On Underdetermination in Self-Referential Portal Puzzles: A Logical Case Study in Minimal Constraint Resolution

Self-referential logical systems can be complete yet underdetermined, sustaining multiple internally consistent solutions when constraints don’t fix uniqueness. This note analyzes a seven-portal puzzle whose inscriptions speak about which portals are safe, under the global rule that exactly three inscriptions are true and exactly three portals lead to the center. A brute-force formulation with classical logic yields six distinct models satisfying both sums, with no invariant safe portal. I argue that injecting a single meta-constraint—formally, an XOR between two dependent inscriptions—collapses the solution space to a unique, stable configuration (Portal 4). I call this the Minimal-Information Closure Principle (MICP): one external bit of information can stabilize a self-referential universe, echoing Gödel/Tarski insights and aligning with Shannon-style information closure. The result clarifies how meta-level intervention converts a Gödelian micro-universe from a multiverse of admissible states into a determinate world, offering a compact case study in logical underdetermination and minimal constraint resolution.