{"id": "c318", "title": "Typed decomposition of the paired458 replacement", "filename": "490d_C318_Typed_458_Source_State_Decomposition_20260924.md", "role": "Research action", "step": 318, "sha256": "85f3051992a42a9cd471f03a5987c5f79ffa5f7d247f4113cb400fbfb9f7ed90", "bytes": 3617, "origins": ["C01–C479/packets/C318/490d_C318_Typed_458_Source_State_Decomposition_20260924.md"], "download": "/sources/85f3051992a42a9cd471f03a5987c5f79ffa5f7d247f4113cb400fbfb9f7ed90.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Mirror / 529"], "excerpt": "490d C318 — Typed458 source-state decomposition\n\n24 September 2026. Step 5 of the ten-step authorization C314–C323.\n\nThe paired458 replacement has a fully typed decomposition on the three rows where C287 is admitted. Its source ingredients are permitted, but the reviewed source passages do not prescribe this exact pair", "content": "# 490d C318 — Typed458 source-state decomposition\n\n24 September 2026. Step 5 of the ten-step authorization C314–C323.\n\nThe paired458 replacement has a fully typed decomposition on the three rows where C287 is admitted. Its source ingredients are permitted, but the reviewed source passages do not prescribe this exact paired selection.\n\n| Side | Explicit sequence | Net coordinate change |\n|---|---|---:|\n|Cumulative|LXX_OFF Shem_i→LXX_ON Shem_i→LXX_ON Shem_(i+2)|−460+2=−458|\n|Regular|SP Flood_g OFF/T0→MT Flood_g OFF/T0→MT ON→MT ON/T60→MT Flood_(g+1), same boundary|−650+130+60+2=−458|\n\nFull430 remains fixed. Each source-to-source step retains the selected start/close boundary. No person-companion is applied to MT or to Flood. The cumulative+2 is annual-member selection; the regular+2 is a permitted Noah/Shem/Flood Gear step. They are distinct operations.\n\nThe sign convention is explicit: cumulative coordinates are signed BC; regular values in this table are positive BC numbers, equivalently reflected coordinates. The corresponding unreflected signed regular changes would have opposite signs. Calling both actions−458 is valid only in this declared comparison frame.\n\nAll five rows admit the initial states and first three regular changes. Both final corners survive only i3/4/5, using SPG1/G2 and restored MTG2/G3. At i6/7 the cumulative final members8/9 and regularGear4 are absent. The formal arithmetic still computes a location there, but no source endpoint is added. Six exact-phase paths are source-admitted; four further component paths are formal failures at these boundaries.\n\nThe controlling identity is `650−130−60=460`. Therefore both sides have the same displacement and retain17K. The new short span is `600+(460−2)=1058=2K`, equivalent to the inherited `598+460`. File_18's regular variant fork, cumulative restoration rule, Flood table and restored-MT examples support the ingredients. This is a transparent relation between two previously admitted configurations, not an independent second derivation of every numeric premise.\n\n## Verification and limits\n\n57 main assertions and 14 frozen-input/predecessor checks passed. A second main replay produced byte-identical DATA and RESULTS. Separate review scope and agreement are recorded in the evidence. All frozen originals and standing controls remain unchanged. These are verification counts, not independent-witness counts.\n\nSigns and source roles remain explicit. Three admissible compositions do not authorize the missing annual members orGear4, nor establish a prescribed selector.\n\nMirror/529 remains the focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction x=1 January AD2; no civil year zero. Regular Gears remain Noah/Shem/Flood only. SP minus-one companion is local to SP Noah/Shem. No cumulative/upstream Gear, new endpoint or phase, fitted anchor, global axis, z-law, canonical or graph edit. Capture now; canonize later. Source state, event identity, generated coordinate and phase ancestry remain distinct. Exact arithmetic and source membership do not establish intention or independent historical witnesses.\n\n## Decision after this result\n\n**C319, proposed and unexecuted:** Hold all source contexts fixed and toggle the cumulative+2 annual selection and regular+2 Gear selection independently in a four-case sensitivity test. Which17K and all-K relations actually require coupling them?\n\nThe decomposition exposes two distinct choices. Testing them independently will distinguish a source-permitted rectangle from the extra selection needed for its2K+15K form.\n", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}]}