PFUSRC-158 Computability Boundaries Under Ontological Topology — Rejecting the Paradigm Obsession of "Rope-Tailoring"

Abstract

The long-term stagnation of frontier theoretical physics does not stem from insufficient computing power or inadequate mathematical tools, as continuous analysis, discrete mathematics and non-perturbative computational methods are all maturely established. The core bottleneck lies in the absence of a fixed ontological topological pivot for mathematical description, leading physical calculation to degenerate into phenomenological fitting with arbitrary free parameters. Taking the triple coaxial 45° biconical ontological architecture as the fundamental substrate of cosmic reality, this paper systematically demonstrates that continuous mathematical systems, discrete mathematical operators and transitional gradient mathematical descriptions are not artificial algorithm classifications, but three intrinsic operating states existing within the hierarchical gradient of biconical topology. All dimensional transformation rules, three-dimensional and four-dimensional physical evolution laws, and complete mathematical formalisms are endogenous products derived from biconical topological structures, fully governed by intrinsic topological constraints. This paper constructs a three-fold mathematical ontology covering continuity, transition and discreteness, correcting the mainstream dualistic separation of continuous and discrete physical descriptions, and establishing a new mathematical epistemology featuring "projection-layer linear result calibration, upward ontological topological tracing, and cross-state intertwined coexistence". Meanwhile, this paper rigorously deconstructs the pervasive "Rope-Tailoring Fallacy" dominating modern physical research: the obsessive pursuit of error-free absolute single-point analytical solutions within the L5 projection layer by stripping all ontological topological background residuals, which is fundamentally unachievable under bicone ontological structure. A five-level solvability operational criterion is proposed to strictly distinguish irreducible ontological rheological residuals from allowable engineering computational precision, and relocate four long-standing core physical puzzles—Yang-Mills mass gap, strongly coupled non-equilibrium quantum field theory, quantum gravity unification, and vacuum fluctuation mechanisms—within the hierarchical biconical topological framework. Finally, this paper puts forward the Apple-and-Worm Principle for physical epistemology: all human observation, cognition and mathematical tools originate from the internal structure of the bicone ontology, and no external absolute observational perspective or neutral universal mathematical system can exist independent of cosmic topological substrates.