Tetralogy I (Scalar Analytics)

Tetralogy I (Scalar Analytics) is an auxiliary program of the multi-dimensional mathematical system (MMS) concerned with metricisation of scalar quantities identified in the scalar coordinate (right-side) field-space of the MMS. Promulgating the coda of the Tetralogue Automaton, Tetralogy I explicates surface integration operations of the triarchic surface set (surface Existemetrics) and population set dynamics of the normal species set.

Tetralogy I (Scalar Analytics) is effectively the geo-biophysical analogue of exploratory particle physics particular to the science of quantum electro-dynamics and quantum chromo-dynamics. To the point, Tetralogy I addresses Tetrad Moments of the tetralogue automaton state for analysis of scalar quantities in t0/tΩ the Primordial Moment and in t3/tz the Quaternary Moment.

Defined by continuous timespace ‘Spectral Protocols’ of the spectral field and spectral modulations, Tetralogy I enumerates the cyclo-symmetric geometries of the cuboid polytrope (hyperoctant) to illustrate complete functor operations (orthogonal rotations) of the function map (population set) in the codomain of the universal set. Tetralogy I is thus explicative of the scalar field 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).

Part A: Axiological System (Matrix Coordinates)

Mathematical methods of the axiological sub-system for matrix coordinate space denotes techniques of matrix spectral-decomposition involving multi-dimensional scaling (MDS) algorithms. This procedure encompasses a rubric of non-linear transformations to the one-dimensional (matrix) coordinate system which involves a sequence of ordination (distancing) techniques in n-dimensional space by means of estimates of coordinates based on classical principal coordinates, followed by metricisation of principal coordinates, error correction functions along the principal coordinate, and topological smoothing (minimum distortion) functions onto the principal coordinates. MDS is a form of numerical classification and taxonomy that prepares the functional array (set elements) for matrix operations.

In the Primordial Moment, the axiological system illustrates the Existemetrics syntactic categorical for the 'Ovulation' stage of reproduction, which is symptomatic of the 'inflationary impulse' radiating from a point of origin (spectral 'crux').

Matrices

The matrix (plural matrices) form is a rectangular array of values (set elements) arranged in rows and columns (grid). Matrix quantities are operant to computational systems that are ‘relational’ in nature, representing generalized linear transformations (affine transformations) of a selectional field-space (categorical set). The matrix form captures and arranges item values for methodic matrix computations, most commonly unitary transformations.

Spectral Determination

As identified in the Spectral Protocol, the t0/tΩ tetrad moment corresponds to a ‘Primordial Moment’ configured by the spectral field in phase-shift at t0, and at cross-section to the sine function wave-shift at tΩ. Spectral modulations of the Primordial Moment are (a) Radiative, meaning that the physical phenomena propagate outwards from the spectral ‘crux’, and (b) Operator, denoting spectral algorithms that disseminate through tetrad moments (phase-shifts) of the tetralogue automaton state.

At a given locus in n-dimensional space, the primordial point emanates as a hypercube-space of dimension zero. Spectral determination is configured by a one perpendicular unit-length timeshift (a point timeshifted to a line segment) which conforms to a unit hypercube of dimension one. Therefore, in the Primordial Moment, the spectral array manifests as a line (in hypercube-space) or ray (in hyperoctant-space).

Orthogonality

At dimension one-space, the Primordial Moment is configured by permutation representation (orthogonality) at a singular point of symmetry (or point subgroup of all isometries).

Quotient Orbital

The quotient orbital denotes the topological group action onto abstract objects in the topological space of the symmetry group (transformation group). In the Primordial Moment, the quotient orbital is given by the factorial moment generating function M_x(t)=E(t^x) manifesting as the E0 Intuition Set of the Normal Species Set (orbital frame), induced by an orthogonal rotation (functor). The resultant is the vector space (function map) of the Population Set (in the codomain of the Universal Set).

Part B: Formal System (Vector Coordinates)

Computational methods for the formal sub-system of vector coordinate space involves orthogonal matrix operations correlated to vector factorisation algorithms of quantum mechanics. Vector factorization is a systematic process of cumulant decomposition to characteristic (polynomial) roots in order to extract linear transformations of the partial integrals (set elements) that qualify the functor of the tetrad moment. Vector factorization enumerates the vector calculus of group actions that are permutation representations and bijective functions on the complex manifold of the Quaternary Moment. The formal sub-system is thus a deductive apparatus of the MMS performed to elucidate affine transformations and continuous functions (periodicity) of the tetrad moments.

Moreover, in the syntax of Existemetrics, vector coordination of the Quarternary Moment is analogous to the ‘Gestation’ stage of reproduction, which is formally specified as the mathematical sine (x axis) function formal argument for the amplitude of electro-dynamic energy compressed as a resultant of 'time latency. This may be transected with the meta-physical cosine (y axis) function actual argument of the 'Ontology gradient' coterminous to 'Existentiality' precepts of Existemetrics.

Fourier Series

Spectral diagnostics of the tetralogue automaton impute Fourier Series analytics that transcribe predominating time-frequency trigonometric (and geometric) functions of the spectral array within the unitary delimiters of the tetralogue automaton state. As indicated in Diagram 3 showing Fourier Analysis of time-shifts of the tetralogue automaton, the Quaternary Moment is confounded by complex conjugates in the summative equation of tan functions. Hence, the Quaternary Moment necessarily involves derivation of the absolute value – Modulus |z| - in order to yield resolutions for the E-3 Executive Profile functor to the desired function map.

