# 490d C445 — Ordered Compositions of the Four Fixed Local Reflections

25 September 2026 UTC. Step 1 of 25, C445–C469.

The frozen local centers are 3188, 3190, 3248 and 3250. For each of their 16 ordered pairs and each of the 373 source coordinates, this census requires both the intermediate and final output to be sourced. Every endpoint retains the list of its original source-record indices.

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.

The affine identity is R_b(R_a(x)) = x + 2(b−a). It supplies candidate translations, but source admission still depends on the intermediate 2a−x. In particular, the equal-center words return only on their respective domains (61, 67, 19 and 61 inputs), not on all 373 coordinates.

| First center | Second center | Shift | Complete paths |
|---|---|---:|---:|
| 3188 | 3188 | +0 | 61 |
| 3188 | 3190 | +4 | 31 |
| 3188 | 3248 | +120 | 1 |
| 3188 | 3250 | +124 | 8 |
| 3190 | 3188 | -4 | 31 |
| 3190 | 3190 | +0 | 67 |
| 3190 | 3248 | +116 | 0 |
| 3190 | 3250 | +120 | 4 |
| 3248 | 3188 | -120 | 1 |
| 3248 | 3190 | -116 | 0 |
| 3248 | 3248 | +0 | 19 |
| 3248 | 3250 | +4 | 17 |
| 3250 | 3188 | -124 | 8 |
| 3250 | 3190 | -120 | 4 |
| 3250 | 3248 | -4 | 17 |
| 3250 | 3250 | +0 | 61 |

The asymmetry between ±116 and ±124 is a finite source-domain result, despite the rectangular arrangement of the four centers. Counting only sourced endpoints would admit additional unsupported words. All excluded endpoint-only inputs are preserved in DATA.json.

## Verification and scope

5969 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 is an exhaustive coordinate census in the frozen register, not a source-prescribed transformation of one biography. Distinct aliases and contexts remain indexed; the path counts are not independent historical witnesses.

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

**C446, proposed and unexecuted:** Compute the complete finite reachability graph of the four fixed reflections on the 373 existing coordinates, retaining center-labelled edges, loops and exact shortest witness paths.

The two-letter census proves partial translation behavior. The next unresolved question is whether longer admitted words enlarge the reachable source families or simply revisit small components.