490d C446 — Finite Reachability Under the Four Fixed Reflections
25 September 2026 UTC. Step 2 of 25, C445–C469.
The four-reflection graph has 373 vertices, 106 edges including four loops, and 272 components. Its 152 active coordinates form 49 nontrivial components plus two loop-only singletons; 221 coordinates have no admitted reflection. The largest components have six coordinates and the maximum shortest-path distance is four.
| Component size | Number |
|---|---|
| 1 | 223 |
| 2 | 25 |
| 3 | 4 |
| 4 | 14 |
| 5 | 4 |
| 6 | 2 |
Every finite admissible reflection word remains inside one of these components. A breadth-first traversal from every vertex records the shortest center-labelled witness for all 191 distinct unordered reachable pairs. This is a closure statement for the four fixed partial comparisons; it does not extend their domains.
Two six-coordinate components occur: 2606, 2610, 2730, 2734, 3766, 3770; and 2667, 2671, 2675, 2795, 3705, 3709. The former contains six ordinary edges, indicating a cycle; the latter has five. The graph and witnesses preserve source-record indices at their vertices.
Finite reachability is now exhausted without choosing a word-length cutoff. It is therefore suitable for a decisive 529 test across all reflection words, including words longer than those enumerated in C445.
Verification and scope
374 calculation checks and 19 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.
Graph connectivity expresses numerical accessibility through sourced coordinates. It does not assert common tradition, role, phase or chronological context along a path.
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
C447, proposed and unexecuted: Test every reachable coordinate pair for nonzero multiples of 529 and locate the established E-path, loop and pivot coordinates in the complete reflection graph.
All reflection words reduce to the computed finite components. A pair census can now settle whether this closure supplies a 529 relation or connects the established nonzero E path.