Abstract
<title>Abstract</title> <p>We examine the hypothesis that fundamental physics is invariant under exact Lorentz transformations together with discrete dilatations x^μ → λ₀ⁿ x^μ, n ∈ ℤ, anchored at the Planck scale (Planck-anchored discrete scale invariance, PDSI). Kinematically the hypothesis is consistent: the widely repeated objection that Lorentz boosts forbid a preferred discrete tower of scales fails at the level of transformation groups, which close exactly as ℝ^{1,3} ⋊ (O(1,3) × ℤ). Unbroken, the symmetry forbids any dimensionful condensate and any isolated nonzero mass; the viable realization is spontaneous breaking, which—being discrete—produces no Goldstone dilaton and thereby evades the fifth-force sector that constrains continuous scale-invariance programs. The genuine obstruction to an exact realization is renormalization-group monotonicity; its known proofs assume a local stress tensor on a fixed background, a hypothesis that fails for quantum gravity, precisely where PDSI is anchored. Discrete self-similarity is moreover native to gravitation: classically at the threshold of black-hole formation, and—the central observation of this paper—in the complex critical exponents θ = θ′ ± iθ″ of the asymptotically safe ultraviolet fixed point, which imply a log-periodic, discretely self-similar approach to the Planck regime. We recompute the exponents from the published Einstein–Hilbert flow (θ = 1.475302 ± 3.043206 i, Litim scheme), exclude exact limit cycles of that flow—analytically in stated regions by index and Bendixson arguments, globally at the numerical level by a Poincaré–Bendixson search—so the discrete scale invariance is emergent, not the shadow of an exact cycle—and derive the inflationary exit map that carries the oscillation to observable scales. The result is a closed master relation, A = S exp[−(θ′/θ″) N ω_log], locking amplitude, frequency, and the log-range N of scales. Its consequences constitute the observational content of the proposal: detectable high-frequency log-oscillations are predicted absent, in accordance with existing null searches; the signal migrates to a scale-dependent contribution to the spectral tilt, bounded by 0.014 S and peaking at the order of the measured 1 − n_s; and the induced running lies at next-generation sensitivity. The background-invariant envelope-to-frequency ratio θ′/θ″ survives as a conditional diagnostic for any future detection. Scheme systematics of the exponents are quantified and identified, together with the amplitude normalization S and the exact-theory cycle question, as the outstanding computations.</p>