490d C401 — Circuit source domain and529-lattice obstruction

24 September 2026. Step 8 of 10 in the C394–C403 authorization.

Every nonempty positive circuit preserves only its selected basepoint in the frozen regular coordinate menu. The admitted path grammar therefore has a local source meaning; its affine extensions do not act on the whole register or preserve the529 lattice.

The six distinct generators are the P–E and J–E circuits centered at each of3031,3033,3035. Testing all373 regular coordinates gives2238 map/input comparisons. Exactly six outputs remain sourced: the fixed basepoint of each map. No other source coordinate survives.

The obstruction is arithmetic, not a search accident. The menu ranges from1741 to3900. Its maximum displacement from these basepoints is smaller than either reduced denominator1587 or6877. An integer input to a reduced-rational dilation about an integer center can remain integral only when that denominator divides its displacement.

For a general circuit with p P-circuits and j J-circuits, the reduced denominator is

3^max(p−j,0) × 13^j × 529^(p+j).

No factor23 occurs in the numerator. Once p+j≥2, the denominator is at least529², far beyond every source displacement. Together with the one-circuit check, this proves the stated source-domain result for all nonempty positive circuits.

Formal span tests also fail to preserve the original529 units:

These are the separations that would result if the same affine map were applied to both endpoints; no missing endpoint is admitted by this calculation.

On the formal lattice c+Kn, the P generator returns a K-multiple only when1587 divides n, and the J generator only when6877 divides n. Both require20631=39K to divide n. This describes a restricted sublattice of possible inputs, not preservation of the entire lattice. Its large derived displacement is an arithmetic diagnostic, not a chronology date.

The positive result from C400 remains intact: admitted paths connect their labelled source endpoints and have a reduced normal form. Applying a path’s affine formula indiscriminately to all other source dates is a different operation, and this test rules it out within the frozen register.

Verification and limits

21 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 lattice calculations supply no new dates or anchors. A coordinate-affine map can be mathematically defined beyond its admitted source path without those additional images becoming source records.

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

C402, proposed and unexecuted: Reconstruct the six source loop configurations from the current table, variant and Flood-boundary rules, and audit which parts of the new operator grammar are explicitly sourced versus algebraically derived. Preserve LXX Cainan-removal and SP companion qualifications.

The algebra and its global limits are now clear. Direct source reconstruction and a bounded warrant audit are needed before the final synthesis can state precisely what has—and has not—been established.