We introduce the Ψ-field (Cohesion Field) as the proposed dynamical substrate of the Inverted Hypersphere Cosmology (IHC) framework on real projective 4-space (RP⁴). The Ψ-field is a conformally coupled real scalar field respecting the antipodal identification x ∼ −x, governed by the Cohesion Field Equation (CFE): □Ψ − (1/3)RΨ + (λ/6)(Ψ²−Ψ₀²)Ψ = 0 with vacuum value Ψ₀ = c⁴/(8πGR_H²) = ρ_crit/3 ≈ 2.56×10⁻¹⁰ J/m³ postulated from the Hubble radius R_H = c/H₀. We prove two exact identities of the 33-shell geometry: the total rotational inertia Σ_{k=0}^{32} φ^{−2k} = φ (golden ratio), and the counter/co-rotating imbalance I_counter − I_co = 1/2. The coherence length ℓ_coh = R_S/N ≈ 428 Mpc is derived from field correlations; the linearised CFE admits a Z₃ standing-wave solution with period λ_{Z₃} = R_H/11 = 404.4 Mpc and fractional amplitude A_{Z₃} = β_coh/β = 0.442%, producing a 90° quadrature modulation of BAO distances D_M and D_H. Applied to all 33 BAO measurements across 7 surveys, the IHC point prediction achieves χ²/dof = 0.916 with zero free parameters (Bayes factor ln B = +4.76 over ΛCDM); on the BOSS DR12 subset alone, residuals fall below 0.13σ with χ²/dof = 0.0110. Observational predictions include: galaxy correlation suppression below ΛCDM at r ≳ ℓ_coh (testable with DESI Year 5 and Euclid), CMB power suppression at ℓ < 33, and Ψ-field Fibonacci resonances at ℓ ∈ {144, 233, 377, 610, 987}. All predictions are parameter-free and falsifiable within the next decade. Keywords for Zenodo: cohesion field; Ψ-field dynamics; real projective space; RP⁴ topology; emergent spacetime; coherence length; large-scale structure; CMB anomalies; cosmological constant; golden ratio