1. In a nutshell: the building blocks 1 & 2

The below building blocks 1 & 2 are supposed to enable a conceptual paradigm change. Each of two proposed building blocks is governed by the conservation law of total (mechanical and dynamic) energy. The mathematical framework for building block 1 is the Krein space. The mathematical framework for building block 2 is the Hilbert space.

The main conceptual change coming along with building block 1 is that the mechanical potential energy based Newton/Coulomb potential concept is replaced by a dynamic potential energy based indefinite norm based potential concept enabled by a system intrinsic potential energy operator.

The construction process for the building block 1 structure is a bottom-up approach from a priori (i.e. meta-physical) „ground state“ & „perfect plasma“ systems.

The proposed energetical dynamic and mechanical systems of building block 1 always deal with quanta types equipped with system intrinsic potential differences. Additionally, there are also potential differences between those systems. The intrinsic potential differences do have the role of a "cohesive Mie pressure". The explicate potential differences between the several quanta systems do have the role of "action causes". 

Building block 2 below provides an appropriate framework to solve the 3D-NSE problem in aligment with the intrinsic Neumann (pressure) problem, (BrK0) p. 26, (BrK11). It is also proposed to be applied to define an integrated mechnical & dynamic Landau equation model. Technically spoken, in case of the 3D-NSE problem this concept provides an dynamic fluid particle. In case of the Landau damping phenomenon it avoids the concepts of a Debye shield and it overcomes the current issue of two different incompatible mathematical Landau damping models (i.e., the Vlasov and the Landau equations) explaining the Landau damping phenomenon by two different underlying governing "forces", (BrK0) p. 63 ff.
 
The construction process for the building block 2 structure is a top-down approach from the (standard variational) domain of the kinetic Laplacian (potential) operator, which is the so-called H(1)-"energy" Hilbert space and a sub-Hilbert space of H(1/2). In this two-layer structure the corresponding (not self-adjoint) dynamic potential operator may be interpreted as a compact disturbance equipped with the domain H(1,ortho); this is the complementary sub-space of H(1) in the newly proposed dynamic H(1/2) Hilbert space.

The physical actions within building block 2 are governed by an implicate potential difference between the standard variational mechanical domain of the Laplacian (potential) operator and its complementary sub-space with respect to the overall H(1/2) Hilbert space norm.

2. Building block 1

   ... is an integrated hierarchical ordered Krein space based scheme of energetical quanta systems accompanied by

- a new dynamic energy type
- Krein space intrinsic concepts of potential and potential differences
- a Krein space intrinsic self-adjoint potential operator
- dynamic energy fields complementary to mechanical (kin./pot.) energy fields
- Krein-space based dynamic fields with related dynamic quanta systems
- 2-component (quanta pair) and 1-component (quanta) systems 
- a priori 2-component „ground state“ & „perfect plasma“ quanta pair systems
- a mechanical 2-component "perfect electromagnetic " quanta pair system
- quanta pair systems governed by the complex Lorentz group SU(2) x SU(2)
- 1-component quanta systems governed by the restricted Lorentz group SU(2)
- implicate intra-quanta system dynamics
- explicate inter-quanta systems dynamics
- dynamic energy generated by intra-quanta potential differences ("pressures")
- mechanical actions triggered by inter-quanta potential differences 
- a cohesive „Mie pressure“ of a generalized „ether physics“.

a. Underlying concepts

The quanta creation process from a priori 2-component „ground state“ & „perfect plasma“ quanta pairs is governed by a mathematical number theoretical "probability" principle. The underlying mathematical rule to build appropriate quanta numbers is based on the different "Schnirelmann densities" of odd (density = 1/2) and even (density = 0) integers. Colloquially spoken, the probability that electrinos or electrons may connect with positrinos or positrons is >0, while the probability of the reverse is 0. 

Quanta compositions from the baseline quanta pairs are acompanied by correspondingly recalculated quanta numbers. The dynamic quantum element types are characterized by quanta numbers < 1. The mechanical quantum element types are characterized by quanta numbers >1. The only quantum element with quanta number =1 is the neutron.

The equalization (entropy) processes of potential differences from mechanical energy systems back to purely "symmetric" dynamic energy systems become a model for related quanta annihilation (decay) processes.

Colloquially spoken, the 2-component systems provide a mathematical model of "the mysterious fabric of our reality", (GaG1), (UnA2). They may be interpreted in the sense of H. Weyl as the whole, which has to be presupposed in order to give meaning to the mechanical particulars, or as a kind of Higgs field.

