{"id":56532,"date":"2025-10-13T08:28:35","date_gmt":"2025-10-13T07:28:35","guid":{"rendered":"https:\/\/4.exstyle.fr\/le-blog-photo\/?p=56532"},"modified":"2025-10-13T08:28:36","modified_gmt":"2025-10-13T07:28:36","slug":"fr-distances-liens-fratries-variante-universelle-f_m","status":"publish","type":"post","link":"https:\/\/4.exstyle.fr\/le-blog-photo\/fr-distances-liens-fratries-variante-universelle-f_m\/","title":{"rendered":"FR – Distances, liens, fratries, variante universelle F_m"},"content":{"rendered":"\n\n\n\n\n

Distances racine\u2194lien, fratries, et famille universelle Fm<\/sub><\/em><\/h2>\n\n\n\n

Cadre.<\/strong> On travaille sur les impairs et l\u2019on recentre toute visite d\u2019un membre de fratrie par l\u2019op\u00e9ration (x-1)\/4<\/code> autant que possible pour revenir \u00e0 la racine minimale<\/em> r<\/em> (i.e. $latex r\\equiv 1,3,7\\ (\\bmod\\ 8)$<\/span>) avant de passer au lien<\/em> Y<\/em>. On appelle distance<\/strong> le saut structurel entre racine et lien :<\/p>\n\n\n\n

$latex d = \\lvert Y-r\\rvert$<\/span>   (toujours pair<\/strong>). On note $latex d^{+}$<\/span> quand $latex r<Y$<\/span> et $latex d^{-}$<\/span> quand $latex r>Y$<\/span>.<\/p>\n\n\n\n

Fratrie<\/strong> issue d\u2019une racine minimale $latex r$<\/span> (ligne du tableau, op\u00e9rateur de colonne x \u21a6 4x+1<\/code>) :<\/p>\n\n\n\n

$latex L_{r,n}=r\\,4^{n}+\\frac{4^{n}-1}{3}=\\frac{(3r+1)\\,4^{n}-1}{3}\\quad(n\\ge 0).$<\/span><\/p>\n\n\n\n\n\n\n\n\n

1) Cas classique 3x+1<\/code> (m=1, C=1)<\/h3>\n\n\n\n

Catalogue par distance.<\/strong> Pour chaque $latex d\\in\\{2,4,6,\\dots\\}$<\/span>, il y a exactement deux liens possibles, qui couvrent tous<\/em> les impairs non multiples de 3 (structurellement 50\/50) :<\/p>\n\n\n\n

    \n
  • $latex d^{+}$<\/span> (saut vers le haut, $latex v_2(3r+1)=1$<\/span>) : $latex r=2d-1,\\quad Y=3d-1$<\/span>  ($latex Y\\equiv 5\\ (\\bmod\\ 6)$<\/span>).<\/li>\n
  • $latex d^{-}$<\/span> (saut vers le bas, $latex v_2(3r+1)=2$<\/span>) : $latex r=4d+1,\\quad Y=3d+1$<\/span>  ($latex Y\\equiv 1\\ (\\bmod\\ 6)$<\/span>).<\/li>\n<\/ul>\n\n\n\n

    Inversion depuis un lien.<\/strong> Chaque lien impair $latex Y\\not\\equiv 0\\ (\\bmod\\ 3)$<\/span> provient d\u2019un unique couple $latex (d,\\pm)$<\/span> :<\/p>\n\n\n\n

      \n
    • Si $latex Y\\equiv 5\\ (\\bmod\\ 6)$<\/span> : $latex d=\\frac{Y+1}{3}$<\/span> (pair), $latex r=\\frac{2Y-1}{3}$<\/span> → branche $latex d^{+}$<\/span>.<\/li>\n
    • Si $latex Y\\equiv 1\\ (\\bmod\\ 6)$<\/span> : $latex d=\\frac{Y-1}{3}$<\/span> (pair), $latex r=\\frac{4Y-1}{3}$<\/span> → branche $latex d^{-}$<\/span>.<\/li>\n<\/ul>\n\n\n\n

