2-Extensions on "Reflections on phenomenic descriptions of the real part 1"

 here is the section for mathematical symbolic proves and formulaic researchs.

unique $Ω$ has unique metric:

$Ω\space parametrizes\space T_pM \rightarrow T_pM\simeq T_qM$(isomorphism),in order to the product be different the vectors must be too(so there's one metric)

one of the other's first's statements declared was about momentum being normal or somewhat normal with tangent parts to it as the curvature has its hybrid structure,sometimes being normal,sometimes not,the following is how the relation goes:

$\space def.\space p \space component \space of \space  phase\space space\space connection\space Ω_{q,p}\in M \space ;Ω_p : Ω_p \propto \delta p_{\forall γ \in \int J_t M}$

$\delta p = \frac{\delta p}{\delta t}\delta t = \frac{\partial p}{\partial q}\cdot \delta q$

 $\parallel$

$\frac{\delta p}{\delta t} = \frac{\partial p}{\partial q}\cdot \frac{dq}{dt}$

note:the partials dont commute in order to have a force in it ,this is due to some hamiltonian like structure,to be sure a lie algebra/poisson algebra(relating to $Ω_M$ and further to $\mathcal L$).

$p= m \frac{\delta q}{\delta t}$, $\dot p =\frac{m\delta ^2q}{\delta t ^2}\rightarrow \delta p \propto \frac{\delta \dot q}{\delta t}\space \text{&&}\space\delta p_{\bot} \propto \frac{\delta \dot q}{\delta s}(Frenet\space Serret \space equations)$

(to approach this the notion, p and q are being assumed to be the complete set of phase degrees of freedom and the default of being all smooth,so $M$ is $N$(as in the original post i said) ,M encompassing all motion is necessary,also frenet equations could be derived not just infered from above calculations in $N$(from a categorical point of view as we saw before reversing directions of proves are a common ideal) ,the parallels are clear,a counterexample could be seen below)

the other case where $m$ (mass) is time dependent there are some others variations to which $\dot p$ is not based on the aceleration,cases like this can be extracted but its good to know that the probably most critical counterexample and the unique is not a basis of fragility to the idea,the mass varying term can be though to be at the other velocity dependent forces,just like the others that add torsion to the system,and  the idea continues.

$Ω_p \propto \delta p_{\forall γ \in \int J_t N} \rightarrow Ω_p \simeq Ω_T +N_n$

$\text{&&}$

$Ω_q \simeq  Ω_T +N_n \rightarrow Ω_{q,p}\simeq Ω_T +N_n$

(same rank)

(thats a consequence of the manifold being Kahler,the coordinate connection mimics the phase connection and vice versa,they are compatible as it should be,and the normal and the tangent frames play an important role on this,and they are complementar being one related to the other(given that phase space mimics the coordinate space,when we reduce the normal to the tangent in coordinates it should happen by covariance in the phase space,and seems obvious that the normal is a dependent typo,normal is normal to something being a trivial degree of freedom(with trivial i dont mean to be deleted but something that is reduced to the tangent,also being a frame to approach,trivial is not null)))

$\Leftrightarrow$

$Ω_{q,p}\simeq Ω_q\simeq \Omega_p \simeq Ω_T \simeq N_n $

this means again and more intrinsically that a consistent dynamical system(q motion/particular or universal spatial behaviours) is given by its phase dynamical system(q,p motion or just p motion),and when one is been treated with less degrees of freedom the same happens to the other,generating degeneracies(time is not covering the space as it should),as i said in the post,is a problem with encompassing time that leads to contradictions and being unable to define a sympletic from,is a problem with external-internal logic that leads to contradictions and being unable to define a sympletic form,making the time generation an universal one,the requiriment so is just to take free of external factors(i'll discuss later if its possible and the overall epistemes to it).

more for who want to see what this really means:

given $n=(q,p) \in N$ && $\Omega_{q,p}=Ω\space \text{3-form}$ $\rightarrow \frac{d\dot q^κ}{dt}=Ω^κ_{i j}n^i n^j$

the geodesics are mixed up in phase space,q and p get too much connected in a order that you can't have one without another variation given the force's constraints to which we are physically and virtually restricted.The post idea was to go from force constraints to the sympletic form and  otherwise too,since our philosophy is constantly entangled with the long term consistence of physical systems by its internal contradictions, being inherent to our ideas go in and and out or on and off,fluctuating in the boolean enriching and its combinations.When we take a look to see whats the phase contributions we get the idea that the system is bouncing in time by an ethereal surface,the ground given by the bare coordinates $q$ are just consistent if they are also $(q,p)$ coordinates,the space is bigger having an interval in the time extended version of the bare coordinates,or better in the $(q,p)$ space($(q,p)$ is more compact,linking to fundamental finiteness(conservation principles)),it is not restricted to virtual forces or conversely, all physical systems are virtually restricted,all possibilities are real,the number of possibilities of the spatial is the essential sample of the spatial,the granular conditional approach of the space is the one that sees the space,getting to know its overall infinity corners by finite amounts part by part in themselfs,without ignoring the singular products of the infinity horizon,so one could fly out from its underground conditions and see it,be able to really gather the conditions of its finite emanance in the folded surface of space  pointing towards time, there's a sense of freedom as you and i can see:

possibilites of $(a,b)=ab$

possibilites of $(q,p)=\mathbb R^n \times \mathbb R^n$ (this is the same as the possibilities of q as we saw the equivalence's made before)

