Birch and Swinnerton-Dyer as Rational-Point Rank Resonance Under L-Function Readout — *Möbius-Fibonacci-Hybrid-Vibration Substrate-Coupling Architecture at the Elliptic-Curve-Substrate-Resolution-Class*

The Birch and Swinnerton-Dyer conjecture (BSD) proposes that, for an elliptic curve E defined over Q, the rank of the group of rational points E (Q) equals the order of vanishing of the associated L-function L (E, s) at s = 1: rank E (Q) = ordₒ=₁ L (E, s). This companion deposit proposes that BSD is the formalism-substrate-resolution-class readout of an underlying Möbius-Fibonacci-hybrid-vibration substrate-coupling-architecture (operator-canonical: "Möbius fibbs hybration") — the same architectural-substance that forms DNA-helix at biological-substrate-resolution-class (HLRP #133 Möbius substrate-architecture + HLRP #147 Planck-Fibonacci hybrid-vibration). Substrate-resolution-class perspective-relativity: > The infinity symbol viewed from above-substrate-formalism-resolution-class IS the Möbius-topology stacked-below-substrate-resolution-class. Two substrate-resolution-class-views of the same architectural-substance. The elliptic-curve-rational-substrate at mathematical-substrate-resolution-class is the formalism-substrate-readout; the Möbius-Fibonacci-hybrid-vibration substrate-coupling-architecture is the substrate-substance underneath. The architectural-mechanism: rational points on E form a substrate-coupling-architecture-recursive-substance (under the elliptic-curve group law) ; the L-function compresses this geometric-substrate-recursion into harmonic-substrate-readout. BSD is the IE-020 compression-equivalence between geometric recursion and harmonic readout. The architectural-junctures at the elliptic-curves have rhythm — probabilistic at the formalism-substrate-resolution-class-surface-edges (per Apex Deposit Corollary 0. 1, "curves don't exist as architectural substance"), architectural at the substrate-coupling-resolution-class. Cross-substrate manifestation: the Möbius-Fibonacci-hybrid-vibration substrate-coupling-architecture renders at five substrate-resolution-classes: BSD (elliptic-curve-mathematical-substrate, this paper), DNA-helix (biological-substrate, HLRP #133 + #147), Riemann (zeta-function-substrate, Companion II — sister architectural-mechanism), Yang-Mills (gauge-field-substrate, Companion III — sister architectural-mechanism), cardiac (HLRP #198 v3 empirical-validation at N = 21, 799, p < 10^-54). The conjectures are stated as internal consequences of the proposed grammar and are offered for proof, refutation, or refinement. Three-tier falsification + replication. Keywords: Birch and Swinnerton-Dyer conjecture; BSD; elliptic curves; rational points; L-function; analytic number theory; substrate-coupling architecture; Hydrogen Lifecycle Research Programme; HLRP; Möbius-Fibonacci-hybrid-vibration; Möbius fibbs hybration; DNA-helix-substrate; Planck-Fibonacci; rank-order resonance; IE-005 RECURSION; IE-020 Compression; cross-substrate perspective-relativity; architectural formalization