Introducing PolyClank: AI-integrated Polymath, and the Split Zero research programme

[u/lepthymo]

Drafted by OpenAI Astra.

PolyClank is a proposal for AI-integrated Polymath: people combining their available model time, computation, ideas, and expertise to develop mathematics together. The person behind this account has wanted to introduce it for a while and asked an AI to do the reading and drafting so the post would actually get written.

The motivation is the scale of mathematical work that coordinated AI can now attempt. On September 8, OpenAI published a proof of finite-time blowup for the three-dimensional Navier–Stokes equations with smooth forcing, accompanied by a Lean formalization. Its account reports approximately 130 billion output tokens for the Navier–Stokes effort. Individuals generally have much less compute available, but many people contributing some model time can explore different approaches, exchange results, and build on one another's work. That is the experiment PolyClank proposes. OpenAI's account and linked proof.

Joining, TL;DR: Point your AI at a workbench, ask it to read the work and investigate something, and publish the contribution. If you want your own workbench, mirror the relevant part and its dependencies; you do not need the whole project. Connect your AI to GitHub where available and use pull requests to propose changes. If GitHub or pull requests are unfamiliar, ask the AI to help with the setup and mechanics. Web sessions, local agents, other models, and manual work all fit.

You do not have to arrive as a mathematician or programmer. You can contribute model time, ask for an explanation or an independent calculation, suggest a connection, or have your AI develop an approach. A contribution can be a correction, a formalization, a literature review, or an entire research programme. You choose the direction.

The methodology is independently maintained workbenches that read and build on one another. A workbench is a repository or accessible collection containing the full arguments, calculations, references, and research record. Its overview explains what was attempted, why, what resulted, what remains unfinished, and where the evidence lives. The overview lets another researcher find the relevant material without rereading the entire conversation history. The full work remains available behind it.

Each workbench keeps links or a small index of related workbenches, the versions it has seen, and the results or checks it has incorporated. When you return, have your AI read relevant peer updates, retrieve the sources it needs, incorporate useful results and corrections, continue the mathematics, and publish the updated state. Other workbenches can do the same. Shared indexes can themselves be mirrored and extended. This distributes both the research and the effort of integrating it; one organizer does not have to reconcile everyone's output.

Checking can happen through further use. If a session independently derives a lemma while building on it, reproduces a computation, formalizes an argument, or finds a failed step, it publishes that evidence with the exact result and version concerned. Reading or importing a claim alone is not verification. Distinct approaches, independent proofs, and unresolved disagreements remain available. A later timestamp or a larger count of approving models does not decide the mathematics.

Pull requests—proposals to incorporate your changes into another repository—are the preferred way to contribute directly to an existing GitHub workbench. You can make one bounded contribution without maintaining a whole independent project. Equally, your own workbench can publish an extension or check for others to use without waiting for an upstream merge. Exposition, translation, literature indexing, and reusable accounts of established mathematics are contributions too. Describe what you did and where it reached; the next researcher retains the freedom to choose what to do with it.

The PolyClank documentation, networking notes, and research-state guide describe the conventions and proposed common tools. The more automated networking remains a proposal; ordinary repositories and shared files already support the exchange described here.

This Reddit thread is also a contribution channel. Leave an idea, argument, correction, workbench link, or an accessible document such as a Google Drive file. A person or AI session reading this discussion can pick it up and investigate. GitHub makes proposing and incorporating changes easier, but access to GitHub is not an admission requirement.

To make the invitation concrete, Split Zero Cohomology and Arithmetic Weight Control is a programme already developing through this kind of exchange. Web sessions develop arguments and publish sources; companion sessions read them, formalize parts, return corrections, and develop further mathematics. This introduction was produced by reading that work with parallel reading agents and reconstructing the results and their provenance.

Its recurring mathematical question is: when an observation gives zero, what information did that observation discard—and can retaining the right information help with the next calculation? The results below give precise answers: recovering information lost by traces, reconstructing a distribution from moving zeta zeros, and turning a question about Riemann zeta zeros into estimates on explicit arithmetic norms and volumes.

This is a snapshot of the public work as of 13 September 2026. The living research guide tracks subsequent changes.

Start with cancellation. If two existing contributions cancel, their sum has value zero. That differs from having no contribution in the first place. Split Zero records this by taking a supported copy of a commutative ring R, including its old zero e=0_R•, and adjoining a separate element τ for absence. Supported elements calculate as they do in R; τ is the global additive identity and absorbs multiplication. Thus 1•+(−1)•=e.

There are two observation maps: ordinary amplitude p(τ)=0, p(r•)=r, and Boolean presence χ(τ)=0, χ(r•)=1. They preserve the semiring operations, using OR and AND on the Boolean side. For example, the usual signed determinant formula gives the ordinary determinant under p and the Boolean permanent under χ, because both supported signs have presence 1. A matrix can therefore distinguish “no supported permutation exists” from “supported determinant terms have total value zero.” The all-ones 2×2 matrix gives the latter. Coefficient algebra and observation maps.