Differential $Q$ $\rightarrow$ Riemannian $M$ $\rightarrow$ Kahler N

(resolution of self compatibility)

Differential $Q$ $\rightarrow$ Kahler $N$

(resolution of phase compatibility)

given that the structure is compatible now,time is the Kahler manifold parameter(as you fills a space length in the bare space you are filling the same and other codependent length in Kahler space being the potential of action),so other way to see:

Geometry $\rightarrow$ dynamics

(time compatibility, variational scheme of geometry earlier correlated to time becomes time itself as predicted)

curvature dynamical structure and topological dynamical structure  are so the phase curvature and topological dynamical structure,including potentials that were marginalized by its velocity dependence.

about the other provable sentence of energy momentum tensor having energy invariant to time variations,seems obvious,time here is not $t$(killing parameter),its $\tau$(the only other curved tensor/scalar that has energy scale is the energy current from the energy tensor momentum):

$E_{curved}=\int_Σ T^{\mu \nu} \cdot dΣ_{\tau\space \mu \nu}$

$\nabla_\mu T^{\mu ν} = 0 \rightarrow \nabla_\tau T^{ττ} =u_{\nu}( u^\mu \nabla_\mu T^{\mu\nu})= 0 \therefore \frac{d}{d\tau}T^{00}=0$(just making a little change of frames)

thats also an important spanning equation,as Euler lagrange equation is ,as hamilton equation is,as Einstein equations is,that's because this lies in the condition that all dynamics is encoded in $\tau$,that one parameter can determine all configurations in configuration space and this parameter is the one that parametrizes PseudoKahler Manifold(something that i forgot to note,in GR we are not talking really about Riemannian manifolds just,we are talking about Pseudo ones to be sure,Kahler Manifold in GR is PseudoKahler Manifold,as i was calling pseudoRiemannian Manifold  an Riemannian Manifold i was calling PseudoKahler... an Kahler...,after doing the post i noted that given all the talk about time to emphasize the negative in the metric not using the Pseudo part is not pedagogical or context-consistent,like if i was forgotting all previous made distinctions),it has the same behaviour as $t$ before ,parametrizing all phase space and having the equation that relates the overall behaviour,the overall symetries of phase space,it completes/encompasses the space using its field sections.In the post after going trough all this parallels (nothing but text going trough the space of lagrangian/noether physical sections that them already made but people seems to not notice,using non phasical constructions),i had one more contradiction to show relating the pole singularity of every smooth scenario,the one that lies in the self-interaction of himself and its coupled brothers and sisters mesmerizing its orbits is chaotic patterns,forming interconnected intricated correlations of non linearity btw all of them.As we know ,with enough conditions gravity is reducted in the plane to newtonian gravity,and the newtonian to schwarzschild gravity,having an singularity in the middle(not necessarily being a black hole):

$\lim_{r \to 0}Φ(\frac{1}{r})\to \infty$

given that the local leafs of coupled equations have mutual gravity dependence,one metric depends on another(GNR-General Relativity),one potential  on another(GR- Galilean Relativity),so the local vibrations come to interact by infinity amounts of quantitative indiferentiated quality(or quantitative indiferentiated local infinitys,relating to a portion of the post that i touched on the local infinity differentiation as an eidetic/signed/genic abstraction):

$\lim_{r_i \to 0}\prod_i Φ_i(\frac{1}{r_i})\to \infty\cdot \infty \cdot\infty...$

or

$F(\lim_{r_i \to 0}\prod_i Φ_i(\frac{1}{r_i}))\to \space const$

(by some interaction of infinitys)

infinity interacting with infinites giving finiteness,thats absurd right?

nah,it happens all the time,and happend before with all kind's of infinity(the infinitely little,the infinitely centered ,the infinitely big,...),these infinitys make them constant by requiring a pattern formation to which the infinitys are relative,all infinitys are the made by the local differentation of the finite,they are mutual and we are looking for the relations that they made so ever,how $F$ becomes finite without having seemly contradictory/liminal objects to our operations,in order to be away from the infinitys of PseudoKahler geometry and Kahler geometry itself in both many body problems,we navigate towards a more concise approach using one unity of aperception(or perception wtv),we want every of these infinites to be regularized requiring unitys that knows the self and the other and is deeply multiconnected with all path solutions(a requiriment necessary due to the nature of co-contra variant tranformations,one would see that requiring no infinites at the core modifys the notion of smoothness itself),one could go back to the post to see the implications of it in and out of information;for now it's this,bye.