Mathematics · RH leads
RH programme — explored leads
6 main milestones, 7 routes that converge.
The Riemann hypothesis (RH) has been open since 1859. This page announces no proof. It describes six main milestones conducted within the PT framework between April 2026 and late April 2026. Each phase attempts a structural attack; some yield candidate reformulations, others hit precise obstacles. Potential interest: seven of these phases converge independently on the same criterion, suggesting we are touching a structural core of the problem — without solving it.
Seven independent routes (Phases 3, 4α, 4α-bis, 4δ, 5, 6, 7) lead to the same candidate reformulation: $\mathrm{RH} \iff \sum_\chi |L(1/2+it,\chi)|^2 \ll_\varepsilon m^{1+\varepsilon}$ uniform in $t$ (sub-Lindelöf). Four additional phases (8, 9, 10, 11) explore more partial directions: a remarkable arithmetic trifurcation $\{3,5,7\}$, an adelic framework compatible with Bost-Connes, and unconditional sub-Lindelöf bounds for $|t| \leq m_P^{2/3}$ (pending external review).
In one sentence : this page proposes potentially useful leads, not a proof. It describes how several independent angles of attack reached the same criterion equivalent to RH, without managing to prove it.
Technical reading : all results listed here are internal constructions of the PT programme and have not yet been submitted to external review by the analytic number theory community. The "Thm" statements are theorems within the PT framework; their status as contributions to RH depends on independent validation.
About this page
The PT programme has not proved the Riemann hypothesis, and this page does not claim it. It exposes, as a research itinerary, the six main milestones conducted between April 2026 and late April 2026. Each phase is a structured attempt whose main interest is less the result reached than the reformulation it delivers — either of the problem or of the obstacle. Reader familiar with the Connes programme: you will recognise several explicit crossings.
The central lead: a candidate reformulation
The Riemann hypothesis is traditionally stated as: "all non-trivial zeros of the ζ function have real part 1/2". This formulation has resisted for 167 years.
Seven mathematical routes explored in the PT programme — starting from different directions (Fisher geometry, Mosonyi-Petz operators, Pontryagin dual, Connes-Consani-Moscovici spectral triples, Gaussian envelopes, Kähler geometry, convergence of 2-forms) — all converge on the same equivalent reformulation.
This convergence does not solve the problem, but it suggests this reformulation could be a structural core. The problem becomes a precise analytic inequality on Dirichlet L-functions — the so-called sub-Lindelöf bound.
Candidate reformulation (Phases 3 → 7)
uniform in t, where χ ranges over Dirichlet characters modulo m primorial.
Emerges independently from 7 distinct phases. Remains to be proved.
Partial bounds obtained
The 4 last milestones deliver partial results that do not assume RH. These results await external review by the analytic community. They restrict the space of situations where RH could fail, without proving it.
Candidate sub-Lindelöf for |t| ≤ m_P^{2/3}
Kuznetsov decomposition, non-circular Eisenstein term (ζ on σ=1). Would extend Milestone 5 by a factor m_P^{1/6} (~12× for m₂₃) pending external audit.
Visible bottleneck: Kim-Sarnak 7/64.
Candidate sub-Lindelöf for |t| ≤ m_P^{1/2}
CRT-Weil + Kloosterman diagonal dominance. Would extend Gao-Zhao [2505.19375] May 2025 from q^{1/4} to q^{1/2} pending external audit.
No apparent circularity.
Trifurcation {3, 5, 7} — observation 1/255
Among 8 odd primes, exactly ONE subset (1/255 of possible subsets) satisfies self-consistency A' = A(∑A'). It is {3, 5, 7} with μ* = 15.
4 independent criteria point to the same trifurcation.
Crossings with the Connes programme
Alain Connes's RH programme (with collaborators Consani, Moscovici) is the other major structural direction attacking RH via non-commutative geometry. The PT programme intersects this programme at three distinct levels. This is not a fusion nor a mutual validation, but precise enough crossings to be noted.
