Nature works in four-dimensional space, which is the combination of three dimensions of space and one dimension of time. This concept is central to the theory of relativity, which describes the behavior of matter and energy in the universe.

In four-dimensional space, events are described using four coordinates: three spatial coordinates (x, y, and z) and one temporal coordinate (t). This allows us to describe the position and motion of objects in space and time.

The laws of physics are the same in four-dimensional space as they are in three-dimensional space. However, the addition of time as a fourth dimension introduces some interesting effects. For example, time dilation occurs when an object moves at high speeds relative to an observer, causing time to appear to slow down for the moving object. This effect has been observed in experiments involving high-speed particles.

Another effect of four-dimensional space is gravitational lensing, which occurs when the path of light is bent by the curvature of spacetime around massive objects. This effect has been observed in astronomical observations, providing evidence for the existence of dark matter and dark energy.

In summary, nature works in four-dimensional space, which includes three dimensions of space and one dimension of time. This concept is central to the theory of relativity and allows us to describe the behavior of matter and energy in the universe, including effects such as time dilation and gravitational lensing.

In a breakthrough technique, B-matrix chains are used in this article to numerically solve single, double, and triple (Hypercube) integrals as a successful preliminary answer to this question. In fact, it is the natural continuation of the success of B-chains in solving the general case of time-dependent partials of differential equations with Dirichlet Arbitrary BC and arbitrary initial conditions that have been explained in previous articles.In this answer we provide specific answers to three of the most persistent questions:i- Do probabilities and statistics belong to physics or mathematics?ii- Followed by the related question, does nature operate in 3D geometry plus time as an external controller or more specifically, does it operate in the inseparable 4D unit space where time is woven?iii-Lagrange multipliers: Is it just a classic mathematical trick that we can do without?The answers to these questions are all interconnected.Here we could say that the modern definition of probability in the transition matrix B is an interconnected thing of the three topics. Imagination is the first important common factor in mathematics, physics and especially in the probability of transition in 4D space.

We propose the following,

i-nature sees the 1D curve as a 2D trapezoidal area.

ii-nature sees the 2D square as a 3D cube or cuboid volume.

iii-nature sees the 3D cube as a 4D Hypercube and evaluates its volume as L^4.

Moreover, in all propositions i-iii, time t is the extra dimension.

However, many people still believe that x-t spacetime is practically separable and that describing x-t as an inseparable unit block is only essential when the speed of the object concerned approaches that of light.

This is the most common error in mathematics.

Brief,

1-It is important to understand that mathematics is only a tool for quantitatively describing physical phenomena, it cannot replace physical understanding.

2-It is claimed that mathematics is the language of physics, but the reverse is also true, physics can be the language of mathematics as in the case of numerical integration and the derivation of the normal/ Gaussian distribution law via the statistical B-matrix chains.

However, in a revolutionary technique, chains of B matrices are used to solve numerically PDE, double and triple integrals as well as the general case of time-dependent partial differential equations with arbitrary Dirichlet boundary conditions and initial arbitrary conditions.

The statement 2 that mathematics is the language of physics always was a given and widely accepted, but the idea that the reverse could be true is quite unexpected.

Most mathematicians believe that all mathematical equations should be applicable to the set of generalized geometric shapes/volumes that emerge from our observations of physically existing "objects".

This is not true since nature does not operate in geometric shapes as observed with our eyes and brain, ie. in x-space alone, but it works in x-t-space now and for eternity.

As a specific example without loss of generality here is  the  results for

1-D numerical statistical integration Is for 7 free nodes and 19  nodes

Equation 1 is the statistical equivalence of Simpson's rule for 7 nodes.Is=[6h2/77]*(6.Y1 +11.Y2 + 14.Y3 +15.Y4 +14.Y5 + 11.Y6 + 6.Y7)  . . . . . . (1)

Is= h2 (0.2571Y1+0.4872Y2+ 0.6902Y3+ 0.8662Y4+1.0150Y5+1.1368Y6+1.2316Y7 +1.2992Y8+1.339Y9+1.3534Y10+1.339Y11+1.2992Y12+1.2316Y13+1.1368Y14 +1.0150Y15+ 0.8662Y16+ 0.6902Y17+0.4872Y18+ 0.2571Y19) . . . (2)

Equation 2 is the statistical equivalence of Simpson's rule for 19 nodes.

It worth mention that equations 1 and 2 were the dream of mathematicians at that time.

Probabilty and statistics are mzthematical tools. Spacetime geometry, at least at scales from subatomic distances to the size of the observable universe, is four-dimensional, described by general relativity, when gravitational effects are relevant; matter is described by the Standard Model.

Lagrange multipliers were invented as a mathematical technique to describe constraints.

One could invoke that energy subdivides between systems by scaling them with parameters, in the same way as Lagrange multipliers are used in variational problems.(?) I just read an article*, where I did not do that, but instead used an entire gradient, similar to 'entire time differentiation'. It would result in the same traveling waves, with one more variable, probably, and need not be reduced to obtain the phenomenon?

*This one ( not so easily read, e.g. whilest tempis denoted H)

https://www.sandermanpub.com/current/aee/140/020103.html