Paper 12: The Geometric Status of c in the Scale-Space Framework: Manifold Radius, Null Propagation, and the Nat as a Unit of Length

The speed of light c appears throughout the scale-space framework — in the scale velocity ˙s= 2GM/(Rc), in the exact identity L˙s= 2c, in the proper time formula, and in both the block-diagonal and corrected 5D metrics. Its role has been treated as inherited from special relativity: c is the speed of light, imported as a known constant. This paper examines what c actually is within the scale-space geometry.
The paper is organised around four items of distinct epistemic status, stated
explicitly at the outset.

    Derived result: the identity L˙s= 2c and its invariant. The product of the curvature radius L and the scale velocity ˙s is exactly 2c for every gravitating body, independent of mass, radius, or composition. c is the unique universal invariant of this relationship. This follows directly from the definitions of L and ˙s in Paper 1 and is the only genuinely new derived result of this paper.

    Definition: c as the manifold radius. We define c as the manifold radius of the scale-space configuration manifold C: the universal propagation constant, equal to the asymptotic null speed of the 5D Lorentzian parent M in the flat-space limit. This is a chosen definition, not a theorem. It is consistent with all established results and illuminates the role of c throughout the series, but it cannot be derived from the geometry without additional structure.

    Conjecture: the nat is a genuine unit of length. The nat appears dimensionless only because s is currently measured indirectly through spatial logarithms. We conjecture that the nat is a genuine unit of length in the s-direction, and that c in nats/s equals c in m/s at the reference scale s= 0. This is a conjecture, not a derivation: no direct s-measuring instrument exists, and the equivalence at the reference scale follows from a reference convention, not from the geometry.

    Interpretive proposal: the electromagnetic c measures the manifold radius. We propose that electromagnetic waves propagate at c because they are null disturbances of M, and that µ0ε0 = 1/c^2 is a geometric statement about the manifold’s base-state response. This is not established: Maxwell’s equations have not been derived from the scale-space geometry, and the electromagnetic field has not been introduced into the framework. This proposal identifies a future programme, not a current result.