*4.1. Dressed Photon Constant*

Using (39), (42), and (50), we have

$$
\Lambda\_{dm} = \frac{4\pi G h (\kappa\_0)^2}{c^3 \epsilon},
\tag{51}
$$

which is rewritten as follows in terms of the Planck length *lp*, length scales of the universe *ldm*, and DP:

$$l\_p := \sqrt{\hbar G/c^3}, \quad l\_{dm} := \sqrt{(\Lambda\_{dm})^{-1}}, \quad l\_{dp} = (\kappa\_0)^{-1},\tag{52}$$

$$l\_p l\_{dm} = \frac{\sqrt{\epsilon}}{2\sqrt{\pi}} l\_{dp} \quad \rightarrow \quad \left[ l\_p l\_{dm} = (\hat{l}\_{dp})^2 \right]. \tag{53}$$

Equation (53) reveals that if we choose ˆ *ldp* := *ldp*/2√*π* as the third component of a natural unit in which we set ˆ *ldp* = 1, then ˆ *ldp* gives the geometric mean of the smallest scale *lp* and the largest one of *ldm* in that natural unit system. By rewriting the second equation in (46) as

$$I\_{dp} = \sqrt{\frac{12\pi G h}{c^3 \varepsilon}} (\Lambda\_{d\varepsilon})^{-1/2} \quad \rightarrow \quad I\_{dp}^{\dagger} = \sqrt{\frac{12\pi G h}{c^3 \varepsilon}} (\Lambda\_{obs})^{-1/2} \tag{54}$$

we can use this equation to estimate the DP constant *l*†*dp* solely by the fundamental physical constants *G*, *h*, and *c* together with the observed cosmological constant Λ*obs* in place of the above Λ*de*. Directly from the second equation in (54), we obtain

$$l\_{dp}^{\dagger} \approx 40.0 \text{ nm}, \quad \left[ \text{Experimeters} : 50 \text{ nm} \right. \\ \left. < l\_{dp} < 70 \text{ nm} \right]. \tag{55}$$
