1070.00 Plurality of Inherent Topological Twonesses
[1070.001077.11 Geometry of Two Scenario]
1071.00 Systematic Character of Prime Thinkability 
1071.10 Prime: Prime means sizeless, timeless, subfrequency. Prime is prehierarchical. Prime is prefrequency. Prime is generalized, a metaphysical conceptualization experience, not a special case. 
1071.24 Radial depth is expressed in frequency of omnidirectional wave propagations per unit of time. 
1071.26 Prime thinkability is inherently systemic. Prime epistemology is generalized thinkability. Epistemology discovers intuition. 
1071.28 Structures are systems and have radial depth; wherefore "surface" triangle structures are always tetrahedra or truncated tetrahedra. 
1072.00 Definability of Structural Systems 
1072.11
Given: definition of prime: Sec. 1071.10 Given: definition of system: Sec. 400.01 Given: definition of structure: Sec. 600.02 Given: definition of structural systems: Sec. 610.20 Given: definition of prime tetra, octa, and icosa: Sec. 1011.30 Given: definition of subfrequency: Sec. 1011.32 Given: definition of nucleus: Secs. 414 and 1012.01 Given: definition of thinkable generalization: Sec. 501 Given: definition of special case realizations: Sec. 504 Given: definition of cosmic inherency: Sec. 1073 Given: definition of two kinds of twoness: Sec. 223.05 
1072.20 Generalized Topological Definability 
1072.30 Wave and Particle Definability 
1072.32
Particle is frequencydefinable special case, and wave
is angularly defined
generalization. The numerically unique condition of
special case 5 (not equal) 4 of generalizations
identifies the dilemma of physics in reconciling the
minimum fourfoldedness of wave
definability by angles and the minimum fivefoldedness^{11}
definability of particle
particularization by unique frequencies. (See Secs.
541.30,
961.46,
973.30, and
986.720).
(Footnote 11: See Sec. 1053.1215.) 
1073.00 Cosmic Inherency 
Fig. 1073.10 
1073.10 Four Kinds of Twoness 
1073.12
All systems are conceptually differentiated out of
Universe.
System + environment = Universe Universe  system = environment 
1073.13
The environment is dual, consisting of the macro and
micro (outsideness and
insideness). Ergo, a fourth twoness of all prime structural
systems is synergetically
accountable as
2 + 2 + 2 + 2 = 8.
(See Fig. 1073.10.) 
1073.14
Integral system is threefold twoness = 6. Integral Universe is fourfold twoness = 8.

1073.15 The Indispensable Center: At the indispensable center of the system convergentdivergent Universe turns itself inside out. The invisible, a priori, multiplicative twoness differentially disclosed in the system's omnitopological hierarchy is manifest of the integrity of the sizeless, timeless nonconceptuality always complementing the conceptual system takeout from nonconceptual Scenario Universe's eternal regenerating. (See Sec. 1006.10.) 
1073.20 Interrelationship Twoness: Third Kind of Twoness 
1074.00 Prime Nuclear Structural Systems 
1074.10 All prime nuclear structural systems have one^{__}and only one^{__} (unity two) interior vertex. 
1074.11 Internally nuclear structural systems internally consist entirely of tetrahedra that have only one common interior vertex: omniconvertex. 
Fig. 1074.13 
1074.13 In all nonredundant, prime, nuclear structural systems the congruently interiorvertexed, omnisystemoccupying tetrahedra of all prime structural systems may all be interiorly truncated by introducing specialcase frequency, which provides chordal as well as radial modular subdivisioning of the isotropicvectormatrix intertriangulation of each radial, frequencyembracing wave layer, always accomplished while sustaining the structural rigidity of the system. (See Fig. 1074.13.) 
1074.20 Omnitopological Domains 
1074.21 Omnitopological domains are defined in terms of the system's unique centralangledefined insideness and its unique surfaceangledefined outsideness. (See Sec. 1006.20.) 
1074.30 Spin Twoness and Duality Twoness 
1074.32

1075.00 Special Case: Energy and Information 
1075.20 Equation of Intellect 
1076.00 Primitive Regeneration 
1076.13
Physics of the 1970s identifies:
2 quantum = 1/2 spin; and 1 quantum = 2 (spin/2)
Conceptionbirth comes with the realization that the aspects of the externally viewed, pluscurvature convexity that are seemingly separate from those of the internally viewed, minuscurvature concavity have no interveningly differentiating, zerocurvature sheath structurally differentiating the only timelessly (or generalized) conceptual coincidence of both the plus and minus curvature. In the alternately plusorminus pulsativeness frequencies of special case time the multiplicative twoness "conception" releases or gives birth to new, coexistent, additive twonesses as independently axially spinnable: special case spin twoness inherently coupled with the duality twoness, producing the individual unity fourness, with its primitive sixfoldedness of integral system interrelatedness and its eightfolded integral Universe environment. 
1077.00 Prime Number Inherency and Constant Relative Abundance 
1077.10 Since the relative interabundance of one nonspin vertex for every two faces and every three edges is a constant topological system condition, we may identify them as a constant interabundance set in terms of the number of vertexes and edges as a constant topological relationship of all symmetrical and triangularly stabilized, modularly unsubdivided, polyhedral systems. 
1077.11
We may substitute T, meaning the number of topological
system sets, for the
bracketed groups:
.: 2 + 2 = T.
And when the symmetrical, omninonredundantly triangulated,
modularly unsubdivided
systems are subjected to symmetrical modular subdivision,
and the number of edge
module subdivisions is represented by F, then:
2 + 2NF^{2} = T.
where the first 2 is the additive spin two; the second
2 is the multiplicative duality two; N
is the prime number uniquely characterizing the system;
F is the frequency of modular
subdividing; and T is the number of topological sets
of one vertex plus two faces equal
three edges (1 + 2 = 3) that exists in the symmetrical
structural (because nonredundant)
triangulated polyhedral system. Q.E.D.: See Sec.
223.

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