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The multi-dimensional mathematic system (MMS) is a complete mathematical device for surface integration operations of the triarchic surface set (surface Existemetrics) and population set dynamics of the normal species set.
MMS embodies an isomorphic (categorical) model correlate to multi-dimensional coordinate geometry specifications for all elements of the population set in the codomain of the universal set. Moreover, MMS elaborates mathematical methods for factorization of x-intercept rational and irrational quantities and mathematical techniques for vectorisation of y-intercept field and particle dynamics.
Table 1 Multidimensional Mathematical System (MMS) [enlarge]
Pivoted on the axiomatic ‘prime unit’ (population) set, and endormorphic tetralogues comprising the ‘power unit’ (quadrant) sub-sets, MMS particularises a time-frequency spectrographic (multi-dimensional) mathematical system. MMS is thus explicative of the mathematics of tetralogy – the logic of the four quadrants – and is intrinsic to universal applications of Existemetrics, the principal super-science postulate of the meta-existential framework (MEF).
The Dual-Form Tetralogue
Quantum mechanisms of the spectrographic MMS illuminates physical shifts of the electro-dynamic waveform in dual states: (a) the geodetic Wave-Shift Tetralogue (T1); and (b) the matrix Phase-Shift Tetralogue (T2).These tetralogues (unit cubes) are instructive of the Euclidean group of isometries which are pre-determinative of cyclo-symmetric and helical propensities endemic to the tetralogues and the quaternary structural boundaries of tetralogy.
Bi-orthogonal System. The triarchic surface set is a complete biorthogonal system {sin(nx), cos(nx)}∞n=0 over R=[-π, π] corresponding to a generalized Fourier series, with weighting function w(x)=1.
Symmetrical Systems. Extant symmetries occur at point, axes and planes of symmetry and are generally invariant under all prime set and power sub-set transformations.
Helical Systems. The quaternary (E-3) iteration of Existemetrics exhibits quaternary system structuration such that at the locus of the Quaternary Moment of the tetralogue automaton the angle of rotation θ required to observe symmetry is irrational rendering the helical system asymmetrical about a non-repeating point group in two dimensions.
Scale Systems. The Universal set is identified to occur at an optimal unit scale otherwise the bi-orthogonal system degenerates and decays.
Homomorphic Systems. The Universal set exhibits characteristic endomorphism (unit analytics), homeomorphism (surface analytics), and diffeomorphism (set transititivity). The Universal set is also automorphic consistently retaining group symmetries under composition of morphisms.
Mathematical Methods
Multidimensional Mathematical System (MMS)
Oblique Geometry
Empirical Mathematics
Quaternary Structuration (4G)
Logic of the Four Quadrants
Diagram 1 Wave-shift Tetralogue (T1)
Diagram 2 Phase-shift Tetralogue (T2)
Surface Integration
Surface integration refers to surface integrals of the bi-orthogonal spectral field which are partial derivatives (surface elements) of the quaternary structuration.
Spectral Protocol
The quaternary structuration is recursive in time latency, epitomising maximal thresholds of the topological frame and cardinal delimiters of the cuboid polytope (hypersphere). Spectral protocols for the tetralogue specify ontological tetrad moments of the spectral modulation that are cyclo-geometric iterations of the scalar quantity (particulate) as a 'projectile' through vector space. These isospectral multi-dimensional transformations substantiate the geo-biophysical propensities constituted by multi-collinear field dynamics and spectral algorithms (transfomation function f(x):x) arrayed to mechanise the geodetic endurant (tx) and perdurant (ty) complexes. The spectral protocol is thus an omniscient automaton, and embodies a unitary axiom that is resonant through continuous scales of time and space.
Table 2 Spectral Protocol
Spectral Analytics
Spectral analytics is derived from a binary examination of the wave-shift unit cube and phase-shift unit cube designated as ‘tetralogues’. Tetralogues capture the unitary moment of the quantum state which is cumulant and structurated by electro-magnetic (geodetic) regularities (time quanta) and photonic (sine function) uniformity (particulate quanta). Accordingly, quanta subunits image density values coincident to temporal markers also known as the argument (t0, t1, t2, t3) and modulus values associated with energy density quadrant markers (tΩ, tx, ty, tz).
