Cairo's Intelligence Techniques: This likely refers to specific methodologies or approaches developed in Cairo, but without context, it’s challenging to define what these techniques entail.

The numerical statistical theory, called "Cairo techniques," dates back to 2020 [1,2,3,4].

It assumes that the spatio-temporal evolution of the energy density field U of the physical system within a closed control volume follows directly from the expression [1,2,3,4]:

U(x,y,z,t+dt)=B. U(x,y,z,t) . . . . . . . (1)

Which implies:

U(x,y,z,t)=D(N). (b+S) + B^N.IC . . . . . (2)

Where B is the statistical transition matrix and N is the elapsed time, expressed as a dimensionless integer of iterations/time steps. Equations 1 and 2 clearly illustrate the mutual entanglement between the system elements, as well as the entanglement between the system elements and the walls. Using Equations 1 and 2, it can be easily demonstrated that the conservation of total energy as a function of time is expressed as follows:

dU/dt)partial = α ∇^2 (U) + S. . . . . . . (3)

Where U(x,y,z,t) is the energy density and α is the thermal diffusivity of this density.

Equation 3 summarizes the whole trick.

The statistical transition matrix B = (Bi,j) is itself well-defined by four statistical assumptions.

The elements B i, j satisfy or are subject to the following conditions:

i- B i, j = 1/2, 1/4, or 1/6 for 1D, 2D, and 3D, respectively.

For i adjacent to j, and B i, j = 0 otherwise.

Equal prior probability.

ii- B i, i = RO,

i.e., the main diagonal consists of constant RO elements.

For the heat diffusion equation, RO can take any value in the closed interval [0,1],

while for Laplace and Poisson PDEs, RO = 0.

i.e., B is a zero main diagonal matrix, which corresponds to the assumption of a zero residue after each time step for all free nodes.

iii- B i, j = B j, i, for all i, j.

The matrix B is symmetrical to respect the detailed physical principle of equilibrium.

iv- The sum of the values ​​B i, j = 1 for all lines distant from the edges and the sum of the values ​​B i, j < 1 for all lines connected to the edges means that the probability of the entire space is equal to 1.

Obviously, the statistical matrix B is very different from the mathematical Laplacian matrix A and the Markov transition matrix.

The physical nature of B is clear and briefly explained above through the four conditions i to iv.

The numerical results are astounding, as they show that the well-established theories of relativity and quantum mechanics are incomplete, partially true, and inherently flawed.

Many of our distinguished contributors and readers often wonder how it is possible to challenge giants like Einstein and Schrödinger and find flaws in their theories.

Does this mean you are smarter than both of them?

The answer is simple: we use the artificial intelligence of the Cairo Techniques and its B-matrix algorithms, which are smarter than A. Einstein and E. Schrödinger, not the author himself.

1- A Numerical Statistical Solution to Laplace and Poisson Partial Differential Equations

2- Foundations of Artificial Intelligence

3- Theory and Practice of Artificial Intelligence

4- BOOK, Foundations of Artificial Intelligence

We assume that neither E. Schrödinger nor N. Bohr understood how quantum mechanics works.

Cairo's AI techniques predict that QM operates according to the square of SE rather than SE itself.

Below, we present the solution for the QM particle in infinite 1D and 2D potential wells, and in a closed control volume box.

The first step is to completely neglect the PDE of SE, as if it never existed, and solve for its squared PDE, supplemented by the AB law proposed by the author:

S(x,y,z,t)=Const.V(x,y,z,t)

The AB law means that the spontaneous potential of SE can be transformed into quantum matter and vice versa.

Here, the PDE matrix of QM for 17 nodes is given by the eigenmatrix equation:

166/2187 244/2187 266/2187 140/2187 35/2187        0         0         0        0         0         0        0        0        0        0        0        0

427/4374 853/4374 434/2187 322/2187 245/4374 35/4374         0         0        0         0         0        0        0        0        0        0        0

133/1458 124/729 343/1458      2/9 205/1458   10/243    5/1458         0        0         0         0        0        0        0        0        0        0

175/4374 230/2187     5/27 373/1458 355/1458 280/2187   55/2187    5/4374        0         0         0        0        0        0        0        0        0

35/4374 70/2187 205/2187 142/729 211/729 194/729 227/2187   26/2187   1/4374         0         0        0        0        0        0        0        0

0   5/1458    5/243   56/729   97/486 505/1458   205/729    50/729   5/1458         0         0        0        0        0        0        0        0

0        0   5/4374 22/2187 227/4374 410/2187 1981/4374 584/2187 43/1458         0         0        0        0        0        0        0        0

0        0        0   1/4374 13/4374 50/2187 292/2187 2849/4374 827/4374         0         0        0        0        0        0        0        0

0        0        0        0        0        0         0         0        1         0         0        0        0        0        0        0        0

0        0        0        0        0        0         0         0 827/4374 2849/4374 292/2187 50/2187 13/4374   1/4374        0        0        0

0        0        0        0        0        0         0         0 43/1458 584/2187 1981/4374 410/2187 227/4374 22/2187   5/4374        0        0

0        0        0        0        0        0         0         0   5/1458    50/729   205/729 505/1458   97/486   56/729    5/243   5/1458        0

0        0        0        0        0        0         0         0   1/4374   26/2187 227/2187 194/729 211/729 142/729 205/2187 70/2187 35/4374

0        0        0        0        0        0         0         0        0    5/4374   55/2187 280/2187 355/1458 373/1458     5/27 230/2187 175/4374

0        0        0        0        0        0         0         0        0         0    5/1458   10/243 205/1458      2/9 343/1458 124/729 133/1458

0        0        0        0        0        0         0         0        0         0         0 35/4374 245/4374 322/2187 434/2187 853/4374 427/4374

0        0        0        0        0        0         0         0        0         0         0        0 35/2187 140/2187 266/2187 244/2187 166/2187

.

