“Does the arrow of time exist? Does the sawtooth pattern exist?”

- the “arrow of time” absolutely really objectively exists – that is “positive direction” in the absolutely universal for everything that changes, including at the creation, absolutely unique for everything “time dimension”:

- at all/every change any/everything absolutely obligatorily moves in the dimensionon on some absolutely fundamentally non-zero time interval only “ahead”.

In this time dimension differ from apace dimensions, where motions ahead and back are possible; but in other points the space dimensions and the time dimension are well similar, including no of the dimensions by no means have some “sawtooth dimensional patterns”.

Though changes of states of material objects states, including at changes of positions in the dimensions, can be “sawtooth”,

More see series of 6 reDzennn comments June 22, 23, 24, 25, 26,27; 5 ,4 , 4, 1, 5, 3 passages in comments line [“Load comments” in the bottom] phys.org article “New theory proposes time has three dimensions, with space as a secondary effect”

https://phys.org/news/2025-06-theory-dimensions-space-secondary-effect.html

- where also a number of other fundamental points that are necessary in this case are clarified, since any point, including “what is time” cannot be clarified without clarifications of all ones.

See also SS post in

https://www.researchgate.net/post/what_is_the_most_important_problem_in_the_theoretical_physics_now/1884 page 1884

Cheers

There are three fundamental universal laws of physics which, sooner or later, will be generally accepted in scientific society:

1- It is true that the Cairo intelligence techniques = natural intelligence = artificial intelligence in the strict sense = unified field theory.

Ignoring or contradicting this law can only lead to erroneous conclusions.

This fundamental universal law has been in force for four years and has given rise to exact numerical solutions for classical/quantum physics and pure mathematics in all fields, such as:

i- Heat diffusion, sound intensity distribution, and reverberation time in audio rooms.

ii- Single, double, and triple integration in an arbitrary closed control volume bounded by a closed surface A and subject to Dirichlet boundary conditions.

iii- Rigorous derivation of stationary distribution laws such as the Gaussian or normal distribution.

iv- Solving the energy density distribution of a 1D, 2D, and 3D quantum system enclosed in infinite free space.

v- Correct and rigorous definition of strict artificial intelligence (AI) for the first time.

..etc. etc.

2- The quantum physical world Q described by the Schrödinger equation for infinite free space is nothing new; it is simply the square root of the classical world B, which means that

Q = √B,

if the square root of B exists.

3- A condition for the square root of Q = √B to exist is that space be curved in a sawtooth pattern due to the energy density distribution, as shown in two dimensional Figure below.

This means that Einstein's general relativity, sometimes called the theory of gravity, should be generalized to the following notion:

The energy density field, not just the gravity field, is the cause of the curvature of space.

📷

Figure 1 - Classical physics imposing no curvature of space.

Figure 2 - Quantum mechanics forcing the sawtooth shape Curvature in 2D📷

Figure 1 - Classical physics imposing no curvature of space.📷

Below, we present three examples of generating 1D, 2D, and 3D quantum matrices from the exterior product of two sawtooth vectors u⊗v.

Or, in index notation:

u⊗vij = [ui vj]

Furthermore, we show that the quantum matrix u⊗v exhibits a pattern reflecting the sawtooth shape of the input vectors. Furthermore, its product with any initial state vector produces sawtooth vector properties.

Example 1,

1D quantum sawtooth matrix with 9 free nodes Q:

Q=

1

2

3

4

5

4

3

2

1

⊗

1 2 3 4 5 4 3 2 1

=

1 2 3 4 5 4 3 2 1

2 4 6 8 10 8 6 4 2

3 6 9 12 15 12 9 6 3

4 8 12 16 20 16 12 8 4

5 10 15 20 25 20 15 10 5

4 8 12 16 20 16 12 8 4

3 6 9 12 15 12 9 6 3

2 4 6 8 10 8 6 4 2

1 2 3 4 5 4 3 2 1

Note that the matrix Q satisfies three conditions or constraints:

1- Q is a double statistical matrix, which means that

ΣQij = 1 for all rows i.

ΣQij = 1 for all columns j.

2- The determinant Q is zero in all 1D, 2D, or 3D spaces.

3- The matrix Q is invertible, which means that Q^-1 exists.

The above 1D quantum matrix, when multiplied by an arbitrary initial quantum state matrix, will produce a sawtooth-shaped quantum matrix, namely,

*1 2 3 4 5 4 3 2 12 4 6 8 10 8 6 4 23 6 9 12 15 12 9 6 34 8 12 16 20 16 12 8 45 10 15 20 25 20 15 10 54 8 12 16 20 16 12 8 43 6 9 12 15 12 9 6 32 4 6 8 10 8 6 4 21 2 3 4 5 4 3 2 1

=100000001

2

4

6

8

10

8

6

4

2

In arbitrary dimensionless units.

What remains is a sawtooth shape.

This is the case for the energy density distribution in a one-dimensional infinite potential well, as shown in Figure 1.

📷