Overview Within the framework of Origin Geometry (OG), the dual-sector architecture H₄ ∪ φH₄ allows the visible (H₄) sector and the phase-shifted (φH₄) sector to coexist within a shared geometric substrate while remaining strongly misaligned at the level of boundary-supported modes. Previous Parts developed the interpretation of dark matter as hidden geometric stress, topological pinning, near-flat-band freezing, reduced electromagnetic accessibility, dark-sector compression, and effective dark collapse 1–5. Part 25 introduced a qualitative mechanism by which antimatter-like excitations sequestered in the (φH₄) sector may leak weakly into the visible sector through extreme curvature regions. The present Part develops a parameterized order-of-magnitude framework for that mechanism. Its goal is not to fit cosmic-ray data, derive a complete transport theory, or calculate absolute fluxes from first principles. Instead, it identifies the mathematical parameter combinations that control whether rare inter-sector leakage could generate observable positron-biased signatures. WKB-Like Tunneling and Mass-Asymmetric Filtering The central ingredient is WKB-like tunneling through an effective topological phase barrier 6. For an excitation species i, the leakage probability is modeled as: Pᵢ ~ exp−Sᵢ where: Sᵢ = (2/ħ) ∫ from x₁ to x₂ √ (2mᵢ, eff Vₚhase (x) − Eᵢ) dx In a uniform-barrier approximation: Sᵢ ≈ (2 Lₑff / ħ) √ (2mᵢ, eff Vₑff) The WKB action scales as √ (mᵢ, eff). Therefore, if the dark-sector effective mass hierarchy satisfies 7: mₚ̄, eff ≫ mₑ+, eff then positron-like modes tunnel far more efficiently than antiproton-like modes under comparable barrier conditions. This produces mass-asymmetric filtering 8, 9. Unlike the earlier qualitative version, the present Part does not assume that a larger dark-sector boundary mass inflation automatically preserves positron dominance. Instead, it writes the effective masses as: mₚ̄, eff = χₚ̄ mₚ mₑ+, eff = χₑ mₑ and states explicitly that positron-dominated leakage requires: χₚ̄ mₚ ≫ χₑ mₑ This condition is central. If it fails, WKB filtering no longer favors positron-like leakage. Parameterized Source Term The leakage source term is also written in dimensionally consistent form. A microscopic tunneling probability Pᵢ is not itself a rate. A leakage rate per source requires an effective attempt frequency νₐtt: Γᵢ, src ~ νₐtt Pᵢ A corresponding source density may then be parameterized as: Qᵢ (r, E) ~ nₛrc (r) Nᵢ, hid (r) νₐtt (r) Pᵢ (r) fᵢ (E) or equivalently in terms of a dark-sector density reservoir: Qᵢ (r, E) ~ ηᵢ ρdark (r) / mᵢ, eff Γᵢ (r) fᵢ (E) Here nₛrc is the density of barrier-weakening environments, Nᵢ, hid is the number of hidden excitations available per source, ηᵢ is an effective leakage efficiency, and fᵢ (E) is an injection spectrum.
The Duy Tan Truong (Tue,) studied this question.