490d C447 — All-Word 529 Test and Established-Path Connectivity
25 September 2026 UTC. Step 3 of 25, C445–C469.
No nonzero multiple of 529 occurs among all 191 reflection-reachable coordinate pairs. The nonzero E-path coordinates 2148, 2020 and 1956 are reflection-isolated. The source pivots 3620/2756 share a four-point component; 3206/3170 share a different four-point component; midpoint3188 is a loop-only singleton.
The exclusion is not limited to a selected maximum word length. C446 exhausted the finite connected components, and this step tested every distinct reachable pair. All gaps are even, since R_c(x)=2c−x preserves parity. Thus odd multiples of529 are already excluded algebraically; the finite census also excludes every possible even multiple within the register.
| Coordinate | Complete reflection component |
|---|---|
| 3620 | 2756, 2760, 3620, 3624 |
| 3206 | 3166, 3170, 3206, 3210 |
| 3188 | 3188 |
| 3170 | 3166, 3170, 3206, 3210 |
| 2756 | 2756, 2760, 3620, 3624 |
| 2148 | 2148 |
| 2020 | 2020 |
| 1956 | 1956 |
The local reflection comparisons do connect the two E-loop stations, but cannot join that station component to its pivot component or to the nonzero2K E path. The exact E-loop and2K observations remain valid in their own source comparisons. This result limits what may be inferred from adding unrestricted word length to the four-reflection menu.
Verification and scope
192 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.
This excludes 529 spans within the four fixed-reflection reachability relation only. It does not exclude direct 529 pairs elsewhere in the source register or paths using other declared operators.
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
C448, proposed and unexecuted: Classify all simple cycles and fixed-point loops in the finite reflection graph, compute their full affine word, and separate formula identities from returns valid only at a particular point.
The 529 closure question is settled. The remaining unclassified feature of the graph is its cycle structure, which may explain repeated sourced returns without creating new 529 spans.