{"id": "c380", "title": "Finite source paths under moving P/J pivots", "filename": "490d_C380_Finite_Common_P_J_Source_Paths_20260924.md", "role": "Research action", "step": 380, "sha256": "e89f7c308f54f651d71eec439b0bc612232652487d89575853c121b09e6289e8", "bytes": 3317, "origins": ["C01–C479/packets/C380/490d_C380_Finite_Common_P_J_Source_Paths_20260924.md"], "download": "/sources/e89f7c308f54f651d71eec439b0bc612232652487d89575853c121b09e6289e8.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Counts & NT", "Mirror / 529"], "excerpt": "490d C380 — Finite common P/J source paths\n\n24 September 2026. Step 7 of 10 in the C374–C383 authorization.\n\nCommon P/J source edges do form short paths when both Flood pivots move with the input. There are exactly four maximal paths in the complete frozen census:\n\n| Source family | Existing coordinate path | Maximum e", "content": "# 490d C380 — Finite common P/J source paths\n\n24 September 2026. Step 7 of 10 in the C374–C383 authorization.\n\n**Common P/J source edges do form short paths when both Flood pivots move with the input.** There are exactly four maximal paths in the complete frozen census:\n\n| Source family | Existing coordinate path | Maximum edges |\n|---|---|---:|\n| Cumulative OFF Shem, even annual members |−5884→−5886→−5888→−5890|3|\n| Cumulative OFF Shem, odd annual members |−5883→−5885→−5887→−5889|3|\n| Regular restored-MT Noah, full430=0 |3031→3033→3035|2|\n| Regular restored-MT Noah, full430=1 |3246→3248→3250|2|\n\nAll ten common coordinate edges are accounted for once. The cumulative paths use the already admitted eight annual members; their terminal edge can lie outside the smaller complete-C287 domain. The regular paths stop at Gear3. No Gear4 or additional cumulative member is introduced.\n\nAt each edge, the oriented input gaps are299h and69h, with signed gain h=−2 cumulative or+2 regular. To use the current output as the next source input, both Flood pivots also change by h. The same598/138 input gaps are thereby restored in the next source configuration.\n\nKeeping the old pivots would miss the next admitted target: the J error is h/299 and the P error h/69. Thus these paths are not iteration of one fixed pair of maps. They are a finite sequence of source-supported comparisons whose pivot records change at each step.\n\nThese individual source paths do not yet show that a paired cumulative/regular path can preserve the original17K cross-field spans. Their coordinate directions are opposite, so that compatibility requires a separate check.\n\n## Verification and limits\n\n34 main checks and 15 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.\n\nMoving-pivot paths are not iteration of a fixed global operator. Cumulative annual members are not Gears, and the source path lengths do not authorize extrapolation.\n\nMirror/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.\n\n## Decision after this result\n\n**C381, proposed and unexecuted:** Test the composability of paired cumulative/regular repairs while retaining the original2K/17K Mirror conditions. Enumerate the admitted two-step pairings and track their source-corner sums and cross-span changes; keep the cumulative annual and regular Gear domains fixed.\n\nThe individual paths run in opposite coordinate directions. Their paired effect on the full Mirror configuration is the remaining compatibility question; it cannot be inferred from local source closure.\n"}