1       1

2      2

3     3

4     4

5     5

6     6

7     7

8     8

9    =  9

8     8

7     7

6     6

5     5

4     4

3     3

2     2

1     1

It is clear that the energy eigenfunction is given by,

[1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1]^T units

And its eigenvalue = 1

Eigenvalue = 1: Corresponds to the fastest path or the principle of minimum action.

The wave function Psi is the SQRT of E, as shown in Figures 1 and 2.

📷 📷

1-Cairo Techniques Solution of Schrödinger's Partial Differential Equation -Time Dependence, ResearchGate, IJISRT review, March 2024.

1- Matrix mechanics in 4D unitary x-t space is much more efficient than the time-dependent PDE in R^4 (3D+t as an external controller).

2- The statistical theory of Cairo techniques, via the B-matrix transition chains of classical physics, proves capable of performing 1D, 2D, and 3D statistical integration and of deriving the Gaussian (normal) statistical distribution law [1].

3- On the other hand, the statistical theory of Cairo techniques, via the Q-matrix transition chains of quantum physics, proves capable of performing 1D, 2D, and 3D statistical differentiation and of deriving the Planck statistical distribution law for blackbody radiation,

B(λ, T) = (2hc^2 / λ^5) * [1 / (exp(hc/λkT) - 1)] . . . . (1)

We assume that Planck's equation 1, viewed in light of current AI, generates incoherent noise and contains a lot of black magic, fragments, or patchwork.

Here is the quantum transition matrix Q for 1D, 19 free nodes.

153/400 -9/16    9/50       0       0       0      0      0      0      0      0      0      0       0       0       0       0      0       0

-1/2 24/25    -3/5    7/50       0       0      0      0      0      0      0      0      0       0       0       0       0      0       0

7/50 -21/40 147/200 -91/200 21/200       0      0      0      0      0      0      0      0       0       0       0       0      0       0

0 21/200 -39/100   27/50 -33/100    3/40      0      0      0      0      0      0      0       0       0       0       0      0       0

0      0    3/40 -11/40     3/8   -9/40   1/20      0      0      0      0      0      0       0       0       0       0      0       0

0      0       0    1/20   -9/50    6/25 -7/50 3/100      0      0      0      0      0       0       0       0       0      0       0

0      0       0       0   3/100 -21/200 27/200 -3/40 3/200      0      0      0      0       0       0       0       0      0       0

0      0       0       0       0   3/200 -1/20   3/50 -3/100 1/200      0      0      0       0       0       0       0      0       0

0      0       0       0       0       0 1/200 -3/200 3/200 -1/200      0      0      0       0       0       0       0      0       0

0      0       0       0       0       0      0      0      0      0      0      0      0       0       0       0       0      0       0

0      0       0       0       0       0      0      0      0 -1/200 3/200 -3/200 1/200       0       0       0       0      0       0

0      0       0       0       0       0      0      0      0 1/200 -3/100   3/50 -1/20   3/200       0       0       0      0       0

0      0       0       0       0       0      0      0      0      0 3/200 -3/40 27/200 -21/200   3/100       0       0      0       0

0      0       0       0       0       0      0      0      0      0      0 3/100 -7/50    6/25   -9/50    1/20       0      0       0

0      0       0       0       0       0      0      0      0      0      0      0   1/20   -9/40     3/8 -11/40    3/40      0       0

0      0       0       0       0       0      0      0      0      0      0      0      0    3/40 -33/100   27/50 -39/100 21/200       0

0      0       0       0       0       0      0      0      0      0      0      0      0       0 21/200 -91/200 147/200 -21/40    7/50

0      0       0       0       0       0      0      0      0      0      0      0      0       0       0    7/50    -3/5 24/25    -1/2

0      0       0       0       0       0      0      0      0      0      0      0      0       0       0       0    9/50 -9/16 153/400

Note that:

1-Q f(x) = d/dx (f(x))

Q^2 . f(x) = d^2/dx^2 (f(x))

Q^3 . f(x) = d^3 / dx^3 (f(x))

. . . . .

etc.

2- The matrix Q can provide a solution to the Schrödinger equation and derive the Planck distribution without resorting to the Planck quantization hypothesis,

E=n h f , n=1,2,3,... infinity.

Which demonstrates that :

Cairo's intelligence techniques = natural intelligence = artificial intelligence in the strict sense = unified field theory?

To be continued.

