Genuine Phantoms: |Sha| = 4 Through the BSD Compiler Copy

Abstract

We run the BSD compiler of Paper 142 on the elliptic curve 27606c1, which has analytic |Sha| = 4. Every ingredient — Fourier coefficients by point counting, L-value by convergent series, Tamagawa numbers by the Tate algorithm, real period by high-precision quadrature — is computed from the Weierstrass coefficients alone. The compiler outputs |Sha| = 4.0000000000: a perfect square. This extends the three-way comparison of Paper 143: genuine curves with |Sha| = 1 produce 1 (a perfect square), genuine curves with |Sha| = 4 produce 4 (a perfect square), and fake controlling functions produce non-integers. The compiler distinguishes genuine phantoms — Sha elements that carry topological content — from fake perturbations that carry none. The period computation for this curve requires particular care: its short Weierstrass form is nearly degenerate, with complex conjugate roots separated by only 1.7 × 10⁻⁴, creating a spike of height 23,000 in the period integrand that standard quadrature misses entirely. The resolution — multi-point splitting through the spike zone with 80-digit intermediate arithmetic — achieves 30-digit agreement with the LMFDB value. All code is provided in Appendix A.Copy

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