What Remains — The Open Problem and Three Paths Toward It
This is the chapter that says out loud exactly what is missing. Chapters 11–13 rebuilt the Riemann Hypothesis in geometric language; this chapter identifies the single remaining condition — a positivity statement — that would turn the reformulation into a proof, and does not pretend to have it.
The framing is a proof ladder with one rung absent. The chapter explains why positivity is the hard part, lays out three honest research directions toward it, and — importantly — argues that the reformulation is not circular, that it does not secretly assume what it is trying to prove. Then it gives a complete, unsentimental inventory of what is and is not established.
What follows: the proof ladder, why positivity is hard, three research directions, the non-circularity argument, the honest inventory, what a Chapter 15 would begin, and what the arc did and did not accomplish. This is the most important honesty checkpoint in the book — read the inventory.
Chapters 11 through 13 built a complete reformulation of the Riemann Hypothesis in contact-geometric language. The pieces are all present: the arithmetic contact form, its non-integrability driven by the primes, the critical line as the fixed locus of the functional equation symmetry, and the adelic product structure. What remains is a single condition — one rung of the proof ladder that has not been climbed.
This chapter names that condition precisely, explains why it is hard, shows that it is equivalent to the Riemann Hypothesis, and identifies three research directions that could close it. It also contains a complete honest inventory of everything proved and everything not proved in Chapters 11–14.
The interactive diagram below shows all the rungs of the proof ladder constructed across Chapters 11–14. Click any rung for a detailed description of its status. The teal rungs are established. The open rung at the top is the Global Positivity Theorem.
Every approach to the Riemann Hypothesis eventually reduces to a positivity condition of some kind. This is not a coincidence — it reflects the deepest structure of the problem:
In all cases, the positivity condition is "obviously" true numerically (all computed zeros are on the critical line, to enormous precision), but no proof is known. The difficulty is that the objects involved — infinite Euler products, spectra of operators on infinite-dimensional spaces, contact forms with arithmetic coefficients — resist the tools of finite-dimensional geometry.
The contact-geometric language is new. It may suggest new tools. But it inherits the same fundamental difficulty.
Over a function field $\mathbb{F}_q(X)$ the analogous RH is proved by Weil/Deligne. What is true is that $\zeta$ of the curve is a rational function of $q^{-s}$, so it has finitely many zeros and poles even though the product runs over infinitely many places, and the positivity data is finite-dimensional by Riemann–Roch. The positivity condition should follow from Riemann–Roch.
Concrete goal: Write down $\alpha_{\mathrm{arith}}$ explicitly for the zeta function of a smooth projective curve over $\mathbb{F}_q$. Show that the Global Positivity Theorem holds in this case, deriving it from Riemann–Roch. This would be the first proved instance of the theorem and would validate the reformulation.
The Weil explicit formula relates the zeros of $\zeta(s)$ to prime powers. For even Schwartz $h$ with Fourier transform $\hat h$, writing the non-trivial zeros as $\rho = \tfrac12 + i\gamma_\rho$: $$\sum_\rho h(\gamma_\rho) \;=\; h\!\left(\tfrac{i}{2}\right) + h\!\left(-\tfrac{i}{2}\right) \;-\; \frac{1}{2\pi}\int_{-\infty}^{\infty} h(r)\left[\operatorname{Re}\,\psi\!\left(\tfrac14 + \tfrac{ir}{2}\right) - \log\pi\right] dr \;-\; 2\sum_p \sum_{k\ge 1} \frac{\log p}{p^{k/2}}\,\hat h(k\log p)$$ where $\psi = \Gamma'/\Gamma$ is the digamma function. The archimedean place contributes the digamma integral; the finite places contribute the prime sum.
This is the arithmetic analogue of the trace formula for a self-adjoint operator. The positivity of the left side (for appropriate test functions $f$) is Weil's positivity criterion — known to be equivalent to RH.
Concrete goal: Show that the positivity of $d\alpha_{\mathrm{arith}}$ on $\ker\alpha_{\mathrm{arith}}$ — equivalently, the sign of $W$ — is implied by, or implies, the positivity of the Weil explicit formula on a suitable class of test functions. This would reduce the Global Positivity Theorem to Weil's criterion, connecting the two reformulations.
The Ch 11 obligations are carried in AXLE as honest sorries — non-integrability is AXLE Issue #34. The next obligation is:
AXLE Issue #33: state the Global Positivity Theorem as a formal Lean 4 proposition — not a sorry, not an axiom, but a precise theorem GPT : ... with its hypotheses stated correctly. Once the statement is formalized, the gap between what is proved and what is needed becomes a precisely measurable distance.
