{"id": "c350", "title": "Composition order of the fixed-endpoint expressions", "filename": "490d_C350_Fixed_Endpoint_Composition_Order_20260924.md", "role": "Research action", "step": 350, "sha256": "ca01d610a0b904b7f2416854fdb0c79ede2f92f638784810c68059f91c581301", "bytes": 3174, "origins": ["C01–C479/packets/C350/490d_C350_Fixed_Endpoint_Composition_Order_20260924.md"], "download": "/sources/ca01d610a0b904b7f2416854fdb0c79ede2f92f638784810c68059f91c581301.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Calendar Keys", "Mirror / 529"], "excerpt": "490d C350 — Composition order of the fixed-endpoint expressions\n\n24 September 2026. Step 7 of 10 in the C344–C353 authorization.\n\nThe three existing fixed-endpoint expressions are order-dependent, and the bounded field contains no input supporting both complete composition orders. Their formal differences are exact and", "content": "# 490d C350 — Composition order of the fixed-endpoint expressions\n\n24 September 2026. Step 7 of 10 in the C344–C353 authorization.\n\n**The three existing fixed-endpoint expressions are order-dependent, and the bounded field contains no input supporting both complete composition orders.** Their formal differences are exact and independent of the chosen input.\n\nFor `f(x)=p+a(x−p)` and `g(x)=q+b(x−q)`,\n\n`f(g(x))−g(f(x))=(a−1)(b−1)(p−q)`.\n\nUse E_N for25/23 fixing SP Noah, J_R for300/299 fixing the original MT Flood, and J_M for300/299 fixing the original MT Noah. The J labels are local diagnostic abbreviations.\n\n| First expression after second, minus reverse order | Exact difference |\n|---|---:|\n| E_N after J_R | **92/299** |\n| E_N after J_M | **40/299** |\n| J_R after J_M | **−2/299** |\n\nBoth primary placements and their matching components give the same results. These are formal coordinate differences, not new endpoints, calendar residues or independent numerical witnesses.\n\nThe source relation table permits some one-sided compositions through a fixed point, but no input has both orders fully supported inside the eleven-node field. Commutativity of scalar multiplication therefore does not authorize exchanging these endpoint operations. This does not exclude larger structures outside the declared domain.\n\nThe regular-side structure is now explicit. The best next comparison returns to the cumulative side: its retained460+598 dissection may admit analogous600 repairs through existing annual members, without assigning cumulative Gears or moving any source lifespan.\n\n## Verification and limits\n\n144 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.\n\nFormal compositions beyond the admitted intermediate domain remain diagnostics. No global action, fitted anchor or canonical graph change.\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**C351, proposed and unexecuted:** Test the cumulative counterpart of the two598→600 repairs on C287’s existing ON-Shem/OFF-Shem/OFF-Flood triples. Check all four complete paths, preserve annual-index and Cainan distinctions, and compare the result with the regular primary repairs.\n\nThe regular source relations and their composition limits are settled. The retained cumulative460+598 dissection offers a directly relevant Mirror/529 comparison that uses no new endpoint search.\n"}