The Higher-Order Underdetermination of Vagueness: Dimensional Nesting, Logical Interlocking, and the Uncomputability of the Absolute Precise Boundary

Traditional sorites paradoxes presuppose an absolute precise boundary C ∗ within a single quantitative dimension. This paper employs spatial occupancy and length-heterogeneity models to demonstrate that identical quantities can yield opposite classifications depending on spatial or length distributions, thereby invalidating the tolerance principle. More fundamentally, rule selection faces a trilemma, resulting in rule underdetermination. To preserve the absolute boundary, epistemicism must compute the corresponding L-Value (the sum of all effective hair lengths). However, this paper proposes "Higher-Order Underdetermination": even when focusing on the single dimension of "length," the computation of the L-Value encounters a "triple concurrence" of infinite horizontal dimensional proliferation, infinite vertical dimensional nesting, and semantic logical interlocking. Furthermore, theunderdetermination of weight allocation and the dynamic flux of the temporal dimension deliver the final blow to the system. The system completely crashes under these concurrent forces. Consequently, C ∗ is untenable not only because it cannot be fixed by rules and dimensions macroscopically, but also because it faces computational bankruptcy microscopically. The root of vagueness lies not in the location of the boundary, but in the higher-order underdetermination caused by dimensional nesting and semantic interlocking.