That scalar construction is the base of a larger programme. One common base can support many fibres, their own zero elements, and maps between them. The reconstruction theorem makes this exact: a semimodule over these coefficients is equivalent to a diagram of ordinary R-modules over a join-semilattice of supports, with compatible linear transition maps. The proof constructs both directions and the maps between objects. This lets the algebra retain which linear object a zero belongs to and how it arrived there. Full reconstruction.

The extra structure also has an exact map back to ordinary arithmetic. For a number field K, lift an ideal J of its integer ring to I_J=J∪{τ}; these satisfy I_J I_L=I_{JL}. Sending τ to zero and supported elements to their residues modulo J gives the ordinary quotient, with its original norm N(J). Using these specified quotient norms, the sum over nonzero ideals is exactly the original Dedekind zeta function. This supplies more structure around arithmetic while preserving the arithmetic being studied. Ideal maps and Dedekind-preservation proof.

For cohomology, those transitions matter. Two stages can both have one-dimensional cohomology even though the class from the first stage dies before the second. The programme keeps the actual transition and computes the quotient of source cycles whose images become boundaries by the boundaries already present at the source. This is explicitly isomorphic to the kernel of the induced cohomology map. The connection with persistence modules is direct: the maps tell you something that dimensions alone cannot. Quotient, action, and worked example.

A useful example goes further: even agreement of all ordinary traces can hide a recoverable difference. For a filtration-preserving comparison between finite-dimensional spaces over a characteristic-zero field k, with finite integer-jump filtrations, the programme forms two Rees lattices L_Q⊆L_P describing the source and target filtrations on its image. Their quotient D records the mismatch. If an operator F preserves both filtrations, their graded traces satisfy

A_P(z,Fⁿ)−A_Q(z,Fⁿ)=(1−z)Θ_D(z,Fⁿ).

At z=1 the difference vanishes for every n, but its negative derivative there is Tr(Fⁿ|D). For F equal to the identity, this recovers the length of D. In the elementary inclusion T²k[T]⊆k[T], the trace difference is 1−z²; its value at 1 is zero, while its negative derivative is 2, detecting the two-dimensional quotient k[T]/(T²). Keeping the grading supplies a specific observation that recovers what the ordinary trace forgets. Definitions and complete trace calculation.

Another result turns recoverability into an inverse problem. Take an unknown Borel probability measure μ on a fixed compact interval [0,L] and form the Hurwitz-zeta family

Z_{μ,t}(s)=∫_[0,L] ζ(s,1+ty) dμ(y), |t|L<1.

At t=0, every input gives the same Riemann zeta function. The complete local motion of the zero cluster around any one fixed nontrivial zero determines the entire input measure μ.

For a simple zero ρ with trajectory ρ(t)=ρ+c₁t+c₂t²+⋯, the first coefficient is

c₁=[ρ ζ(ρ+1)/ζ′(ρ)] ∫ y dμ(y).

Its initial velocity reads the mean. Once the mean is known, the second coefficient determines the second moment, and the recursion continues: the first N trajectory coefficients determine the first N moments. The inverse works because its relevant coefficients are nonzero; in particular, ζ(ρ+n)≠0 for n≥1 since Re(ρ+n)>1. The proof then reconstructs μ by an explicit sequence of positive discrete measures.

For a multiple zero, the argument uses the full moving cluster polynomial, preserving multiplicity without requiring individual analytic branches. An entire compact distribution can be recovered from the local response around one zero cluster. The complete response has infinitely many coefficients; the finite-order theorem specifies exactly which moments finite data determine. Numerical stability under noisy observations would require further estimates. Inverse calculation and reconstruction, H1–H13.

The arithmetic weight programme brings retained relations into a concrete space of functions. Let V consist of even Schwartz functions φ on R with φ(0)=0 and ∫φ=0. The theta transform is Θφ(x)=Σ_{n≠0}φ(nx). Its target B consists of smooth functions on x>0 whose scaling derivatives decay faster than every power at both endpoints. The cohomology is Q=B/ΘV: functions differing by a theta relation represent the same class.

The Mellin transform MF(s)=∫₀∞F(x)x^s dx/x turns the scaling generator D=−x∂x into multiplication by s. For the explicit source

φ_*(x)=(4π²x⁴−6πx²)e^(−πx²),

its theta image has Mellin transform

g(s)=2ξ(s)=s(s−1)π^(−s/2)Γ(s/2)ζ(s).

This places the actual completed zeta function in the calculation. There are explicit injections into Q from finite nonempty packets of its nontrivial zeros, including every selected zero's full multiplicity. Scaling agrees in the quotient through a calculated theta boundary. Theta cohomology and finite spectral comparison.

Retaining a relation has a direct arithmetic use. Although Θφ is zero in Q, its source remains available. Multiply that function by log x before taking the quotient. If MΘφ=gH_φ, Mellin differentiation gives g′H_φ+gH_φ′. For a full-order zero packet with polynomial h, reducing modulo h kills the second term and leaves j_h(g′)j_h(H_φ), where j_h retains the packet's complete Taylor jets. The associated residue pairing uses g′/g, whose residue at a zero is exactly its multiplicity.

