In this work we revisit the axial third-nearest-neighbor Ising chain and examine in detail some aspects of its phase behavior ensuing from competing interactions and resulting frustration. We probe the phase behavior with two complimentary tools: a microscopic two-point correlation function $\mathcal{C}\left(n\right)$, which we carefully construct after appropriate spin transformations, and a macroscopic, thermodynamic curvature $R$ which we obtain from the free entropy. We report observations of phenomena such as hyperscaling breakdown and intermediate-ranged modulations among others. The zero-field thermodynamic curvature $R_0$ is shown to systematically subdivide the ground-state phases into regions of attractive or repulsive effective interactions of varying strength. Furthermore, $R_0$ brings forth the significance of some third-order moments in describing the effects of frustration, including the multiphase lines. Combined use of both the tools in the short-ranged modulated order regime confirms and further clarifies the discussion in [S. Khatua et al., Phys. Rev. E 111, L022105 (2025)] regarding the appropriate measure of high temperature correlation length in this regime.

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• Critical phenomena
• Entropy
• Phase diagrams
• 1-dimensional spin chains
• Antiferromagnets
• Ferromagnets
• Geometry