      Fratries param\u00e9tr\u00e9es par d.<\/strong><\/p>\n\n\n\n

        \n
      • Depuis $latex d^{+}$<\/span> ($latex r=2d-1,\\ Y=3d-1$<\/span>) : $latex L_{+,d}(n)=\\frac{2(3d-1)\\,4^{n}-1}{3}$<\/span>.<\/li>\n
      • Depuis $latex d^{-}$<\/span> ($latex r=4d+1,\\ Y=3d+1$<\/span>) : $latex L_{-,d}(n)=\\frac{(3d+1)\\,4^{n+1}-1}{3}$<\/span>.<\/li>\n<\/ul>\n\n\n\n
        \n
        d<\/th>d+<\/sup> : r \u2192 Y<\/th>d\u2212<\/sup> : r \u2192 Y<\/th><\/tr><\/thead>
        2<\/td>3 \u2192 5<\/td>9 \u2192 7<\/td><\/tr>\n
        4<\/td>7 \u2192 11<\/td>17 \u2192 13<\/td><\/tr>\n
        6<\/td>11 \u2192 17<\/td>25 \u2192 19<\/td><\/tr>\n
        8<\/td>15 \u2192 23<\/td>33 \u2192 25<\/td><\/tr>\n<\/tbody><\/table><\/figure>\n\n\n\n\n\n\n\n\n

        2) Famille universelle Fm<\/sub><\/em> (constante $latex C=2^{m}-1$<\/span>)<\/h3>\n\n\n\n

        On g\u00e9n\u00e9ralise via $latex T_m(x)=\\frac{3x+C}{2^{\\,v_2(3x+C)}}$<\/span> avec $latex C=2^{m}-1$<\/span> (impair). Le catalogue par distance $latex d$<\/span> (toujours paire) devient :<\/p>\n\n\n\n

          \n
        • $latex d^{+}$<\/span> ($latex v_2(3r+C)=1$<\/span>) : $latex r=2d-C,\\quad Y=3d-C$<\/span>.<\/li>\n
        • $latex d^{-}$<\/span> ($latex v_2(3r+C)=2$<\/span>) : $latex r=4d+C,\\quad Y=3d+C$<\/span>.<\/li>\n<\/ul>\n\n\n\n

          Inversion depuis un lien.<\/strong> Tout lien impair se met de fa\u00e7on unique sous l\u2019une des deux formes $latex Y=3d-C$<\/span> ou $latex Y=3d+C$<\/span> avec $latex d$<\/span> pair. Poser $latex d_{+}=\\frac{Y+C}{3}$<\/span> et $latex d_{-}=\\frac{Y-C}{3}$<\/span> :<\/p>\n\n\n\n

            \n
          • Si $latex d_{+}$<\/span> est pair → $latex Y=3d_{+}-C$<\/span> et $latex (r,Y)=(2d_{+}-C,\\ 3d_{+}-C)$<\/span> (branche $latex d^{+}$<\/span>).<\/li>\n
          • SINON $latex d_{-}$<\/span> est pair → $latex (r,Y)=(4d_{-}+C,\\ 3d_{-}+C)$<\/span> (branche $latex d^{-}$<\/span>).<\/li>\n<\/ul>\n\n\n\n

            Fratrie pour Fm<\/sub>.<\/strong> L\u2019op\u00e9rateur de colonne reste affine (car $latex 3(4x+C)+C=4(3x+C)$<\/span>) et l\u2019on a :<\/p>\n\n\n\n

            $latex L^{(m)}_{r,n}=r\\,4^{n}+\\frac{C\\,(4^{n}-1)}{3}=\\frac{(3r+C)\\,4^{n}-C}{3}\\quad(n\\ge 0).$<\/span><\/p>\n\n\n\n

            Remarque mod 3.<\/em> Si $latex m$<\/span> est impair ($latex C\\equiv 1\\ (\\bmod\\ 3)$<\/span>), alors $latex Y\\equiv -1\\ (\\bmod\\ 3)$<\/span> caract\u00e9rise la branche $latex d^{+}$<\/span> et $latex Y\\equiv +1\\ (\\bmod\\ 3)$<\/span> la branche $latex d^{-}$<\/span>. Si $latex m$<\/span> est pair ($latex C\\equiv 0\\ (\\bmod\\ 3)$<\/span>), on a $latex Y\\equiv 0\\ (\\bmod\\ 3)$<\/span> dans les deux cas : on utilise alors directement l\u2019algorithme $latex d_{\\pm}=\\frac{Y\\pm C}{3}$<\/span> et (dans tes posts) le p\u00e9rim\u00e8tre S<\/em> adapt\u00e9.<\/p>\n\n\n\n\n\n\n\n\n

