The Riemann Hypothesis as Non-Integrability of an Arithmetic Contact Structure on the Adele Class Space

The Riemann Hypothesis as Non-Integrability of an Arithmetic Contact Structure on the Adele Class SpaceVersion 1 — 2026 Pablo Nogueira Grossi · G6 LLC, Newark NJ · ORCID: 0009-0000-6496-2186 V1 (this version): https://doi.org/10.5281/zenodo.22179684Concept DOI (resolves to latest): https://doi.org/10.5281/zenodo.22179683Repositories: https://github.com/TOTOGT/GTCT (Lean, this deposit) ·https://github.com/TOTOGT/geometry (Book 3 of the Principia Orthogona series / intro chapters from Vol 3)Chapters this paper formalises: Book 4 Ch 11 and Ch 12, public since 2026-06-09 —https://totogt.github.io/geometry/book4/ch12.html What this paper does It reformulates RH as a non-vanishing condition on an arithmetic contact 3-form on the adeleclass space. The zeta function is lifted from the plane to an extended phase space by promotingthe height t from a parameter to a coordinate; the resulting 1-form αₐrithcarries the von Mangoldt–Dirichlet series as its coefficient; the non-integrability conditionα dα ≠ 0, automatic in the smooth ODE setting, becomes a global positivitystatement on the idele class group action. It is a translation dictionary, not a proof. §4.6 states, claim by claim, what is proved,what is machine-checked, what is classical, and what is numerical only. What is new here, and what is not **New.** Two reflection laws for the form's coefficients under the functional equation: g(σ,t) − g(1−σ,t) = Im[(χ′/χ)(σ+it)] difference law c(σ,t) + c(1−σ,t) = −Re[(χ′/χ)(σ+it)] sum law verified numerically to 30 significant digits. That the simple pole of ζ′/ζ at each zero falls**entirely into one component** of the form while the other stays analytic (Proposition 4.3).And a **refutation** of the conjecture, stated in Book 4 §12.2, that the functional equationacts as a contactomorphism of αₐrith — it does not, off the critical line, andthe obstruction is an explicit gamma-factor term carrying no von Mangoldt data (Proposition 4.6). **Not new, and credited as such in the text.** The corollary c(½,t) = ϑ'(t), withϑ the Riemann–Siegel theta function, is equivalent to the standard fact thatZ(t) = eiϑ(t)ζ(½+it) is real-valued. Its role here is only to identify thed U coefficient on the critical wall with the density of the Riemann–von Mangoldtcounting formula N(T) = ϑ(T)/π + 1 + S(T). ### What is machine-checked, exactly One lemma. `lseries_vonMangoldt_eq_neg_Zlog` in `ZetaReflection.lean` — that∑ Λ(n) n⁻ˢ = -ζ'/ζ(s) for s > 1, hence that c and g are the realand imaginary parts of one meromorphic object — proved against **Mathlib v4.32.0**, resting on`[propext, Classical.choice, Quot.sound]` and nothing further. **Two statements are admitted, and carry `sorryAx`:** `reflection_law` and`chiLog_real_on_critical_line`. They are in the file as statements of record, numericallyconfirmed and not proved. The file's header says so. Running`leancheck.sh --audit ZetaReflection.lean` reports 4 declarations, 2 trusting `sorryAx`. Nothing in this deposit bears on RH itself. ### On the figure Figure 1 shows the lift: the same three trajectories without and with the third coordinate. Theprinted version fixes one viewpoint and loses the rotation, which is most of what makes theseparation legible. The interactive version is Figure 12.1 of Book 4 Chapter 12, linked above,and readers who want the geometry rather than a picture of it should start there. ### Files in this deposit | File | What it is ||---|---|| `rh_arithmetic_contact_v1.pdf` | The paper, 13 pages || `RH_arithmetic_contact_structure.md` | Source of the above || `ZetaReflection.lean` | Lean 4 / Mathlib v4.32.0 — one lemma proved, two admitted || `verify_reflection_laws.py` | Reproduces all four numerical claims; exits non-zero on failure || `figures.py` | Regenerates Figure 1 || `fig1_rh_lift.pdf`, `.png` | Figure 1 | ### Build instructions **Numerics** — reproduces every figure quoted in §4.4–§4.6: pip install mpmath python3 verify_reflection_laws.py # ~2 min; exit 0 = all checks passed **Figure:** pip install mpmath matplotlib python3 figures.py **Paper:** pandoc RH_arithmetic_contact_structure.md -o rh_arithmetic_contact_v1.pdf \ --pdf-engine=pdflatex -V geometry:"margin=1in" -V fontsize=11pt \ -V colorlinks=true --toc --toc-depth=2 **Lean** — from a checkout with a complete Mathlib v4.32.0 build: lake env lean ZetaReflection.lean ### Open obligations 1. Prove `reflection_law` rather than admit it. Mathlib has `Complex.digamma` and `completedRiemannZeta_one_sub`; no contact geometry is required.2. Prove `chiLog_real_on_critical_line` — conjugate-symmetry of digamma.3. Determine the transformation of U and V themselves, without which the graded form of Proposition 4.6 cannot be made precise.4. The corresponding law for the d U component off the critical line.5. Global positivity (§6). Open, and untouched by this deposit. Version history V1 (this deposit, 30 August 2026). First deposit. The manuscript was written 10 June 2026 (deposited to github, dated)and revised 30 August 2026 to add §4.4–§4.6, Figure 1, and the status-of-claims table. --- MSC codes: 11M26 · 53D10 · 11R56 · 81Q10 **Keywords:** Riemann Hypothesis · contact geometry · adele class space · von Mangoldt function ·Riemann–Siegel theta function · Weil explicit formula · Connes spectral triple · p-adicdifferential forms · Lean 4 · formal verification · Principia Orthogona **License:** CC BY-NC-ND 4.0 (paper) · MIT (code)