Take a prime p and compute the powers of 10 modulo p: 10, 100, 1000, … Eventually they cycle. When the cycle has length exactly p − 1—the maximum possible—we say 10 is a primitive root mod p. The decimal expansion of 1/p then has the longest possible repeating period.
Artin conjectured in 1927 that this happens for infinitely many primes. Nearly a century later, it remains unproved for any single value of a. Hooley proved it conditionally on GRH (1967). Heath-Brown showed at most two values of a can fail (1986), but cannot identify which. No unconditional proof exists.
The multiplicative order of 10 mod p grows without bound as p → ∞. One might hope this unbounded growth forces ord10(p) = p − 1 infinitely often. It does not. C10 proves that growth and maximality are logically independent: the first is a consequence of the prime number theorem, the second requires control of the divisor lattice of p − 1. Any viable strategy toward Artin must exploit this lattice structure.
For each prime q, there is an obstruction event: 10 fails to be a primitive root mod p when 10(p−1)/q ≡ 1 (mod p). C11 proves these events are exactly independent across every finite block—the splitting fields Kq are linearly disjoint. The Artin constant CArtin ≈ 0.3739 emerges as the natural product. The conjecture reduces to the absence of infinitely many Siegel zeros.
The sieve for Artin's conjecture has dimension κ = 0. This eliminates the parity barrier—the obstruction that blocks the twin prime sieve (κ = 2). The lower sieve yields an unconditional bound. The surplus—primes that survive the sieve but have defect index d(p) > 1—is decomposed into a controlled range and a residual range reduced to a single mean-value hypothesis (Hypothesis A), which is a weakening of a conjecture of Erdős. Verified computationally for k ≤ 104.
The Artin trilogy is self-contained: it cites no SDL paper, assumes no SDL axiom, and can be read by any number theorist with no exposure to constrained generative systems. But the papers would not exist without the laboratory practice that SDL built. Here is what transferred—not as content, but as method.
SDL trains the reflex of identifying the minimal independent conditions before seeking results. The Separation Theorem (C10) is a direct application: instead of attacking Artin directly, isolate growth from saturation and prove they are independent. The problem becomes two problems, each tractable.
The defect index d(p) = (p−1)/ord10(p) is the structural invariant of the problem. SDL habituates to asking: what is the quantity that does not depend on the agent, only on the structure? In Artin, the agent is the sieve; the invariant is d(p). The entire trilogy pivots on this single object.
The surplus in C12—primes that survive the sieve but are not primitive roots—is structurally analogous to the residual in a saturated CGS: the system has consumed all resources it can reach, but the outcome is not the target configuration. Recognizing this analogy guided the decomposition strategy: partition the surplus by the size of the smallest obstruction, exactly as regime analysis partitions trajectories by their dominant constraint.
Section 9 of C12—"What this paper establishes and does not establish"—is SDL methodology applied to mathematical writing. The discipline of stating the negative result (we do not prove Artin unconditionally; we do not prove Hypothesis A) with the same precision as the positive results is not standard practice. It comes from the principle that the boundary of what is known is as informative as the interior.
The trilogy's arc is one of progressive reduction. C10 reduces the problem space by separating two logically independent phenomena. C11 reduces the residual obstruction to Siegel zeros. C12 reduces it further to a single mean-value hypothesis. At each step, something is removed from the problem—not added to the toolbox. This is the SDL principle togliendo, non aggiungendo operating inside number theory.
Sieve dimension κ = 0 (no parity barrier). Unconditional lower sieve bound. Exceptional Siegel zero controlled via Goldfeld–Gross–Zagier. Surplus reduced to Hypothesis A. Hypothesis A verified computationally for k ≤ 104 (mean ω*(k) ≈ 0.67).
An unconditional proof of Artin's conjecture. Hypothesis A is the residual obstruction. It is connected to the Erdős conjecture and to the absence of a Bombieri–Vinogradov type theorem for the non-abelian family {Kq}.
The defect index d(p) is a quotient, not a product. The question of which structures are realizable as (p−1)/orda(p) for infinitely many primes is a question about the geometry of quotients—a perspective the existing literature has not explored.