The Inflationary Parameters as Geometric Signatures of the Universal Cascade Law: A Derivation from Stellar-Scale Measurements via Inter-Scale Geometric Architecture

Description / Abstract: The Universal Cascade Law (Randolph, 2026a) predicts identical geometric architecture at every scale of the self-application hierarchy. Paper III (Randolph, 2026c) predicted that inflationary parameters should be derivable from the geometry of the dual attractor basin transition — stated as a specific, falsifiable prediction. This paper delivers that prediction. 50,000 stars from Gaia DR3, already confirmed as organizing in dual attractor basins at p < 10⁻³⁰⁰, are analyzed for three transition zone parameters: the departure from self-similarity (stellar nₛ), the ratio of lateral to radial dispersion (stellar r), and the number of resolved sub-harmonic levels (stellar N). Two Feigenbaum family constants govern the inter-scale coupling: δ(z=6) = 9.2962 for spatial geometry (N and r), ln(δ) = 1.5410 for temporal geometry (nₛ). The distinction arises because N and r are snapshot geometric quantities while nₛ records a sequential process that unfolded while time itself was emerging. Predicted inflationary parameters from stellar measurements: N = 65.1 e-folds (8.5% from standard ~60), r = 0.0072 (below Planck upper bound of 0.036), nₛ = 0.9656 (within 0.17σ of Planck 2018 value 0.9649 ± 0.0042). Three predictions, three confirmations, all within observational constraints. The back-computed stellar nₛ implied by the ln(δ) hypothesis sits at 1/6 of one standard deviation from the measured center. The time emergence factor ln(δ) — the same constant that encodes Feigenbaum's universal constant as a meta-system slope (Randolph, 2026b) — governs how time emergence amplifies spectral tilt between hierarchical scales. The inflaton field is not required. The inflationary parameters are geometric necessities of the dual attractor basin transition, derivable from measurements at the adjacent scale. This is the first derivation of CMB inflationary parameters from a non-cosmological data source. Keywords: inflationary parameters; spectral index; tensor-to-scalar ratio; e-folds; Universal Cascade Law; Feigenbaum constant; inter-scale architecture; dual attractor basins; time emergence; inflaton; cosmic microwave background; Planck satellite; Gaia DR3; geometric necessity; Resonance Theory; cosmological parameters; basin transition; period-doubling universality Notes: Fourth paper in the Universal Cascade Law series. Delivers the prediction stated in Paper III (The Full Extent of the Universal Cascade Law, DOI: 10.5281/zenodo.18818011) that inflationary parameters should be derivable from dual attractor basin transition geometry. Framework paper: "The Universal Cascade Law" (DOI: 10.5281/zenodo.18818007). Derivation methodology established in: "The Geometric Necessity of Feigenbaum's Constant" (DOI: 10.5281/zenodo.18818009). All computational code publicly available. All results derived from public data (Gaia DR3, Planck 2018). All predictions falsifiable — specific r prediction (r ≈ 0.007) testable by CMB-S4 experiment.