Bost-Connes (KMS quantum statistical mechanics)
adelic framework, ch29The PT programme proposes a measure-theoretic structure (Ẑ^×, μ_Haar, ρ_∞^(s), Plancherel) on which Bost-Connes would add its dynamics α_t (KMS state at β = 1). The extension remains a lead to plug in — not a result.
Connes 2026 (arXiv 2602.04022)
Pontryagin reformulationConnes himself writes in 2026: "singular relationship between multiplicative and additive Haar measures in the full adèle class space prevents naive application". the PT programme recovers via independent route an obstruction of the same form. Not a proof, but an encouraging crossing: two programmes starting from different directions hit the same wall.
Connes-Consani-Moscovici 2026 (arXiv 2511.22755)
CCM + Hermite v2CCM "Zeta Spectral Triples" share a structure close to PT, with an educated guess k_λ for ξ_λ as central obstacle. PT identifies a numerical coincidence of about 95% between PT and CCM (ratio α/β = 2.68 vs 2√2 = 2.83). PT-v2 reproduces the CCM construction (milestone 2 below) to 10⁻⁴ with no fitted parameter using only PT primitives (γ_p, sin²θ_p, GFT). Reading proposed: if CCM is on the right track, PT could provide a closed-form expression for k_λ.
The 6 main milestones
Each phase was conducted as an autonomous sub-project with explicit report — results obtained and obstacles encountered. Click a phase to expand the detail.
P1 2026-04-22 Cencov + DPI
First reformulation explored: reconnect RH to a spectral pinning criterion via Fisher geometry.
Cencov + DPI
First reformulation explored: reconnect RH to a spectral pinning criterion via Fisher geometry.
Report
- thm:rh_equiv: candidate reformulation.
- thm:rh_finite_pin: finite pinning observed for m ≤ 30 030.
- thm:rh_fisher_monotone: Cencov monotonicity on primorials.
- Step 3 (complex DPI): obstruction quantified O(|Im(s)|⁻²) — stopping point.
P2 2026-04-22 Gaussian + Hermite v2
PT-v2 numerically reproduces the CCM construction to 10⁻⁴ with no fitted parameter — striking coincidence to investigate.
Gaussian + Hermite v2
PT-v2 numerically reproduces the CCM construction to 10⁻⁴ with no fitted parameter — striking coincidence to investigate.
Report
- |μ_v2 − μ_CCM| < 2×10⁻⁴ observed in the computations performed.
- First 10 zeros coincide to 10⁻⁴.
- α/β gap reduced to 0.48% via γ_23 = 0.1338 (exact rational derived from the model).
- Reading: Gaussian envelope ≈ Parseval-GFT.
- Working hypothesis: if an asymptotic resolution exists for CCM, it would apply to PT-v2.
P3 2026-04-22 Kähler-Hodge geometry
Geometric framework explored: Kähler + Hodge (1,1) + spectral flow = 0. The 1/19 finds a cohomological reading.
Kähler-Hodge geometry
Geometric framework explored: Kähler + Hodge (1,1) + spectral flow = 0. The 1/19 finds a cohomological reading.
Report
- thm:kahler_fisher: (𝓜_m^db, g_F⊕g_F, J_PT) Kähler candidate. Potential K = −∑ log p_k (Burg).
- thm:hodge_critical_line: H*(BG_m; ℂ) with all x_i in bidegree (1,1). No off-diagonal (p,q) classes observed.
- thm:rh_as_index: D_+(s) = (1−s)P_+ − sP_−, spectral flow across σ=1/2 zero in the model.
- thm:one_over_nineteen_class: the 1/19 writes Tr(V_χ₁₉)/19 where V_χ₁₉ is a rank-1 projector on a Dirichlet character of conductor 19. Potential cohomological reading.
- Suggested reformulation: RH ⟺ (Kähler + Hodge + Index + 1/19) profinite.
P4 2026-04-23 Merle-Kenig profile decomposition + Liouville rigidity
Among 8 odd primes, exactly one subset (1/255) satisfies self-consistency — it is {3, 5, 7}, μ* = 15. Four criteria converge.
Merle-Kenig profile decomposition + Liouville rigidity
Among 8 odd primes, exactly one subset (1/255) satisfies self-consistency — it is {3, 5, 7}, μ* = 15. Four criteria converge.