1- Statistical integration and differentiation, ResearchGate, IJISRT review.

We all know that the Schrödinger equation is very effective in explaining the observed experimental phenomenon of quantum particle tunneling.

Let's analyze a misleading error in a Wikipedia-Google search.

"Tunneling is a consequence of the wave nature of matter. The quantum wavefunction describes the state of a particle or other physical system, and wave equations, such as the Schrödinger equation, describe their behavior.

This is called quantum tunneling!"

But this is only an illusion, because the artificial intelligence of Cairo's techniques allows us to deduce the formula for quantum tunneling from the square of the Schrödinger equation, using classical statistics.

In reality, the artificial intelligence of Cairo's techniques allows for the superposition of total energy in the classical sense. Therefore, quantum tunneling can be rigorously described from the square of the Schrödinger equation, using the statistics of Cairo techniques.

Step 1

Find the allowed stationary solution to the proposed quadratic equation for a quantum particle in a one-dimensional infinite potential well using the statistical chains of matrix B, i.e., 1/n^2, 1, 1/4, 1/9, 1/16, etc. (as shown in Figure 1).

Step 2

The above values ​​correspond to the eigenenergies: 1, 4, 9, 16, etc.

Step 3

Apply the superposition principle, i.e., E = (1, 4, 9, 16, etc.)

with probabilities P1, P2, P3, etc.

Step 4

If the quantum system (or particle) has a total energy Etotal, then:

Etotal = P1 * E1 + P2 * E2 + P3 * E3 +...PnEn

The tunneling problem is reduced to solving the algebraic equation:

X is a function of the total energy such that:

for x .LT.1

X + X^2 + X^3 + . . = Etotal

Step 5

In exponential form, for x .LT. 1:

e^x + e^2x + e^3x . . = Etotal

OR,

1/(1-e^x)=Etotal

Obvious example: for x = 2/3, then Etotal = 3 units.

Step 6

Transmission or quantum tunneling part:

Transmission or quantum tunneling part:

e^ (- Alpha b).Etotal.e^-1/(1-x).GE.Uo (tunnel edge)

Fig. 2

Where l is the barrier width and Alpha is the exponential decay coefficient.

The tunneling ratio Etransmitted/Eincident is called Transmission T=e^-Alpha . l e^- (Etotal/Uo)

Where x = Etotal/Uo . . . (1)

Formula 1 is equivalent to that derived from solving the Schrödinger partial differential equation.

It should be noted that the artificial intelligence approach of the Cairo techniques is fundamentally different from that used to solve the Schrödinger PDE.

To be continued.

1-A statistical numerical solution for the time- independent Schrödinger equation, ResearchGate, IJISRT  journal, November 2023.

SIMPLE CHALLENGE

The simple challenge to our distinguished contributors and readers is the statistical transition matrix B for a closed control volume of surface A subject to Dirichlet boundary conditions,

U(x,y,z,t+dt)=B. U(x,y,z,t) . . . (1)

Equation 1 is not only universal, but it also describes and solves all time-dependent phenomena in the entire universe (partial differential equations of classical physics in its most general form, quantum physics, statistical distributions, integration and differentiation, etc.).

The challenge is as follows:

Name a single physical phenomenon from the four above that does not belong to Equation 1.

Thank you.

Hi Ismail Abbas,

Thanks for sharing this elegant and thought-provoking formulation. I really like the way you’ve framed time evolution through the matrix equation:

It's clean, compact, and captures a wide range of physical systems beautifully.

That said, I had one question that came to mind: How would this framework handle wave function collapse as described in the Copenhagen interpretation of quantum mechanics?

Collapse isn’t just another form of evolution—it’s abrupt, non-unitary, and probabilistic. It's not governed by any deterministic operator BBB, but instead involves a kind of “jump” triggered by measurement, often modeled as projection onto an eigenstate.

So I’m wondering:

Do you consider wave function collapse an exception to your framework? Or do you view it as something that can eventually be embedded within a more general matrix-based or statistical formalism?

I’d genuinely love to hear your thoughts. This kind of unified viewpoint is fascinating, and I’m curious how far it can really be pushed.

Warm regards

In response to our distinguished friend from IRAN:

The statistical theory of Cairo techniques allows us to explain this complex situation in more detail:

The time-dependent diffusion equation (like the Laplacian operator equation) represents or models the natural binary collision relaxation process in a closed-surface control volume subject to Dirichlet boundary conditions.

1- In both time-dependent and time-independent processes, the information waves are always contained within the same control volume.

2- This means that the diffusion process is associated with and controlled by the information waves filling the control volume, both in macroscopic classical physics, such as the macroscopic heat diffusion equation, and in microscopic subatomic quantum physics.

Information waves are the driving force: no diffusion without permission.

3- This means that the Schrödinger partial differential equation is not a pure wave equation, but a partially diffusion and partially wave equation.

4- When we attempt to extract information from a quantum system through measurement, the information wave diminishes and the system transforms into a pure scattering system, a system of binary interacting particles.

In other words the integrated quantum system cannot withstand and would collapse when undergoing measurements coming from outside the system.

This is sometimes called quantum wave collapse.