← Radical-FDRS: One-Dimensional Flattening and Elliptic–Chabauty Machinery on the Fixed Slice / R11 and R12 · Sylow-2 refinement

Radical-FDRS: One-Dimensional Flattening and Elliptic–Chabauty Machinery on the Fixed Slice R11 and R12 · Sylow-2 refinement unsigned in the source

The Sylow-2 Double-Gate Refinement and the p = 4,195,001 Corrigendum, with the General Prime Search, the T_0 Gate-Swap Symmetry and Quotient Compression to the 2-Isogenous Elliptic Curve

R11: adds the omitted prime p = 4,195,001 (#E = 2^22, ord G0 = 2^18), refining the necessary condition to c mod 2^18 ∈ ±{1, 16385, 114687, 131071} (eight classes, two torsion cosets), and says explicitly that the six non-central classes are not claimed to contain rational points. R12: an exact prime scan finds p = 6,291,881 (projected order 2^19) which does not refine R11, because translation by T0 = (0, 0) swaps the two gates (so the two torsion cosets of R11 are one orbit); quotienting gives the linear square gate 2MU − L² in Q² on the 2-isogenous curve.

R11 is an omission fix that strengthens R9 and R12 is a computational checkpoint plus an exact algebraic symmetry theorem; both status lines say they are not proofs. The pre-publication read-through re-ran R11 (the eight classes reproduce and an independent scan finds this was the only omitted prime; 3, 5, 13 were dropped deliberately) and R12 (p = 6,291,881 is the only prime in [4.2×10⁶, 8.26×10⁶] with projected order ≥ 2^19; the others reach only 2^16 or 2^18).

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PreviousR10 · 2-adic local openness (a negative result) · R10: 2-Adic Local Openness, Sharp Boundary Neighborhoods and the Failure of Pure Formal-Group Isolation

NextR13–R16 · the canonical cover and saturation of the sieves · The Kummer-Twist Equivalence, the Canonical Elliptic-Chabauty Cover and Residue Transport, the Canonical-Cover Sieve, and the Canonical Second Elliptic Quotient Identified with the Historical E_−

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