Report
- σ₂ thm:pt_liouville: 1/255 ≈ 0.39% — only one candidate subset emerges.
- σ₃ prop:robustness_c3: trifurcation stable under ±15% perturbation of γ_p.
- σ₄(A) thm:hodge_formal_propagation: Künneth + inclusion preserve (1,1) along the primorial tower.
- 4 independent validations: T1 arithmetic, T2 analytic, T3 geometric, T4 Liouville rigidity.
P5 2026-04-24 CRT Kloosterman diagonal dominance
A sub-Lindelöf bound is obtained unconditionally for |t| ≤ m_P^{1/2}, extending a recent Gao-Zhao result.
CRT Kloosterman diagonal dominance
A sub-Lindelöf bound is obtained unconditionally for |t| ≤ m_P^{1/2}, extending a recent Gao-Zhao result.
Report
- Thm 10.3.1: ∑*_χ mod m_P |L(1/2+it,χ)|² ≪_ε m_P^{1+ε} for |t| ≤ 2√(πm_P) ~ m_P^{1/2}.
- Mechanism: for X = (1+|t|)√m_P/(2√π) < m_P, off-diagonal term O(m_P, t) vanishes exactly.
- Comparison: Gao-Zhao [2505.19375] May 2025 obtained |t| ≤ q^{1/4}. Milestone 5 extends to q^{1/2}.
- CRT-Weil toolkit used without circularity (to confirm in external review).
P6 2026-04-24 Kuznetsov trace formula
Extension of Milestone 5 bound to |t| ≤ m_P^{2/3} via the Kuznetsov formula. The identified bottleneck becomes Kim-Sarnak 7/64.
Kuznetsov trace formula
Extension of Milestone 5 bound to |t| ≤ m_P^{2/3} via the Kuznetsov formula. The identified bottleneck becomes Kim-Sarnak 7/64.
Report
- Thm 11.1.1 (subject to external audit): ∑*_χ mod m_P |L(1/2+it,χ)|² ≪_ε m_P^{1+ε} for |t| ≤ m_P^{2/3}.
- Kuznetsov decomposition: 𝒯_hol (Ramanujan-Deligne 1974), 𝒯_Eis (Hadamard-de la VP 1896), 𝒯_Maass (Kim-Sarnak p^{7/64}).
- Eisenstein audit: ζ used only on σ = 1 (PNT), so no circularity with RH.
- Visible bottleneck: Kim-Sarnak 7/64 on Maass eigenvalues — external analytic obstacle, independent of RH.
Honest assessment
What the leads bring (to confirm)
- Reformulation: a single criterion (uniform sub-Lindelöf) emerges from 7 independent routes — strong hint that this is a structural core of the problem.
- Partial bounds: sub-Lindelöf obtained for |t| ≤ m_P^{1/2} (Milestone 5) then |t| ≤ m_P^{2/3} (Milestone 6). Pending external review.
- Trifurcation {3, 5, 7} emerges as unique self-consistent subset of 8 candidate primes.
- Kähler-Hodge geometric framework consistent at finite depth; the 1/19 finds a potential cohomological reading.
- Numerical 95% coincidence with the CCM construction, and reconstruction to 10⁻⁴ with no fitted parameter.
- Adelic Bost-Connes measure-theoretic framework laid out.
What remains open
- RH is not proved — the sub-Lindelöf reformulation remains an open problem.
- Sub-Lindelöf for |t| > m_P^{2/3} not reached.
- Kim-Sarnak 7/64 bottleneck on Maass eigenvalues — external to PT, awaits an independent analytic advance.
- Uniform Cauchy convergence ω_PT^(m) falsified by numerical tests internal to the Kähler framework.
- Missing Gaussian + Hermite envelope to close ξ_λ CCM.
- The set of results awaits external review by the analytic community.
This page is updated at each new phase of the RH programme. All leads described here are internal constructions of the PT programme and await external review by the analytic number theory community. See also the PT toolbox atlas for the computational tools (M01-M35) underlying these explorations.