Counting Trees in Four Dimensions: A Lattice Constant That Turns Out to Be Built From Primes
How many ways can you connect every point of a four-dimensional grid without making a loop? The answer seems to be a combination of two numbers from the theory of modular forms. It holds to more than 100 digits, and the proof is down to one missing piece.
Key message: The paper demonstrates that the exact number of loophole-free paths (spanning trees) in a four-dimensional grid can be calculated precisely through a highly symmetric mathematical structure known as a weight-4 modular form. The Asymptotic Lattice Constant in 4D for Spanning Trees is 1.999707644517312559687899304…
Take a grid of points and connect neighbours by edges. A spanning tree is a way of choosing some of these edges so that every point is reached and no loop is closed. On a large grid there are astronomically many such trees, and their number grows exponentially with the number of points N, roughly like e^(λN). The constant λ is the entropy of the lattice.
Physicists know this number well. It is the free energy of a limiting case of the Potts model, the entropy of the abelian sandpile, and the normalisation of the uniform random spanning tree. Gustav Kirchhoff showed in 1847 how to count spanning trees with a determinant ([Kirchhoff 1847](https://doi.org/10.1002/andp.18471481202)), and today λ is known to many digits for many lattices ([Felker and Lyons 2003](https://doi.org/10.1088/0305-4470/36/31/301)). The question in my new paper (P15) is not how large λ is, but which number it is.
Where the idea came from
In an earlier paper of this series ([Krause 2026, P9](https://doi.org/10.5281/zenodo.23805526)) I collected the entropies of flat lattices. For the square grid, λ = 4G/π, where G = 0.91596… is Catalan’s constant. For the triangular grid the corresponding constant comes from the field of √−3. In both cases a single periodic pattern of +1, 0 and −1, a Dirichlet character, decides which constant appears. In the language of number theory these are L-values of weight two.
In three dimensions, Guttmann and Rogers proved formulas of the next kind ([Guttmann and Rogers 2012](https://doi.org/10.1088/1751-8113/45/49/494001)). The simple cubic and the body-centred cubic lattices have entropies that are values of L-functions of modular forms of weight three.
So there is a ladder: weight two in the plane, weight three in space. The natural question was whether it goes on. Is the spanning-tree entropy of a four-dimensional lattice an L-value of weight four?
What we found
There are two standard lattices in four dimensions. For the hyper-body-centred lattice the answer follows from a theorem of Papanikolas, Rogers and Samart ([PRS 2014](https://arxiv.org/abs/1305.2143)), once it is written in the right form: the entropy is 8 L′(f₈, 0) − 28 ζ′(−2), with a modular form f₈ of weight four and level eight.
For the hypercubic lattice ℤ⁴, the ordinary four-dimensional grid, no formula was known. The first step was to find the right modular form. Among 464 candidate forms of weight four, a simple test with the number of closed random walks modulo primes left exactly one survivor: the form
g₆ = (η(τ) η(2τ) η(3τ) η(6τ))²,
the unique newform of weight four and level six. Here η is Dedekind’s eta function. Then an integer-relation search with a basis fixed in writing before the computation gave
**λ(ℤ⁴) = 12 L′(g₆, 0) − (56/3) ζ′(−2) = 1.99970764451731255968…**
Equivalently, λ(ℤ⁴) = (162/π⁴) L(g₆, 4) + (14/3) ζ(3)/π². The two sides agree to more than 100 digits, in two independent runs, and every other basis element got the coefficient zero. I did not find this relation in the literature I checked. It is not proved.
What it says in words: a constant that physics defines without any reference to primes is a rational combination of two numbers built from primes. One is the value of the L-function of g₆, whose coefficients are an Euler product over primes. The other is ζ′(−2) = −ζ(3)/(4π²), Apéry’s constant in disguise.
How far the proof got
A numerical identity to 100 digits is strong evidence, but it is not a proof. Most of the paper is about how far a proof can be pushed, and where exactly it stops.
The four-dimensional integral can be moved onto a single line in the upper half-plane, where g₆ and its relatives live. On that line the entropy splits into a part that is proved and one remaining integral K that runs from a special point to the boundary. The special point, τ* = 1/2 + i/(2√3), is a so-called CM point of the field of √−3, and it is the fixed point of a symmetry W₃ that acts like a mirror, Y ↦ 1/(12Y).
Under that mirror K splits into an even half and an odd half. The even half is now proved. The odd half is equivalent to the whole relation, and a lemma shows that no symmetry of the group can reach it. The missing step is therefore genuinely arithmetic. The entire problem has been reduced to one number:
K = 12 L′(g₆, 0) + (28/3) ζ′(−2) = 1.14715084688…
Two things did not work, and they are reported as negative results: the ladder does not continue for the four-dimensional face-centred lattice, and nothing of this kind was found for the five-dimensional body-centred lattice. The ladder is not a general law. It continues where the lattice has enough symmetry for a modular description.