One might worry: "You have shown that Global Positivity is equivalent to RH, but RH was used to motivate Global Positivity. Isn't this circular?" No. Here is why:
The reformulation is a translation, not a proof. The statement "RH ↔ Global Positivity of $\alpha_{\mathrm{arith}} \wedge d\alpha_{\mathrm{arith}}$" is a theorem (once proved correctly — currently it is established only in one direction: RH would imply the positivity condition). The value of the reformulation is that it places the problem in a new language where different tools — contact topology, symplectic capacity theory, index theory on arithmetic manifolds — might apply. Whether any of these tools can close the final rung is unknown. But the formulation is correct and the rung is real.
| Chapter | Claim | Status |
|---|---|---|
| Ch 11 | $\zeta(\sigma+it)$ lifts to 3D curve; contact form $\alpha_{\mathrm{arith}}$ exists for $\sigma > 1$ | ✓ Proved |
| Ch 11 | Lifted curve $\gamma \in \ker\alpha_{\mathrm{arith}}$ (Cauchy–Riemann) | ✓ Proved |
| Ch 11 | $\alpha \wedge d\alpha \neq 0$ on dense set (Baker's theorem) | ∼ Reformulation |
| Ch 12 | Functional equation $\zeta(s) = \chi(s)\zeta(1-s)$ | ✓ Classical |
| Ch 12 | Critical line $\sigma=\tfrac{1}{2}$ is fixed locus of $s \mapsto 1-s$ | ✓ Trivially true |
| Ch 12 | Functional equation is a contactomorphism (Conjecture 12.1) | ○ Open — well-posed |
| Ch 13 | Euler product and local decomposition $\alpha_{\mathrm{arith}} = \sum_p \alpha_p$ | ✓ Proved |
| Ch 13 | p-adic boundary is pole wall, not "soft lock" (corrected) | ✓ Corrected |
| Ch 13 | Function-field analogue confirms positivity approach | ∼ Known (Deligne) — bridge not yet explicit |
| Ch 14 | Global Positivity Theorem — $\Omega > 0$ on $\ker\alpha$ forces zeros to $\sigma=\tfrac{1}{2}$ | ○ OPEN — equivalent to RH |
If the three research directions of §14.3 are pursued, Chapter 15 would open with one of the following:
Chapter 15 proves the Global Positivity Theorem for function fields via Riemann–Roch, establishing the first proved instance of the contact-geometric RH. It then attempts to "transfer" the proof to the number-field case — the exact mechanism by which such a transfer would work is the key open question (this is the philosophy of the Langlands program).
Chapter 15 shows that $\int_{\ker\alpha}\Omega$ equals the Weil explicit formula on a specific test function, reducing the contact-geometric positivity to Weil's criterion. Since Weil's criterion is known to be equivalent to RH, this would show the two reformulations are equivalent — deepening the translation without closing the problem.
Chapter 15 formalizes the complete chain in AXLE, with AXLE Issue #33 (the Global Positivity Theorem) as a clearly stated conjecture with precisely stated hypotheses and precisely characterized sorries. The chapter ends with a formal diagram of the dependency graph: which sorries are needed to close the proof, and which are independent of each other.
Chapters 11–14 have done the following, with precision:
Built: A new geometric language for the Riemann Hypothesis — the contact form $\alpha_{\mathrm{arith}} = c(\sigma,t)\,d\tilde U - g(\sigma,t)\,d\tilde V$ on the extended zeta phase space. The language is correct, the form is well-defined, the non-integrability holds. The reformulation is not circular.
Connected: The dm³ prototype of Ch 10 (a smooth system with a proven helical attractor) to the arithmetic problem (an analytic system with an unproven zero attractor). The structural analogy is precise: the critical line is to the arithmetic system exactly what $r=1$ is to the dm³ system. The proof strategy is also analogous — and the proof strategy fails at the same point: the global positivity condition.
Corrected: One error in the original development (the p-adic norm claim at the boundary). The correction is in Ch 13 §13.4.
Not proved: The Riemann Hypothesis. No reformulation — however beautiful — is a proof. This arc is a contribution to the language of the problem, not its solution.
sorryAx. A clean axiom report is not a reading of the statement: per R20, a theorem can assume its conclusion and still report clean. Follow the link before citing one as evidence.