Forests as Heyting spectra: representation criteria, constructions, and obstructions

Our main result shows that an Esakia space whose order is a downward forest has a locally finite clopen-upset algebra whenever its maximal boundary is scattered. More precisely, failure of local finiteness forces a nonempty perfect principal maximal boundary. We also construct locally finite representations of immediately splitting trees without assuming well-foundedness, and classify the cardinal multiplicities of a uniform family of binary canopies. The countable-forest inverse-sequence characterization and failure of infinitary order axiomatizability are consequences of these results, using established inverse-limit and model-theoretic methods. We give finite-partition certificates as explicit forms of the existing duality and compactification machinery. The unrestricted intrinsic structural classification of forest spectra remains unresolved here.

GitHub repository: https://github.com/eulogica/heyting-forest-spectra