The invariant quantities of the above three 2-component layers are governed by the two isomorphic normal subgroups of the group SO(4), which are directly related to the complex Lorentz group SU(2)xSU(2), (BrK0) p. 40.

The 1-component atomic dynamics system is governed by the restricted Lorentz group SU(2) (isomorphic to the unit quaternions) accompanied by the concept of a Maxwell-Mie-pressure enabling links to the SRT-Minkowski space and to "Einstein's lost key", (UnA1). In other words, the creation of the 1-component atomic system from the perfect 2-component electromagnetic system is accompanied by a symmetry breakdown from the complex Lorentz group (the main tool to prove the CPT theorem, StR)) down to the real restricted Lorentz group.

The 1-component atomic dynamics system is governed by the restricted Lorentz group SU(2) (isomorphic to the unit quaternions). In other words, the creation of the 1-component atomic system from the perfect 2-component electromagnetic system is accompanied by a symmetry breakdown from the complex Lorentz group (the main tool to prove the CPT theorem, StR)) down to the real restricted Lorentz group.

b. Opportunities

Note: The "perfect plasma" quanta pair system "explains" the decay of a neutron into an electron and a proton.

Note: The a priori 2-component dynamic „ground state“ & „perfect plasma“ quanta pair systems provide an appropriate modelling framework for the hypothesis of "vacuum flucuations" in line with the observed "Casimir effect" phenomenon.

Note: Plasma physics is about classical statistical fluid mechanics and classical fluid dynamics. The Landau damping phenomenon is a characteristic of collisionless plasma dynamics. It is a wave damping without energy dissipation by elementary particle collisions. Therefore, the a priori 2-component dynamic „perfect plasma“ quanta pair system provides an appropriate model enabling plasma dynamics without energy dissipation by elementary mechanical (!) particle collisions.

Note: The mechanical 2-component "perfect electromagnetic" quanta pair system provides an appropriate modelling framework for the CMBR phenomenon and Ehrenhaft's photophoresis phenomenon.

Note: The CMBR phenomenon and the physical mechanism of oceans generating microwave background, ((RoP2), can be interpreted as an (interaction) echo caused by system intrinsic potential differences of dynamic 2-component systems like the ground state, the perfect plasma, the perfect electromagnetic, and the hydroxyl-hydrogen systems.

Note: The mechanical electroton quanta system solves the "self-energy problem" of Dirac's electron system providing an explanation of the beta decay process.

Note: The atomic nucleus Dirac 2.0 systems are accompanied by three types of atomic nuclei, N(+), N(-), and N(0): 

"The Meissner effect shows that a bulk superconductor behaves as if the magnetic field inside the specimen vanishes. ....from Ohm’s law one may concluded that the flux through the metal cannot change on cooling through the transition. The Meissner effect suggests that perfect diamagnetism (external magnetic field and an induced intrinsic magnetic field) is an essential property of the superconducting state“, (KiC) pp. 262/263. 

What if, the three atomic nucleus types of the Dirac 2.0 systems enable a new concept to differentiate between superconductors (--> diamagnetism), insulators, and bulk conductors (--> external magnetic field and an induced intrinsic electric field; replacing the Maxwell displacement current) ?). This would provide as a new basis for a quantum theory of superconductivity replacing the BCS theory.

3. Building block 2

  ... is the energetical H(1/2) Hilbert space approximation system of the system scheme of building block 1 accompanied by dynamic fluid particles and

- a new "mass element" as distributional function of the H(-1/2) Hilbert space
- a well-posed NSE system aligned with Plemelj's enhanced Newton potential
- a properly defined Prandtl operator (incl. domain) for the Neumann problem
- a resolved d'Alembert "paradox", in fact the failure of the Euler equation (the    model of an ideal incompressible fluid) as a model for fluid-solid interaction
- a nonlinear dynamic potential operator interpreted as compact disturbance of     the mechanical Laplacian potential operator, (BrK0) p. 11
- mechanics / dynamics governed by Fourier waves / Calderon wavelets
- an alignment with the global nonlinear stability of the Minkowski space
- ...
- ... (BrK10).   

a. Underlying concepts 

Building block 2 provides a purely Hilbert space based modelling framework accompanied by the concept of a dynamic fluid particle. It may be interpreted as an approximation framework to the Krein space based quanta field scheme. Technically spoken, the H(1/2) Hilbert space is an extension of the standard variational mechanical energy Hilbert space H(1), which becomes a compactly embedded sub-Hilbert space of H(1/2) = H(1) x H(1,ortho). The complementary closed sub-space may be interpreted as dynamic fluid energy space.

