All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques -- effectively generates only countable fragments within a closed formal system. We formalize this limitation as the "fractal boundary of constructivity", the asymptotic limit of all constructive extensions under syntactically enumerable rules. A central theorem establishes the impossibility of fully capturing the structure of the continuum within any such system. We further introduce the concept of "fractal countability", a process-relative refinement of countability based on layered constructive closure. This provides a framework for analyzing definability beyond classical recursion without invoking uncountable totalities. We interpret the continuum not as an object constructively realizable, but as a horizon of formal expressibility.

翻译：现代数学中采用的所有构造性方法，即使旨在超越可数性，也只能产生可数集。我们证明，任何关于不可数性的构造性论证——排除对角线技巧——实际上仅在封闭形式系统中生成可数的片段。我们将这一局限性形式化为“构造性分形边界”，即所有构造性扩展在语法可枚举规则下的渐近极限。一个核心定理确立了在此类系统中完全捕捉连续统结构的不可能性。我们进一步引入“分形可数性”概念，这是一种基于分层构造闭包的过程相对可数性细化。这为分析超越经典递归的可定义性提供了框架，而无需援引不可数总体。我们将连续统诠释为形式可表达性的视界，而非构造上可实现的对象。