The Geometric Sieve for Artin's Conjecture: Multi-Base Energy Inequalities, the Index Moment Obstruction, and the Spectral Bound on Non-Primitive-Root Density

This paper develops a geometric sieve for Artin's conjecture on primitive roots by exploiting the multi-base structure of the digit polygon framework. For each integer base b ≥ 2, the coset decomposition identity M₁, ₏ = Ab (p) + CEb (p) separates the unconditionally evaluated per-prime moment into a normalized polygon area and a non-negative coset energy that vanishes if and only if b is a primitive root mod p. The paper introduces the multi-base energy tensor — the joint coset energy across m bases — and proves a cross-base energy inequality bounding the sum of coset energies via a joint convexity constraint controlled by the index of the joint subgroup generated by two multiplicatively independent bases. Additional results include the spectral index multiplicity theorem (unconditionally sharpening the trivial count of primes of prescribed multiplicative order by a logarithmic factor), the 4-lag sieve reduction (encoding the full Artin obstruction in four autocorrelation values of the residue orbit), a spectral Selberg identity relating the second moment of coset energy to these four values, two conditional positive-density theorems stated purely in terms of digit polygon invariants, and a p-adic amplification principle providing an exponentially precise index detector via the tower of polygons for 1/pⁿ. All main identities are verified numerically for all primes below 50, 000 in bases 2, 3, 5, 7, and 10.