In a nutshell, more complete math details to follow,

The empirical data from Pioneer 10/11 probes out at 20-70 AU to appears to show a CONSTANT acceleration ~ cH ~ 10^-7 cm/sec^2 radially directed back toward the Sun, i.e. the approximate region of origin of the probes.

It is trivial to model this approximately as a post-inflationary seed shorter-scale dark energy effect consistent with the Galactic Halo observations.

We do not need any fancy math of fractal geometries, Clifford algebras, Finsler spaces etc, nor do we we have to posit max and min accelerations at this point for this particular set of observations. Battle-tested General Relativity is sufficient and we want more from less with a minimum of excess theoretical baggage, no matter how beautiful, consistent with the principle of parsimony. This is not to say, however that the approaches taken by Castro, Smith, Notalle et-al may not be useful in terms of practical empirical physics. I am not ruling that out. However, I do not see any compelling argument in Carlo's latest paper explaining WHY Pioneer data is showing the minimum acceleration in the sense of his Ansatz of fractal scale-invariant force = INVARIANT STRING TENSION

mpc^2/Lp = Muc^2/R = c^4/G = STRING TENSION = SPACE-TIME STIFFNESS = MACRO-QUANTUM GOLDSTONE PHASE RIGIDITY

Where Mu is "mass" of Universe needed to agree with Inflationary spatial flatness Omega = 1. Of course. only 0.04 of that 1 is from real on mass shell particles i.e. mostly nucleons - 0.96 of that Omega = 1 is from VIRTUAL exotic vacuum, i.e. UNCOMPENSATED random micro-quantum zero point vacuum fluctuation energy density and pressure with w = -1 from INFLATIONARY SEED FLUCTUATIONS in the MACRO-QUANTUM Higgs Ocean "vacuum condensate" out of which Einstein's local smooth geometrodynamic field equations from 1915 naturally and beautifully emerge. Generalizations to Gennady Shipov torsion compensating gauge fields, conformal gravity compensating gauge fields in sense of Tony Smith, resolution-dependent wavelet frame shifts (Nottale), deformations of quantum group algebra with q as in Carlos Castro's latest paper can all be done later from my paradigm IF observations, like the Pioneer anomaly, demand it - as well they may. But for now we can get a lot of insight into the Pioneer anomaly without all that excess baggage. The Pioneer data may be as important as the bending of light by gravity with the non-Newtonian factor of 2 that comes from w = +1/3 for real photons as distinct from virtual zero point fluctuations of all quanta that obey w = -1 because of General Relativity.

There are gaps in the knowledge of how the galaxies and stars form in the early universe. I think now I have a simple general insight into how that happens in general that I will illustrate with a simple model giving the observed Pioneer anomaly with a direct simple explanation of the DIRECTION of the anomaly that does not require a year long graduate course in Geometric Algebra to quickly understand. ;-)

CONJECTURE: Every massive body in the Universe is in an exotic vacuum effective gravity well or landscape attractor left over from the pre -> post inflationary false -> true vacuum phase transition from the unstable CONFORMAL globally flat micro-quantum Dirac Sea of virtual MASSLESS fermion-antifermion pairs and their MASSLESS virtual gauge bosons, to the macro-quantum coherent "Higgs ocean" with MASSIVE fermion-antifermion quasi-particles and MASSIVE W bosons and of course Einstein's gravity emergent as a coherent ODLRO effect well known in soft-condensed-matter physics as part of the "More is different" Ansatz of emergent complexity formulated by P.W. Anderson in 1967 that adds essential details missing in Andrei Sakharov's (also 1967) notion of connecting Einstein's gravity to zero point energy.

OK let's cut to the chase with a simple model for the Pioneer 10/11 Anomaly.

Imagine the INFLATIONARY SEED BIG BANG REMNANT exotic vacuum spherical halo centered at the Sun's center with the functional static local field dependence

/\zpf = [Lp(1 + R(t))]^-1[Lp + r']^-1

where r' = radial distance to Sun. R(r) is the dimensionless FRW cosmological scale factor of precision cosmology. We can use Newtonian potential theory here to a good approximation over the 100 AU scale. The Green's function G(r - r')solution to Newton's Poisson field equation for the universal potential energy per unit test mass

V(r) = c^2Integral{G(r - r')/\zpf(r')d^3r']

for this spherically symmetric exotic vacuum halo centered at the Sun is pure S-wave in the multipole expansion into Legendre polynomials gives when R(t) >> 1 as it is today and r >> Lp = 10^-33 cm

V(r) ~ +c^2r/LpR(t)

This is LINEAR in r the radial distance of Pioneer back to the Sun.

At this moment in cosmic time t, LpR(t) ~ 10^28 cm.

The ATTRACTIVE QUASI-CONSTANT Newtonian force from this POSITIVE exotic vacuum zero point energy density of NEGATIVE pressure is the negative gradient i.e.

a_P = -c^2/LpR(t) ~ 10^-7 cm/sec^2 directed back to the center of the Sun.

This is basically Physics 101 in the Cornell Tradition from the late 1950's of getting important physical results with as little math as possible as taught by Hans Bethe, Phil Morrison, Ed Salpeter et-al.

The power law dependence is positive, which is why the DARK ENERGY SOLAR HALO is an attractive well speeding up the early concentration of matter in the formation of the Sun. If the power law dependence was with a negative exponent, we would get ANTI-GRAVITY universal repulsion instead - an exotic vacuum instability.