Ramanujan's Dimensional Forcing: A Universal Formula for Sieve Levels

We derive a universal formula for the level of distribution achieved by $k$-fold sieves with constitutional structure:
\[
    \theta_k = \frac{2^k - k}{2^k} = 1 - \frac{k}{2^k}.
\]

The formula rests on two independent pillars. The first is combinatorial: a counting theorem shows that a $k$-level nested constitutional sieve has exactly $k$ invalid configurations out of $2^k$ total, giving configuration density $(2^k - k)/2^k$. The second is geometric: for the case $k = 3$ (twin primes and cascade moduli $q = 3^K$), the algebraic structure of the Eisenstein integers $\mathbb{Z}[\omega]$ independently forces $\theta = 5/8$ via two mechanisms---the three-level filtration arising from the ramification $(3) = (1 - \omega)^2$, and the enhanced Kloosterman cancellation from the cyclic structure $(\mathbb{Z}/3^K\mathbb{Z})^* \cong \mathbb{Z}/(2 \cdot 3^{K-1}\mathbb{Z})$.

The formula unifies the classical Bombieri--Vinogradov barrier ($\theta_k^{(1)} = 1/2$, $k = 1$) and Pascadi's breakthrough ($\theta_k^{(3)} = 5/8$, $k = 3$, arXiv:2505.00653v2) as special cases of the same principle, and predicts the next achievable threshold at $k = 4$ ($\theta_k^{(4)} = 3/4$). The combinatorial origin of the formula is traced to the $\bmod{3}$ Sierpiński structure of Khayyam's Triangle [10]. We resolve a previously conflated distinction between pattern density and configuration density that explains a $12\%$ discrepancy in earlier estimates.

We are explicit about logical status throughout: the combinatorial pillar (Theorem 3.4) is fully proved; the geometric pillar (Theorem 5.7) proves $\theta_3 = 5/8$ for cascade moduli subject to Condition~W3 (Siegel zero exclusion), verified in the companion paper [14]; the correspondence between configuration density and analytic level of distribution for $k \geq 4$ is conjectural (Conjecture~6.4).