Mordred

Really lets look at this equation quite frankly your describing an ideal gas. The critical density formula is an ideal gas solution. Where is there any term in those equations that does not describe a homogenous and isotropic fluid ? You stated were not to argue ontology or philosophy how about straight up mathematics ? typical response perhaps You can understand my frustration when I receive responses such as this. Why is it anytime I so much as mention anything outside your papers you respond negatively is beyond me when you also claim a desire to improve your applications of simplifying physics yet choose to ignore a huge bulk of those physics relations or react negatively when they are mentioned. That is what has been the real frustration in trying to discuss your papers. What you choose to do with your work is your choice I'm simply describing what I see and feel should be included in better detail if they are included.

flour, baking powder with a bit of salt and enough water or milk to make a soft dough then fry it with a bit of butter or oil add other ingredients to suit your taste I particularly like adding blueberries

look directly at many of your derivatives that uses critical density The formula itself is derived with a homegenous and isotropic fluid and applying Jeans instability. The FLRW equations and treatments you have shown I don't recall seeing the curvature equation of state nor have I seen a gradient operator. Did I miss some form of gradient operator ? Do you have some form of parallel transport ? I don't recall seeing anything along those lines. I do recall one of your lemmas on maximally symmetric which isn't curvature. Its against your minimalization to employ tangent vectors as that is a form of curve fitting along with affine connections.

I find the term traditional scientist a bit confusing. One of the earliest formal lessons I learned is never rely on any given theory. Its one of the reasons I strive to learn as many different theories as I can. However no matter how outstanding tried and true any theory is all it takes is sufficient evidence to overturn any theory. Regardless if its considered a law or not. The other lesson I learned is any viable theory has some likely-hood of being correct. So if flexibility is traditionalist then I for one will continue in that manner as it maintains an open mind to other possibilities.

for quick easy snacks I often make bannock and add various berries, chocolate chips or whatever is at hand I think would go well with it lol.

let me ask you a couple of questions it may help understand why I was pushing for equations beyond scalar relations. I will be 100% honest with you, whenever I look at your relations I see no involvement of spacetime curvature itself. you can take every equation and relation you have posted here and if you were to parallel transport 2 or more separate laser beams your equations don't include how those beams will converge or diverge due to curvature. The simplest example is the center of mass. Take 2 objects and drop em. Yes they will fall at precisely the same rate however the distance between them decreases as they approach the center of mass. In regards to CCP ( Cosmological coincidence problem this directly relates as the Rayleigh equations are considered a solution). Second example take your Mercury Sun system. Do you show how we can see Hades star system directly behind the sun at specific locations and times ? in regards to pressure itself. Yes the Gamma law equations pertaining to Boltzmann are straightforward even when applying Van de Walls corrections if your spacetime is flat they work great but what if the area your examining is not homogeneous and isotropic where the thermodynamic state will not expand in an adiabatic and isentropic thermodynamic state ? The 3P relation specifically equates to 3 dimensional flux in a given area and will deviate if say the x axis component has a density difference from the Y and Z components. The critical density formula only works in situations where the cosmological principle applies accurately. As our universe is becoming more and more anistropic sometime in the future it will no longer be accurate. Take Doppler type shifts Doppler, gravitational, and cosmological redshift. they each naively look the same but when you get to higher examinations they become rather distinctive especially in causation. One excellent example is late time integrated Sache-Wolfe effect. Your signals are subjective to additional blueshift and red-shifting as the signal crosses underdense and overdense regions. I could go on and on describing situations but you should see the point above. Fundamentally though its your work, your ontology, your philosophical approach. How far you choose to take it is your choice and not mine to make. There isn't any FAULT with your mathematics they simply do not show the above. This is where your minimalization places restrictions on what dynamics you can describe. So the choice is simple ( ignore the situations where they can't ) or find ways to expand your treatments to encompass factors that equate to vector relations. Your call it is your work not mine. edit with some of the above one can apply a gradient much of your work also relates to the following. Principle of general Covariance. (all physics theories must account for this) https://en.wikipedia.org/wiki/General_covariance with CCP a fundamental question one can ask is " if the fundamental constants so easy to understand and develop relations to why is unification so hard ? "

I realize you likely have me blocked or simply refuse to respond to me but for others even if you do not respond ( which is fine) the above looks remarkably familiar with the cosmological coincidence problem. If you do see this then my recommendation is that you pursue that direction of research. Your approach on the last post is better than many I have read. So +1 for that. Edit should be more specific the fine structure constant , gravitational constant as well as numerous other fundamental constants are also related. This includes DM and DE in some examinations. Granted it likely wasn't your goal but it would work well into that line of research.. Truth be told though it was @KJW earlier comment that brought the above to mind

Got a solid good laugh today one of my nieces when told to do her math homework stated math is " mental abuse to humans". Couldn't argue that logic lmao

Just a friendly FYI the fine structure constant has been suggested in terms of the cosmological constant in the Machian universe as well as BSBM ( a form of TeVeS aka MOND) as well been looking into current constraints as it is interesting. ( I seem to recall a variation of LQC that also ties it in but I could be wrong on that). BSBM paper relating to above https://arxiv.org/pdf/1202.0069

Lmao thanks Must have been half asleep lmao

thanks was worried on title length at the time forgot to edit it after initial post. Corrected above

