What question did this study set out to answer?

To define and analyze Discrete Geometric Spaces (DGS) and identify the unique invariant π■.

June 23, 2026Open Access

The Minimal Discrete Geometry

Puntos clave

To define and analyze Discrete Geometric Spaces (DGS) and identify the unique invariant π■.
Introduced a class of integer-valued metric spaces on Z² following four structural axioms.
Defined the discrete invariant π■ and established its properties via independent invariants (combinatorial, algebraic, metric).
Proved that π■=1 corresponds to a unique metric, with its implications on dimensionality.
Demonstrated that the unique minimum value of π■ among all DGS is 1, achieved by the metric d■(P,Q)=|Δx|+2|Δy|.
Established that this realization is unique up to isomorphism, emphasizing its significance in the structure of DGS.
Proved that π■=1 necessitates the space to be two-dimensional, extending the concept of π in a discrete setting.

Resumen

We introduce the class of Discrete Geometric Spaces (DGS): integer-valued metric spaces on Z² satisfying four structural axioms. To each DGS we associate a discrete invariant π■, defined as the asymptotic ratio of sphere cardinality to radius, generalising the classical constant π≈3.14159 to discrete settings. Main Theorem: among all DGS, the value π■=1 is the unique minimum. It is achieved by the metric d■(P,Q)=|Δx|+2|Δy|, and this realisation is unique up to isomorphism. We establish this through three independent invariants (combinatorial, algebraic, metric), show that π■=1 forces dimension 2, and prove that π■ is a strict generalisation of the Euclidean π. This article is fully proven (12 theorems, no conjectures) and self-contained. It is the mathematical foundation of the π■=1 programme.

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