Fig.4. Model: The planet four physical spheres. 1. Earth - Lithosphere.

2. Water - Hydrosphere. 3. Air - Atmosphere. 4. Sun - Heliosphere (incl. stars, space)

         The Pythagorean aphorism "Quaternion is an inexhaustible source of life" fundamental meaning was preserved, understood and raised to the utmost height the thousand years later by the Pythagorean Hierocles (5th century AD). He explains the Pythagorean tetrad/quaternion deep meaning in this way, extending it to society and human. The quaternion is "a depository of the eternal world order, it is the same as God the creator".... “Quaternion has volume. It contains an outline of a simple form of a pyramid ... Also, live creatures have four cognitive abilities: sense, knowledge, opinion, sentiment ... Overall, quaternion comprehends all things existent: the number of elements (earth, water, air, fire/sun -- L.S.), seasons, ages, estates; and there is no evidence that anything exists that does not depend o­n quaternion as the root and the beginning. Quaternion is ... the creator and the cause of everything; it is god fathomable by sense; it is the cause of ... God". [Pythagorean Golden Verses. M.1995, p.30, 96-98.]

         Such is the spheral structure of inorganic, physical nature within the framework and o­n the basis of which living, organic nature arises - the planet biosphere, within which an artificial social nature arises, created by human, homo sapiens wisely. The human does not limit his life sources by the given natural resources in its spheres. o­n their basis and from their material he produces fundamentally new, artificial social resources in their artificial social spheres, generating its special, social nature and life, in which the biological source of human life is preserved – his/her birth and the physical/ecological spheres as his/her eternal, cosmic, God-created maternal womb.

         The Earth inorganic, physical spheres, as well as the society spheres, still do not have a single, integral and holistic science. Knowledge about the planet today is united in "Earth Science", Geonomy or Geoscience, covering hundreds of relevant, narrow disciplines, the exact number of which cannot be found in any encyclopedia of our time. They provide a scientific understanding of the Earth different segments, details but are unable to provide a holistic, scientific understanding/vision of our planet within their traditional, fragmentary/disciplinary thinking. Here they are like the blind men from the famous Indian parable who feel an elephant, falsely identifying its different parts with it in a whole, as in Fig. 2.

         This is a fundamental, unresolved problem and limitation of modern science, which is determined by its holistic underdevelopment and insufficiency, which does not allow its disciplines to see, explain and control the natural harmony of the planetary spheres in general, as well as social spheres that gives rise to multi-thousand-year confrontation and wars. Their fragmentation gives rise to a confrontation in the Earth disciplinary science regarding the ecological, holistic problem: some scientists recognize the ecological crisis o­n the basis of some scientific facts, while others reject it o­n the basis of other scientific facts from the same sectoral positions, not possessing and not suspecting spheral, holistic thinking. A similar confrontation and unhealthy competition between different groups of disciplinary sciences arises not o­nly in ecology, but also in other segments: in energy (carbon-green-nuclear), in medicine, pharmaceutics (vaccinations-harmful-useful), etc., which is again identical to the conflict between the blind men at the elephant in Fig. 2.

         V. Vernadsky first showed the spheral thinking example by the spheral category “noosphere” (see paragraph-2), linking it and similar o­nes with the spheral scientific revolution in the way of thinking, the necessity of which as a “substantially new manner of thinking” was stated by Einstein that we examined in the previous paragraph. This millennial cognitive problem is the fundamental epistemological source, cause of all confrontations and wars o­n the planet in history, which is revealed in detail in the following paragraphs. The spheral scientific understanding of these causes will allow humanity to quickly, peacefully and effectively solve both the ecological problem and the perpetual, global peace problem, along with all other planetary problems, leaving human and society with o­nly endless local and temporary problems.

         The spheral thinking within broad ecological context was considered in a few joint works by the GGHA coauthors: Timi Ecimovic (http://peacefromharmony.org/?cat=en_c&key=749), John Avery (https://peacefromharmony.org/?cat=en_c&key=672) and Leo Semashko. o­ne of them, 2017, is published here: Ecimovic T., Avery J., Semashko L. The Climate Change System 2017. Pages 28: http://peacefromharmony.org/docs/ECIMOVIC_CLIMATE%20CHANGE_18-04-17_Final.pdf

2. Fact-11. The planetary law of "Equally Necessary and Sufficient

Spheres" (ENSS) of physical and living nature being

         The Pythagorean fundamental truth of the "quaternion" of physical nature spheres as the source of all planet life, including social life, contains a deep understanding of the undeniable fact that these spheres determine the immanent planetary law of "Equally Necessary and Sufficient Spheres" (ENSS) of physical, organic and social nature existence. o­nly the equal necessity of each of these spheres for any life, and o­nly their joint sufficiency for it constitute the "source of inexhaustible life" o­n earth. Indeed, each person is convinced of the equal necessity of each sphere for life, because the absence, "zeroing" or disappearance of any of them makes the life of human, society, as well as all living nature, the biosphere together with the entire planet as a whole IMPOSSIBLE. Each person cannot live in the absence of any o­ne of them. There is scientific data o­n how quickly a person, people die in the absence of air, water, food and sun in different conditions.

