Is it the Gravitation circle here? I guess, so no better  place where to get attracted non earth  strings  attached, ops I almost dismissed the other … holy strings the supadupa strings of the unifying Theory  Quest ….
No joking to much, I hope.
The subject:

1) the Newtonian gravitation law supposes a intertial absolute frame, plus Cavendish experiment s and so on gave an experimental verification of F= k*(M*m)/(R^2) not too faar from human touch and normal sight …. 2) This is the ontic side of the issue, lets go at the ontological side:a) why interacting mass do not add up, but multiply their action? b) isn’t it possible hat the Newtonian law contains already enough  information aiming to quanto-gravitational theories?

The questions are linked, in my mind, of course, by a series of neural strings connecting: the product of masses to a statistical sight (probability of independent but related events – the coins & Binomial – multiply, 2a) by equating M to m --> m^2 woul be a perfect square of a probable mass operator”, c) where to disguised the hidden wave function? The best place were to look is a the Gravitational field, calculating the potential U= - k* (Mm/R).

The results requiring heavy critics and suggestions:

1) since equating the masses, in the sense of considering a single mass and its "virtual" double, is improper in Newtonian theory frame, could be that labeling somehow the equated mass by “I” the imaginary unity would have help. So I had done it.

U= - (k im*m)/R  -> (U/m) = -(k i m)/R. -> (U/m) + (k i m)/R =?.

Here is the critical point, peraphs since I deliberatly I tought that what =0 for real values, could = z in the complex frame i’m trying to get out  of / or fit in (?) the classic formula. I then equated  the upper right side to a possibly mock complex variable Z,  followed by rising the formula as a pover of “e”-> e^Z=e^(U/m) * e^(k I m)/R,
Recalling that Z=a+iB is equivalent to e^Z=e^a *(cos(b) +I sin(b)) (cfr Wunsh, Complex Variables p 96.97)
I proceeded conformably  

e^Z=e^(U/m) * (cos((km)/R) + I  sin ((km)/R)

The equation identify clearly a real and an imaginary component of the potential energy, and what most intrigues is the wave form of both:

                       Re                                           Img
e^Z = e^(U/m) * (cos((km)/R) + I e^(U/m)* sin ((km)/R)

Reonsidering the initial condition Z=0 the equation becomes
 

(e^-(U/m))= (cos((km)/R) + I e^(U/m)* sin ((km)/R)

Practically we are facing two orthogonal waves, the cos Real valued wave, and rotate of 90° by I multiplication sin wave, with a difference in period  of Pi/2.
The waves are mainly concentrated within the unit disk. It's there the maximal density and wide change in period spectra, because of R at denominator at the cos and sin arguments.

Though very clumsy, and bound to a thin hope about the legitimacy of introducing I when equating masses, guess the scenario offered at the end, is rather fascinating. Somehow it relates to the inner behavior of the mass its-self, and possibly with its peculiar way of originating as “a double vitual-real”. In my view m^2 instead of M*m, is the Unitarian subject of the gravitational law. though it must be intended analogically because of the dynamic double). The orthogonal waves, (vector  basis of a possible mass field?), oscillating asynchronously must show further consequences in terms of gravitational energy /mass distribution and mass motion and interaction: as a “particle” &/or as a “2 waves packets”.

Hope finding many critics to all this, and some help  to get things done properly.
Thanks for your attention.
Federiclet