λ-Universality Across Scales establishes that the URT recursion-efficiency factor λ is not an adjustable parameter and that irreversible updates across physical domains obey a single universal recursion–efficiency law controlled by one dimensionless ratio: informational stiffness divided by local thermal bandwidth, x = σ/(k_B T_loc). Unified Recursion Theory (URT) relates energy cost and informational entropy change for admissible irreversible compression via the proportionality law ΔE = λ k_B T_loc ΔH. Paper 7 shows that URT’s operator structure (Ψ_cons for reversible propagation, Ψ_comp for compressive evolution, and ORM for admissibility selection), together with dimensional invariance and free-energy admissibility, restrict λ to a universal functional dependence on x. In the idealized negligible finite-time dissipation limit, this yields the canonical master curve λ(x) = λ0 exp(−x), with finite-time effects entering multiplicatively through a dissipation factor λ_t. A central result is the near-equilibrium efficiency constant λ0: the low-stiffness (σ → 0) asymptote of irreversible recursion efficiency. Paper 7 clarifies that λ0 is not fixed by algebra alone; instead it is empirically extrapolated from reversible fluctuation statistics and converges across validated domains to λ0 ≈ 0.78. This cross-domain convergence is presented as a nontrivial empirical finding supporting λ0 as a structural constant of recursion rather than a domain-specific fit parameter. The manuscript develops the universality claim in three layers: Statistical and admissibility basis for an upper bound on low-stiffness irreversible efficiency (λ0), Canonical factorization λ = λ0 exp(−σ/(k_B T_loc)) λ_t and the universality condition that λ depends only on σ/T_loc, A free-energy landscape argument showing that monotonic suppression with increasing stiffness-to-bandwidth ratio and admissibility (λ 0) are consistent with the exponential form under URT constraints and observed behavior. Paper 7 connects the universal λ-curve to prior URT developments and limits, including: Quantum measurement admissibility and outcome selection thresholds (ORM), Thermal and fluctuation-dissipation scaling, Chemical barrier/transition-state regimes, Biological efficiency bands (folding funnels, catalysis, fidelity), Strong-curvature recursion suppression and freeze behavior (IFT-SC), Damping behavior in stiffness-wave dynamics (ISW). Crucially, the paper specifies falsifiable cross-domain tests. The strongest test is curve collapse: if systems from different physical domains cannot be rescaled by x = σ/(k_B T_loc) to fall on the same λ(x) = λ0 exp(−x) master curve with a single λ0 ≈ 0.78, λ-universality is falsified. Additional falsifiers include failure of exponential suppression at large x, failure of linear small-x behavior, and mismatch of predicted crossover scales. This paper provides the bedrock for forthcoming URT work on free-energy landscape geometry, biological applications, and cosmological extensions by defining λ(x) as the master function governing admissible irreversibility across scales. URT PAPER FAMILY This work forms part of the Unified Recursion Theory (URT) research program, which develops a cross-domain framework for physical evolution based on constrained informational recursion and an energy–entropy proportionality law. Each paper in the series is self-contained, while collectively establishing the theoretical structure across quantum, geometric, biological, cosmological, and particle-level domains. Related URT works available on Zenodo: FOUNDATIONAL PAPERS 1. Unified Recursion Theory — Core Framework (URT Core) DOI: 10.5281/zenodo.17642761Record: https://zenodo.org/records/17642761 2. Discrete Admissible Regimes in Unified Recursion Theory: Operator Closure, Constraint Topology, and the Necessity of Five Operators DOI: 10.5281/zenodo.18148192Record: https://zenodo.org/records/18148193 3. Informational Field Theory in Strong Curvature (IFT-SC) DOI: 10.5281/zenodo.17850379Record: https://zenodo.org/records/17850379 4. Dynamical Evolution of the Informational Stiffness Field (ISW Theory) DOI: 10.5281/zenodo.17860533Record: https://zenodo.org/records/17860533 RESOLUTION PAPERS (PHYSICAL PARADOXES) 5. Informational Recursion and the Dissolution of the Black Hole Information Paradox DOI: 10.5281/zenodo.17868662Record: https://zenodo.org/records/17868662 6. ORM and the Quantum Measurement Problem (ORM) DOI: 10.5281/zenodo.17881944Record: https://zenodo.org/records/17881944 BRIDGING / CONSTRAINT PAPER 7. Distinguishability Geometry in Informational State Space DOI: 10.5281/zenodo.17957062Record: https://zenodo.org/records/17957062 Provides the geometric foundation for informational state space.Underpins the emergence of spacetime, efficiency universality, and landscape geometry. THEORETICAL EXPANSION PAPERS 8. Emergent Spacetime from Informational Recursion DOI: 10.5281/zenodo.17885555Record: https://zenodo.org/records/17885555 9. λ-Universality Across Scales (λ-UAS) DOI: 10.5281/zenodo.17934065Record: https://zenodo.org/records/17934065 10. Free-Energy Landscape Geometry in Unified Recursion Theory DOI: 10.5281/zenodo.17940995Record: https://zenodo.org/records/17940995 BIOLOGY / COMPLEXITY PAPER 11. URT in Biology: Efficiency, Folding Funnels, Replication Fidelity, and Molecular Motor Dynamics DOI: 10.5281/zenodo.17945209Record: https://zenodo.org/records/17945209 COSMOLOGICAL EXTENSIONS 12. Cyclic Cosmology from Informational Recursion DOI: 10.5281/zenodo.17955043Record: https://zenodo.org/records/17955043 13. Antimatter as Inverse Recursion: Temporal Operator Asymmetry and Matter–Antimatter Imbalance in Unified Recursion Theory DOI: 10.5281/zenodo.17955043Record: https://zenodo.org/records/17955625