         Not o­nly a person, but all life o­n the planet as a whole cannot exist in the absence of any o­ne of the EWAS spheres. And if any o­ne of them is absent or disappears, then the planet becomes dead, devoid of life that makes each of these spheres EQUALLY necessary for life. Every thinking person can imagine this sad result for himself and all living things with the "disappearance-zeroing" of any of the EWAS spheres o­n our planet in his/her thought experiment. Therefore, among these spheres there is no "main, defining, primary" and the like. Each of them is EQUALLY NECESSARY for life and o­nly together, in mutual penetration, universal interconnection, systemically and holistically they constitute a SUFFICIENT source of eternal, inexhaustible and holistic life o­n the planet.

         Therefore, the EQUAL NECESSITY AND SUFFICIENCY OF SPHERES (ENSS) of the EWAS is the eternal and enduring law of every and all life o­n our planet, which is simultaneously the fundamental scientific law of humanity for its entire life. It is the same for all people and nations without exception, in the truth of which for each people and individual they are forced to agree with its "iron" necessity, since its violation threatens death, extinction to each person and nation, humanity as a whole. According to this law, they cannot have confrontations and wars across spheres, and they do not exist between spheres, but o­nly between their time segments, classes, workshops, nations, elites, etc. – “partons”, in the spheral thinking terminology, in contrast to spheres as “holons”. This law of the elemental, spheral and harmonious reality of our planet as a whole, cognized by humanity, becomes the eternal law of its scientific knowledge, which it consciously uses to ensure its peaceful life, to master all the EWAS spheres of its natural God-given resources-sources.

         This law, like all scientific laws/truths without exception, unites humanity into o­ne family of nations and individuals with o­ne source of life and o­ne destiny o­n the planet as a single home for all without exception. The EWAS planetary spheres are studied by many natural sciences, which penetrate ever deeper into the nature of each of them and their interrelations. But no matter how deep and extensive the disciplinary sciences knowledge, they will never be able to master their constant law ENSS, because it requires adequate, spheral thinking with its transdisciplinary integrity of its spheral substance. Without the categories of "spheres" and derivatives it is impossible. Disciplinary thinking is not capable to shake it, nor to master this law and its spheral substance in all its planetary truth. This is an indisputable historical fact. It has been so throughout past history up to this day, and it will remain so in the future.

         Another extremely important attribute of the ENSS law is its spheral, fourdimensional structure, construction and architecture, which is preserved in all its spatio-temporal locations o­n the planet, o­n any square centimeter of its area and in any second of its historical time. Whatever changes in the EWAS occur in their locations, their structure remains unchanged, fourdimensional-spheral. This is the extremely complex and contradictory, dialectical feature of the fourspheral structure: the constant of its eternal integral stability/harmony is ensured by the continuous variability/“straggle” of its sectoral parts/partons in space/time. People constantly observe both the EWAS continuous changes in the entire spectrum of their internal partons, and their deep stability, the preservation of their eternal spheral structure in all of them.

         This makes the EWAS spheres existence self-evident, not requiring any special evidence, and therefore is not particularly discussed by people, o­nly philosophers try to penetrate into the invisible depth of the EWAS interrelations in their different conditions. Naturally, people's main attention is occupied by changes in these spheres that can be dangerous to their lives: earthquakes, floods, hurricanes, excessive solar activity and the like, which obscure and push into the background of the fourdimensional structural stability fact of these spheres as something self-evident.

3. Spheres of living nature

         We will not dwell here o­n the specifics of biological life and its spheres, we will o­nly emphasize the fourspheral classification of living organisms, in which four of their kingdoms or spheres are distinguished: Bacteria, Fungi, Plants (Flora), Animals (Fauna). These spheres/kingdoms include many structural levels of living organisms, an infinite number of their species. It can be assumed that the ENDS law is preserved and operates at the level of living nature, but in a specific and extremely complex form. This is a question of biology, not sociology, so we digress from it and move o­n to a general consideration of the social spheres, humanity in their single social nature.

4. Spheres of social nature

         The first question that arises here: what are those EQUALLY NECESSARY AND SUFFICIENT SPHERES (ENSS) of social, artificial resources, without which the life of an individual, as well as any society, and humanity as a whole is impossible? The official macrosocial science still does not have an answer to this fundamental question. It can provide evidence of o­ne to 12 social spheres, of which o­nly the fourdimensional spheral paradigm is true, adequate to the ENSS planetary law, to which all others do not correspond and, at best, can o­nly be approximations, steps towards it. However, thousands of types of sectoral, disciplinary knowledge of society, social or humanitarian sciences, still do not provide a single, general and integral scientific knowledge of society and its social nature. Both disciplinary natural science and disciplinary social science are powerless in understanding their integral, spheral nature, accessible o­nly to spheral thinking with its categorical network of spheral concepts and its spheral logic. Their holistic, spheral knowledge was first presented in the "Peace Science", "Spheronics" and a general and brief, without unnecessary details and at the same time popular introduction to it is offered in our "Primer".

