{"id": "c414", "title": "Complete 2K source-pair census under fixed 500 translation", "filename": "490d_C414_Complete_2K_Source_Pair_Census_under_Fixed_500_Translation_20260924.md", "role": "Research action", "step": 414, "sha256": "5c8d4b01cfa2acb0993cdeb0eefb2ed5dbd25d08779f0eb93273c3250c3c65c6", "bytes": 3446, "origins": ["C01–C479/packets/C414/490d_C414_Complete_2K_Source_Pair_Census_under_Fixed_500_Translation_20260924.md"], "download": "/sources/5c8d4b01cfa2acb0993cdeb0eefb2ed5dbd25d08779f0eb93273c3250c3c65c6.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Mirror / 529"], "excerpt": "490d C414 — Complete 2K source-pair census under fixed 500 translation\n\n24 September 2026. Step 1 of 10 in the C414–C423 authorization.\n\nThe frozen regular register has26 coordinate pairs exactly1058=2K apart. Six retain both sourced endpoints after one fixed−500 translation; none retains a second consecutive applicati", "content": "# 490d C414 — Complete 2K source-pair census under fixed 500 translation\n\n24 September 2026. Step 1 of 10 in the C414–C423 authorization.\n\n**The frozen regular register has26 coordinate pairs exactly1058=2K apart. Six retain both sourced endpoints after one fixed−500 translation; none retains a second consecutive application.**\n\nThe26 coordinate pairs carry46 original endpoint-label pairings. All528 records and373 coordinates remain fixed. No alternative width or translation was searched.\n\n|Source2K pair|Pair after−500|Second application|\n|---|---|---|\n|3299, 2241|2799, 1741|Not source-complete|\n|3491, 2433|2991, 1933|Not source-complete|\n|3493, 2435|2993, 1935|Not source-complete|\n|3706, 2648|3206, 2148|Not source-complete|\n|3708, 2650|3208, 2150|Not source-complete|\n|3899, 2841|3399, 2341|Not source-complete|\n\nThus the induced coordinate graph on2K pairs consists of six one-arrow components and fourteen isolated pairs. The longer individual-point chains in C410 do not supply a two-step paired2K chain. Translation preserves the width arithmetically at every generated level; availability of both source endpoints is a separate requirement.\n\nThe original primaryi3/i5 rectangles are among the six one-step survivors. Keeping their cumulative corners fixed gives cross-spans8993 before translation and8493 afterward, retaining C409. A second formal shift would give7993, but its regular endpoint pair is not sourced and is not adopted.\n\nFour survivors reproduce the familiar primary Noah/Flood-to-Shem comparisons. The3299/2241 and3899/2841 pairs require a direct label audit: the raw census deliberately retains all co-located records instead of assigning a source identity by coordinate alone. That audit is the next bounded step.\n\n## Verification and limits\n\n110 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.\n\nCoordinate survival is not yet a selected source-record path. Generated second-shift endpoints and formal cross-spans remain unadopted; no new shift, width, phase or axis was searched.\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**C415, proposed and unexecuted:** Audit continuous record-labelled lifts of the six paired2K translations using C411's role-defined500 arrows, with its extra bridge tested separately. Identify the two additional coordinate pair types, reject unsupported alias substitutions, and compare the surviving source contexts with the existing Noah/Flood-to-Shem construction.\n\nThe paired coordinate domain is now complete. Its two additional pair types may carry a different source structure, but endpoint identities must be established before interpreting them.\n"}