J source-path normal form

24 September 2026. Step 7 of 10 in the C394–C403 authorization.

The two source loops join at the identical MT G2 Noah record, giving a three-vertex graph with an exact path normal form. Its vertices are3031,3033,3035; it has two forward P edges, two forward J edges and two E return edges. No new source node or arrow was added.

For any admitted path from source u to source v, let e,p,j count its E,P,J arrows. Then its entire affine map is

F(t)=v+E^e P^p J^j(t−u),

and its net Gear displacement obeys

(v−u)/2=p+j−e.

The endpoints fix the intercept once the slope is known. Thus different routes with the same endpoints and operator counts give exactly the same affine map. Conversely, prime7 determines p, prime13 determines j, and prime5 then determines e. Different count triples cannot have the same slope. Source and pivot histories remain separately recorded even when the resulting affine maps agree.

At the central vertex3033, the circuit through the lower neighboring Gear and the corresponding circuit through the higher neighboring Gear agree as affine maps: both P circuits have multiplier1750/1587, and both J circuits7500/6877. Therefore the two-generator commutative positive circuit system is consistent across the joined graph. All three source vertices have the same circuit multipliers, each centered at its own point.

The general claim follows from affine composition and prime valuations. A finite enumeration of every admitted path of zero through six edges gives402 paths and78 distinct affine classes when source/target vertices are retained. This includes the identity paths and the previously checked circuits; the counts are not independent observations.

Two successive E returns are now a valid source path3035→3033→3031, with slope625/529. They use the different existing held Noah pivots3058 and3056. This does not reopen the failed second E application at one fixed Noah pivot, and it does not supply a fourth Gear.

This is a reduced grammar for coordinate-affine comparisons on an existing finite source graph. A path’s selected source endpoints are admitted; the formula alone does not guarantee that applying the same affine map to every other source date stays within the register. That is the next required distinction.

Verification and limits

732 main checks and 16 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 normal form is a coordinate-affine quotient of admitted paths; distinct source routes and tradition aliases keep their provenance. It does not imply global source closure or source-prescribed iteration.

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

C401, proposed and unexecuted: Test the closed-circuit generators against every frozen regular coordinate and against529-spaced displacements. Determine the exact source-preserving domain and whether the original2K/17K spans or a529 lattice are preserved.

The reduced grammar is now exact on admitted paths. Its interpretation depends on separating that local source-path structure from a single affine action on the whole chronology or529 lattice.