We introduce a multiplicative geometric model built prior to the classical notion of prime. The model starts from a free multiplicative monoid of irreducible directions, endowed with a logarithmic height function. A critical regime is identified by the asymptotic law λᵢ ∼ log i (or, under mild regularity, an equivalent counting law N_λ (T) ∼ eT), and this regime selects the line ℜ (s) = 1/2 as the natural equilibrium line of the geometry. The associated spectral function transforms the study of cancellations into a problem of phase interference. The correct fine object is not the primary spectrum itself but the spectrum of differences. In the rigid model λᵢ = log i, that difference spectrum can be computed explicitly and is naturally indexed by positive rational quotients. We develop the local geometry near the diagonal, the dyadic multiscale decomposition of differences, partial quantitative derivations of the central lower bound H1 and the oscillatory loss H2 in the rigid model, and an angular anti-bipolarization principle for the central mass. We then add a new exact layer: the rigid difference measure admits a continuous effective quotient profile of Laplace type, with strictly positive Fourier transform, and the transfer from that continuous profile to the exact discrete model is reduced to a single weighted sawtooth-discrepancy problem over geometric progressions. We then add a complementary residual route internal to the model: a geometric sieve interpretation of the irreducible directions, a resonance-energy formalism on the difference spectrum, and a smoothed local-moment closure criterion showing that the exact discrete transfer is equivalent to a single short-window moment theorem in the rigid model. The final alignment theorem is stated as an internal theorem of the class of multiplicative geometric critical stable models. The manuscript is explicit about the logical status of each layer: fully proved identities, structurally motivated local results, partial rigid derivations of H1 and H2, a continuous effective positivity theorem, and a single sharply isolated analytic frontier for the exact discrete transfer.
Jaume Gotanegra (Mon,) studied this question.