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\title{\Large\textbf{Combinatorial Phase Separability and Congruence}\\\large\textbf{Physical Laws and Observations}
\normalsize\author{(C) Paris (s.) Miles-Brenden}\\ Albuquerque, NM, United States of America\\ \it{paris.smb.0@gmail.com}}

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\maketitle

\abstract{The clearest distinction found in that of the domain of matter, sound, light, and energy is the congruency of the reflective, absorptive, transmissibility and birefringent refractory properties of crystals and optics, as well as microphones, for that of material specificity and the mode to eigenvalue procedure of breakdown of all but quasicrystals and certain colloid to nematic-crystal phases.  Here, we will explain the properties of these materials in terms of the phase-congruency between cooperative phases of crystal and fluid, transparent, and opaque for that of the atomic (sound - phonon) and electronic (charge - photon) modes of these crystals.  Phase-separation, with, the phenomena of agreement between the mode driven, or free; and separated by a weak link of the alternative differential equation, prescribes a manner in logical foundation of geometric description of crystals, and analytical description.  This new nature of broken-duality is an example of energy-eigenvalue-geometric specturm compared to the varietal phases of matter in group and phase.  Thus, it is learned in this paper that the differential anharmonicity wit harmonic conditions is a resonance phenomenon of which is a formerly understood material property in relation to theory, the concealed state of matter for which it is akin a counter-process to a quasicrystal.  The process of the secondary eigenvalue and primary first eigenvalue is then reversed, to-which the Eiophantane equation governing behaviors in chart and catalog is related of Pell's equation, the KP equation, Boussineq Equation, Schroedinger equation, Sine-Gordon, Korteg d'veg de'Vries, and Laplace equation in the nonlinear sciences with the application of optimization theory.  The differential equation then - very much illustrates a secondary relationship of the eigenvalue off-center, and off-basis of the diagonal of the differential on-diagonal elements of the various dimensions.
The secondary dimension as it stands as a basis of condition on blind-zero-on-sight is a disinclusive basis of neutralization of the primary initial second dimension of the matrix describing the system.}

\normalsize
\section*{Introduction}
\noindent Whence beginning (preliminarily) with the KP-equation, and the Lorentz-Maxwell equations for that of the non-linear oscillation, and; including the normal simple harmonic oscillation, in transformation there are off-center terms.  Thus, the inclusion of a Brownian bridge, or similar, of the form of intermediate neutral phase as understood in a Mott insulator, furnishes a threshold condition, phase congruency in the differential equation, and phase separability in that of physical phenomena, of which proportionately via an eigenvalue and geometry of phase a given group-phase acoustic-optical bridge with slits that can be controlled is-formed.  This, can then be utilized to create an operational-calculational derivational machine, by operating the vertex-cell of dots, with the domain-cell over line-like aperature of optical and electromagnetic bridges.  Thus, it is similar in \textbf{E} and \textbf{B} to a telescope.\\

\noindent By discharging, charging, and symbolically replacing, carrying, and prompting characters; computational process is possible with dots and strips, to do processing of optical-sound-electrical-topologically in-build hybrid processes.  As a consequence of the Mott-insulator and Kondo-effect bridges, then, it can be seen; certain materials illustrate doping-dependent crossover and transition, similar in nature to transistors, for that of a Gunn diode or-similar.  These behaviors depend on the secondary-linear ($\alpha-\beta z$) dependency term; and congruency between phases and separation, to which breaking in-translation and chart reference, the Affine-Abelian (Artin algebraic terms and units of the field) refer to the Godelian hypothesis (correspondently), in syntax and semantic queue of the relationship of bracket symbolism.  Despite surprise, there is no universal key, and no universality but of the algebraic limits of power, memory, energy, and entropy.\\

\noindent Dimensional 'union and subtraction of concave and convex set', in-graph, then relate of miniature and gross Casmir energy flow in power, time, and inertia, to which the effect in principle of translation of a symbolic equation into form is a unique physical theory as summarized in Hawking radiation, the theory of the Mott insulator and Semiconductors, Super Insulators, and Super Conductors.  Secondly, the Mendelev chemistry table, and optimization theory in Mathematics relate of a grand sort of numerical patterns.  When this process is of one hierarchy, and the virtual is distinct the process divisionally is quantified by a multiplier-basis and 2-1 on diode-channel, thus mutual-phase of two mutual phenomenena responsible (for a Mott insulator) phases of insulator and partial metal and partial conductor.\\

\noindent Essentially a channelled-upper process and lower process form an elliptic unitary translational modular group with non-unitary hyperbolic point, separating phases of the phase-diagram in union of covalent and valent sea; into, charge and polariton channels of flow, which form, regulatively and flow to two complimentary spaces, and conjoin at an ancillary ($\pi/2$,$\pi$)-point, relating of valent and non-covalent space.  The differential equation of a zero and non-zero binary category form and separate and the singly charged electron advances and slows in orbit around the ellipoidal separation of plane of projective divergence.  As a result the formation of the resonance on the two-channels forms an elliptic centroid of it's off axis motion in resumptively held angle with respect to off-coordinate basis, on KP-differential integral constant and Sine-Gordon of translation of the one-form, to the channel of the unitary electron aprojectively 'hidden' electron with a polariton of opposite hidden positive charge.   This is the off diagonal KP equation of a low amplitude shallow wave, of medium-translational-inertial center and axis, to-which the effective mass of the electrons becomes exponentially and geometrically massive in colombic repulsion.  Therefore the algebraic symbolic agreement between particle wave axis and valency is a sigma center, of which carries weight of measure comparable the scale of homogeneous mathematical support and physical definition.\\

\noindent Generalizing this behavior establishes a 2-1 heirarchy of phenomena in relation to electronic principles.

\section*{Prerequisites}
\noindent Catching the relationship of a two-folded algebraic elliptic is as simple, \textit{as the argumentative basis of}:
\begin{equation}
\iota\rightarrow B:[(p(l),q(l)\;\eta)(a(z),b(z))]\rightarrow [(L(\pi,\tau),T(\pi,\tau))]
\end{equation}
\noindent When these equations are cast in doubt, they are to be recollected, as the $\{A,B,C\}$ as inherited by the $\{\Pi(\omega),\Theta(t)\}$ of the algebraic versus literal-worded \textit{phenomena} of the \text{phenomenon} of varietal intercongruent \textit{phases} of matter, in relation to sequences and flows of relationship, for which; the amended-differential (wed relationship in fact and tool) is a relationship of equation and the agreed reprudiation of a 'non-consequeative bias' on flow, for factual measure, for a given-tool.\\

\noindent When* (\textit{then}) as-such the differential equation is related of the third step in one-manner of dialog, it is formative a hypothesis is it's alternative comparative another (non-tool) co-example of demonstration to, then the second to the first partition, of the tantamount relationship of a secondary peir, comparable a primary peir.  Thus, appertaining to a dialog for the concession to a student, of a teacher, amends the process of two discredited student(s) on behalf of addressing a class, as in type, or manner of disposition to their respective parts.  Thus, including the basis of imaginary hypothetical real observable is a non-counter-factual bias or root of the differential eigenfunction, as it is concluded.

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