We present spherical cow philosophy—a methodological framework for mathematical discovery that complements classical proof-by-contradiction by accepting empirical constitutional constraints as foundational and deriving their forced consequences. Named after the physics heuristic of modelling complex systems through essential simplification, the approach inverts the usual hierarchy: instead of deriving special cases from universal axioms, we identify structural constraints that are empirically universal within a domain, derive what they force rigorously, and validate predictions across independent domains. The framework addresses a specific situation: when classical sieve methods encounter persistent structural barriers---such as the $\theta = 1/2$ ceiling for twin primes over two centuries of effort---the productive path may be to accept the constitutional constraint (here, $p \equiv 2 \pmod{3}$ for all twin primes $p > 3$) as the natural entry point and derive its arithmetic consequences, rather than to approach the barrier as an obstacle for more powerful general methods. Applied to twin primes, this methodology motivates the cascade moduli framework proved in the companion paper [12], achieves level of distribution $\theta_W = 5/8$ for cascade moduli $q = 3^K$, and connects the arithmetic result to a structural parallel in information theory documented in [14]. The combinatorial foundation is traced to the $\bmod{3}$ fractal of Khayyam's Triangle [10]. The framework is explicitly falsifiable: it predicts a confirmed non-instance (Sophie Germain primes at $\theta = 1/2$) and an open prediction ($k = 4$ constellation at $\theta = 3/4$) via the $k$-hierarchy of [14]. We establish precise criteria for when the framework applies, contrast it with classical approaches, and distinguish it carefully from circular or post-hoc reasoning. Logical status labels are attached to every claim.