# 490d C442 — Source Bounds of Reflection Conjugacy and Reversed E Order

25 September 2026 UTC. Step 8 of 10, C435–C444.

The local reflection conjugacy is exact, but its **nonzero2K path has no sourced reflected counterpart** in the fixed register. Only the zero-net loop survives in both orders.

For p=c+432 and q=c−432,

`R_c E_p R_c = E_q`,

and therefore

`R_c (E_q∘E_p) R_c = E_p∘E_q`.

The forward word fixes c+18; the reversed word fixes c−18. Their slopes are625/529, but the maps differ by the nonzero constant−3456/529 (forward minus reverse). They do not commute.

All1,492 reversed-word trials at the four sourced centers were checked. The only complete source path is **3170→3206→3170** at center3188, using2756 then3620. It is the reflected order of the C435 zero-net loop.

The original nonzero path2148→2020→1956 reflects about3188 to **4228→4356→4420**. Every one of those coordinates lies above the frozen register's maximum3900. The reflected arithmetic is exact, but none of its three stations is sourced in this domain. No enlarged source menu is introduced.

A primitive-arrow audit gives the same limit. There are five E_p arrows whose inputs, outputs and reflected counterparts are all sourced: four zero-gap fixed-pivot arrows, one at each center, plus the nonzero3206→3170 loop leg at3188. Its reflected leg is3170→3206 about the opposite pivot.

For the reversed word, integer outputs require `x≡c−18 (mod529)`, since625 and529 are coprime. This arithmetic condition does not ensure source membership; the full census retains both filters.

Thus the newly derived reflection profile supplies a sourced two-direction zero-net loop, but not a source-closed action carrying the192-year2K path. Prior seasonal/component restrictions remain; exact integer-coordinate conjugacy is not a calendar-operation prescription.

## Verification and scope

2991 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.

Only a zero-net loop is fully sourced in both orders. The nonzero2K reflection and any global/calendar action remain unsupported; missing stations are not added.

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

**C443, proposed and unexecuted:** Apply the two inherited fixed Mirror corner exchanges to the complete five-point3188 profile and the current local-reflection source domains. Test membership in the frozen cumulative register and preserve the original17K corner controls; do not fit a new exchange or cumulative source.

The local reflection/E relation now has an exact source boundary. The remaining integration check is whether this new local structure connects to the original fixed Mirror, rather than treating a local midpoint as a global axis.