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                "Problème des trois producteurs"
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                "The “Three Producers Problem”"
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                "La nécessité de définir une monnaie commune étant admise au sein de la communauté des individus malgré leurs désaccords fondamentaux quant à ce qu'il peuvent juger comme étant de la valeur ou pas, il reste un problème fondamental de définition de cette monnaie. Il se réduit à la résolution du problème spatio-temporel des trois producteurs que voici :"
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                "The necessity to define a common currency being given inside the community of individuals, despite their fundamental disagreements regarding what constitutes value or not, a fundamental problem remains for the definition of this currency. It can be reduced to solving the following time-space “Three Producers Problem”:"
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                "X, Y, et Z, produisent respectivement des valeurs Vx Vy et Vz."
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                "X souhaite obtenir Vy, Y souhaite obtenir Vz, et Z souhaite obtenir Vx."
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                "On le voit immédiatement, les échanges ne peuvent pas se faire bilatéralement, mais circulairement. Comme par ailleurs il est tout à fait possible que X n'accorde aucune valeur à Vz, Y à Vx, et Z à Vy (principe de relativité), aucun des biens ou services produits ne peut servir de mesure commune. C'est bien là le point fondamental, qui implique que la monnaie devra être définie sur une base indépendante des valeurs produites par chacun."
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                "We can see that the exchanges cannot be bilateral, but must be circular. Furthermore as it is perfectly possible that X grants no value to Vz, Y to Vx, and Z to Vy (relativity principle), none of the produced goods and services can be used as a common metric. This is the fundamental argument that implies that the currency must be defined on a basis that is independent of the produced values by each individual."
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                "Le problème existe aussi dans le temps où, les individus, les productions, les services et les besoins évolueront dans leur nature, et seront progressivement amenés à disparaître ou à être remplacés. Il n'en devient pas moins nécessaire d'avoir, à tout instant, la possibilité d'échanger de façon convenable la production des uns et des autres, afin de satisfaire leurs besoins évolutifs respectifs."
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                "The problem also exists in time, where the individuals, productions, services and needs will evolve in nature and will progressively be replaced or disappear. It is not less necessary for individuals to be able, at all times, to trade appropriately each other's production in order to satisfy their respective changing needs."
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                "*Pour une petite unité de temps « dt »* *les individus X Y et Z co-existent,* *produisent et échangent des valeurs stables. (Luc Fievet TRM 2.0)*"
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                "*For a small “dt” time unit, individuals X, Y and Z coexist, produce and exchange stable values. (Luc Fievet RTM 2.0)*"
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                "Donc, non seulement *« dans l'espace »* (pour un temps d'évolution court «dt») les valeurs ne sont pas communément reconnues par les producteurs et font pourtant l'objet d'échanges circulaires, mais *« dans le temps »* les individus et les valeurs produites changent du tout au tout."
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                "Il n'en reste pas moins, que pour un temps suffisamment court, on constate une certaine stabilité. Il y a donc une évolution continue des paramètres économiques, dont la monnaie que nous voulons définir, qui permet aux producteurs présents à tout instant, et au moins pour ce temps court, de s'accorder sur la stabilité de leur outil d'échange circulaire."
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                "Nevertheless, for a time that is short enough, we observe some stability. So, there is a continuous evolution of economical parameters, including the currency we want to define, which allows the present producers at any moment, and at least for this short period of time, to agree on the stability of their circular exchange tool."
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                "Aussi comme nous allons le démontrer dans ce qui suit, et pour être cohérent avec nos fondements, seule une quantification purement mathématique des échanges, indépendante de tout bien ou service de référence est acceptable par les acteurs du problème des trois producteurs."
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                "Also, as we shall demonstrate in what follows, and to be coherent with our fundamentals, only a purely mathematical quantification of exchanges, independent of all references, goods and services, is acceptable for our actors in the “Three Producers Problem”."
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                "Ce résultat n'enlève en rien de la valeur à la monnaie, puisque quoique purement mathématique, sa quantité totale est limitée à tout instant. Le potentiel d'achat qu'elle recèle demeurera donc limité par les prix au delà desquels les producteurs ne pourraient pas échanger leurs productions par manque de monnaie."
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                "This result doesn't reduce the value of the money because its total quantity, although purely mathematics, is limited in all instant. The purchasing potential of this money is limited by the prices beyond which the producers would not be able to exchange their productions because of a lack of money."
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                "*Le problème des trois producteurs se complique* *lorsqu'au bout d'un temps suffisamment long ils sont morts et remplacés (Luc Fievet TRM 2.0)*"
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                "*The “Three Producers Problem” gets more complicated when after some time, they die and they are replaced. (Luc Fievet TRM 2.0)*"
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                "Le problème est posé, nous allons maintenant balayer les solutions envisagées et les soucis qu'elles posent avant d'aborder la solution relativiste proprement dite."
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