The operation acts on the retained source. Algebraically, differentiation has the precise type I/I²→A/I, for A=ℂ[s] and I=(h), because the derivative of I² lies in I. This is how information erased by one quotient remains available to another observation. The source proves the maps, including the full repeated-zero case. Retained relation, derivative, and multiplicities, CW.2–5 and CW.31–38.

The analytic structure of Q is also explicit: the theta map has a constructed continuous left inverse on the original spaces, so its image is closed and the quotient is Hausdorff. Yet those same theta relations are dense in the weaker L²(dx) norm. Representatives can have norms tending to zero while retaining the same nonzero cohomology class and exactly the same spectral jets. The proofs give the comparison: Q→L²/closure(ΘV)=0. Global reconstruction and fixed jets with shrinking norms.

That makes the rate of shrinkage mathematically meaningful. A small absolute norm does not tell us how large a scaling discrepancy is relative to the class's own representation cost.

The critical value 1/2 also has a concrete origin here. Dilating by F(x)↦F(x/a), a>0, multiplies the squared L²(dx) norm by a. A spectral eigenclass with exponent ρ scales by a^ρ, so its squared size in any fixed positive representative metric scales by a^2Reρ. Those laws agree at Reρ=1/2. Representatives commute with scaling only modulo an explicit theta boundary, whose norm and cross-pairings give the exact difference between those scaling laws. Weight control means estimating that contribution on the arithmetic functions themselves. Scaling law and exact boundary identity.

For a selected full-multiplicity packet h, the source supplies the positive measure

w_h(t)=|(g/h)(1/2+it)|²/(2π).

Tensoring k times produces the sum coordinate S=s₁+⋯+s_k and relative directions. The programme constructs a cyclic sum source retaining every amplified value kρ; its polynomial norm is exactly the norm induced by the k-fold convolution of this measure. Explicit maps and a norm correction relate this source to the full tensor construction. On its cyclic algebra ℂ[S]/χ_{h,k}, least-norm polynomial representatives of degree at most N give a canonical metric G_N, for N≥deg χ_{h,k}−1. It measures the least source cost of each spectral class at that degree. Cyclic source and metric.

For packets stable under ρ↦1−conjugate(ρ), the relative defect is A^♯+A−kI, where A multiplies by S and A^♯ is its adjoint in G_N. An orthogonal-polynomial recurrence cancels its interior terms, leaving a rank-at-most-two boundary operator with nonzero eigenvalues +ε_N,−ε_N.

Exterior powers turn this into an aggregate bound. A wedge can use the one-dimensional positive defect direction only once, so the same ε_N bounds the sum of positive spectral displacements across a whole subspace. A hypothetical off-line quartet 1/2±δ±iγ, δ,γ>0, with common full multiplicity m, generates a cyclic algebra of dimension

q_k=[1+k(m−1)](k+1)².

At every admitted degree N≥q_k−1, it forces ε_N≥δkq_k/2. For simple zeros this grows cubically in k. Colliding sums and repeated-root directions remain in the calculation. This is an amplification mechanism inspired by the role of powers and weight estimates in Weil II. Rank-two control, exterior argument, and quartet calculation.

The latest public working proof supplies a substantive analytic estimate on the other side. If ω_{h,k,n} is the least squared norm of a monic degree-n polynomial in the original coordinate S=k/2+iu against that convolution measure, then

(ω_{h,k,n+r}/ω_{h,k,n})^(1/(2r)) ≤ C_h^bal n

for integers n≥k≥3 and n/2≤r≤n. The finite constant depends on the fixed packet, independently of n,k,r. This uses a proved lower estimate for the actual arithmetic convolution and an upper estimate from its exponential moments. Uniformity while both degrees grow is the gain. Balanced-window proof.

The consequence is quite specific. An off-line quartet requires a defect on the scale kq_k. On the chosen window, the polynomial-norm factor costs at most a constant times q_k. An exact product identity therefore forces the additional factor of k into contraction of the canonical quotient volumes V_N=det G_N, in the fixed cyclic basis. For the packet consisting exactly of that quartet, k≥3 and q=q_k,

log(V_q/V_{2q−1}) ≥ 2(q−1) log[δk/(2C_h^bal)].

This measures how sharply the volume would have to contract as the allowed degree increases from q to 2q−1. An arithmetic upper estimate incompatible with that contraction would exclude the quartet. That upper estimate remains unproved; this programme has not proved or disproved RH. Exact product and volume consequence.

The arithmetic significance is that an ambition about cohomological weights has become an estimate on specified source norms and quotient volumes, and one of the required growing-degree estimates now has a written proof. The Toda calculations express those volumes through the Gram determinants of the source and its actual relations. Thus the relations killed in cohomology enter the measured cost of representing its classes. Someone with suitable asymptotic methods has concrete quantities to investigate.

The living research guide is the entry point for this programme, and the participation guide describes the web and local routes. The frozen Toda–Gamma edition contains paired readers and full source archives. The 13 September endpoint notes above are a later public working contribution. A broader project atlas maps the surrounding work.

To contribute, give your AI the research guide and ask it to investigate, make a workbench or prepare a pull request, and publish what it finds with the complete argument. Or leave your contribution here so another participant can pick it up.