Prime Numbers as Elementary Modes
The prime numbers 2, 3, 5, 7, 11, … are the multiplicative atoms of the integers: every positive integer factors uniquely into primes. The Euler product formula expresses this as an identity between a sum over integers and a product over primes:
Within the GPP framework, primes are the elementary graviton modes: a prime p corresponds to an indecomposable loop on the multiplicative group (ℝ⁺, ×), composite numbers to multi-graviton Fock states. The Riemann Hypothesis — that all non-trivial zeros of ζ(s) have Re(s) = ½ — is then T-invariance of all graviton resonances.
Primes and the Bost–Connes System
The Bost–Connes Hamiltonian HBC acts on the Hilbert space of the quantum statistical mechanical system whose partition function is ζ(s). Its spectrum contains log p for each prime p, mirroring the Euler product. The conjecture — not yet proved within the framework — is:
This would make the KMS states of the Bost–Connes system the thermal states of the celestial CFT at inverse temperature β = Re(s), and the phase transition at β = 1 the crossing of the principal series.
Tate's Thesis and Haar Measure
Tate's 1950 doctoral thesis reformulated the Riemann zeta function as a Fourier transform on the adèle ring 𝔸 = ℝ × ∏p ℚp. The key object is the Haar measure on the idèle class group 𝔸×/ℚ×.
The completed zeta function
satisfies ξ(s) = ξ(1−s). This functional equation is the Fourier-analytic consequence of Haar self-duality dμ(x−1) = dμ(x) on (ℝ+, ×). Under Δ = 2s this becomes the shadow symmetry Δ ⇔ 2−Δ of the celestial CFT. The Riemann Hypothesis is the statement that the fixed-point set of this involution — Re(s) = ½ — contains all zeros.
Five Routes Under Study
| Route | Key Mechanism | Status |
|---|---|---|
| Geometric (Haar) | Self-dual measure + Peter–Weyl compactness | Argued in manuscript; central L² step open in the formalization |
| Spectral (Meyer) | Spectral realisation of the zeros (Meyer, Duke Math. J. 2005) | Literature result used as a tool; positivity not supplied |
| Probabilistic (BPY) | Brownian bridge + Kuiper statistic (Biane–Pitman–Yor, Bull. AMS 2001) | Literature result used as a tool |
| Physical (celestial) | BMS₄ unitarity + principal series | Conjectural dictionary; the symmetry is verified, the positivity is open |
| Completeness (Yakaboylu) | Beurling–Malliavin + Weil–Bombieri | Hilbert–Pólya-type realisation attempt; not verified here |
Routes that fail are kept as theorems about why they fail. See the Riemann Hypothesis status page for what is verified and what is open.
The geometric gap — positivity of V̂R,ε only on its form domain — is honestly assessed. Test vectors leave the form domain for small ε. The programme is an advance, not a completed CMI proof.
The Explicit Formula
The Riemann explicit formula expresses the prime-counting function π(x) in terms of the zeros ρ = ½ + iγ of ζ(s):
In the GPP framework this formula is the spectral decomposition of the prime-counting function in terms of the graviton resonances (zeros) of the celestial CFT. The primes are peaks in the spectral density; the zeros are the resonant frequencies at which the density concentrates.