For the set of complex coodinates, the modulus |z| is obtained from the complex equation form |z|=√x^2+√iy² which may also be calculated for the complex analogue of the polar form z=re^iθ. The resultant positive reals comprise a subgroup of the complex numbers under multiplication and are endomorphic of the multiplicative group of the complex set (quotient space). The Quaternary Moment therefore interpolates adjustment to an imperfect orthogonal rotation (functor) of the circular (sin²θ, cos²θ, tan²θ) functions, in order to derive sectional vector spaces (function map) which are aetiological frames of the preferential population subset.

Time Latency

Time latency refers to a stochastic trend associated with gamma residuals. In the physical sciences, the time latency is a consequence of exiguous velocity and dissipation of the electro-dynamic potential. A time latency switch (operator) is activated when physically instantiated (embodied) in a tetralogue automaton (physical system) ideated by spatial dimensions greater than zero. Given that velocity is always less than or equal to the speed of light, time latency tends to accrue through sequential tetrad moments (temporal intensification) of the tetralogue automaton state in time and space. Time latency in the Quaternary Moment is therefore epitomised by the quantum physics’ ‘Curse of Dimensionality’ which refers to an inverse law of decreasing availability of quanta stimulants with exponentially increasing volume space.

Cross Graining (Physiology)

The phase-shift tetralogue (T1) exhibits three planes of symmetry: (a) Prime Diagonal (top left and right bottom); (b) Secondary Diagonal (bottom left and right top) and (c) Transversal (top and bottom). Phase-shifts of the photonic quanta therefore consolidate in ‘orthoform’, a physical sciences analogue of the spheroid ‘mind’.

The wave-shift tetralogue (T2) emerges from three planes of symmetry: (a) Sagittal (front and back); (b) Coronal (left and right), and (c) Transversal (top and bottom). Wave-shifts of the electro-magnetic quanta therefore congeal in ‘somatoform’, a physical sciences analogue of the octoid ‘body’.

Coarse Graining (Psychology)

The octant squared (hyperoctant) quaternary structuration is ‘subjacent’ compared to the cube squared (octant) tertiary structuration of the Tertiary Moment. This condensation is configured by the orthogonal dimensions of the hyperoctant s-count (axes of symmetry) which is twelve for the 64 subunit-cube (tetralogue) and twelve for the 8 subunit-cube (octant). The Quaternary Moment is thus cyclo-static compared to the 'supersymmetry' of the Tertiary Moment. Explicable as cadences of ‘granularity’, the Quaternary Moment is perplexed by ‘coarse graining’ – from fine grain to coarse grain – which is accretive toward maximal entropy and the second law of thermodynamics.

Spectral Determination

As identified in the Spectral Protocol, the t3/tz tetrad moment corresponds to a 'Quaternary Moment' configured by the spectral field in phase-shift at t3, and at cross-section to the tangent function wave-shift at tz. Spectral modulations of the Quaternary Moment are (a) Time Latent, whereby the physical phenomena disseminate latent residuals toward the centre of curvature of the spectral ‘crux’, and (b) Capacitant, denoting spectral absorptivity that condenses electro-dynamic energy (radiation) through phase-shifts of the tetralogue automaton state.

At a given locus in n-dimensional space, the hypercube (or glome) emanates as a hyperoctant-space of dimension four. Spectral determination is configured by a one perpendicular unit-length timeshift (an octant timeshifted to a hypercube) which conforms to a unit hyperoctant of dimension four. Notwithstanding, in the Quaternary Moment, the spectral array is complex and indistinguishable from the spacetime continuum of the Tertiary Moment. As a result, the Quaternary Moment features one degree of freedom and manifests as a hypercube (in hypercube-space) and hyperoctant (in hyperoctant-space).

Orthogonality

At dimension four-space, the Quaternary Moment demonstrates ‘antipodal symmetry’ and is configured by permutation representation (orthogonality) at a point reflection through the spectral ‘crux’ connoting the subgroup of all isometries.

Quotient Orbital

The quotient orbital typifies the topological group action onto abstract objects in the topological space of the symmetry group (transformation group). In the Quaternary Moment, the quotient orbital is given by the cumulant ('supersymmetry') moment generating function M_x=E(e^tx) manifesting as the Existemetric E-2 Socio-Political Orientation (orbital frame), powered by an orthogonal rotation (functor). The resultant is the E-3 Executive Profile vector space (function map) of the Population Set (in the codomain of the Universal Set).

Quotient Helical

The quotient helical refers to the rotational symmetry consonant with translation along the axis of rotation, the spectral ‘crux’ (screw axis). The coiling angle of simultaneous rotation and translation occurs at a degenerating (entropic) linear speed through phase-shifts of the tetrad moments homologous to the quaternary structuration of the tetralogue automaton. As shown in the Diagram 4, Existemetric phase-shifts through tetrad moments of the tetralogue automaton terminate at t3/tz Quadrant 4 where cumulative degeneracy of the helical structure obtains critical mass. At this locus, the angle of rotation θ required to observe the symmetry is sufficiently devolved and irrational such that it fails to repeat. Hence, the quotient helical exhibits universal and invariant endomorphism and recursionism (perennialism) of the tetralogue automaton.

How to cite this material:

MEF (2014) Tetralogy I (Scalar Analytics). The Meta-Existential Framework. MEF e-Publishing.