How could Planck length be actually equal to much larger CP_{2} radius?!
The following argument stating that Planck length l_{P} equals to CP_{2} radius R: l_{P}=R and Newton's constant can be identified G= R^{2}/ℏ_{eff}. This idea looking nonsensical at first glance was inspired by an FB discussion with Stephen Paul King.
First some background.
 I believed for long time that Planck length l_{P} would be CP_{2} length scale R squared multiplied by a numerical constant of order 10^{3.5}. Quantum criticality would have fixed the value of l_{P} and therefore G=l_{P}^{2}/ℏ.
 Twistor lift of TGD led to the conclusion that that Planck length l_{P} is essentially the radius of twistor sphere of M^{4} so that in TGD the situation seemed to be settled since l_{P} would be purely geometric parameter rather than genuine coupling constant. But it is not! One should be able to understand why the ratio l_{P}/R but here quantum criticality, which should determine only the values of genuine coupling parameters, does not seem to help.
Remark: M^{4} has twistor space as the usual conformal sense with metric determined only apart from a conformal factor and in geometric sense as M^{4}× S^{2}: these two twistor spaces are part of double fibering.
Could CP_{2} radius R be the radius of M^{4} twistor sphere, and could one say that Planck length l_{P} is actually equal to R: l_{P}=R? One might get G= l_{P}^{2}/ℏ from G= R^{2}/ℏ_{eff}!
 It is indeed important to notice that one has G=l_{P}^{2}/ℏ. ℏ is in TGD replaced with a spectrum of ℏ_{eff}=nℏ_{0}, where ℏ= 6ℏ_{0} is a good guess. At flux tubes mediating gravitational interactions one has
ℏ_{eff}=ℏ_{gr}= GMm/v_{0} ,
where v_{0} is a parameter with dimensions of velocity. I recently proposed a concrete physical interpretation for v_{0} (see this). The value v_{0}=2^{12} is suggestive on basis of the proposed applications but the parameter can in principle depend on the system considered.
 Could one consider the possibility that twistor sphere radius for M^{4} has CP_{2} radius R: l_{P}= R after all? This would allow to circumvent introduction of Planck length as new fundamental length and would mean a partial return to the original picture. One would l_{P}= R and G= R^{2}/ℏ_{eff}. ℏ_{eff}/ℏ would be of 10^{7}10^{8}!
The problem is that ℏ_{eff} varies in large limits so that also G would vary. This does not seem to make sense at all. Or does it?!
To get some perspective, consider first the phase transition replacing hbar and more generally hbar_{eff,i} with hbar_{eff,f}=h_{gr} .
 Fine structure constant is what matters in electrodynamics. For a pair of interacting systems with charges Z_{1} and Z_{2} one has coupling strength Z_{1}Z_{2}e^{2}/4πℏ= Z_{1}Z_{2}α, α≈ 1/137.
 One can also define gravitational fine structure constant α_{gr}. Only α_{gr} should matter in quantum gravitational scattering amplitudes. α_{gr} wold be given by
α_{gr}= GMm/4πℏ_{gr}= v_{0}/4π .
v_{0}/4π would appear as a small expansion parameter in the scattering amplitudes. This in fact suggests that v_{0} is analogous to α and a universal coupling constant which could however be subject to discrete number theoretic coupling constant evolution.
 The proposed physical interpretation is that a phase transition hbar_{eff,i}→ hbar_{eff,f}=h_{gr} at the flux tubes mediating gravitational interaction between M and m occurs if the perturbation series in α_{gr}=GMm/4π/hbar fails to converge (Mm∼ m_{Pl}^{2} is the naive first guess for this value). Nature would be theoretician friendly and increase h_{eff} and reducing α_{gr} so that perturbation series converges again.
Number theoretically this means the increase of algebraic complexity as the dimension n=h_{eff}/h_{0} of the extension of rationals involved increases fron n_{i} to n_{f} and the number n sheets in the covering defined by spacetime surfaces increases correspondingly. Also the scale of the sheets would increase by the ratio n_{f}/n_{i}.
This phase transition can also occur for gauge interactions. For electromagnetism the criterion is that Z_{1}Z_{2}α is so large that perturbation theory fails. The replacement hbar→ Z_{1}Z_{2}e^{2}/v_{0} makes v_{0}/4π the coupling constant strength. The phase transition could occur for atoms having Z≥ 137, which are indeed problematic for Dirac equation. For color interactions the criterion would mean that v_{0}/4π becomes coupling strength of color interactions when α_{s} is above some critical value. Hadronization would naturally correspond to the emergence of this phase.
One can raise interesting questions. Is v_{0} (presumably depending on the extension of rationals) a completely universal coupling strength characterizing any quantum critical system independent of the interaction making it critical? Can for instance gravitation and electromagnetism are mediated by the same flux tubes? I have assumed that this is not the case. It it could be the case, one could have for GMm<m_{Pl}^{2} a situtation in which effective coupling strength is of form (GmMm/Z_{1}Z_{2}e^{2}) (v_{0}/4π).
The possibility of the proposed phase transition has rather dramatic implications for both quantum and classical gravitation.
 Consider first quantum gravitation. v_{0} does not depend on the value of G at all!The dependence of G on ℏ_{eff} could be therefore allowed and one could have l_{P}= R. At quantum level scattering amplitudes would not depend on G but on v_{0}. I was happy of having found small expansion parameter v_{0} but did not realize the enormous importance of the independence on G!
Quantum gravitation would be like any gauge interaction with dimensionless coupling, which is even small! This might relate closely to the speculated TGD counterpart of AdS/CFT duality between gauge theories and gravitational theories.
 But what about classical gravitation? Here G should appear. What could the proportionality of classical gravitational force on 1/ℏ_{eff} mean? The invariance of Newton's equation
dv/dt =GM r/r^{3}
under h_{eff}→ xh_{eff} would be achieved by scaling v→ v/x and t→ t/x. Note that these transformations have general coordinate invariant meaning as transformations of coordinates of M^{4} in M^{4}×CP_{2}. This scaling means the zooming up of size of spacetime sheet by x, which indeed is expected to happen in
h_{eff}→ xh_{eff}!
What is so intriguing that this connects to an old problem that I pondered a lot during the period 19801990 as I attempted to construct to the field equations for Kähler action approximate spherically symmetric stationary solutions. The naive arguments based on the asymptotic behavior of the solution ansatz suggested that the one should have G= R^{2}/ℏ. For a long time indeed assumed R=l_{P} but padic mass calculations and work with cosmic strings forced to conclude that this cannot be the case. The mystery was how G= R^{2}/ℏ could be normalized to G=l_{P}^{2}/ℏ: the solution of the mystery is ℏ→ ℏ_{eff} as I have now  decades later  realized!
See the chapter Some Questions Related to the Twistor Lift of TGD or the article About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant.