            3) Structure 50\/50 vs fr\u00e9quences dynamiques<\/h3>\n\n\n\n

            Structurellement<\/strong> (catalogue par $latex d$<\/span>) : exactement moiti\u00e9 des liens sont de la forme $latex 3d-C$<\/span> (branche $latex d^{+}$<\/span>) et moiti\u00e9 de la forme $latex 3d+C$<\/span> (branche $latex d^{-}$<\/span>).<\/p>\n\n\n\n

            Dynamiquement<\/strong> (le long d\u2019une trajectoire), pour le cas classique, les racines minimales $latex r\\equiv 3,7\\ (\\bmod\\ 8)$<\/span> (branche $latex +$<\/span>) se pr\u00e9sentent typiquement environ deux fois plus que $latex r\\equiv 1\\ (\\bmod\\ 8)$<\/span> (branche $latex -$<\/span>) : on observe souvent ~2\/3<\/span> de pas \u201c+\u201d pour ~1\/3<\/span> de pas \u201c\u2212\u201d. Cela n\u2019affecte pas le 50\/50 structurel.<\/p>\n\n\n\n\n\n\n\n\n

            4) Translation inter-fratries $latex \\Delta=r’-r$<\/span><\/h3>\n\n\n\n

            Apr\u00e8s le saut $latex r\\to Y$<\/span>, on recentre $latex Y$<\/span> par (x-1)\/4<\/code> autant que possible jusqu\u2019\u00e0 la racine minimale $latex r’$<\/span>. Posons $latex q=\\left\\lfloor \\frac{v_2(3Y+C)}{2}\\right\\rfloor$<\/span>, alors<\/p>\n\n\n\n

            $latex r’=\\frac{\\,3Y+C\\,-\\,C\\,4^{q}\\,}{\\,3\\cdot 4^{q}\\,}$<\/span>,   d\u2019o\u00f9 $latex \\Delta=r’-r$<\/span>.<\/p>\n\n\n\n

              \n
            • Branche<\/strong> $latex d^{+}$<\/span> ($latex r=2d-C,\\ Y=3d-C$<\/span>) :\n$latex \\ \\Delta=\\frac{\\,9d-2C\\,-\\,C\\,4^{q}\\,}{\\,3\\cdot 4^{q}\\,}-(2d-C)$<\/span>, avec $latex q=\\left\\lfloor \\frac{v_2(9d-2C)}{2}\\right\\rfloor$<\/span>.<\/li>\n
            • Branche<\/strong> $latex d^{-}$<\/span> ($latex r=4d+C,\\ Y=3d+C$<\/span>) :\n$latex \\ \\Delta=\\frac{\\,9d+4C\\,-\\,C\\,4^{q}\\,}{\\,3\\cdot 4^{q}\\,}-(4d+C)$<\/span>, avec $latex q=\\left\\lfloor \\frac{v_2(9d+4C)}{2}\\right\\rfloor$<\/span>.<\/li>\n<\/ul>\n\n\n\n

              Cas g\u00e9n\u00e9rique<\/em> $latex q=0$<\/span> (pas de recentrage suppl\u00e9mentaire apr\u00e8s le saut) : $latex \\Delta=+d$<\/span> pour la branche $latex d^{+}$<\/span>, et $latex \\Delta=-d$<\/span> pour la branche $latex d^{-}$<\/span>. Si $latex q\\ge 1$<\/span>, les recentrages (x-1)\/4<\/code> rabaisseront $latex r’$<\/span> et feront d\u00e9vier $latex \\Delta$<\/span> de $latex \\pm d$<\/span>.<\/p>\n\n\n\n\n\n\n\n\n

