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Estrutura Tensorial Multi-Dimensional da Árvore Reversa de Collatz: Uma Proposta Ontológica via Matriz Multidimensional Fundamental

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Autore principale:	Terêncio de Bastos, Carlos Alberto
Natura:	Recurso digital
Lingua:	portoghese
Pubblicazione:	Zenodo 2026
Soggetti:	Conjectura de Collatz, problema 3n+1, árvore reversa, estrutura tensorial multi-dimensional, foliação aritmética, distribuição diádica, princípios geradores, ontologia matemática, fatoração canônica, sistemas dinâmicos sobre N
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_version_	1866901860573511680
author	Terêncio de Bastos, Carlos Alberto
author_facet	Terêncio de Bastos, Carlos Alberto
contents	<h3 class="text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold">RESUMO (PT)</h3> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Apresentamos uma reformulação tensorial multi-dimensional da árvore reversa do problema de Collatz <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span>, organizada em torno de uma matriz construtiva 8-dimensional cuja diagonal é a sequência dos ímpares positivos crescentes. Cada inteiro ímpar é mapeado a um ponto único em <span class="katex"><span class="katex-mathml">Z8\mathbb{Z}^8 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">Z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span> via coordenadas algébricas e dinâmicas: <span class="katex"><span class="katex-mathml">(n,σ,q,ν3,ν2(3n+1),freq,Pmax⁡,pmin⁡)(n, \sigma, q, \nu_3, \nu_2(3n+1), \mathrm{freq}, P_{\max}, p_{\min}) </span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mord mathnormal">q</span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord"><span class="mord mathrm">freq</span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">i</span><span class="mtight">n</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mclose">)</span></span></span></span>, todas geradas por princípios aritméticos sobre inteiros. A contribuição central é estrutural: mostramos que a árvore reversa de Collatz é redutível a um sistema de geradores aritméticos simples sobre inteiros, em número suficiente para capturar a aritmética específica do problema. Por gerador aritmético simples entendemos uma operação elementar definida diretamente sobre inteiros (somar, multiplicar, dividir, valorar <span class="katex"><span class="katex-mathml">pp </span><span class="katex-html"><span class="base"><span class="mord mathnormal">p</span></span></span></span>-adicamente, contar iterações), sem necessidade de extensão a estruturas auxiliares. Geradores deste tipo são aceitos como definicionais em fundamentos da matemática: <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> é gerado pela regra <span class="katex"><span class="katex-mathml">+1+1 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">1</span></span></span></span> (Peano), os ímpares positivos pela regra <span class="katex"><span class="katex-mathml">+2+2 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">2</span></span></span></span>, sem que se exija prova formal de cobertura. Mostramos que a árvore reversa de Collatz se articula em 8 geradores desta mesma natureza, organizados na matriz tensorial 8-dimensional. Identidades individuais que conectam as dimensões aparecem em formas equivalentes na literatura clássica de Collatz [Terras 1976, Lagarias 1985, Wirsching 1998] — a contribuição não é redescobri-las, mas reorganizá-las de modo que a redução estrutural se torne visível. Apresentamos a matriz com seus geradores, demonstramos identidades estruturais conectando as dimensões, comparamos sistematicamente com princípios geradores universalmente aceitos para <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> e <span class="katex"><span class="katex-mathml">Nıˊmpar+\mathbb{N}^+_{\text{ímpar}} </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord accent mtight"><span class="vlist-t"><span class="mord latin_fallback mtight">ı</span><span class="accent-body">ˊ</span></span></span>mpar</span></span></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>, e formulamos uma proposta ontológica (Princípio 7.1) que expressa o racional estrutural compartilhado entre nossa formulação e geradores aceitos como Peano. Argumentamos por simetria de tratamento: objeções genéricas como o contraexemplo trivial aplicam-se uniformemente a todas formulações geradoras, e a aceitação de Peano como tautológica reflete convenção fundacional, não demonstração formal de não-existência de exclusões. A verificação computacional das identidades em 17.501 ímpares de magnitude <span class="katex"><span class="katex-mathml">&sim;109\sim 10^9 </span><span class="katex-html"><span class="base"><span class="mrel">&sim;</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span> é apresentada juntamente com a verificação histórica da conjectura por Barina (2020) em <span class="katex"><span