Table 3 Tetralogue Phase-shift Analysis
Wave-Shift Tetralogue (T1)
The wave-shift tetralogue exhibits definite endomorphism of the electromagnetic radiation which corresponds to RGB (red,green,blue) patterning of the visual spectrum
The wave-shift tetralogue also illuminates precision quadrantisation of matrix-space, internal regularisation of quadrants (sub-sets), and perfect inversion of the core section
Phase-Shift Tetralogue (T2)
The phase-shift tetralogue depicts patterns of symmetry and symmetry groups correspondent to perfect harmonisation and complementarity of the four quadrants (sub-sets) which are in the functional domain of the principal determinant (prime diametric)
The phase-shift tetralogue further evinces bipedalism of matrix space, and latent and residual stochastic elements prevalent in the core section
Spectral Diagnostics
Table 4 Tetralogue Spectral Diagnostics [enlarge]
Spectral diagnostics is the ‘cytology’ of composite instantiation of the tetralogues along Cartesian (temporal) and Euclidean (modulus) coordinate space. Resultant diagnostics are concatenations of the summative quantisations of the 4x4x4 tetralogue (64 subunit-cube space). Resultant diagnostics impute Fourier Series analytics that transcribe predominating time-frequency trigonometric shift identities sin2θ+cos2θ=sin2t+1/2π+cos2t+1/2π =cos2t+sin2t=1 (and geometric) functions of the spectral array within the unitary delimiters of quantum state.
Quadrant 1: The Sine function is predominant, and quadrant space corresponds to a sub-positioning (transitive) of Existemetric E0. Resultant diagnostics indicate an original Species Set bias for logical/Rational-Mathematical reasoning.
Quadrant 2: The Cosine function is predominant, and quadrant space corresponds to a sub-positioning (transitive) of Existemetric E1. Resultant diagnostics indicate a Second Moment Species Set bias for Genic Pull
Quadrant 3: The Sinusoid function is predominant, and quadrant space corresponds to a sub-positioning (transitive) of Existemetric E2. Resultant diagnostics indicate a Third Moment Species Set bias for Genic Normativity
Quadrant 4: The Tangent function is predominant, and quadrant space corresponds to a sub-positioning (transitive) of Existemetric E3. Resultant diagnostics indicate a Fourth Moment Species Set bias for Bipedalism
Mathematical Formulation
The mathematical formulation of the MMS refers to the constellation and multiplicity of mathematical objects such as geometries, groups, fields, topologies, orders, relations, differentials and categories that facilitate multi-variable functions and higher-order theoretical constructs of MMS.
The axiological sub-system instigates an annexure of logic and instructive axioms assembled to coherently derive the tetralogues, the tetralogue automaton (spectral protocols) and mathematical principles of tetralogy. Correspondingly, the axiological sub-system conforms to mathematical structures - Euclidean, coordinate geometry, matrices, set theory, quantum mechanics – to ascertain general precepts, validity of proofs and consistency of MMS concepts.
Contingent to the axiological sub-system, the formal sub-system impels the quaternary mechanization of surface integration and is inherently formulaic of a complete bi-orthogonal and periodic formal sub-system. The formal sub-system denotes a deductive apparatus of the MMS circumstantiated by antecedent axiological systems and corollary rules of inference. Therefore, the formal sub-system preserves direct lines of derivation that maintain syntactic concordance such as to convey deductive integrity.
Overview of Tetralogy
Tetralogy, meaning the applied science of the tetralogues and the spectral protocol, is consolidated in the duality of model theory which is both semantic concerned with logic and truth, and syntactic bound by constructivist formulas and proofs.
The theoretical frame of MMS methodically incorporates the circumfusion of vector fields (T1) and scalar fields (T2) that self-assort and align consonant with prescribed mathematical sub-systems.
Axiological Sub-systems: matrix methods of factorization and indicialisation
Formal Sub-systems: functional algorithms for techniques of vectorisation
Combined these mathematical methods specify a complete exposition of the stochastic partial differential systems of Tetralogy.
Tetralogy I: Scalar Analytics. Explains the scalar field space through point (1 dimensional), and time-space (4 dimensional) coordinate geometry of the multiset of the orthogonal matrix.
Tetralogy II: Vector Analytics. Explicates the vector field space through axial (2 dimensional) and planar (3 dimensional) coordinate geometry of the multiset of the orthogonal matrix.
How to cite this material:
MEF (2014) Tetralogy (Synopsis). The Meta-Existential Framework. MEF e-Publishing.
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