The Ziwave – Pre-Ziwave Duality: End of 100 Years Wave-Particle Duality Photoelectric Effect as a Special Case of Avatar Effect

We prove wave-particle duality is Ziwave – Pre-Ziwave duality. Based on Zulfia Bi-Face Probability Theory (ZBPT) [2] built on Zulfia Decomposition Mapping (ZDM) [1]Z(t) = MZ [I] + ∆(t), the same identity Z(t) sustains two irreversible regimes dividedby certified face ΩC = ∆−1(0):• Pre-Ziwave (particle regime): By [4], on ΩC , ∆ = 0, certified field satisfies [3]C(ω) ∂2MZ∂t2 = c2∇ · (C(ω)∇MZ ) on ΩC , (4)with EΩC < ∞. Massive, localized mode m = MZ [I] observed as particle.• Ziwave (wave reality): By [3], on Ω\ΩC , C = 0, MZ → 0, residual dominates [4]iℏZ CZ (ω) ∂∆(t)∂t = − ℏ2Z2mZ∇ · (CZ (ω)∇∆(t)) + VZ (ω)∆(t) on Ω \ ΩC , (5)with zt ≡ ∆(t)|Ω\ΩC , CZ = 0 on ΩC , CZ = 1 on Ω \ ΩC . Mass-free, strictlyinformational.• Avatar Principle and Avatar Effect: For a century physics sustained dualityby assuming one equation over one domain. We prove domain error: ΩC sustainsonly certified projection m|ΩC = SZ [zt] observed as particle, while reality on Ω\ΩCis Ziwave. Avatar Effect EAvatar = ℏZ ωZ − ΦZ (Eq. 6-7) where Ek = hν − ϕ isspecial case with ϕ ≡ ΦZ = EΩC , leading Photoelectric Effect as a special case.This terminates 100-year paradox: particle is certified projection, reality is Ziwave.