class="katex-mathml">n≤268n \leq 2^{68} </span><span class="katex-html"><span class="base"><span class="mord mathnormal">n</span><span class="mrel">≤</span></span><span class="base"><span class="mord">2<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">68</span></span></span></span></span></span></span></span></span></span>. Não afirmamos prova: oferecemos arcabouço estrutural cuja avaliação cabe à comunidade. Em apêndice ilustrativo, aplicamos a metodologia a três sistemas dinâmicos análogos sobre <span class="katex"><span class="katex-mathml">N+\mathbb{N}^+ </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span> — um convergente em 6D demonstrável por indução, outro em 6D com forma fechada via estrutura binária, e um terceiro em 7D onde a cobertura falha demonstravelmente — evidenciando que a formulação é dimensão-adaptativa e não constitui construção ad hoc para o caso de Collatz.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Palavras-chave:</strong> Conjectura de Collatz, problema <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span>, árvore reversa, estrutura tensorial multi-dimensional, foliação aritmética, distribuição diádica, princípios geradores, ontologia matemática.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Classificação MSC 2020:</strong> 11B83 (primária), 11A07, 37A45, 37P05, 03A05.</p> <h3 class="text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold">ABSTRACT (EN)</h3> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">We present a multi-dimensional tensor reformulation of the reverse tree of the Collatz <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span> problem, organized around a constructive 8-dimensional matrix whose diagonal is the sequence of increasing positive odd integers. Each odd integer is mapped to a unique point in <span class="katex"><span class="katex-mathml">Z8\mathbb{Z}^8 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">Z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span> via algebraic and dynamical coordinates: <span class="katex"><span class="katex-mathml">(n,σ,q,ν3,ν2(3n+1),freq,Pmax⁡,pmin⁡)(n, \sigma, q, \nu_3, \nu_2(3n+1), \mathrm{freq}, P_{\max}, p_{\min}) </span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mord mathnormal">q</span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord"><span class="mord mathrm">freq</span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">i</span><span class="mtight">n</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mclose">)</span></span></span></span>, all generated by arithmetic principles over integers. The central contribution is structural: we show that the reverse Collatz tree is reducible to a system of simple arithmetic generators over integers, in number sufficient to capture the specific arithmetic of the problem. By simple arithmetic generator we mean an elementary operation defined directly on integers (addition, multiplication, division, <span class="katex"><span class="katex-mathml">pp </span><span class="katex-html"><span class="base"><span class="mord mathnormal">p</span></span></span></span>-adic valuation, iteration count), without need for extension to auxiliary structures. Generators of this type are accepted as definitional in the foundations of mathematics: <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> is generated by the rule <span class="katex"><span class="katex-mathml">+1+1 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">1</span></span></span></span> (Peano), the positive odd integers by the rule <span class="katex"><span class="katex-mathml">+2+2 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">2</span></span></span></span>, without requiring any formal proof of coverage. We show that the reverse Collatz tree articulates into 8 generators of this same nature, organized into the 8-dimensional tensor matrix. Individual identities connecting the dimensions appear in equivalent forms in the classical Collatz literature [Terras 1976, Lagarias 1985, Wirsching 1998] — the contribution is not to rediscover them, but to reorganize them so that the structural reduction becomes visible. We present the matrix with its generators, demonstrate structural identities connecting the dimensions, systematically compare with universally accepted generating principles for <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> and <span class="katex"><span class="katex-mathml">Nodd+\mathbb{N}^+_{\text{odd}} </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight">odd</span></span></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>, and formulate an ontological proposal (Principle 7.1) that expresses the structural rationale shared between our formulation and accepted generators such as Peano. We argue from symmetry of treatment: generic objections such as the trivial counterexample apply uniformly to all generating formulations, and the acceptance of Peano as tautological reflects foundational