         The exposition of this science cannot go historically, covering millennia of its formation in the works of tens of thousands of scientists and philosophers of different countries, continents and eras, starting from ancient times. A brief exposition of this science can o­nly be logical, not from parts to the whole, but from the whole, "from a bird's eye view" of its panoramic vision by descending to the definition and consideration of its numerous underlying levels and their details. This science has an integration nature, uniting in synergy, i.e. in animation, thousands of different rational social ideas and achievements, regardless of space, time, source and author of their origin into a single, true paradigm, based o­n historical facts and spheral verification of the ENSS universal law. In o­ne of the following paragraphs, everyone will be offered a method for its simple spheral verification in 2 hours, accessible to every literate person and schoolchild.

         Our "Peace Science", following Pythagoras and by analogy with him defines the humanity social nature formula of being: "The Humanity Four Spheres Quaternion is Its Inexhaustible Life Source in Perpetual Peace" that is revealed in the following paragraphs.

         The scientific macrosociological structure of 16 (4x4) social ENSS of fractal architecture, built o­n the four of its generating resources PIOT, is systematically presented in their model in Figure 1 (paragraph-1) in its center. The following paragraphs of our primer are devoted to the consideration of these societal spheres blocks, determining the humanity eternal social genome and its constant cell at all levels, starting with the individual and family. Here, in conclusion of this paragraph, we will briefly touch upon o­nly the hypothesis of measurement, qualimetry of spheres.

5. Hypothesis of spheral qualimetry, spheral mathematics and spheral statistics for AI

         The fundamental digitalization and qualimetry of humanity of all sorts and all times begins since the Pythagorean quaternion, potentially containing ten – the whole numbers. Spheral qualimetry, embodied in universal spherical mathematics and statistics for AI, constitutes a quantitative toolkit of the highest, spheral way of scientific thinking of the future, which is being born today, therefore it can be identified o­nly as a fundamental hypothesis, nothing more. Pythagoras determined the source of this hypothesis. Hierocles explained it, in the spirit of his time a thousand years later. God for the Pythagoreans is the number of numbers. All numbers are contained in ten. The ten comprehends all the numbers, while "the ten's capacity is a quaternion ... because adding the numbers from o­ne to four we get ten ... Quaternion is also the mean of o­ne and seven ... Quaternion has volume. It contains an outline of a simple form of a pyramid." [Pythagorean Golden Verses. M.1995, p.30, 96-98.].

         This fundamental hypothesis is superimposed by another fundamental ancient hypothesis. The spheres penetrate and mutually include each other according to the principle of Anaxagoras "all in all", forming a spheral fractal structure of being. Therefore, the fourdimensional external structure is identical to the four-dimensional immanent, internal structure of each of them with the priority of o­ne, determining the quality and specific laws of being of each sphere. The structural fourdimensionality of spheres serves as the basis for the fundamental scientific hypothesis of their immanent spheral qualimetry, which expresses the quantitative measure of harmony of the spheres among themselves at all levels and in all objects of existence. It is expressed by the matrices of spheral indices of 4x4ⁿ dimension, where ‘n’ is a series of natural numbers, or any other in the possible aspects of measuring and calculating spheres.

         In the spheral, fractal matrices, each of the four spheres is infinitely divided into four parts for each other in an infinite chain of historical transformations of these spheres in their infinite space and time that is expressed by its index “n”. To begin with, our hypothesis of spheral qualimetry, spheral mathematics (the mathematics of spheral fractals), spheral statistics and spheral programming as necessary, innovative and technological tools of spheral thinking is defined. It opens up an infinite mental space of their development in an extremely complex search for common units of their measurement, common mathematical and statistical laws, proportions and measures of their harmony for their calculation in spheral AI from a personal smartphone to the humanity AI in endless locations, the consideration of which goes beyond the boundaries of our primer. In o­ne of its paragraphs further, we will touch upon, to illustrate the reality of this hypothesis, the paradigm of spheral social Global Statistics, Globstat below, which we have long been developing, o­n the basis of which the architecture of spheral AI (SAI) is constructed.

          The questions of spheral thinking and its qualimetric tools (mathematics, statistics and programming) for SАI, touched upon in the paragraph, have been posed by me since 1976 and were reflected in my first book of 1992: “The Spheral Approach. Philosophy, Democracy, Market, Man”: https://peacefromharmony.org/file/5162/1992-semashko-sferniy_podhod.pdf. It concerned all spheres of society, especially political democracy in Russia orgsphere.

6. Appendices

         (The appendices of this paragraph, as well as others, will be selected jointly with the primer coauthors from the blocks of quotations proposed by them after the completion of all its paragraphs.)