The Geometric Sieve for Artin's Conjecture: Exact Four-Lag Reductions, Multi-Base Structural Constraints, and the Universal p-Adic Attractor

This paper applies the exact digit-polygon theory to Artin's primitive root conjecture by constructing an unconditional geometric sieve that decomposes the normalized digit-polygon area into a universal main term and an explicit orbit-restricted deficiency. The digit polygon of 1/p in base b carries a signed area whose normalization by p minus 1 is the central analytic object; when b is a primitive root modulo p the order equals p minus 1 and the normalized area is known to approach the universal constant (b² minus 1) /24. The paper begins from the exact four-lag formula, which expresses twice p-squared times the signed area as a linear combination of four orbit autocorrelations with explicit base-dependent coefficients. It then defines an unconditional full-group comparator by replacing the orbit-restricted autocorrelations, which sum only over the orbit of b in the multiplicative group, with full-group correlations that sum over all nonzero residues. The main theorem establishes that the full-group comparator equals (b² minus 1) /24 plus an error of order 1/p, unconditionally for every prime not dividing b, by computing two moment sums using a grid-counting lemma for digit functions and the equidistribution of fractions a/p in digit intervals. The exact geometric sieve identity then expresses the normalized orbit area as the full-group comparator minus a four-lag deficiency, which is itself an explicit linear combination of the gaps between full-group and orbit correlations at lags 0, 1, 2, 3. When b is a primitive root modulo p the orbit equals the full group, all four gaps vanish, and the deficiency is zero. For non-full-reptend primes the deficiency measures precisely how much the restricted orbit correlations deviate from their full-group counterparts. The unrestricted main term therefore accumulates over primes at the expected Artin density rate: the sum of the full-group comparator over primes up to x equals the Artin constant sigma times the prime-counting function plus an error of order log log x. The Artin conjecture thus reduces to proving sign and averaging control for the four-lag deficiency across primes. The exact joint-index formula for several bases is proved in the cyclic group: the index of the subgroup generated by r bases b₁ through bᵣ in the multiplicative group modulo p equals the gcd of their individual indices. This gives an exact structural constraint for multi-base problems, though it implies no cross-base energy inequality for the digit-polygon areas. The paper then integrates exact results from the companion digit-polygon series: the exact CRT area law for products pq, expressing the area of the digit polygon of 1/ (pq) as a combination of the individual areas, a cross-term governed by the polarization of the area form, and an exact CRT defect; the spectral orthogonality criterion showing the cross-term vanishes when the gcd of the individual periods is at most 2; and the exact order-lifting theorem and block-average decomposition for prime-power denominators. A worked counterexample demonstrates that the naive scaling relation claiming the area of 1/pⁿ equals p^ (n-1) times the area of 1/p is false. The universal p-adic attractor theorem states that for every odd prime not dividing b, the normalized area of the digit polygon of 1/pⁿ converges to exactly (b² minus 1) /24 along the p-adic tower, with an explicit error rate of order p^ (- (n-t) ) where t is the Wieferich index. Since this limit is the same regardless of whether b is a primitive root modulo p, the raw normalized tower areas cannot serve as an asymptotic detector for the Artin index. The genuine remaining open problems are identified precisely: proving the nonnegativity of the four-lag deficiency, establishing its vanishing as a characterization of full-reptend primes, and controlling the average of the orbit-correlation gaps over primes.