The Logarithmic Phase Coordinate of a Prime: Generational Sieve Structure and an Exact Second-Order Refinement of the Generalized Benford Law for Primes

For an integer base b >= 2, we attach to each prime p a coordinate Psi_b(p) = (g(p), theta(p)), its generation g(p) = floor(log_b p) and its logarithmic phase theta(p) = log_b(p) - g(p) in [0,1). We prove: (i) A single interval computation on this coordinate (the Window Theorem) determines the exact range of a prime's co-factors at first sieve activation; its two cases recover, as corollaries, a "closure" property when a prime's phase is below 1/2 and a "straddle" property when it is at or above 1/2, together with the exact phase sub-interval in each case. Verified computationally with zero exceptions across five bases (b = 2, 3, 5, 7, 10; 3,150 primes for closure, a further 1,392 for the window formula itself). (ii) The elementary Generation Sum Law -- that two primes' generations, multiplied, land in one of only two possible generations -- is the carry rule for adding two phases. (iii) The density of primes with phase below any fixed threshold theta, within a single generation I_k^(b), admits the expansion F_k(theta) = (b^theta-1)/(b-1) + D1(theta,b)/k + D2(theta,b)/k^2 + O(1/k^3) with D1 and D2 exact closed forms (D1 given explicitly; D2 given via six elementary moment integrals), derived by direct calculus -- integration by parts and an explicit, algebraically verified remainder identity -- with no assumed parametric family and no fitted constant. We show that (iii), evaluated at digit-boundary values of theta in base 10, reproduces at leading order the constant found by Hurlimann (2015, Corollary 4.1) for the size-dependent exponent of the classical Generalized Benford Law for primes, and that our second-order term is, by contrast, EXACT: we show by direct computation that the corresponding second-order matching condition in the original analytic argument of Luque and Lacasa (2009, eq. 4.23) has no real solution, so that no choice of their single fitted parameter can achieve second-order agreement, whereas our expansion achieves it unconditionally by construction. We further show, by an extensive literature search, that the continuous- theta formulation of (iii) and its explicit validity for every base b (not only b=10) do not appear in the surveyed literature, though we state this with the epistemic caveats appropriate to a literature search rather than a completeness proof. A distinct statistic -- the fraction of two-prime products landing on one versus the other admissible generation-sum value -- is shown to require the joint distribution of two phases rather than a projection of one, and is left explicitly open.