Your findings about EDoF, Landauer limit, and GR scaling with ln(4) caught my attention. In our research, we have been working with a segmented spacetime model, where space naturally divides into four fundamental segments. We have described a deeper relationship between Pi and Phi as scaling factors in our model, which govern the natural segmentation of space.

You mention that multiplying E_Landauer or EDoF by the number of Planck areas recovers the energy from GR × ln(4).

In our approach, the factor ln(4) emerges naturally from a four-segmented space, which might suggest that gravity is an emergent process from fundamental information segmentation.

If gravity is tied to a four-bit emission process, could it mean that gravitational energy is inherently quantized through discrete informational steps rather than continuous curvature?

Your observation that larger black holes have lower EDoF energies suggests a logarithmic scaling of gravitational energy emission.

This aligns with findings in our research, where logarithmic spirals naturally emerge in the description of spacetime segmentation and gravitational structure formation.

Interestingly, the scaling of Phi in our segmented spacetime approach also leads to a natural emergence of logarithmic growth laws, which might indicate a deeper structural connection to gravity.

Have you considered whether the scaling of GR × ln(4) could be linked to a deeper logarithmic geometric structure? And do you see a connection between your four-bit emission model and logarithmic encoding of information in curved spacetime?

If you are interested, you can read our papers here on RG. They build on each other, which is why I recommend reading them one after the other.

Article Segmented Spacetime and the Natural Boundary of Black Holes:...

Preprint Segmented Spacetime -Solution to the paradox of singularities

Preprint Segmented Spacetime - A New Perspective on Light, Gravity an...

I see ln(4) as a four bit emission process. Showing that the constrained answers, forbidden answers see constrained by the position of the gravity bit carried by A’s counterfactual spin state. Guest B contains the Hawking photon which spin locks B for the next measurement. I am thinking that Higgs is a spin boson and is what adds the degree of freedom to the entangled system until symmetry breaks. I’m still considering g that, but it makes some sense. Take a look on the entanglement presentation. It has a good explanation of how entanglement potentiates virtual gravity bits and first particles possess the virtual Hawking photons. It’s much easier to see how it works with the Host Guest relations. Also, it strongly suggests we live in a super-determinstic universe. The hidden info is the fixture or almost like 3-2-1 nest created by the wave function. First a fixture must be loaded and then it’s measured into existence erasing the fixture until next interaction, aka superposition. Next interaction loads the fixture preparing the measurement and the next measurement brings the fixture into reality. Thanks for replying! Feel free to ask any questions. I think the model I posted is a complete story how gravity emerges first from cosmic horizon entropy management as relative gravity and turn how entangled spin measurements cause local gravity to emerge. Gravity is Nature’s expression for causal spin uncertainty.

The short answer is: Dimensional analysis.

If one starts with an area (in this case a ``Planck area''), by definition it's a length squared. Now if one has some other length scale-in this case the wavelength of some electromagnetic wave, aka that corresponding to the energy of a photon-it's clear that that area will be proportional to the square of that length, with some numerical coefficient(s).

This is, just, dimensional analysis. The physics lies in understanding the numerical coefficients-the numbers that can't be guessed by dimensional analysis and understanding what this has to do with gravity at all.

Any area is proportional to the square of any length, up to numerical coefficients. The real question is which area does it make sense to associate to the square of which length.

As Richard Feynman stressed, when asking a ``why...?'' question, one must first spell out what is assumed to be known and what isn't.

William Manuel This raises some questions about the role of information and spin in gravitational emergence. Your idea of ln(4) as a four-bit emission process strongly aligns with our work on segmented spacetime, where space naturally divides into four fundamental segments. If gravity truly emerges from entangled spin measurements, could it be that the fundamental segmentation of space itself determines the scale at which these processes occur?

The idea that the Higgs boson is a spin boson that adds degrees of freedom before symmetry breaking: Do you see a deeper mathematical link between the Higgs mechanism and the ln(4) scaling factor? Could the Higgs field itself be a consequence of a deeper segmented background?

A superdeterministic 3-2-1 nesting structure suggests that reality is revealed in discrete informational steps. Could this structure be an expression of a natural segmentation of space? We have observed Fibonacci-like cascading effects in our work on gravitational scaling. Do you think the emergence of gravity from spin uncertainty could be linked to a logarithmic encoding of information?

Your explanation that cosmic horizon entropy management drives "relative gravity" while local gravity emerges from entanglement makes a lot of sense. In our segmented spacetime approach, we also see a natural emergence of gravity through discrete quantized information steps.

If gravity is Nature’s expression of causal spin uncertainty, then perhaps its quantization follows a deeper logarithmic structure, similar to what we have found in our work.

We’d love to explore these ideas further with you!

Maybe you should have a look at Buckminster Fuller Synergetics, since the fine structure constant seems to directly emerge for vector sums of its tetrahedral constructs:

Preprint Correlation of the electromagnetic mechanics of elementary p...

It's always possible to establish a relation between an area, chosen beforehand, for no apparent reason, and the square of some length, also chosen for no apparent reason. They will be proportional with some numerical coefficients. This process doesn't-and can't-address the question of why such an attempt to relate that particular area with that particular length has any meaning, beyond being an exercise in dimensional analysis.

This can be done in an infinite number of ways.

The reason why it's useful to focus on the area of the horizon of a black hole at all is due to the so-called laws of black hole mechanics, that were formulated in the 1970s and led to the conclusion that it makes sense to assign to a black hole an entropy that would be proportional to the area of its event horizon. However there's no particular reason to relate the entropy and this area to the square of the wavelength of any electromagnetic wave in particular.