490d C397 — Commuting affine circuits in sourced E/P/J loops

24 September 2026. Step 4 of 10 in the C394–C403 authorization.

The closed P–E and J–E circuits commute at each source vertex, although E, P and J themselves do not. This supplies an exact local composition structure on the two fully sourced MT loops.

The source graphs have lower/higher MT Noah vertices3031/3033 and3033/3035. Their held E Noah pivots are3056 and3058; their J Flood pivots2433/2435 and P Flood pivots2893/2895 are all existing records. The E held Noah is a distinct source point from the SP/LXX outer Noah of the original529 rectangle.

At a lower vertex c, the generators are E after P and E after J. At the corresponding higher vertex, they are P after E and J after E. In either case the two complete circuits are dilations about the same existing source c:

Consequently they commute as whole affine maps. A circuit word containing p P-circuits and j J-circuits has the form

t ↦ c+(1750/1587)^p(7500/6877)^j(t−c).

The order of complete circuits does not matter. Different count pairs do matter: prime7 has valuations1 and0 in the two multipliers, while prime13 has valuations0 and−1. These force equality of multipliers only when both counts agree. The resulting positive circuit system is a free commutative monoid on two generators at each source vertex. This terminology describes the exact comparison maps; it does not establish a prescribed ancient operation.

The individual operator defects remain nonzero: E after P minus P after E is326/1587; E after J minus J after E is1246/6877; P after J minus J after P is20/897. Thus circuit commutation must not be shortened to “E, P and J commute.” No nonempty positive circuit is the identity because both circuit multipliers exceed1.

All56 circuit words of one through three circuits across the four source vertices were checked against the general formula. Each repeats existing source edges and vertices. Repetition does not create a fourth Gear or extend a forward source path beyond its boundary. These checks support the algebraic classification, not a count of independent witnesses.

Verification and limits

99 main checks and 14 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 monoid consists of exact affine comparison maps on admitted source circuits. It is not a group of inverse source operations, a global action on all dates, or a source-prescribed composition rule.

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

C398, proposed and unexecuted: Test whether the complete E/P/J loops transport through the already recorded+215 branch translation to the original full43017K configurations. Solve the unique translated E pivot and verify source and phase availability without fitting an anchor.

The positive circuit structure is located on the lower-full430 common P/J branch. The original Mirror/529 question requires determining whether the entire structure reaches the original17K branch.