# 490d C431 — All sourced realizations of the thirteen squared-E maps

24 September 2026. Step 8 of10, C424–C433.

The wider same-map pivot search finds **one additional sourced two-E coordinate path**. It realizes the already known squared-E map held at Shem3206, while changing the two actual pivots:

**2148 →2020 →1956, using E pivots3620 then2756.**

This is not the absent fixed-pivot intermediate2056. The whole affine map is nevertheless identical because

`(25×3620 +23×2756)/48 =3206`.

Across all13 sourced squared-E target maps and373 possible first pivots each, the4849 trials give39 sourced pivot-pair solutions. Of these,26 use distinct pivots. Only five solutions have a sourced intermediate: the four earlier complete fixed-pivot Noah ladders, plus this new factorization. One of the nine formerly endpoint-only maps is therefore repaired as a coordinate path; eight remain without one in the tested menu.

For any held target H, every integer two-pivot solution has

`p=H+23t`, `q=H−25t`, `intermediate=H−1150−2t`.

The new solution uses t=18: p=3206+414=3620; q=3206−450=2756; intermediate=2056−36=2020. Its two gap expansions are1472→1600 and736→800, or64×23→64×25 followed by32×23→32×25. The gains128+64 equal the same192 net gain as the ordinary92+100 split.

The source-label question now matters. The search deliberately allowed every existing regular coordinate as a possible pivot. A positive coordinate path must still retain the actual roles and contexts of3620,2756 and2020; it is not automatically a prescribed chronology operation. That is the next bounded check.

The earlier negative results retain their scopes. The fixed3206 intermediate2056 is still absent. The restricted return-graph comparisons did not search every regular pivot pair. The held3899 map still has only its missing2749 intermediate. Prime7/13/5 slope valuations continue to force every positive E/P/J word equal to one of these complete squared-E maps to exactly two E factors; longer positive words do not create another slope-compatible case.

## Verification and scope

4850 calculation checks and 14 source/predecessor authentication checks passed. A separate main replay is byte-identical. Root-agent analysis and replay; no new independent-agent review is claimed. Verification counts are not independent historical witnesses.

The positive result is a coordinate path in the deliberately generous existing-source pivot menu. No source-prescribed pivot grammar, new endpoint, new phase or global chronology operator is inferred.

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.

## Decision after this result

**C432, proposed and unexecuted:** Audit the actual source identities and contexts of the new2148→2020→1956 factorization and its pivots3620/2756 directly from current passages. Check its+2 counterpart and inherited quarter components, keeping coordinate-path validity separate from a prescribed source operation or one Noah biography.

The all-pivot search found a positive exception. Its source and Gear qualifications now take priority over a generic completeness summary, before C433 consolidates the batch.