# 490d C454 — Direct 529 Pairs Versus Operator Reachability

25 September 2026 UTC. Step 10 of 25, C445–C469.

The frozen register contains60 direct nonzero529-multiple coordinate pairs:34 at529 and26 at1058, with119 labelled source assignments. No pair is connected in either direction by the fixed mixed E/reflection menu. Thus the graph exclusion does not mean the source register lacks529 spans.

All69,378 distinct coordinate pairs were screened by exact divisibility. There are no3K or4K pairs in this register. Every hit retains both endpoint source-index lists, and all119 label combinations are archived without merging aliases.

The34 odd-K pairs cannot be connected by these parity-preserving operators. The26 even-K pairs pass parity but fail the exhaustive reachability test. They include2148/3206 and2150/3208, as well as the known corner pairs2648/3706 and2650/3708.

This makes a necessary distinction in the continuing529 work. A direct gap between two source events, an E map’s effective pivot-to-input gap, and the difference between a word’s input and output are separate comparisons. The established2148→2020→1956 path has effective pivot3206: its input gap is1058, while its endpoint displacement is192.

The next census therefore examines effective E pivots explicitly, using all ordered pairs from the already fixed eight-pivot menu. A formal affine fixed point will be admitted as a sourced pivot only when it is an actual record coordinate.

## Verification and scope

60 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.

This is a finite inventory and operator comparison. It makes no claim that these60 pairs are globally new to the project or historically independent.

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

**C455, proposed and unexecuted:** Census all64 ordered two-E words at the eight fixed pivots, classify their effective fixed points as sourced or merely formal, and test complete paths for nonzero529-multiple input gaps from a sourced effective pivot.

The direct-gap inventory confirms that529 is present but inaccessible by this operator menu. The relevant E question is instead whether any other source-qualified effective pivot supports a complete529-gap expansion.