BSD as reciprocity, written as a product with one term per rung

Recovered from the frozen log, where it existed only as an unfiled reading — the exact loss the foundation-from-a-digest finding was about (docs/CHARTER.md).

The Birch–Swinnerton-Dyer formula,

L^(r)(E,1)/r! = Ω · Reg · (∏_p c_p) · |Ш| / |E(ℚ)_tors|²,

is reciprocity for an elliptic curve: it connects the local half to the global half completely, and each factor is one rung's contribution to the product.

• r, the analytic rank (order of vanishing) — the free global fixed points, the global structure the local data does not see directly.
• Ω, the real period — the archimedean/magnitude rung: a period, placeable but not locally derivable.
• Reg, the regulator — the metric on the Néron–Tate lattice of global points.
• ∏_p c_p, the Tamagawa numbers — the local defect rung, one factor per bad prime (the singular fibres).
• |Ш|, the Tate–Shafarevich order — the anonymous obstruction: everywhere-locally-trivial classes with no global witness, of square order (a residue of a group action, [[N-012]]).
• |E(ℚ)_tors|² — the finite global torsion rung.

The local Frobenius counts a_p determine L(E,s) by modularity; continuation to s = 1 and the equality with the product is the reciprocity — the assertion that the local invariants, assembled over all places, reconstruct the global invariants exactly, with no residue left over. Written this way BSD is the product formula's shape at the level of an elliptic curve ([[EX-13]]): one term per rung, the local half and the global half forced to agree.