Extensions to "A discrete spectral triple of metric dimension 2 from prime numbers": (I) Analytic properties of the shifted prime zeta function (II) KMS states and comparison with Bost–Connes

This work develops two independent extensions of the spectral triple constructed in the author’s companion paper (Part II). Part I – Shifted prime zeta function. We study the Dirichlet series Q (s) =∑p∈P (p+1) −sQ (s) =∑p∈P (p+1) −s (shifted prime zeta function). We prove that Q (s) Q (s) and the ordinary prime zeta function P (s) =∑pp−sP (s) =∑pp−s share the same singularity structure for ℜs>0ℜs>0; their difference R (s) =P (s) −Q (s) R (s) =P (s) −Q (s) is holomorphic on ℜs>0ℜs>0 and is expressed via an integral representation. A binomial expansion gives Q (s) =∑k≥0 (−sk) P (s+k) Q (s) =∑k≥0 (k−s) P (s+k). The spectral zeta function ζD~ (s) =Tr⁡ (∣D~∣−s) ζD~ (s) =Tr (∣D~∣−s) of the adelic Dirac operator D~D~ factorises as ζ (s/2) Q (s) ζ (s/2) Q (s). Its dominant singularity is a simple pole at s=2s=2 with residue 2Q (2) =2∑p (p+1) −22Q (2) =2∑p (p+1) −2, confirming the Dixmier trace computation of the companion paper. Part II – KMS states and comparison with Bost–Connes. For the adelic spectral triple, we consider the time evolution σt (a) =eitD~ae−itD~σt (a) =eitD~ae−itD~. We show that the Gibbs state ϕβ (a) =Tr⁡ (e−βD~a) /Z (β) ϕβ (a) =Tr (e−βD~a) /Z (β) exists and is unique for every inverse temperature β>0β>0 (the partition function Z (β) =Tr⁡ (e−βD~) Z (β) =Tr (e−βD~) is finite for all β>0β>0). The critical inverse temperature βc=2βc=2 is identified via the spectral partition function Tr⁡ (∣D~∣−β) Tr (∣D~∣−β), which converges for β>2β>2 and diverges for β≤2β≤2. This marks a spectral-geometric singularity (∣D~∣−2∈L1, ∞∖S1∣D~∣−2∈L1, ∞∖S1) rather than a thermodynamic phase transition. We compare these features with the Bost–Connes system, highlighting the differences in Hilbert space, eigenvalue spectrum, zeta function, critical temperature, and symmetry structure. A partial symmetry group Θ=∏pZ/2ZΘ=∏pZ/2Z (involutions θpθp acting by sign changes) is shown to commute with D~D~ and σtσt. We propose a larger profinite candidate GP+1⊊Z^∗GP+1⊊Z^∗ (defined by invariance of the set of shifted primes modulo nn) as the possible full symmetry group, leaving its explicit determination as an open problem. The index Ind⁡ (u) =∣P∣Ind (u) =∣P∣ from the companion paper is reinterpreted as a winding number of the symbol of uu in the Toeplitz extension. These results connect noncommutative geometry, analytic number theory (prime zeta functions), operator algebras (KMS states), and the Bost–Connes system.