A discrete spectral triple of metric dimension 2 from prime numbers: Dixmier trace, Wodzicki residue, and index via Toeplitz extension

We construct an explicit unbounded spectral triple (A,H,D~)(A,H,D~) where H=ℓ2(P×N)H=ℓ2(P×N) with PP the set of prime numbers. The Dirac operator is D~ep,k=(p+1)k1/2ep,kD~ep,k=(p+1)k1/2ep,k, obtained by reduction from an adelic quantum field model using the Kozyrev wavelet basis (the shift p↦p+1p↦p+1 follows from the geometry of the Bruhat–Tits tree). A toy model with pp instead of p+1p+1 is also studied. We prove that the triple is regular in the sense of Connes–Moscovici and has spectral dimension 22 via the Weyl law N(Λ)≍Λ2N(Λ)≍Λ2. The operator ∣D~∣−2∣D~∣−2 belongs to the Dixmier ideal L1,∞L1,∞ and is measurable; its Dixmier trace equals the residue of the spectral zeta function at s=2s=2, giving Tr⁡ω(∣D~∣−2)=2∑p∈P(p+1)−2.Trω(∣D~∣−2)=2p∈P∑(p+1)−2. The toy model yields 2∑pp−22∑pp−2, and we show that the two operators are inequivalent modulo L01,∞L01,∞ (the difference has a non‑vanishing logarithmic Cesàro mean). For the index of a truncated partial isometry u=eSe+(1−e)u=eSe+(1−e) associated with a finite set of primes PP and a cutoff KK, we observe that D>0D>0 forces the standard odd‑pairing formula to give zero. The correct index is obtained from the Toeplitz extension generated by the shift SS; the K‑theory boundary map yields Ind⁡(u)=∣P∣Ind(u)=∣P∣. This integer also appears as a finite boundary term in the Connes–Chamseddine spectral action. The construction provides the first example of a regular spectral triple of metric dimension 22 based on prime numbers, with explicit invariants linking noncommutative geometry, operator ideals and arithmetic.