Form as the Densification of Spatial Dimension: A Single-Axiom Cosmological Framework Built on the Dimensional-Flow Mechanism and the Principle of Mutual Compatibility (ΨD)

This work treats cosmological history within a unified framework built upon a single minimal axiom — the background space is three-dimensional (D = 3) — and structural assumptions that either derive from this axiom or are read as preconditions of its physical consistency. In the framework, n is the spatial dimensional density of the form residing in D = 3 space and takes values in the open interval n ∈ (0, 4). Four active regimes — Planck-floor (d = 0), string (d = 1), surface/energy (d = 2), volumetric matter (d = 3) — are organized by the form-energy ledger Eₙ = xⁿ; the Z₂ symmetry n ↔ 4 − n is centered at n = 2. The extreme values n = 0 and n = 4 are physically inaccessible; these are not a separate assumption but the kinematic consequence of the single axiom. The dynamical mechanism is a cyclic dimensional flow: in the surface regime area accumulation exceeds the n = 3 threshold; in the volumetric matter regime, since no new upper dimension can open, energy-matter conversion is forced into single-channel matter accumulation and gravitational backreaction internally produces black holes. The descent continues down to the n → 0⁺ Planck-floor, where bouncing takes over. With the canonical potential V (φ) = Λ⁴ coshβ (φ − φ★) /M and the modified Friedmann equation H² = (ρ/3M²) 1 − ρ/ρb, ρb = 2M⁴/cosh (u), the structure supports double-bouncing. Dark energy is the late-time manifestation of the transformation energy released at form-transitions (ΣQ = x⁴ − 1) ; dark matter is not a separate entity but the gravitational appearance of local n deviation (Δn). The cycle-residue principle accounts for mature structures in the early universe. The central prediction w (z) = −1 + ε₀ (1 + z) ^ (2βₑff²) is testable with DESI DR2, Euclid and LSST; the structural w ≥ −1 bound of the framework is consistent with the late-time (z ≲ 0. 5) quintessence-like behavior indicated by DESI DR2, while the high-redshift phantom-crossing signal is discussed as a parametrization-dependent appearance (Section 13. 1).