Chanced upon this rather interesting application of gravitational waves to determine Hubble constant. Thats definitely one uses I would never have thought of lmao. Measuring intervening density ( underdense, overdense) I have thought of a few times but not Hubble constant. https://arxiv.org/pdf/2503.01997 It is nice to see that this also applies the evolutionary density to Hubble constant via E(z)=\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z^2)+\Omega_\Lambda} which is used in a wide range of measurement related equations for far field measurements. (Beyond Hubble horizon ). Note use of E here is not energy, it is expansion rate at a given Z instead of using H_z and avoiding confusion with H_o) Hubble constant today on latter with prior as Hubble constant as function of redshift. For late times such as universe in Lambda dominant era and curvature term k=0 the above expression simplifies to E_z=\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}

Top quark pairing observed. https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2025-008/ Relevant arxiv copy of first linked article of above news article https://arxiv.org/abs/2601.11780

One useful time saver though is to check if equations originated elsewhere this is about the few time saver aspects I could see AI useful in this regard

Expansion is largely a thermodynamic process. Here to check your numbers \[{\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&Scale (a)&S&T (Gyr)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&V_{now}/c&V_{then}/c&H(t)&Temp(K)&rho(kg/m^3)&OmegaM&OmegaL&OmegaR&OmegaT \\ \hline 1.09e+3&9.17e-4&1.09e+3&3.71e-4&6.25e-4&4.53e+1&4.15e-2&5.67e-2&8.38e-4&3.13e+0&6.64e+1&1.56e+6&4.59e-18&2.97e+3&7.56e-1&1.29e-9&2.44e-1&1.00e+0\\ \hline 9.70e+2&1.03e-3&9.71e+2&4.52e-4&7.55e-4&4.52e+1&4.65e-2&6.36e-2&1.03e-3&3.13e+0&6.16e+1&1.29e+6&3.15e-18&2.65e+3&7.77e-1&1.88e-9&2.23e-1&1.00e+0\\ \hline 8.63e+2&1.16e-3&8.64e+2&5.48e-4&9.11e-4&4.51e+1&5.22e-2&7.14e-2&1.26e-3&3.12e+0&5.73e+1&1.07e+6&2.17e-18&2.36e+3&7.97e-1&2.74e-9&2.03e-1&1.00e+0\\ \hline 7.68e+2&1.30e-3&7.69e+2&6.65e-4&1.10e-3&4.50e+1&5.85e-2&8.01e-2&1.53e-3&3.11e+0&5.34e+1&8.91e+5&1.49e-18&2.10e+3&8.15e-1&3.97e-9&1.85e-1&1.00e+0\\ \hline 6.83e+2&1.46e-3&6.84e+2&8.06e-4&1.32e-3&4.49e+1&6.56e-2&8.98e-2&1.87e-3&3.11e+0&4.97e+1&7.41e+5&1.03e-18&1.87e+3&8.32e-1&5.75e-9&1.68e-1&1.00e+0\\ \hline 6.08e+2&1.64e-3&6.09e+2&9.75e-4&1.59e-3&4.48e+1&7.36e-2&1.01e-1&2.28e-3&3.10e+0&4.63e+1&6.16e+5&7.13e-19&1.66e+3&8.48e-1&8.31e-9&1.52e-1&1.00e+0\\ \hline 5.41e+2&1.84e-3&5.42e+2&1.18e-3&1.91e-3&4.47e+1&8.24e-2&1.13e-1&2.78e-3&3.09e+0&4.32e+1&5.13e+5&4.94e-19&1.48e+3&8.62e-1&1.20e-8&1.38e-1&1.00e+0\\ \hline 4.81e+2&2.07e-3&4.82e+2&1.42e-3&2.29e-3&4.46e+1&9.24e-2&1.27e-1&3.38e-3&3.08e+0&4.04e+1&4.27e+5&3.43e-19&1.31e+3&8.75e-1&1.73e-8&1.25e-1&1.00e+0\\ \hline 4.28e+2&2.33e-3&4.29e+2&1.71e-3&2.74e-3&4.44e+1&1.03e-1&1.42e-1&4.11e-3&3.07e+0&3.77e+1&3.56e+5&2.38e-19&1.17e+3&8.88e-1&2.48e-8&1.12e-1&1.00e+0\\ \hline 3.81e+2&2.62e-3&3.82e+2&2.06e-3&3.29e-3&4.43e+1&1.16e-1&1.59e-1&4.99e-3&3.06e+0&3.52e+1&2.97e+5&1.66e-19&1.04e+3&8.99e-1&3.57e-8&1.01e-1&1.00e+0\\ \hline 3.39e+2&2.94e-3&3.40e+2&2.49e-3&3.94e-3&4.41e+1&1.30e-1&1.79e-1&6.05e-3&3.05e+0&3.29e+1&2.48e+5&1.16e-19&9.27e+2&9.09e-1&5.12e-8&9.12e-2&1.00e+0\\ \hline 3.02e+2&3.30e-3&3.03e+2&2.99e-3&4.72e-3&4.40e+1&1.45e-1&2.00e-1&7.33e-3&3.04e+0&3.08e+1&2.07e+5&8.07e-20&8.25e+2&9.18e-1&7.34e-8&8.20e-2&1.00e+0\\ \hline 2.68e+2&3.71e-3&2.69e+2&3.59e-3&5.64e-3&4.38e+1&1.63e-1&2.24e-1&8.88e-3&3.03e+0&2.88e+1&1.73e+5&5.64e-20&7.34e+2&9.26e-1&1.05e-7&7.36e-2&1.00e+0\\ \hline 2.39e+2&4.17e-3&2.40e+2&4.31e-3&6.75e-3&4.36e+1&1.82e-1&2.51e-1&1.07e-2&3.02e+0&2.70e+1&1.45e+5&3.94e-20&6.53e+2&9.34e-1&1.50e-7&6.61e-2&1.00e+0\\ \hline 2.12e+2&4.69e-3&2.13e+2&5.17e-3&8.07e-3&4.34e+1&2.04e-1&2.81e-1&1.30e-2&3.00e+0&2.52e+1&1.21e+5&2.76e-20&5.81e+2&9.41e-1&2.15e-7&5.92e-2&1.00e+0\\ \hline 1.89e+2&5.27e-3&1.90e+2&6.20e-3&9.64e-3&4.32e+1&2.28e-1&3.15e-1&1.57e-2&2.99e+0&2.36e+1&1.01e+5&1.93e-20&5.17e+2&9.47e-1&3.07e-7&5.31e-2&1.00e+0\\ \hline 1.68e+2&5.92e-3&1.69e+2&7.43e-3&1.15e-2&4.30e+1&2.54e-1&3.53e-1&1.89e-2&2.98e+0&2.21e+1&8.49e+4&1.35e-20&4.60e+2&9.53e-1&4.38e-7&4.75e-2&1.00e+0\\ \hline 1.49e+2&6.65e-3&1.50e+2&8.90e-3&1.38e-2&4.28e+1&2.84e-1&3.95e-1&2.28e-2&2.96e+0&2.07e+1&7.11e+4&9.49e-21&4.10e+2&9.58e-1&6.24e-7&4.25e-2&1.00e+0\\ \hline 1.33e+2&7.47e-3&1.34e+2&1.07e-2&1.64e-2&4.25e+1&3.18e-1&4.42e-1&2.75e-2&2.94e+0&1.93e+1&5.95e+4&6.66e-21&3.65e+2&9.62e-1&8.89e-7&3.80e-2&1.00e+0\\ \hline 1.18e+2&8.40e-3&1.19e+2&1.28e-2&1.96e-2&4.22e+1&3.55e-1&4.94e-1&3.31e-2&2.92e+0&1.81e+1&4.99e+4&4.67e-21&3.25e+2&9.66e-1&1.27e-6&3.40e-2&1.00e+0\\ \hline 1.05e+2&9.44e-3&1.06e+2&1.52e-2&2.34e-2&4.20e+1&3.96e-1&5.52e-1&3.98e-2&2.90e+0&1.69e+1&4.18e+4&3.28e-21&2.89e+2&9.70e-1&1.80e-6&3.03e-2&1.00e+0\\ \hline 9.33e+1&1.06e-2&9.43e+1&1.82e-2&2.79e-2&4.17e+1&4.42e-1&6.18e-1&4.79e-2&2.88e+0&1.58e+1&3.50e+4&2.31e-21&2.57e+2&9.73e-1&2.57e-6&2.71e-2&1.00e+0\\ \hline 8.29e+1&1.19e-2&8.39e+1&2.18e-2&3.33e-2&4.14e+1&4.93e-1&6.90e-1&5.76e-2&2.86e+0&1.48e+1&2.94e+4&1.62e-21&2.29e+2&9.76e-1&3.65e-6&2.42e-2&1.00e+0\\ \hline 7.37e+1&1.34e-2&7.47e+1&2.60e-2&3.97e-2&4.10e+1&5.49e-1&7.71e-1&6.92e-2&2.84e+0&1.38e+1&2.46e+4&1.14e-21&2.04e+2&9.78e-1&5.20e-6&2.16e-2&1.00e+0\\ \hline 6.55e+1&1.50e-2&6.65e+1&3.11e-2&4.73e-2&4.07e+1&6.12e-1&8.61e-1&8.32e-2&2.81e+0&1.29e+1&2.07e+4&8.01e-22&1.81e+2&9.81e-1&7.39e-6&1.92e-2&1.00e+0\\ \hline 5.82e+1&1.69e-2&5.92e+1&3.71e-2&5.65e-2&4.03e+1&6.81e-1&9.61e-1&9.98e-2&2.79e+0&1.21e+1&1.73e+4&5.64e-22&1.61e+2&9.83e-1&1.05e-5&1.72e-2&1.00e+0\\ \hline 5.17e+1&1.90e-2&5.27e+1&4.43e-2&6.73e-2&3.99e+1&7.57e-1&1.07e+0&1.20e-1&2.76e+0&1.13e+1&1.45e+4&3.96e-22&1.44e+2&9.85e-1&1.49e-5&1.53e-2&1.00e+0\\ \hline 4.59e+1&2.13e-2&4.69e+1&5.29e-2&8.02e-2&3.95e+1&8.42e-1&1.20e+0&1.44e-1&2.73e+0&1.05e+1&1.22e+4&2.79e-22&1.28e+2&9.86e-1&2.12e-5&1.36e-2&1.00e+0\\ \hline 4.07e+1&2.40e-2&4.17e+1&6.31e-2&9.56e-2&3.90e+1&9.35e-1&1.33e+0&1.72e-1&2.70e+0&9.78e+0&1.02e+4&1.96e-22&1.14e+2&9.88e-1&3.02e-5&1.22e-2&1.00e+0\\ \hline 3.61e+1&2.69e-2&3.71e+1&7.53e-2&1.14e-1&3.85e+1&1.04e+0&1.48e+0&2.06e-1&2.67e+0&9.11e+0&8.58e+3&1.38e-22&1.01e+2&9.89e-1&4.29e-5&1.08e-2&1.00e+0\\ \hline 3.20e+1&3.03e-2&3.30e+1&8.98e-2&1.36e-1&3.80e+1&1.15e+0&1.65e+0&2.47e-1&2.63e+0&8.47e+0&7.20e+3&9.73e-23&9.00e+1&9.90e-1&6.09e-5&9.65e-3&1.00e+0\\ \hline 2.84e+1&3.40e-2&2.94e+1&1.07e-1&1.62e-1&3.75e+1&1.28e+0&1.84e+0&2.96e-1&2.59e+0&7.88e+0&6.04e+3&6.85e-23&8.01e+1&9.91e-1&8.64e-5&8.60e-3&1.00e+0\\ \hline 2.52e+1&3.82e-2&2.62e+1&1.28e-1&1.93e-1&3.69e+1&1.41e+0&2.04e+0&3.55e-1&2.55e+0&7.31e+0&5.07e+3&4.83e-23&7.13e+1&9.92e-1&1.23e-4&7.66e-3&1.00e+0\\ \hline 2.23e+1&4.30e-2&2.33e+1&1.52e-1&2.30e-1&3.63e+1&1.56e+0&2.27e+0&4.25e-1&2.51e+0&6.78e+0&4.25e+3&3.40e-23&6.35e+1&9.93e-1&1.74e-4&6.82e-3&1.00e+0\\ \hline 