Inflation from the τ √2 Potential: No Inflaton Required

Can the Spectral Fractal-Temporal Theory (STFT) produce inflation without an inflaton field? We first establish a no-go result: standard slow-roll inflation on the STFT confining potential U₀ (σ) = Λ₀e^−λσ + Λb e^νσ is excluded. The consistency relation r = 8 (1 − nₛ) is a hard floor for exponential potentials, giving r ≈ 0. 28 for nₛ = 0. 965 — incompatible with BICEP/Keck (r 1 (time flows faster in weak gravitational fields). The causal horizon d = ∫ c τ (t') dt' is enlarged by the factor ⟨τ⟩, reducing the required number of e-folds from ~60 to ~60/⟨τ⟩. No inflaton needed. (2) Quasi-exponential expansion: produced by the multi-mode post-rebound cascade. Following the elastic quantum bounce (from the negative vacuum pressure Pᵥac = −A (L) ρ^√2 at ρcrit = ρP/8), each fractal level n of the √2 texture tower contributes a decaying mode with rate Γₙ = 2ⁿ/t₀. The superposition of Nₘax modes produces destructive interference in the slow-roll parameter: ε ~ 2^−Nₘax. This is analogous to an explosion in a negative-pressure chamber: the medium assists the expansion, the confining walls provide restoring force, and energy cascades through the √2 hierarchy. (3) Primordial perturbation spectrum: generated by progressive mode freeze-out. As σ → σ_* (stabilization), the texture modes with mₙ > H freeze out sequentially, reducing the effective mode count. This Kibble-Zurek mechanism produces: nₛ − 1 = −α ln 2, r = 16 × 2^−Nₘax where α is the mode freeze-out rate per e-fold and Nₘax is the total number of active modes. These two observables are decoupled — there is no consistency relation r = f (nₛ). This is structurally distinct from all single-field inflation models. For nₛ = 0. 9649 (Planck 2018): α ≈ 0. 051, meaning approximately one mode freezes every ~20 e-folds — physically reasonable. For Nₘax = 10–13: r = 0. 016–0. 002, within reach of LiteBIRD (σᵣ ~ 10⁻³). Log-periodic modulations at ωf = 2π/ln√2 ≈ 18. 1 in the CMB power spectrum are an intrinsic prediction, not an ad hoc addition. The same √2 tower that drives inflation also provides dark matter (partially frozen modes, Chronostasis mechanism, 47/50 SPARC fit) and dark energy (stabilized branch vacuum energy, w > −1). This creates cross-epoch consistency tests: the modes generating perturbations at z ~ 10³⁰ are the same modes shaping galaxy rotation curves at z ~ 0. Five falsification criteria are provided: (1) detection of a consistency relation r = f (nₛ), (2) r at a value not matching 16 × 2^−N for integer N, (3) absence of log-periodic modulations at ωf ≈ 18. 1, (4) significant spectral running, (5) phantom dark energy (w 5σ). Note: This paper supersedes the earlier version (v1), which explored a temporal gradient mechanism based on slow-roll. The present version (v2) proves that slow-roll is excluded on the v4 potential and develops the multi-mode cascade mechanism instead. The core insight — "no inflaton required" — is preserved and strengthened.