The standard variational domain of the mechanical (self-adjoint) potential operator is the H(1) energy space. Its complemetary sub-space of the newly proposed extended energy Hilbert space H(1/2) with respect to the H(1/2) norm (the energy invariant scalar function) provides the domain of a complementary dynamic potential operator. This operator is not self-adjoint, however, in the corresponding variational representation it may be interpreted as a compact disturbance of the mechanical potential operator enabled by the underlying coerciveness (Garding type) inequality, (BrK0) p. 26.

There is a mathematically correspondence between the norm (the invariant of a Hilbert space based energy system) of the proposed extended energy Hilbert space H(1/2) of building block 2 and the "wave energy" norm of the baseline energy Hilbert space of building block 1, (BrK0) p. 17.

Building block 2 provides an appropriate framework to solve the 3D-NSE problem in aligment with the intrinsic Neumann (pressure) problem, (BrK0) p. 26, (BrK11):

it turned out that the non-linear energy term of the 3D-NSE system is bounded with respect to the  H(1/2) energy norm as a simple consequence of the Sobolevskii inequality (BrK11), (GiY) lemma 3.2. At the same time, the dynamic fluid H(1/2) energy concept is in line with a well-defined Plemelj’s double layer potential function, (BrK11). The related Prandtl operator accompanied by a Hilbert scale domain H(r) (where  ½ smaller or equal than r smaller than 1) provides a unique solution of the underlying Neumann boundary value problem for the pressure p(x,t), (BrK7), (BrK11), (LiI) p. 95 ff., (PlJ).

The concept of a H(1/2) energy field based dynamic fluid element is accompanied by a well-posed exterior Neumann problem, a well defined related Prandtl operator, (BrK7), and Plemelj's double layer potential, where the mass density (du/ds)(s) of the single layer potential is replaced by the differential du(s), (BrK11), (PlJ).

The inner product of H(1/2) is isometric to an inner product in the form (Qx,Px), where Q resp. P denote Schrödinger‘s position & momentum operators. The combination with the Riesz transform operator provides the link to the considered operators in (BrK0), and is in line with the alternatively proposed Schrödinger operator in (BrK6) and related early thoughts on a new ground state energy model in (BrK8).

b. Opportunities

Note: The Prandtl operator enables a concept of a H(1/2) energy system intrinsic potential difference. It „solves“ the source of the d’Alembert “paradox”. This is about the failure of the Euler equation as a model for fluid-solid interaction, as in incompressible fluids there are no frictional forces. In other words, the Prandtl operator enables a dynamic fluid accompanied by frictional forces, which is applicable as mathematical tool for exterior space problems in physics.

Note: Building block 2 is also supposed to define an integrated mechanical & dynamic Landau equation model avoiding the concept of a Debye shield. An integrated mechanical & dynamic Landau system will overcome the current issue where two different Landau damping models (i.e., the Vlasov and the Landau equations) explain the Landau damping with two different underlying governing "forces", (BrK0) p. 63 ff.  

Note: The 1-component atomic dynamics system of building block 1 is governed by the restricted Lorentz group SU(2) (isomorphic to the unit quaternions). In building block 2 this concept may be adapted to Mie's original approach enabling links to the SRT-Minkowski space and to "Einstein's lost key", (UnA1).

Note: Exterior space problems in physics primarily revolve around the limitations and inconsistencies between general relativity and quantum mechanics, particularly when dealing with extreme conditions and the nature of space-time itself. Key areas include the nature of dark matter  and dark energy, the unification of gravity with other forces, and the behavior of space-time at the Big Bang and black holes. Additionally, the concept of quantum gravity  and the potential for emergent space-time are active areas of research, Wikipedia.

Note: There is no direct observations neither of the pressure nor of the density in the entire atmosphere of the sun, (UnA4) p. 59. … The energy production process of the sun is the fusion process from hydrogen to helium. The hydrogen molecules shows only two electrons, however there is an enormous bounding energy of the electron in a hydrogen molecule, (UnA4) p. 64. … Metals appear incompressible, but exposed to an exterior pressure they can reduce their volume. ... According to de Broglie's wave length formula for a electron such a volume reduction would lead to a reduction of the wave lengths of all electrons (i.e. basically the inverse of the mechanical momentum of those electrons) in the metal, (UnA4) p. 66. … There is an interesting phenomenon when water is subjected to massive pressure: water close by a nuclear explosion gets black and opaque, (UnA4) pp. 71/72.

Note: The Stefan-Boltzmann law states that "the total energy radiated per unit surface area of a black body (e.g., main sequence stars from the Herzsprung-Russell diagram) across all wavelengths per unit time is directly proportional to the fourth power of the black body's absolute temperature", Wikipedia.