The Dimensional Collapse Mechanism in the HDC

Abstract

https://www.hdc-theory.comEarlier papers of the Hierarchical Dimension Cascade (HDC) characterised the dimensional collapse by V″(ε_c) < 0. With the bare potential V(ε) = λε² this condition cannot be met, and the matter density does not appear in it. This paper shows that the condition holds for the effective potential V_eff(ε) = (λ − kρ)ε² and establishes three results. Channel bound: under two declared carrier premises, which formalise the published assignment of the strain field to D3 and of the coherence field to D2, the state φ_3 at every point with ε ≠ 0 has exactly three active projection channels, n_a(φ_3) = 3. Conversely, where fewer than three channels are active, ε = 0 and ψ = 0. Threshold and instability: at locally given ρ and fixed ψ, the strain equation is exactly linear in ε and becomes unstable for ρ > ρ_c = λ/k, where the effective strain mass m_ε² = 2(λ − kρ)/β turns negative. This provides the derivation on which premise (P) of the linear perturbation analysis of the series rests. Collapse criterion: by a declared structural identification, the instability above ρ_c is channel failure, rank(Π(φ_3)) < 3; the failure of one channel suffices. The location then lies outside the realization domain R_3 = {x | ε(x) > 0}, in the structureless state ε = 0 ⟺ ψ = 0. Space is not compressed. Within the HDC, the question of what becomes of information at a location with rank(Π) < 3 cannot be answered from D3, since the structure beyond the realized level is the permanent epistemological boundary of the framework. The standard formulation of the information paradox therefore has no admissible counterpart in the HDC. Every statement carries an explicit status: formally derived, declared premise, structural identification, structural interpretation, or speculation. Open items are listed in the paper.