Prime States and Spectral Transitions: A Configuration Space Temporality Approach to the Riemann Hypothesis via Multi-Force Hamiltonians on ℓ²(ℙ)

Prime States and Spectral Transitions: A Configuration Space Temporality Approach to the Riemann Hypothesis via Multi-Force Hamiltonians on ℓ²(ℙ) Frédéric David Blum · Catalyst AI Research · Tel Aviv, IsraelORCID: 0009-0009-2487-2974 · March 2026 — v2 Summary We construct an explicit Hamiltonian H_CST on ℓ²(ℙ) — the Hilbert space over prime numbers — encoding three simultaneous coupling forces between prime states: additive gap decay (F₁), multiplicative log-ratio coupling (F₂), and spectral resonance from the zeta oscillation (F₃). This operator implements the three-rope principle from Configuration Space Temporality (CST): a state is stable only when multiple independent forces balance simultaneously, mirroring destructive interference in the Riemann zeta function. Core Thesis In the CST framework, time is not a fundamental axis — reality consists of discrete, irreducible states. We propose that: Prime numbers index the stable states of a CST system (irreducible elements of ℕ) Zeta zeros are the spectral modes of state-transition dynamics (via the explicit formula of Riemann–von Mangoldt) The critical line Re(s) = 1/2 is the locus of multi-force equilibrium RH is a regularity condition on the density of accessible states Numerical Results Spectral statistics of H_CST (500 primes) compared to the first 500 non-trivial zeros of ζ: Model KS → GUE NNS Overlap with ζ Σ² distance to ζ Graph Laplacian (baseline) 0.373 0.508 4.09 Chebyshev Hamiltonian 0.306 0.527 2.24 CST three-force (optimal) 0.226 0.616 1.26 ζ zeros (target) 0.050 — — Key findings: 39% improvement in KS distance to GUE over the graph Laplacian 62% NNS overlap with the Riemann zeta zeros 3.2× reduction in number variance distance to the ζ spectrum Optimal coupling: weak additive (β₁ ~ 0.1), moderate multiplicative (β₂ ~ 0.5) — confirming the primacy of Euler product structure Complementary test: Witten Laplacian H_W = −Δ + α²/r² on the Sinai billiard shifts spectral statistics toward ζ-like structure (4.9% Σ² improvement) NEW in v2 — Independent Convergence (Section 5.4) Two recent independent results, discovered after submission of v1, arrive at strikingly similar structural conclusions from entirely different starting points: Hartnoll & Yang (Cambridge, JHEP 2025) show that near a spacetime singularity, the BKL dynamics maps onto a particle in the modular domain. The semiclassical quantization yields wavefunctions proportional to L-functions along the critical axis — vanishing at the non-trivial zeros — and a partition function equal to a primon gas of oscillators labeled by primes. Perlmutter (2025) attaches a degree-4 L-function to every 2D CFT partition function and shows that convergence of the Dirichlet series implies random matrix (GUE) behavior of level spacings — the same universality class we test numerically. The key distinction: these works observe that primes emerge as natural labels in gravitational and conformal physics. CST explains why — in a framework where time is emergent and reality consists of irreducible states, primes are not a convenient label but the unique canonical choice forced by the multiplicative structure of ℕ. The analogy is Kepler vs Newton: Kepler observed elliptical orbits; Newton explained why they must be ellipses. This triangle of independent convergence — gravitational singularities (Hartnoll-Yang), conformal field theory (Perlmutter), and emergent time (this work) — strongly suggests that the role of primes in fundamental physics is structural, not metaphorical. Theoretical Framework Section 2.1: CST prime states — primes as irreducible state indices, PNT as deceleration, epistemic limitation on state identification Section 2.2: Three-rope principle — Penning trap, ALPHA/CERN, Dirac quantization as physical precedents Section 2.3: ζ as state-space function — additive/multiplicative duality, explicit formula as spectral decomposition Section 2.4: Critical line as equilibrium — functional equation symmetry as force balance Section 3: Construction of H_CST on ℓ²(ℙ) with three-force coupling; connection to Witten Laplacian Section 5.2: Prime gaps reinterpreted as non-uniform time — PNT, RH, twin primes, Cramér's conjecture become physical statements Section 5.3: Falsifiable experimental predictions in multi-force confinement systems Section 5.4 (NEW): Independent convergence with Hartnoll-Yang and Perlmutter — from observation to explanation Experimental Predictions The framework predicts that in physical systems with three or more simultaneously coupled forces (Penning traps, SQUID magneto-gravitational devices, cold-atom optical lattices), the spacing distribution of discrete stable configurations should exhibit GUE-compatible statistics with a prime-gap signature. Relation to Other Work This paper connects to three existing programmes by the author: CST (Zenodo DOI: 10.5281/zenodo.18859602) — the parent framework; H_CST is a discrete analogue of the Witten Laplacian H_W FDBC/ESG (submitted to CMP, CIMP-D-26-00460) — operator-theoretic approach to RH via the BDC equation and Li coefficients SDSQ (Zenodo DOI: 10.5281/zenodo.18779189; GitHub: github.com/davidangularme/sdsq) — Schrödinger–Marchenko reformulation of RH as spectral condition And now to two independent programmes: Hartnoll & Yang (JHEP 2025, arXiv: 2502.02661) — conformal primon gas from BKL gravity Perlmutter (arXiv: 2509.21672) — L-function framework for 2D CFT with GUE universality Together, these six programmes form a convergent network attacking the prime-physics interface from complementary directions. What This Paper Does NOT Claim It does not claim to prove RH It does not claim the CST prime Hamiltonian is the Hilbert–Pólya operator It does provide the first numerical evidence that a physics-motivated multi-force Hamiltonian on ℓ²(ℙ) quantitatively approaches the spectral statistics of ζ zeros It does identify the direction for closing the remaining spectral gap It does (v2) show that this direction is independently corroborated by peer-reviewed results in gravitational and conformal physics Version History v2 (March 2026): Added Section 5.4 on independent convergence with Hartnoll-Yang (JHEP 2025) and Perlmutter (2025). Updated conclusion and references [12,13]. v1 (March 14, 2026): Initial preprint. Data and Reproducibility All computations performed in Python (NumPy, SciPy, mpmath). Source code available from the author upon request. Keywords: Riemann Hypothesis · Configuration Space Temporality · Witten Laplacian · prime numbers · spectral statistics · GUE · Hilbert–Pólya conjecture · multi-force coupling · prime gaps · number variance · Sinai billiard · Penning trap · Dirac quantization · primon gas · BKL dynamics · conformal field theory · L-functions · independent convergence Contact: freddavidblum@catalystais.com · catalystais.com