{"id": "c382", "title": "Moving-pivot mixed-square law for P and J", "filename": "490d_C382_Moving_Pivot_P_J_Mixed_Square_Law_20260924.md", "role": "Research action", "step": 382, "sha256": "fa7af1207bbb1d95ac57822958a7dfd502da038ffe39b9bd7a8099d920949562", "bytes": 3873, "origins": ["C01–C479/packets/C382/490d_C382_Moving_Pivot_P_J_Mixed_Square_Law_20260924.md"], "download": "/sources/fa7af1207bbb1d95ac57822958a7dfd502da038ffe39b9bd7a8099d920949562.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Mirror / 529"], "excerpt": "490d C382 — Moving-pivot P/J mixed-square law\n\n24 September 2026. Step 9 of 10 in the C374–C383 authorization.\n\nP and J obey an exact mixed-order law when their pivots move by the common source gain. All six admitted consecutive source-edge pairs support the identity as an affine map, not merely at the tested endpoint.", "content": "# 490d C382 — Moving-pivot P/J mixed-square law\n\n24 September 2026. Step 9 of 10 in the C374–C383 authorization.\n\n**P and J obey an exact mixed-order law when their pivots move by the common source gain.** All six admitted consecutive source-edge pairs support the identity as an affine map, not merely at the tested endpoint.\n\nLet J_A(x)=A+(300/299)(x−A), P_B(x)=B+(70/69)(x−B), and let both next pivots shift by h. Then\n\n`P_(B+h) ∘ J_A − J_(A+h) ∘ P_B = (B−A−230h)/20631`.\n\nThe two compositions have the same slope; the displayed difference is constant. They are equal precisely when `B−A=230h`. The sourced cases have h=+2 regular or−2 cumulative and pivot separation±460. This is the same equal-gain condition that produced the primitive230m/529m structure.\n\nWith fixed pivots, h=0, the order defect is±20/897. Therefore the result is not commutation of the original fixed-pivot P and J maps.\n\nA simple coordinate description explains the moving-pivot identity. At step t, subtract the accumulated shift th from the input frame and(t+1)h from the output frame. Both maps then become dilations about the same existing initial source point\n\n`x₀=A+299h=B+69h`.\n\nTheir ratios remain300/299 and70/69. Dilations about a common center commute. This is a re-expression of the finite source comparisons, not a new date anchor or a change to the global Mirror axis.\n\nThe complete finite path check contains24 P/J word routes over the four maximal source paths. A length-n word with p P-steps has affine form\n\n`x ↦ xₙ + P^p J^(n−p)(x−x₀)`.\n\nChanging the order while keeping the operator counts fixed leaves the whole affine map unchanged. Different counts still give different slopes, even though the one admitted starting source x₀ reaches the same ending source xₙ in every route. This distinguishes common source transport from equality of all operators.\n\nThe reduced composition law does not remove the established limits: source paths stop at their recorded boundaries; fixed-center phase transport remains distinct from matched components; and paired cumulative/regular continuation changes17K to17K+4. The result supplies a precise local composition structure, without establishing a global529-preserving action.\n\n## Verification and limits\n\n78 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\nThis is a finite moving-frame structure, not fixed-pivot commutation or a global physical-time action. Neither source-domain closure nor17K preservation is inferred from the mixed-square identity.\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**C383, proposed and unexecuted:** Consolidate C374–C382 in C383, verify the principal source joins and algebra using a separately written root calculation, replay the sealed packets, and publish the synthesis, checkpoint and recovery files. Leave C384 unexecuted.\n\nThe batch now has a bounded source census, a clear17K closure obstruction and an exact surviving moving-frame composition law. Consolidation and a verified recovery point are the best final authorized step.\n"}