Fundamental Theorem of Calculus as Fundamental Physics

Recent posts have presented New Newtonian Gravity as a new way of viewing the connection between gravitational potential $\phi (x)$ and gravitational mass density $\rho (x)$ through the assignment $\rho (x)=\Delta\phi (x)$, where $\Delta$ is the Laplacian differential operator acting with respect to $x$ as an Euclidean space coordinate. Gravitational force at $x$ is given as $\nabla\phi (x)$ with $\nabla$ the gradient with respect to $x$. 

The gravitational potential $\phi (x)$ is here viewed to be primordial somehow generating mass density $\rho (x)$ by differentiation as a process of local instant action, to be compared with the classical view with mass density primordial generating gravitational potential by instant action at distance lacking physics.

The connection/assignment $\rho (x)=\Delta\phi (x)$ is the result of:

1. Observing that gravitational force $F$ is conservative (work independent of path) shows that $F=\nabla\phi$ for some $\phi$.
2. Conservation of gravitational flux/force $F$ of the form $\nabla\cdot F = S$ with $S$ source of $F$.
3. Assignment $\rho = S$. 

Here 2 can be seen to express the Fundamental Theorem of Calculus in the form of Gauss Theorem:

• $\int_\Gamma F\cdot n ds =\int_\Omega \nabla\cdot F dx$,

with $\Omega$ a domain with boundary $\Gamma$ with outward unit normal $n$, which expresses

• total flux out of a domain = total source inside domain 

as a law of conservation. 

In the case of one space dimension Gauss Theorem reduces to the Fundamental Theorem in the form 

• $F(b) - F(a) = \int_a^b\frac{dF}{dt}dt = \sum dF$  

which expresses that the total change of $F(x)$ as $F(b)-F(a)$ is equal to the integral/sum of little changes $dF$ or that "the whole is the sum of its parts".

The connection $\rho (x)=\Delta\phi (x)$ thus is the result of: 

1. Observation that gravitational force is conservative.
2. Conservation of gravitational flux.
3. Assignment of mass density. 

Is it possible that 1 or 2 as conservation laws can be violated? This connects to the question of existence of dark matter arising from observing gravitational flux/force for which the corresponding source appears to be missing because it cannot be seen. Insisting on conservation means that the source must be there as an invisible source = dark matter. Giving up conservation means giving up science with a resort to mysticism.

A conservation law reflects a scientific principle as a form of book-keeping where everything is recorded and balanced so that In = Out. With this view 2 is true a priori without need of experimental verification:

If something is missing, then that something must exist upon more careful inspection. There must be some dark energy explaining to observation of gravitational force/flux. Or it is just a mystery.

There is also very good reason to believe that Nature requires conservation in order to exist. Constant input without corresponding output leads to blow-up, and constant output without input leads to extinction. Total energy must be conserved. 

Recall that the 1st Law of Thermodynamics states that total energy is conserved. In particular, kinetic energy is transformed into heat energy by friction and turbulent dissipation accounted for as losses as parts  of an energy budget. Whatever disappears in one column of the budget as loss appears as gain in another column.    

Altogether, there seems to be very good reasons to believe that New Newtonian Gravity captures real physics and so little/no reason to believe that Einsteinian Gravity can offer something better.   

Einstein questioned Newtonian Gravity and wave nature of light, but he never questioned Conservation of Energy. Why?