# 490d Mirror checkpoint after C456

25 September 2026 UTC. **C456 complete; C457 unexecuted.** 13 steps remain in the current authorization. Current decisions override archived next-step text.

## Current result

For integer input and effective pivot, a two-E output is integral exactly when the input gap is divisible by529. The13 sourced-pivot words have17 integer-output cases,15 sourced outputs and11 complete paths. Among nonzero gaps the gates narrow4→2→1; only2148→2020→1956 survives.

## Earlier results in this batch

**C445**: All 5,968 ordered two-reflection trials give 330 complete paths: 208 zero-shift returns and 122 nonzero paths. Sourced shifts are ±4 (48 each), ±120 (5 each) and ±124 (8 each); both ±116 have zero complete paths.

**C446**: 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.

**C447**: 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.

**C448**: The reflection graph has one ordinary simple cycle, 2610→3766→2730→3770→2610, through centers3188,3248,3250,3190. Its four-reflection affine word is the identity because3188+3250=3248+3190. The other four cycles are single-center fixed-point loops, not identity maps.

**C449**: All 8,952 four-distinct-center trials produce exactly eight complete words, all returns: the two directions and four starts of one cycle. Each of the eight identity orders has only one sourced input. The cycle has eight source-label assignments and none keeps one unchanged source context.

**C450**: The eight existing pivots yield37 sourced E arrows in2,984 trials: eight pivot self-arrows and29 distinct nonzero arrows. Another2,854 outputs are fractional and93 are integer coordinates absent from the frozen register.

**C451**: The combined fixed E/reflection graph has245 operator-labelled arrows and522 reachable directed distinct pairs,140 beyond reflection-only reachability. None has a nonzero529-multiple gap. One merged strong component has eight coordinates; all other strong components retain the reflection partition.

**C452**: The only enlarged strong component joins two former reflection components. Its unique simple E-bearing directed cycle is2731→3765→2735→2733→2731. The two E legs each expand a signed−23 pivot gap to−25, moving the Noah-death coordinate down one Gear. The full word fixes2731 with slope625/529 and is not an identity.

**C453**: 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.

**C454**: 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.

**C455**: Across23,872 input/word trials,64 ordered E pairs yield15 complete paths. Thirteen words have a sourced effective fixed point;11 complete paths use one. The only nonzero529-multiple input gap is the established1058 gap atH3206, with2148→2020→1956. Four additional paths have rational, unsourced effective pivots.

## Baseline and standing scope

C444 is the authenticated baseline. Its checkpoint is frozen in sources/C444_CHECKPOINT.md. The fixed local reflection centers are 3188, 3190, 3248 and 3250, each an existing MT Noah birth record. The 528 labelled Noah/Shem/Flood records occupy 373 integer BC coordinates. These local source comparisons do not replace the January-AD1 global Mirror. The only complete forward 2K E path remains 2148→2020→1956. The SKL appendix and corrected LXX8464 observations remain supplemental; no redirection follows. C432’s E excerpt locator was corrected in C444 to File_46 L3, lines117–133; prior arithmetic already used25/23.

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.

## Limits and verification

The absence claims are restricted to the frozen source register. No missing2022,2655,2641,2657 or2643 record is generated or admitted.

4866 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.

## Best next bounded step

C457: Starting from all60 actual529/1058 source pairs, compute full pair reachability when the same admitted E or reflection is applied to both endpoints, preserving endpoint ancestry and every intermediate source pair.

Single-vertex reachability excludes529 endpoint connections, while E still transports intervals. A full shared-word pair closure tests the meaningful interval question directly and has a finite stopping condition.

Continue within the authorized batch and reassess after the next result.