1.97e+1&4.83e-2&2.07e+1&1.82e-1&2.74e-1&3.57e+1&1.72e+0&2.52e+0&5.08e-1&2.47e+0&6.28e+0&3.57e+3&2.39e-23&5.65e+1&9.94e-1&2.47e-4&6.08e-3&1.00e+0\\ \hline 1.74e+1&5.42e-2&1.84e+1&2.17e-1&3.26e-1&3.50e+1&1.90e+0&2.80e+0&6.08e-1&2.42e+0&5.81e+0&3.00e+3&1.69e-23&5.03e+1&9.94e-1&3.51e-4&5.41e-3&1.00e+0\\ \hline 1.54e+1&6.09e-2&1.64e+1&2.58e-1&3.89e-1&3.43e+1&2.09e+0&3.10e+0&7.27e-1&2.37e+0&5.37e+0&2.52e+3&1.19e-23&4.47e+1&9.95e-1&4.99e-4&4.82e-3&1.00e+0\\ \hline 1.36e+1&6.85e-2&1.46e+1&3.08e-1&4.63e-1&3.35e+1&2.29e+0&3.43e+0&8.70e-1&2.32e+0&4.95e+0&2.11e+3&8.37e-24&3.98e+1&9.95e-1&7.07e-4&4.29e-3&1.00e+0\\ \hline 1.20e+1&7.69e-2&1.30e+1&3.67e-1&5.52e-1&3.27e+1&2.51e+0&3.79e+0&1.04e+0&2.26e+0&4.56e+0&1.77e+3&5.90e-24&3.54e+1&9.95e-1&1.00e-3&3.82e-3&1.00e+0\\ \hline 1.06e+1&8.64e-2&1.16e+1&4.37e-1&6.57e-1&3.18e+1&2.75e+0&4.18e+0&1.24e+0&2.20e+0&4.19e+0&1.49e+3&4.16e-24&3.15e+1&9.95e-1&1.42e-3&3.40e-3&1.00e+0\\ \hline 9.29e+0&9.71e-2&1.03e+1&5.21e-1&7.83e-1&3.09e+1&3.00e+0&4.61e+0&1.48e+0&2.14e+0&3.84e+0&1.25e+3&2.93e-24&2.81e+1&9.95e-1&2.02e-3&3.02e-3&1.00e+0\\ \hline 8.16e+0&1.09e-1&9.16e+0&6.20e-1&9.32e-1&2.99e+1&3.27e+0&5.08e+0&1.77e+0&2.07e+0&3.51e+0&1.05e+3&2.07e-24&2.50e+1&9.94e-1&2.86e-3&2.69e-3&1.00e+0\\ \hline 7.15e+0&1.23e-1&8.15e+0&7.39e-1&1.11e+0&2.89e+1&3.55e+0&5.58e+0&2.12e+0&2.00e+0&3.20e+0&8.81e+2&1.46e-24&2.22e+1&9.94e-1&4.06e-3&2.39e-3&1.00e+0\\ \hline 6.26e+0&1.38e-1&7.26e+0&8.80e-1&1.32e+0&2.78e+1&3.83e+0&6.12e+0&2.53e+0&1.93e+0&2.90e+0&7.40e+2&1.03e-24&1.98e+1&9.92e-1&5.75e-3&2.12e-3&1.00e+0\\ \hline 5.46e+0&1.55e-1&6.46e+0&1.05e+0&1.57e+0&2.67e+1&4.13e+0&6.70e+0&3.02e+0&1.85e+0&2.63e+0&6.22e+2&7.28e-25&1.76e+1&9.90e-1&8.14e-3&1.89e-3&1.00e+0\\ \hline 4.75e+0&1.74e-1&5.75e+0&1.25e+0&1.87e+0&2.55e+1&4.43e+0&7.32e+0&3.61e+0&1.76e+0&2.37e+0&5.23e+2&5.14e-25&1.57e+1&9.87e-1&1.15e-2&1.67e-3&1.00e+0\\ \hline 4.11e+0&1.96e-1&5.11e+0&1.49e+0&2.22e+0&2.42e+1&4.73e+0&7.97e+0&4.30e+0&1.67e+0&2.13e+0&4.40e+2&3.64e-25&1.39e+1&9.82e-1&1.63e-2&1.48e-3&1.00e+0\\ \hline 3.55e+0&2.20e-1&4.55e+0&1.77e+0&2.64e+0&2.28e+1&5.01e+0&8.65e+0&5.14e+0&1.58e+0&1.90e+0&3.71e+2&2.58e-25&1.24e+1&9.76e-1&2.29e-2&1.31e-3&1.00e+0\\ \hline 3.05e+0&2.47e-1&4.05e+0&2.10e+0&3.12e+0&2.14e+1&5.28e+0&9.37e+0&6.13e+0&1.48e+0&1.69e+0&3.13e+2&1.84e-25&1.10e+1&9.67e-1&3.22e-2&1.16e-3&1.00e+0\\ \hline 2.61e+0&2.77e-1&3.61e+0&2.50e+0&3.70e+0&1.99e+1&5.51e+0&1.01e+1&7.30e+0&1.37e+0&1.49e+0&2.64e+2&1.31e-25&9.83e+0&9.54e-1&4.51e-2&1.02e-3&1.00e+0\\ \hline 2.21e+0&3.12e-1&3.21e+0&2.97e+0&4.36e+0&1.83e+1&5.69e+0&1.09e+1&8.70e+0&1.26e+0&1.30e+0&2.24e+2&9.43e-26&8.74e+0&9.36e-1&6.28e-2&8.87e-4&1.00e+0\\ \hline 1.86e+0&3.50e-1&2.86e+0&3.52e+0&5.13e+0&1.66e+1&5.81e+0&1.16e+1&1.04e+1&1.15e+0&1.13e+0&1.91e+2&6.82e-26&7.78e+0&9.12e-1&8.68e-2&7.69e-4&1.00e+0\\ \hline 1.54e+0&3.94e-1&2.54e+0&4.17e+0&6.00e+0&1.49e+1&5.84e+0&1.24e+1&1.23e+1&1.03e+0&9.74e-1&1.63e+2&4.98e-26&6.93e+0&8.80e-1&1.19e-1&6.60e-4&1.00e+0\\ \hline 1.26e+0&4.42e-1&2.26e+0&4.93e+0&6.98e+0&1.30e+1&5.77e+0&1.31e+1&1.47e+1&9.03e-1&8.26e-1&1.40e+2&3.69e-26&6.16e+0&8.39e-1&1.61e-1&5.60e-4&1.00e+0\\ \hline 1.01e+0&4.97e-1&2.01e+0&5.80e+0&8.05e+0&1.12e+1&5.55e+0&1.38e+1&1.74e+1&7.73e-1&6.90e-1&1.22e+2&2.77e-26&5.49e+0&7.86e-1&2.14e-1&4.67e-4&1.00e+0\\ \hline 7.91e-1&5.58e-1&1.79e+0&6.80e+0&9.18e+0&9.27e+0&5.18e+0&1.44e+1&2.06e+1&6.42e-1&5.64e-1&1.06e+2&2.13e-26&4.88e+0&7.22e-1&2.78e-1&3.81e-4&1.00e+0\\ \hline 5.94e-1&6.27e-1&1.59e+0&7.94e+0&1.04e+1&7.35e+0&4.61e+0&1.50e+1&2.44e+1&5.09e-1&4.46e-1&9.45e+1&1.68e-26&4.34e+0&6.46e-1&3.53e-1&3.04e-4&1.00e+0\\ \hline 4.19e-1&7.05e-1&1.42e+0&9.22e+0&1.15e+1&5.44e+0&3.83e+0&1.55e+1&2.87e+1&3.76e-1&3.33e-1&8.50e+1&1.36e-26&3.87e+0&5.63e-1&4.37e-1&2.36e-4&1.00e+0\\ \hline 2.63e-1&7.92e-1&1.26e+0&1.06e+1&1.26e+1&3.55e+0&2.81e+0&1.59e+1&3.38e+1&2.46e-1&2.23e-1&7.76e+1&1.13e-26&3.44e+0&4.76e-1&5.24e-1&1.77e-4&1.00e+0\\ \hline 1.24e-1&8.90e-1&1.12e+0&1.22e+1&1.36e+1&1.73e+0&1.54e+0&1.63e+1&3.96e+1&1.20e-1&1.14e-1&7.19e+1&9.72e-27&3.06e+0&3.90e-1&6.09e-1&1.29e-4&1.00e+0\\ \hline 0.00e+0&1.00e+0&1.00e+0&1.38e+1&1.45e+1&0.00e+0&0.00e+0&1.66e+1&4.62e+1&0.00e+0&0.00e+0&6.77e+1&8.60e-27&2.73e+0&3.11e-1&6.89e-1&9.18e-5&1.00e+0\\ \hline -1.09e-1&1.12e+0&8.91e-1&1.55e+1&1.52e+1&1.61e+0&1.81e+0&1.68e+1&5.36e+1&1.11e-1&1.19e-1&6.45e+1&7.82e-27&2.43e+0&2.42e-1&7.58e-1&6.37e-5&1.00e+0\\ \hline -2.06e-1&1.26e+0&7.94e-1&1.73e+1&1.57e+1&3.11e+0&3.91e+0&1.70e+1&6.21e+1&2.15e-1&2.49e-1&6.22e+1&7.26e-27&2.16e+0&1.85e-1&8.15e-1&4.33e-5&1.00e+0\\ \hline -2.92e-1&1.41e+0&7.08e-1&1.91e+1&1.62e+1&4.49e+0&6.34e+0&1.71e+1&7.16e+1&3.10e-1&3.92e-1&6.05e+1&6.87e-27&1.93e+0&1.38e-1&8.62e-1&2.88e-5&1.00e+0\\ \hline -3.69e-1&1.58e+0&6.31e-1&2.10e+1&1.65e+1&5.74e+0&9.10e+0&1.72e+1&8.23e+1&3.97e-1&5.52e-1&5.93e+1&6.60e-27&1.72e+0&1.02e-1&8.98e-1&1.90e-5&1.00e+0\\ \hline -4.38e-1&1.78e+0&5.62e-1&2.29e+1&1.68e+1&6.88e+0&1.22e+1&1.72e+1&9.44e+1&4.76e-1&7.31e-1&5.84e+1&6.40e-27&1.53e+0&7.43e-2&9.26e-1&1.23e-5&1.00e+0\\ \hline -4.99e-1&2.00e+0&5.01e-1&2.48e+1&1.69e+1&7.91e+0&1.58e+1&1.73e+1&1.08e+2&5.48e-1&9.32e-1&5.77e+1&6.26e-27&1.37e+0&5.38e-2&9.46e-1&7.96e-6&1.00e+0\\ \hline -5.53e-1&2.24e+0&4.47e-1&2.68e+1&1.71e+1&8.84e+0&1.98e+1&1.73e+1&1.23e+2&6.12e-1&1.16e+0&5.73e+1&6.16e-27&1.22e+0&3.87e-2&9.61e-1&5.10e-6&1.00e+0\\ \hline -6.02e-1&2.51e+0&3.98e-1&2.88e+1&1.72e+1&9.67e+0&2.43e+1&1.74e+1&1.40e+2&6.69e-1&1.42e+0&5.70e+1&6.09e-27&1.09e+0&2.77e-2&9.72e-1&3.25e-6&1.00e+0\\ \hline -6.45e-1&2.82e+0&3.55e-1&3.08e+1&1.72e+1&1.04e+1&2.94e+1&1.74e+1&1.60e+2&7.21e-1&1.70e+0&5.67e+1&6.04e-27&9.67e-1&1.98e-2&9.80e-1&2.07e-6&1.00e+0\\ \hline -6.84e-1&3.16e+0&3.16e-1&3.27e+1&1.73e+1&1.11e+1&3.51e+1&1.74e+1&1.81e+2&7.67e-1&2.03e+0&5.66e+1&6.01e-27&8.62e-1&1.41e-2&9.86e-1&1.31e-6&1.00e+0\\ \hline -7.18e-1&3.55e+0&2.82e-1&3.47e+1&1.73e+1&1.17e+1&4.14e+1&1.74e+1&2.05e+2&8.08e-1&2.39e+0&5.64e+1&5.98e-27&7.68e-1&1.00e-2&9.90e-1&8.32e-7&1.00e+0\\ \hline -7.49e-1&3.98e+0&2.51e-1&3.67e+1&1.73e+1&1.22e+1&4.86e+1&1.74e+1&2.32e+2&8.45e-1&2.80e+0&5.64e+1&5.97e-27&6.85e-1&7.10e-3&9.93e-1&5.27e-7&1.00e+0\\ \hline -7.76e-1&4.47e+0&2.24e-1&3.87e+1&1.74e+1&1.27e+1&5.67e+1&1.74e+1&2.63e+2&8.78e-1&3.26e+0&5.63e+1&5.95e-27&6.10e-1&5.04e-3&9.95e-1&3.33e-7&1.00e+0\\ \hline -8.00e-1&5.01e+0&2.00e-1&4.07e+1&1.74e+1&1.31e+1&6.57e+1&1.74e+1&2.97e+2&9.07e-1&3.78e+0&5.63e+1&5.94e-27&5.44e-1&3.57e-3&9.96e-1&2.10e-7&1.00e+0\\ \hline -8.22e-1&5.62e+0&1.78e-1&4.27e+1&1.74e+1&1.35e+1&7.58e+1&1.74e+1&3.36e+2&9.33e-1&4.36e+0&5.62e+1&5.94e-27&4.85e-1&2.53e-3&9.97e-1&1.33e-7&1.00e+0\\ \hline -8.42e-1&6.31e+0&1.58e-1&4.47e+1&1.74e+1&1.38e+1&8.72e+1&1.74e+1&3.79e+2&9.56e-1&5.01e+0&5.62e+1&5.93e-27&4.32e-1&1.79e-3&9.98e-1&8.39e-8&1.00e+0\\ \hline -8.59e-1&7.08e+0&1.41e-1&4.67e+1&1.74e+1&1.41e+1&1.00e+2&1.74e+1&4.27e+2&9.77e-1&5.75e+0&5.62e+1&5.93e-27&3.85e-1&1.27e-3&9.99e-1&5.30e-8&1.00e+0\\ \hline -8.74e-1&7.94e+0&1.26e-1&4.87e+1&1.74e+1&1.44e+1&1.14e+2&1.74e+1&4.81e+2&9.96e-1&6.57e+0&5.62e+1&5.93e-27&3.43e-1&9.00e-4&9.99e-1&3.34e-8&1.00e+0\\ \hline -8.88e-1&8.91e+0&1.12e-1&5.07e+1&1.74e+1&1.46e+1&1.30e+2&1.74e+1&5.42e+2&1.01e+0&7.49e+0&5.62e+1&5.93e-27&3.06e-1&6.37e-4&9.99e-1&2.11e-8&1.00e+0\\ \hline -9.00e-1&1.00e+1&1.00e-1&5.27e+1&1.74e+1&1.48e+1&1.48e+2&1.74e+1&6.10e+2&1.03e+0&8.52e+0&5.62e+1&5.93e-27&2.73e-1&4.51e-4&1.00e+0&1.33e-8&1.00e+0\\ \hline -9.11e-1&1.12e+1&8.91e-2&5.48e+1&1.74e+1&1.50e+1&1.69e+2&1.74e+1&6.87e+2&1.04e+0&9.69e+0&5.62e+1&5.93e-27&2.43e-1&3.20e-4&1.00e+0&8.40e-9&1.00e+0\\ \hline -9.21e-1&1.26e+1&7.94e-2&5.68e+1&1.74e+1&1.52e+1&1.91e+2&1.74e+1&7.73e+2&1.05e+0&1.10e+1&5.62e+1&5.93e-27&2.16e-1&2.26e-4&1.00e+0&5.30e-9&1.00e+0\\ \hline -9.29e-1&1.41e+1&7.08e-2&5.88e+1&1.74e+1&1.53e+1&2.17e+2&1.74e+1&8.69e+2&1.06e+0&1.25e+1&5.62e+1&5.92e-27&1.93e-1&1.60e-4&1.00e+0&3.35e-9&1.00e+0\\ \hline -9.37e-1&1.58e+1&6.31e-2&6.08e+1&1.74e+1&1.55e+1&2.45e+2&1.74e+1&9.77e+2&1.07e+0&1.41e+1&5.62e+1&5.92e-27&1.72e-1&1.13e-4&1.00e+0&2.11e-9&1.00e+0\\ \hline -9.44e-1&1.78e+1&5.62e-2&6.28e+1&1.74e+1&1.56e+1&2.77e+2&1.74e+1&1.10e+3&1.08e+0&1.59e+1&5.62e+1&5.92e-27&1.53e-1&8.03e-5&1.00e+0&1.33e-9&1.00e+0\\ \hline -9.50e-1&2.00e+1&5.01e-2&6.48e+1&1.74e+1&1.57e+1&3.13e+2&1.74e+1&1.24e+3&1.09e+0&1.80e+1&5.62e+1&5.92e-27&1.37e-1&5.68e-5&1.00e+0&8.41e-10&1.00e+0\\ \hline -9.55e-1&2.24e+1&4.47e-2&6.68e+1&1.74e+1&1.58e+1&3.54e+2&1.74e+1&1.39e+3&1.09e+0&2.03e+1&5.62e+1&5.92e-27&1.22e-1&4.02e-5&1.00e+0&5.30e-10&1.00e+0\\ \hline -9.60e-1&2.51e+1&3.98e-2&6.88e+1&1.74e+1&1.59e+1&3.99e+2&1.74e+1&1.56e+3&1.10e+0&2.29e+1&5.62e+1&5.92e-27&1.09e-1&2.85e-5&1.00e+0&3.35e-10&1.00e+0\\ \hline -9.65e-1&2.82e+1&3.55e-2&7.08e+1&1.74e+1&1.60e+1&4.50e+2&1.74e+1&1.75e+3&1.10e+0&2.58e+1&5.62e+1&5.92e-27&9.67e-2&2.02e-5&1.00e+0&2.11e-10&1.00e+0\\ \hline -9.68e-1&3.16e+1&3.16e-2&7.28e+1&1.74e+1&1.60e+1&5.07e+2&1.74e+1&1.97e+3&1.11e+0&2.91e+1&5.62e+1&5.92e-27&8.62e-2&1.43e-5&1.00e+0&1.33e-10&1.00e+0\\ \hline -9.72e-1&3.55e+1&2.82e-2&7.48e+1&1.74e+1&1.61e+1&5.71e+2&1.74e+1&2.21e+3&1.11e+0&3.28e+1&5.62e+1&5.92e-27&7.68e-2&1.01e-5&1.00e+0&8.41e-11&1.00e+0\\ \hline -9.75e-1&3.98e+1&2.51e-2&7.68e+1&1.74e+1&1.61e+1&6.43e+2&1.74e+1&2.48e+3&1.12e+0&3.69e+1&5.62e+1&5.92e-27&6.85e-2&7.16e-6&1.00e+0&5.30e-11&1.00e+0\\ \hline -9.78e-1&4.47e+1&2.24e-2&7.88e+1&1.74e+1&1.62e+1&7.23e+2&1.74e+1&2.79e+3&1.12e+0&4.15e+1&5.62e+1&5.92e-27&6.10e-2&5.07e-6&1.00e+0&3.35e-11&1.00e+0\\ \hline -9.80e-1&5.01e+1&2.00e-2&8.08e+1&1.74e+1&1.62e+1&8.14e+2&1.74e+1&3.13e+3&1.12e+0&4.67e+1&5.62e+1&5.92e-27&5.44e-2&3.59e-6&1.00e+0&2.11e-11&1.00e+0\\ \hline -9.82e-1&5.62e+1&1.78e-2&8.28e+1&1.74e+1&1.63e+1&9.15e+2&1.74e+1&3.51e+3&1.13e+0&5.25e+1&5.62e+1&5.92e-27&4.85e-2&2.54e-6&1.00e+0&1.33e-11&1.00e+0\\ \hline -9.84e-1&6.31e+1&1.58e-2&8.48e+1&1.74e+1&1.63e+1&1.03e+3&1.74e+1&3.94e+3&1.13e+0&5.91e+1&5.62e+1&5.92e-27&4.32e-2&1.80e-6&1.00e+0&8.41e-12&1.00e+0\\ \hline -9.86e-1&7.08e+1&1.41e-2&8.68e+1&1.74e+1&1.63e+1&1.16e+3&1.74e+1&4.43e+3&1.13e+0&6.64e+1&5.62e+1&5.92e-27&3.85e-2&1.27e-6&1.00e+0&5.30e-12&1.00e+0\\ \hline -9.87e-1&7.94e+1&1.26e-2&8.88e+1&1.74e+1&1.64e+1&1.30e+3&1.74e+1&4.97e+3&1.13e+0&7.46e+1&5.62e+1&5.92e-27&3.43e-2&9.01e-7&1.00e+0&3.35e-12&1.00e+0\\ \hline -9.89e-1&8.91e+1&1.12e-2&9.08e+1&1.74e+1&1.64e+1&1.46e+3&1.74e+1&5.58e+3&1.13e+0&8.39e+1&5.62e+1&5.92e-27&3.06e-2&6.38e-7&1.00e+0&2.11e-12&1.00e+0\\ \hline -9.90e-1&1.00e+2&1.00e-2&9.28e+1&1.74e+1&1.64e+1&1.64e+3&1.74e+1&6.26e+3&1.14e+0&9.42e+1&5.62e+1&5.92e-27&2.73e-2&4.51e-7&1.00e+0&1.33e-12&1.00e+0\\ \hline \end{array}}\]

