixed pivots. The remaining arrow is MTNoah2841→2843 about2818, explained by950−650−215−60−2=23.
For one tradition, role and rail, let A be C0/T0/F1/Gg and B be C1/T1/F0/G(g+1), g=1 or2. A−B=215−130−60−2=23. Expanding B about pivot A moves B down one Gear; expanding A about pivot B moves A up one Gear. The construction stays within the permitted three-Gear domain.
All44 eligible base records supply both signs. The source comparison therefore has88 labelled templates and64 coordinate instances. These additional template pivots are recorded for explanation only; the active operator menu still contains the same eight pivots.
At the fixed pivots the negative templates are2735→2733 about2758 and2733→2731 about2756, each with SP-primary and LXX source assignments. The positive template is SP-companion Noah birth3705→3707 about3682.
The other fixed one-Gear arrow has a different source equation. MTNoah birth2841 lies23 above SP-primary/LXX-OFF Noah death2818: the950-year Noah lifespan minus650 tradition offset,215 full430 addition,60 Terah addition and2 Gear shift leaves23. Its output2843 is MTNoah birthG2. This cross-tradition, cross-role comparison is kept separate from the same-role template.
These equations explain the unit-Gear E arithmetic. They do not imply a529-year endpoint gap; the complete mixed graph has already excluded such a gap. A register-wide529 pair census will identify exactly which direct source comparisons remain outside that graph.
Verification and scope
176 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.
The88 templates are not88 independent witnesses or an expansion of the fixed operator menu. Every shift is limited to Noah/Shem/Flood source records.
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
C454, proposed and unexecuted: Census every nonzero529-multiple coordinate pair in the frozen373-coordinate register and compare it with mixed E/reflection reachability, retaining source identities and parity classes.
The operator mechanism is now source-resolved. Comparing its closure with the entire finite inventory of actual529 gaps will show whether the negative result reflects missing pairs or missing admissible connections.