Information Manifolds and Pencil Projections: Structural Rigidity and Symmetry Invariants in the Modulo-12 Prime Lattice

This paper introduces a deterministic geometric formalization of prime numberdistribution through the ”Pencil Projection” methodology applied to a Modulo-12(M12) information manifold. Departing from traditional probabilistic frameworks,we characterize prime numbers as topological singularities emerging within a discrete lattice saturated by a sub-linear family of linear projectors. These projectorsare anchored at a fixed focal origin, F(−1/2, −1/2), which establishes the fundamental symmetry of the system.We demonstrate that the structural rigidity of the M12 lattice imposes threeprimary symmetry invariants: (i) the Riemann Hypothesis is identified as a necessary spectral reflection of the focal offset onto the Re(s) = 1/2 critical line; (ii)the Twin Prime Conjecture is formalized as a proximity invariance resulting fromthe geometric incommensurability of generating lines within modular corridors; (iii)Goldbach’s Strong Conjecture is redefined as a reflexive vector-sum invariant acrossthe manifold’s primary columns {5, 7, 11, 13}