convention, not formal demonstration of non-existence of exclusions. Computational verification of the identities in 17,501 odd integers of magnitude <span class="katex"><span class="katex-mathml">&sim;109\sim 10^9 </span><span class="katex-html"><span class="base"><span class="mrel">&sim;</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span> is presented together with the historical verification of the conjecture by Barina (2020) for <span class="katex"><span class="katex-mathml">n≤268n \leq 2^{68} </span><span class="katex-html"><span class="base"><span class="mord mathnormal">n</span><span class="mrel">≤</span></span><span class="base"><span class="mord">2<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">68</span></span></span></span></span></span></span></span></span></span>. We do not claim a proof: we offer a structural framework whose evaluation rests with the community. In an illustrative appendix, we apply the methodology to three analogous dynamical systems over <span class="katex"><span class="katex-mathml">N+\mathbb{N}^+ </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span> — one convergent in 6D demonstrable by induction, another in 6D with closed form via binary structure, and a third in 7D where coverage demonstrably fails — evidencing that the formulation is dimension-adaptive and does not constitute an ad hoc construction for the Collatz case.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Keywords:</strong> Collatz Conjecture, <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span> problem, reverse tree, multi-dimensional tensor structure, arithmetic foliation, dyadic distribution, generating principles, mathematical ontology.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>MSC 2020 Classification:</strong> 11B83 (primary), 11A07, 37A45, 37P05, 03A05.</p>
format	Recurso digital
id	zenodo_https___doi_org_10_5281_zenodo_19868187
institution	Zenodo
language	por
publishDate	2026
publisher	Zenodo
record_format	zenodo
spellingShingle	Estrutura Tensorial Multi-Dimensional da Árvore Reversa de Collatz: Uma Proposta Ontológica via Matriz Multidimensional Fundamental Terêncio de Bastos, Carlos Alberto Conjectura de Collatz, problema 3n+1, árvore reversa, estrutura tensorial multi-dimensional, foliação aritmética, distribuição diádica, princípios geradores, ontologia matemática, fatoração canônica, sistemas dinâmicos sobre N <h3 class="text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold">RESUMO (PT)</h3> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Apresentamos uma reformulação tensorial multi-dimensional da árvore reversa do problema de Collatz <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span>, organizada em torno de uma matriz construtiva 8-dimensional cuja diagonal é a sequência dos ímpares positivos crescentes. Cada inteiro ímpar é mapeado a um ponto único em <span class="katex"><span class="katex-mathml">Z8\mathbb{Z}^8 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">Z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span> via coordenadas algébricas e dinâmicas: <span class="katex"><span class="katex-mathml">(n,σ,q,ν3,ν2(3n+1),freq,Pmax⁡,pmin⁡)(n, \sigma, q, \nu_3, \nu_2(3n+1), \mathrm{freq}, P_{\max}, p_{\min}) </span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mord mathnormal">q</span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord"><span class="mord mathrm">freq</span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">i</span><span class="mtight">n</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mclose">)</span></span></span></span>, todas geradas por princípios aritméticos sobre inteiros. A contribuição central é estrutural: mostramos que a árvore reversa de Collatz é redutível a um sistema de geradores aritméticos simples sobre inteiros, em número suficiente para capturar a aritmética específica do problema. Por gerador aritmético simples entendemos uma operação elementar definida diretamente sobre inteiros (somar, multiplicar, dividir, valorar <span class="katex"><span class="katex-mathml">pp </span><span class="katex-html"><span class="base"><span class="mord mathnormal">p</span></span></span></span>-adicamente, contar iterações), sem necessidade de extensão a estruturas auxiliares. Geradores deste tipo são aceitos como definicionais em fundamentos da matemática: <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> é gerado pela regra <span class="katex"><span class="katex-mathml">+1+1 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">1</span></span></span></span> (Peano), os ímpares positivos pela regra <span class="katex"><span class="katex-mathml">+2+2 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">2</span></span></span></span>, sem que se exija prova formal de cobertura. Mostramos que a árvore reversa de Collatz se articula em 8 geradores desta mesma natureza, organizados na matriz tensorial 8-dimensional. Identidades individuais que conectam as dimensões aparecem