This paper reports a set of empirical scaling relations spanning 35 orders of magnitude and proposes observer compilation as a unifying perspective. We find that characteristic lengths including the proton charge radius (Rp≈0.84 fm), the reduced Compton wavelength of the electron (λe≈0.386 pm), and the Bohr radius of the hydrogen atom (a0≈52.9 pm) can all be uniformly expressed asL=C⋅mmP⋅ℓPwhere ℓP is the Planck length, mP is the Planck mass, m is the mass of the corresponding particle, and C is a simple geometric factor (e.g., 1,4,α−1, with α being the fine-structure constant).
Taking the proton radius as an example,Rp=4⋅mpmP⋅ℓP=0.8412 fmwhich deviates by less than 0.04% from the experimental value 0.84087(39) fm measured via muonic hydrogen spectroscopy.
Furthermore, we propose a discrete geometric picture: the intrinsic radius of a quark is 1ℓP; three quarks form a nucleon through close packing, with an intrinsic repulsive core radius of 2ℓP; the geometric contact distance between two nucleons is 4ℓP; the force range extends to 8ℓP; and the interaction decays to a minimum at 9ℓP and switches to electromagnetic dominance.
This integer sequence (n=1,2,4,5,6,7,8,9) is in precise agreement with asymptotic freedom, quark confinement, the repulsive core, the attractive peak, transitional steps, the force-range boundary, and the electromagnetic switching point of the nucleon–nucleon potential well.
Based on the above observations, we put forward a working hypothesis: all fundamental interactions at apparent distances originate from the same Newtonian gravitational force GmP2/R2, and their strength differences are determined solely by geometric contact factors (the strong force corresponds to 1, the electromagnetic force to α, the weak force to 34πα∣1−k∣, where the value of 34πα coincides exactly with the decimal part of the reciprocal of the fine-structure constant, ε≈0.036; gravity corresponds to (M/mP)2).
This paper is framed as phenomenological reasoning and does not claim to be a complete theory. Rigorous dynamical derivations will be provided in a separate article. It does not seek to replace QCD or the electroweak theory, but offers a complementary geometric perspective, presenting only these numerical relations, their geometric interpretations, and falsifiable experimental predictions (the discrete steps of nuclear force).
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