https://yangresearch.weebly.com/uploads/5/6/3/4/5634250/correlation_functions.pdf For above will help to correlate how early DM needs to form for LSS structure formation Monte Carlo applications of above (also pertinant to Planck datasets and DESI etc) includes wcdm relations for deviation searches from LCDM https://arxiv.org/pdf/2508.20971v1

Useful information on 2 point correlation employing integrated late time Sache Wolfe effect for DM distribution detection. https://ned.ipac.caltech.edu/level5/Sept19/Cooray/paper.pdf This article will take me some time to fully grasp lmao been awhile since I looked at 2 point correlation Kertosis 4 point correlation template https://arxiv.org/abs/2509.05419 Structure factor S(k) Fourier transform of the radial distribution function. g(r) \[S(k)=\frac{1}{N}\langle \rho k\rho-k\rangle\] \[S(k)=\frac{1}{N}\langle\sum^N_{i=1}\sum^N_{j=1} exp(-ikr_i)exp(ikr_j)\rangle\] using Kronecker Delta \(\delta\) distributions S(k) \[S(k)=1+\frac{1}{n}\langle\int\int exp(-k(r-\acute{r})\sum^N_{i=1}\sum^N_{j=1,i\neq j} \delta(r-r_i)\delta(r-r_j)dxd\acute{x}\] homogeneous and isotropic fluid case \[S(k)=1+\frac{\rho^2}{N}\int\int exp(-ik(r-\acute{r})g(r,\acute{r})dxd\acute{x}\] radial distribution only depends on \( |r-\acute{r}\) resulting Fourier transform only depends on modulus k of wavevector |k| \[S(k)=1+2\pi \rho\int r^2g(r)\int^\pi_0 exp(ikr cos(\theta) sin(\theta) d(\theta) dr\] \[S(k)=1+4\pi \rho\int^\infty_0 r^2 g(r)\frac{sin(sr)}{kr}dr\]

