Universal Internal Coherence: pi, c, and the Geometry of Light

Here, I propose a reformulation of physical invariants grounded on a new principle: the Universal Internal Coherence (UIC). This principle asserts that apparent constants in physics, such as pi and c, are locally invariant but globally relational quantities—emergent from the geometry and causal structure of spacetime itself. We show that pi encodes the curvature-dependent geometry of space, while c represents the causal coherence of time. Their product, pi·c, remains invariant across frames, linking geometric curvature and temporal causality through a single, covariant coherence condition.

This framework unifies the constancy of pi and c under a shared geometric origin and implies that all so-called “constants” are phase-dependent relational invariants of the spacetime manifold.

Figure 1. Curvature-dependent geometry of $\pi(x)$.

The function $\pi(x)$, defined as the ratio between circumference and diameter on a spatial slice of curvature $R(x)$, decreases in positively curved regions and increases in negatively curved ones. Locally, every observer measures a constant $\pi_0$ within their tangent space, but globally $\pi(x)$ encodes the curvature of spacetime. This illustrates how geometric constancy is an artifact of local flatness under the Universal Internal Coherence principle.

Figure 2. Causal coherence $c(x)$ across spacetime curvature.

The local causal speed $c(x)$, interpreted as the limit of light propagation along null geodesics, varies inversely with curvature. Where space bends ($R>0$), causal coherence increases to maintain global invariance. The inverse relationship between $\pi(x)$ and $c(x)$ ensures that their product $\pi(x)c(x)$ remains constant, preserving coherence between geometry and causality.

Figure 3. Big Bang–Black Hole duality under Universal Internal Coherence.

Both cosmological and gravitational singularities correspond to extremal coherence states. At the Big Bang, spatial geometry vanishes ($\pi \to 0$) and causal connectivity diverges ($c \to \infty$). At a black hole horizon, the same limit occurs locally. The constancy of $\pi c$ across these regimes reveals a hidden symmetry: creation and collapse are dual manifestations of the same coherent structure of spacetime.