Toward a Geometric-Dynamical Reformulation of the Goldbach Partition Function

The strong Goldbach conjecture asserts that every even integer greater than 2 is the sum of two primes. Rather than presenting a proof of this conjecture, this paper proposes a geometric-dynamical reformulation of the associated Goldbach partition function.

For a finite cutoff X, let P_X be the set of primes p ≤ X, and define the finite configuration space C_X = P_X × P_X. The finite Goldbach map G_X sends a prime pair (p, q) to p + q. For even target values 2n under consideration, the Goldbach partition function can be written as r_G(2n) = #G_X^(-1)(2n).

The main proposal of the paper is to reinterpret G_X^(-1)(2n) as a geometric multiplicity: the number of prime-labelled trajectories, intersections, layers, or finite-time hitting events reaching a target region R_(2n) associated with the even integer 2n.

A more concrete finite prototype is proposed through a two-particle billiard model. For a cutoff X, each unordered pair of odd primes (p, q) is assigned a billiard table carrying two balls labelled by p and q. Even integers are represented by target regions or even area labels, and Goldbach representations are interpreted as finite-time hitting events by two-particle billiard trajectories.

As an initial toy geometry, the paper considers rectangular billiard tables. A rectangle with integer side lengths a and b has area ab, and different rectangular shapes may represent the same even area label. For example, 1 × 20 = 2 × 10 = 4 × 5 = 20. Thus the model separates the numerical target from the particular geometric shape: the goal is to cover all even area labels, independently of which rectangular realization represents them.

Because a billiard trajectory on a fixed rectangle does not move from one table to another, the appropriate finite phase space is viewed as a disjoint union over all admissible even area labels and all rectangular realizations of those labels. In this global phase space, the covering problem means that for every even area label, at least one component contains a two-particle prime-labelled trajectory that reaches the corresponding target region in finite time.

Inspired by the dynamics of translation surfaces and rigidity phenomena for the SL(2,R)-action on moduli spaces, the paper formulates a conditional covering principle. Rectangular billiards are used only as a first toy model and visualization device; their unfoldings are flat tori, so any deeper use of Eskin-Mirzakhani-type rigidity would likely require moving beyond rectangles to more general polygonal billiards or translation-surface families with nontrivial moduli.

The contribution of this paper is not a proof of Goldbach's conjecture. Rather, it is a conditional research program: to identify an intrinsic geometric encoding of primality and an independent, non-circular covering principle that would force every even target region or even area label to be reached in finite time. If such missing ingredients can be supplied, the conditional framework would imply the strong Goldbach conjecture.