The Geo-Matter Duality (GMD) of the Theory of Entropicity (ToE)

1. Geometry: 

The Fisher–Rao information metric gI, constructed from S(r), generates the physical spacetime metric gS via the emergence map gS = λ gI. The Schwarzschild geometry – encoded in A(r) and B(r) – is thus a manifestation of the amplitude structure of the entropic field.

2. Matter (mass): 

The parameter S₁ in the entropic profile S(r) = S₀ + S₁ / r is interpreted, through the weak–field potential Φ(r), as the mass M of the Schwarzschild solution. The mass is therefore an emergent dynamical attribute of the same entropic field, not an independent ontological input.

In this sense, the Schwarzschild solution shows explicitly how, in ToE, what general relativity treats as “geometry” (the metric) and “matter” (the mass parameter M) both arise from a single entropic structure. Geometry and matter are complementary manifestations of the entropic field, realizing the geometry–matter (geo-matter) duality at the level of a familiar classical solution.