The Φ-Process: Operator–Algebraic Embeddings of Possibilities, Transfinite Stabilization, and a Quantitative Application to Sensory Depletion

Part (a) of the paper’s Theorem 5.3—monotone inequality X^circ ⪯ X^intact—follows by a standard induction and logical contraction, and is correct. However, the strictness claim in (b) is false under the stated hypotheses (positive E injective on nonnull supports; B monotone, 1-Lipschitz, logically contractive) even when μ(F)>0 and the inputs hit F infinitely often. A simple linear example with B(x)=αx∈(0,1), E=Id, F=S, and s_n=2^{-n} yields X^intact=X^circ=0, contradicting the theorem’s strict inequality claim. This directly refutes the proof’s step that a positive per-step gap “persists in the limit under event-indexed contraction” without a uniform lower bound or strict incremental responsiveness. The paper itself later introduces additional conditions (periodic/dense events and incremental lower bounds for B) to secure quantified positive gaps, aligning with the model’s proposed fixes, but these are not assumed in Theorem 5.3’s statement. Therefore, (b) as stated is incorrect, while the model’s diagnosis and counterexamples are valid. See Theorem 5.3 and its proof, and the later gap-quantification results Lemma 5.6/Proposition 5.7 for the necessary strengthenings . Logical contraction (Axiom 3.1/Theorem 3.2) ensures convergence but not strict positivity of the limit gap absent those extra hypotheses .