From Scale Cancellation to Continuum Reconstruction: Dimensionless Glueball Ratios in SU(N) Yang–Mills Theory

This work presents a systematic numerical study of dimensionless observables in pure SU(N) Yang–Mills theory, developed as an observable-level numerical layer within the broader Zsa-G continuum-reconstruction program. The analysis is organized around scale-free ratios in which the common lattice scale cancels, including the scalar and tensor glueball masses relative to the string tension and the fourth root of the topological susceptibility relative to the string tension. Rather than identifying finite-β plateaus with the continuum limit, the study explicitly distinguishes finite-β scaling regimes from continuum-sensitive extrapolation. For SU(3), the analysis combines finite-β scaling, constant-fit diagnostics, O(a²σ) and higher-order continuum-sensitive fits, alternative lattice realizations of the topological susceptibility, and the E⁺⁺ and T₂⁺⁺ lattice representations of the tensor glueball. These analyses provide cross-observable, cross-discretization, and cross-representation consistency checks while keeping fit-form dependence and incomplete covariance information explicit. The study then examines the large-N behavior of continuum-extrapolated glueball mass ratios using SU(N) inputs for N = 2, 3, 4, 5, 6, 8, 10, and 12. Competing parametrizations, c_N = c∞ + a/N + b/N² and c_N = c∞ + b/N², are compared quantitatively. The results show that a simple 1/N description is inadequate for the analyzed data, while the extended parametrization provides an improved description of both the scalar 0⁺⁺ and tensor 2⁺⁺ channels. The analysis also demonstrates that large-N functional-form dependence is channel dependent and should be treated as an explicit source of extrapolation uncertainty. Robustness is examined through systematic leave-one-N-out analyses. The scalar and tensor channels exhibit different sensitivities to individual gauge-group inputs, separating data-point robustness from functional-form robustness. Finally, the analysis produces genuinely withheld, blind out-of-sample predictions for N = 16 and N = 20 using only the N ≤ 12 training dataset. For the extended large-N parametrization, the predicted ratios are m₀⁺⁺/√σ = 3.0862 ± 0.0251 and 3.0827 ± 0.0281 at N = 16 and N = 20, respectively, and m₂⁺⁺/√σ = 4.6341 ± 0.0203 and 4.6367 ± 0.0228. These values are prospective prediction targets rather than confirmed lattice results. No N = 16 or N = 20 measurements are used in their construction. Their purpose is to provide falsifiable future tests of the competing large-N extrapolation structures. The resulting numerical architecture proceeds from dimensionless scale cancellation through finite-β scaling, continuum-sensitive extrapolation, cross-observable consistency, large-N structure, leave-one-N-out robustness, and blind out-of-sample prediction. The analysis therefore provides quantitative observable-level evidence and constraints for the broader Zsa-G continuum-reconstruction program, while maintaining a clear distinction between numerical observable-level consistency and the deeper mathematical reconstruction of the continuum Yang–Mills theory. Scope The present work does not treat finite-β scaling as a proof of continuum existence, does not establish a universal pure 1/N² law, and does not interpret the N = 16 and N = 20 predictions as confirmations. The remaining uncertainties include finite-β coverage, continuum fit-form dependence, incomplete covariance information, finite-volume control, heterogeneity of the published large-N inputs, and functional-form dependence in the large-N extrapolation. Version note This release represents the current numerical continuation of the Zsa-G research program and extends the preceding versions with a systematic observable-level analysis of finite-β scaling, continuum-sensitive extrapolation, multi-observable consistency, large-N structure, leave-one-N-out robustness, and blind out-of-sample predictions at N = 16 and N = 20. The present release supersedes the preceding version as the current research version of this work while preserving the complete Zenodo version history.