# 490d C409 — Fixed Mirror-corner test of the Shem loop translation

24 September 2026. Step 6 of 10 in the C404–C413 authorization.

**The new source-qualified regular translation does not extend to the original Mirror rectangle under either existing corner exchange.** No exchange constant was changed.

|Original state|Exchange C in C−x|Translated regular2K endpoints|Unchanged cumulative2K endpoints|New cross-spans|
|---|---:|---|---|---|
|i3|−2639|3206,2148|−6345,−5287|8493,8493|
|i5|−2635|3208,2150|−6343,−5285|8493,8493|

Both short sides remain1058=2K. Both old8993=17K cross-spans decrease by500. Under the fixed reflection R_C(x)=C−x, the identity R_C∘T_−500=T_+500∘R_C explains the source requirement: translating a regular corner down500 moves its reflected cumulative image up500. The required new cumulative corners are absent.

The same test was applied to all52 labelled endpoint/pivot slots of the newly translated Shem loops under both exchanges. None of the104 reflected slots belongs to the frozen48-record cumulative menu. All208 inherited±¼ component checks, with reversed ancestry offset under reflection, also fail membership.

These counts include repeated labels and coordinates; they measure the audit, not independent witnesses. The negative result is confined to the fixed source menus and the two already recorded corner exchanges. It does not introduce a new axis or treat signed cumulative coordinates as civil dates with a year zero.

The regular loop correspondence remains exact. Its limitation is now precise: preserving source roles and2K within the regular field is insufficient to preserve the original17K cumulative/regular connection. The next useful question is the complete source domain of the fixed500 translation itself.

## Verification and limits

112 main checks and 20 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. These checks and the source interpretation were performed by the root agent; no new independent-agent review is claimed. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.

The absence claim is limited to the frozen cumulative menu and existing two corner exchanges. Source roles, labels and phases remain distinct; no new global axis is fitted.

Mirror/529 focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction January AD2; no civil year zero. Regular Gears are Noah/Shem/Flood only. SP minus-one person companions remain local to SP Noah/Shem; Flood close is a separate boundary. No upstream/cumulative Gear, new endpoint or phase, fitted anchor, global axis, primer reopening, canonical or graph edit. Capture now; canonize later. Source identity, context, modal binding, generated coordinates and phase ancestry stay distinct.

## Decision after this result

**C410, proposed and unexecuted:** Enumerate the fixed−500 comparison on all373 existing regular coordinates, preserving every endpoint label. Classify the source-supported arrows, determine maximal finite chains, and distinguish the proven Noah/Flood-to-Shem role correspondence from additional coordinate coincidences. Do not search other shifts or introduce new endpoints.

Mirror closure fails for a specific source reason. A complete finite domain census of the already motivated500 translation can show what regular source structure it actually supports before considering any broader transport claim.