em formas equivalentes na literatura clássica de Collatz [Terras 1976, Lagarias 1985, Wirsching 1998] — a contribuição não é redescobri-las, mas reorganizá-las de modo que a redução estrutural se torne visível. Apresentamos a matriz com seus geradores, demonstramos identidades estruturais conectando as dimensões, comparamos sistematicamente com princípios geradores universalmente aceitos para <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> e <span class="katex"><span class="katex-mathml">Nıˊmpar+\mathbb{N}^+_{\text{ímpar}} </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord accent mtight"><span class="vlist-t"><span class="mord latin_fallback mtight">ı</span><span class="accent-body">ˊ</span></span></span>mpar</span></span></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>, e formulamos uma proposta ontológica (Princípio 7.1) que expressa o racional estrutural compartilhado entre nossa formulação e geradores aceitos como Peano. Argumentamos por simetria de tratamento: objeções genéricas como o contraexemplo trivial aplicam-se uniformemente a todas formulações geradoras, e a aceitação de Peano como tautológica reflete convenção fundacional, não demonstração formal de não-existência de exclusões. A verificação computacional das identidades em 17.501 ímpares de magnitude <span class="katex"><span class="katex-mathml">&sim;109\sim 10^9 </span><span class="katex-html"><span class="base"><span class="mrel">&sim;</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span> é apresentada juntamente com a verificação histórica da conjectura por Barina (2020) em <span class="katex"><span class="katex-mathml">n≤268n \leq 2^{68} </span><span class="katex-html"><span class="base"><span class="mord mathnormal">n</span><span class="mrel">≤</span></span><span class="base"><span class="mord">2<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">68</span></span></span></span></span></span></span></span></span></span>. Não afirmamos prova: oferecemos arcabouço estrutural cuja avaliação cabe à comunidade. Em apêndice ilustrativo, aplicamos a metodologia a três sistemas dinâmicos análogos sobre <span class="katex"><span class="katex-mathml">N+\mathbb{N}^+ </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span> — um convergente em 6D demonstrável por indução, outro em 6D com forma fechada via estrutura binária, e um terceiro em 7D onde a cobertura falha demonstravelmente — evidenciando que a formulação é dimensão-adaptativa e não constitui construção ad hoc para o caso de Collatz.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Palavras-chave:</strong> Conjectura de Collatz, problema <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span>, árvore reversa, estrutura tensorial multi-dimensional, foliação aritmética, distribuição diádica, princípios geradores, ontologia matemática.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Classificação MSC 2020:</strong> 11B83 (primária), 11A07, 37A45, 37P05, 03A05.</p> <h3 class="text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold">ABSTRACT (EN)</h3> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">We present a multi-dimensional tensor reformulation of the reverse tree of the Collatz <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span> problem, organized around a constructive 8-dimensional matrix whose diagonal is the sequence of increasing positive odd integers. Each odd integer is mapped to a unique point in <span class="katex"><span class="katex-mathml">Z8\mathbb{Z}^8 </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">Z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span> via algebraic and dynamical coordinates: <span class="katex"><span class="katex-mathml">(n,σ,q,ν3,ν2(3n+1),freq,Pmax⁡,pmin⁡)(n, \sigma, q, \nu_3, \nu_2(3n+1), \mathrm{freq}, P_{\max}, p_{\min}) </span><span class="katex-html"><span class="base"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mord mathnormal">σ</span><span class="mpunct">,</span><span class="mord mathnormal">q</span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">ν</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mopen">(</span><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord"><span class="mord mathrm">freq</span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">a</span><span class="mtight">x</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">m</span><span class="mtight">i</span><span class="mtight">n</span></span></span></span></span><span class="vlist-s"></span></span></span></span></span><span class="mclose">)</span></span></span></span>, all generated by arithmetic principles over integers. The central contribution is structural: we show that the reverse Collatz tree is reducible to a system of simple arithmetic generators over integers, in number sufficient to capture the specific arithmetic of the problem. By simple arithmetic generator we mean an elementary operation defined directly on integers (addition, multiplication, division, <span