              5) Unit\u00e9 universelle B | u<\/code> (rappel concis, m=1)<\/h3>\n\n\n\n

              Tout impair $latex x$<\/span> s\u2019\u00e9crit $latex x=8B+u$<\/span> avec $latex u\\in\\{1,3,5,7\\}$<\/span>. Pour $latex d=2k$<\/span> :<\/p>\n\n\n\n

                \n
              • Branche<\/strong> $latex d^{+}$<\/span> ($latex r=4k-1,\\ Y=6k-1$<\/span>) : \n$latex u_r=3$<\/span> si $latex k$<\/span> impair, $latex u_r=7$<\/span> si $latex k$<\/span> pair; et\n$latex u_Y=7,5,3,1$<\/span> pour $latex d\\equiv 0,2,4,6\\ (\\bmod\\ 8)$<\/span>.<\/li>\n
              • Branche<\/strong> $latex d^{-}$<\/span> ($latex r=8k+1,\\ Y=6k+1$<\/span>) : \n$latex u_r=1$<\/span>; et $latex u_Y=1,7,5,3$<\/span> pour $latex d\\equiv 0,2,4,6\\ (\\bmod\\ 8)$<\/span>.<\/li>\n<\/ul>\n\n\n\n

                Dans une fratrie $latex L_{r,n}$<\/span> : l\u2019unit\u00e9 vaut $latex u_r$<\/span> en $latex n=0$<\/span>, puis 1<\/strong> en $latex n=1$<\/span>, puis 5<\/strong> pour tout $latex n\\ge 2$<\/span>.<\/p>\n\n\n\n\n\n\n\n\n

                6) R\u00e9sum\u00e9 express<\/h3>\n\n\n\n
                  \n
                • Distance paire<\/strong> $latex d$<\/span> → deux liens et seulement deux : \n$latex d^{+}:(r,Y)=(2d-C,\\ 3d-C)$<\/span>,\n$latex d^{-}:(r,Y)=(4d+C,\\ 3d+C)$<\/span>.<\/li>\n
                • Inversion unique<\/strong> depuis $latex Y$<\/span> par $latex d_{\\pm}=\\frac{Y\\pm C}{3}$<\/span> (exactement l\u2019un est pair).<\/li>\n
                • Fratrie<\/strong> : $latex L^{(m)}_{r,n}=r\\,4^{n}+\\frac{C(4^{n}-1)}{3}=\\frac{(3r+C)\\,4^{n}-C}{3}$<\/span>.<\/li>\n
                • Structure vs dynamique<\/strong> : 50\/50 entre $latex 3d\\pm C$<\/span> au niveau catalogue; \u22482\/3 \u201c+\u201d et \u22481\/3 \u201c\u2212\u201d le long d\u2019une trajectoire classique.<\/li>\n
                • Translation inter-fratries<\/strong> : g\u00e9n\u00e9riquement $latex \\Delta=\\pm d$<\/span> (si $latex v_2(3Y+C)=1$<\/span>), sinon d\u00e9viation gouvern\u00e9e par $latex q=\\left\\lfloor \\frac{v_2(3Y+C)}{2}\\right\\rfloor$<\/span>.<\/li>\n<\/ul>\n\n","protected":false},"excerpt":{"rendered":"

                  Distances racine\u2194lien, fratries, et famille universelle Fm Cadre. On travaille sur les impairs et l\u2019on recentre toute visite d\u2019un membre de fratrie par l\u2019op\u00e9ration (x-1)\/4 autant que possible pour revenir \u00e0 la racine minimale r (i.e. $latex r\\equiv 1,3,7\\ (\\bmod\\ 8)$) avant de passer au lien Y. On appelle distance le saut structurel entre racine […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-56532","post","type-post","status-publish","format-standard","hentry","category-non-classe"],"_links":{"self":[{"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/posts\/56532","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/comments?post=56532"}],"version-history":[{"count":1,"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/posts\/56532\/revisions"}],"predecessor-version":[{"id":56533,"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/posts\/56532\/revisions\/56533"}],"wp:attachment":[{"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/media?parent=56532"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/categories?post=56532"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/4.exstyle.fr\/le-blog-photo\/wp-json\/wp\/v2\/tags?post=56532"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}