Decided not to bother

\[ds^2=\sum_{i,j}g_{i,j}dx^dx^j\] \[\begin{array}\rightarrow &\rightarrow &A_{331}&A_{332}&A_{333}\\\rightarrow &A__{231}&A_{232}&A_{233}&\rightarrow\\A_{131}&A_{132}&A_{133}&\rightarrow&\rightarrow\end{array}\]

I seem to recall that a fair while back it not uncommon, took me years to finally try learning string theory for much the same reasons. QFT for me makes far more sense. We all have our preferred mathematics some prefer differentials others prefer integrals etc.

I'll let you judge that for yourself. Quite frankly in my viewpoint step one of a minimal geometry is the fewest number of variables to uniquely identify each start position. initial distribution, same applies to the final distribution is the simplest portions. This obviously includes to tool to choose a reference frame. Describing causation between initial and final distribution is where the real fun is.

That raises a fundamental question " what is a minimal system" -Define a start point to every location ( how the original distribution is located point by point) - describe how any changes occur in the starting distribution with minimal conversions and calculations to any given frame of reference. -identifying the cause and how much the cause of change results in the final distribution from the starting distribution. Good luck on those criteria without a geometry as that's really what a geometry is used to describe ( distribution)

Thats an accurate description

If it was I would have had zero issues following it. Lol brings to mind a rule I tend to follow. Look at the math applied and adherence to math rigor and ignore verbal explanations except to identify the labels. Lol though I do that with professional peer review articles as well

I will have to be careful here in answering your last point and great explanation @joigus . The reason being is a large part of the answer will involve the probaility nature described by Joigus who also mentioned the particle number density. The higher the energy as well as the known properties one can define a state used for the particle identity factors such as mass spin energy momentum etc. This is the initial state, the more energy in the beam (intensity as mentioned above) gives us step or ladder operator based on the a classical formula used (likely in your Feymann lectures ) \[E=\hbar \omega\] for photons the number of quanta derived by that formula is the number of photons. given the above a higher intensity beam will generate more photons giving a higher probability of photons that will make it through the slit without interference. The topic of location of a wave often comes up. QM has a rather handy probability wavefunction for this called the Dirac Delta function it acts much like a switch when a hit occurs and in essence gives us relations to the billiard ball like aspects a particle. The wavefunction under graph is a sharp spike to infinity easily (localized) . Other wavefunctions are not easily localized such as the sinusoidal wavefunction. Keep in mind much of the graph of any function will include those probabilities into the graph itself. once an interaction or measurement is performed the wavefunction changes as you now have more details to better localize a particle. The Dirac Delta function is often used for localization of a particle another closely related is the Heaviside step function. I only mention these as one can readily pull up images of each and they give a better sense of a localized wavefunction. cross posted with KJW good answer