class="katex"><span class="katex-mathml">pp </span><span class="katex-html"><span class="base"><span class="mord mathnormal">p</span></span></span></span>-adic valuation, iteration count), without need for extension to auxiliary structures. Generators of this type are accepted as definitional in the foundations of mathematics: <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> is generated by the rule <span class="katex"><span class="katex-mathml">+1+1 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">1</span></span></span></span> (Peano), the positive odd integers by the rule <span class="katex"><span class="katex-mathml">+2+2 </span><span class="katex-html"><span class="base"><span class="mord">+</span><span class="mord">2</span></span></span></span>, without requiring any formal proof of coverage. We show that the reverse Collatz tree articulates into 8 generators of this same nature, organized into the 8-dimensional tensor matrix. Individual identities connecting the dimensions appear in equivalent forms in the classical Collatz literature [Terras 1976, Lagarias 1985, Wirsching 1998] — the contribution is not to rediscover them, but to reorganize them so that the structural reduction becomes visible. We present the matrix with its generators, demonstrate structural identities connecting the dimensions, systematically compare with universally accepted generating principles for <span class="katex"><span class="katex-mathml">N\mathbb{N} </span><span class="katex-html"><span class="base"><span class="mord mathbb">N</span></span></span></span> and <span class="katex"><span class="katex-mathml">Nodd+\mathbb{N}^+_{\text{odd}} </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight">odd</span></span></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span><span class="vlist-s"></span></span></span></span></span></span></span></span>, and formulate an ontological proposal (Principle 7.1) that expresses the structural rationale shared between our formulation and accepted generators such as Peano. We argue from symmetry of treatment: generic objections such as the trivial counterexample apply uniformly to all generating formulations, and the acceptance of Peano as tautological reflects foundational convention, not formal demonstration of non-existence of exclusions. Computational verification of the identities in 17,501 odd integers of magnitude <span class="katex"><span class="katex-mathml">&sim;109\sim 10^9 </span><span class="katex-html"><span class="base"><span class="mrel">&sim;</span></span><span class="base"><span class="mord">1</span><span class="mord">0<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span> is presented together with the historical verification of the conjecture by Barina (2020) for <span class="katex"><span class="katex-mathml">n≤268n \leq 2^{68} </span><span class="katex-html"><span class="base"><span class="mord mathnormal">n</span><span class="mrel">≤</span></span><span class="base"><span class="mord">2<span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">68</span></span></span></span></span></span></span></span></span></span>. We do not claim a proof: we offer a structural framework whose evaluation rests with the community. In an illustrative appendix, we apply the methodology to three analogous dynamical systems over <span class="katex"><span class="katex-mathml">N+\mathbb{N}^+ </span><span class="katex-html"><span class="base"><span class="mord"><span class="mord mathbb">N</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist"><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span></span> — one convergent in 6D demonstrable by induction, another in 6D with closed form via binary structure, and a third in 7D where coverage demonstrably fails — evidencing that the formulation is dimension-adaptive and does not constitute an ad hoc construction for the Collatz case.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>Keywords:</strong> Collatz Conjecture, <span class="katex"><span class="katex-mathml">3n+13n+1 </span><span class="katex-html"><span class="base"><span class="mord">3</span><span class="mord mathnormal">n</span><span class="mbin">+</span></span><span class="base"><span class="mord">1</span></span></span></span> problem, reverse tree, multi-dimensional tensor structure, arithmetic foliation, dyadic distribution, generating principles, mathematical ontology.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]"><strong>MSC 2020 Classification:</strong> 11B83 (primary), 11A07, 37A45, 37P05, 03A05.</p>
title	Estrutura Tensorial Multi-Dimensional da Árvore Reversa de Collatz: Uma Proposta Ontológica via Matriz Multidimensional Fundamental
topic	Conjectura de Collatz, problema 3n+1, árvore reversa, estrutura tensorial multi-dimensional, foliação aritmética, distribuição diádica, princípios geradores, ontologia matemática, fatoração canônica, sistemas dinâmicos sobre N
url	https://doi.org/10.5281/zenodo.19868187