PARABOLIC NONCOMMUTATIVE GEOMETRY

The paper’s Theorem 4.1 establishes that for any strictly tangled spectral triple (ST2) with bounding matrix ε, the operator Dt = Σj Dtj j yields a higher order spectral triple (HOST) for t ∈ Ω(ε) ∩ (0,1]I (and for all t ∈ Ω(ε) if the ST2 is regular), using a key interpolation Lemma 4.2 and standard functional calculus facts (including Lemmas 3.7–3.8) . The candidate model solution proves the same statement by an alternative route: resolvent integral representations for fractional powers/sign(Di), an ideal-closure argument to propagate local compactness to all t, and explicit commutator estimates leading to [Dt,a](1+|Dt|ε′)−1 bounded for any ε′ with supij εij ti/tj < ε′ < 1. Apart from a minor phrasing in the domain invariance step (easily corrected), the model proof aligns with the paper’s result and assumptions. The paper additionally quantifies an order bound (4.1), which the model does not compute explicitly; otherwise the conclusions coincide.