490d C421 — Source-path obstruction for the new squared-E endpoint
24 September 2026. Step 8 of 10 in the C414–C423 authorization.
No sourced two-E path realizes the new held3899 squared-E map. This remains true even if either pivot is allowed to be any of the373 existing regular coordinates.
For two E actions with first pivot p and second pivot q, their effective squared-E pivot is (25p+23q)/48. Equality with the held3899 map therefore requires25p+23q=48×3899. Scanning every sourced first pivot gives only one sourced second-pivot solution: p=q=3899. Its intermediate is2749, absent from the source menu.
There is also a short integer proof. Every integer solution is p=3899+23t, q=3899−25t. The source menu's maximum is3900. The first upper bound forces t≤0 and the second t≥0, hence t=0. This does not invent a broader admissible pivot protocol: allowing all existing coordinates is a generous exclusion test, and even that larger menu fails.
The obstruction covers every finite positive E/P/J word that equals this whole affine map. The target slope is625/529. Prime7 in the slope forces the P count to zero; prime13 forces the J count to zero; prime5 then forces exactly two E factors. There are no other positive words with the target slope. With only the repeated3899 pivot available, the absent2749 intermediate cannot be bypassed in this grammar.
The exact endpoint comparison2841→2649 remains valid under E². The result excludes a sourced realization of that same map, not all different-slope maps that might happen to agree at one input, inverse operators, or future source extensions.
Verification and limits
378 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 positive-word proof concerns equality of complete affine maps and sourced intermediate paths. It does not broaden J into a calendar operator or claim that every unrelated endpoint route has been excluded.
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
C422, proposed and unexecuted: Reconstruct the profile, boundary controls and new squared-E endpoint directly from current source excerpts, and check inherited quarter-offset components. Separate source-prescribed relations from derived comparisons, fixed-pivot seasonal action and jointly shifted endpoint comparisons before final synthesis.
The algebraic path question is closed. A focused source and component audit is now the most useful final research step before C423 synthesis and recovery packaging.