On the Physical Inadequacy of Singularities and the Unified Frequency-Governed Framework of Extended Classical Mechanics (ECM)

The classical black-hole singularity remains one of the most mathematically accepted yet physically unresolved constructs in modern theoretical physics. Within General Relativity (GR), the internal core of a black hole is formally represented by the limiting condition r = 0, implying L = 0, d = 0, and therefore V = 0. This raises a foundational physical contradiction: if a black hole is claimed to obey the relativistic mass–energy relation E = Mc², then the associated mass (M) must remain physically meaningful, which ordinarily requires nonzero spatial support.

This paper argues that a strictly zero-dimensional singular core cannot be regarded as a legitimate physical object because it forfeits both spatial existence and localized mass meaning. Once the Planck boundary (ℓ_P = √(ℏG/c³)) is recognized as the lower limit of reliable physical interpretation, the singularity must be understood not as a real object, but as a signal of formal breakdown in the classical geometric framework.

To address this inadequacy, the paper introduces an alternative interpretation through Extended Classical Mechanics (ECM). In ECM, sub-Planck collapse does not terminate in geometric divergence, but transforms through a deterministic, frequency-governed energetic transition:

ΔPE_ECM ↔ M^app ↔ ΔM_M ↔ ΔKE_ECM

governed by the intrinsic frequency relation:

f_source = f_observed + Δf_source

Under this framework, the black hole is reinterpreted not as an inaccessible zero-volume paradox, but as a dynamic energetic transformer whose internal frequency evolution continuously maps to its external gravitational footprint. ECM therefore restores physical continuity, preserves spatial interpretability, and offers a conceptually unified alternative to classical singularity-based black-hole ontology.