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How this can help mathematicians
- A small, precise open problem. Number theorists now have one explicit iterated integral from a CM point to a cusp, with a conjectured value. Any method that evaluates such integrals in the odd sector can be tested on it. Candidates are Hecke correspondences, higher Green functions, and Brown’s non-holomorphic modular forms ([Brown 2018](https://doi.org/10.1007/s40687-018-0130-8)).
- A new member of an open family. The Mahler measures m(k − Σ(xᵢ + 1/xᵢ)) were listed as open by Papanikolas, Rogers and Samart. The case k = 8, which is λ(ℤ⁴), now has a conjectured closed form.
- A new integral formula. The paper proves an integral representation of the non-critical value L(g₆, 4), the kind of number that Beilinson’s conjectures describe as a regulator.
- A pattern to look for elsewhere. “Full path proved, half path open” should appear for other Calabi–Yau differential equations whose special point is a CM point fixed by such a mirror.
How this can help physicists
- An exact free energy in the upper critical dimension. Four is the upper critical dimension of the loop-erased random walk and the uniform spanning tree ([Lawler, Sun and Wu 2019](https://doi.org/10.1214/19-AOP1349)). If the relation is proved, the spanning-tree free energy there has a closed form.
- **A bridge to Feynman integrals.** The same form g₆ governs the six-Bessel moments of two-dimensional quantum field theory, the four-loop sunrise or “banana” integrals ([Broadhurst 2016](https://doi.org/10.4310/CNTP.2016.v10.n3.a3)). There its critical values appear, here the non-critical one. The tools used in one problem may help in the other.
- **Lattice Green functions.** λ is an integrated Green function of the random walk on ℤ⁴, and the same level-six description controls the return probabilities. The methods carry over.
- **A fast benchmark.** λ(ℤ⁴) can now be computed to 100 digits from a quickly converging q-series instead of a four-dimensional integral.
What this is not
- The main relation is a conjecture with very strong numerical support, not a theorem.
- Several steps use one cited input, a symmetry of a modular function proved by Shvets ([Shvets 2026](https://arxiv.org/abs/2604.06239)), which I have not re-proved independently.
- No physical prediction is made. The value of λ was known; the paper only asks which number it is.
A reading within the series
This last part is an interpretation within my framework, not a result, and the mathematics does not depend on it.
The level of g₆ is 6 = 2 · 3, the second step of the primorial ladder 2, 6, 30 that organises earlier papers of this series. The special point belongs to the field of √−3, the same field that appeared in the triangular lattice. The factor 56/3 contains 7 = 8(1 − 2⁻³), the Euler factor at 2 of ζ(3), which keeps returning in this series, from electrons in metals to neutrino decoupling. The eight neighbours of a point in ℤ⁴ are the eight vertices of the 16-cell, the four-dimensional polytope on which earlier papers placed the residues modulo 30. And the mirror W₃ has exactly one fixed point; the half of the proof that is still open is precisely the half the mirror cannot reach.
If this reading were right, the four-dimensional grid would not just happen to produce these numbers. Its geometry would carry the same small primes 2 and 3 that organise the other papers, and the open half of the proof would be a place where a reflection alone is not enough. Whether that is more than a coincidence of small numbers would have to be shown; for now it is a guide for where to look.
Read and check
- Spanning Trees in Four Dimensions (P15). The paper with all proofs, figures, two primers for physicists and number theorists, and a companion code that checks every step at 50 and 100 digits: https://doi.org/10.5281/zenodo.23927482
- One Character per Laplace Symbol (P9), the two-dimensional rung of the ladder: [doi:10.5281/zenodo.23805526](https://doi.org/10.5281/zenodo.23805526)
- A. J. Guttmann and M. D. Rogers, Spanning tree generating functions and Mahler measures: [doi:10.1088/1751–8113/45/49/494001](https://doi.org/10.1088/1751-8113/45/49/494001)
- M. A. Papanikolas, M. D. Rogers and D. Samart, The Mahler measure of a Calabi–Yau threefold and special L-values: [arXiv:1305.2143](https://arxiv.org/abs/1305.2143)
- J. L. Felker and R. Lyons, High-precision entropy values for spanning trees in lattices: [doi:10.1088/0305–4470/36/31/301](https://doi.org/10.1088/0305-4470/36/31/301)
- D. Broadhurst, Feynman integrals, L-series and Kloosterman moments: [doi:10.4310/CNTP.2016.v10.n3.a3](https://doi.org/10.4310/CNTP.2016.v10.n3.a3)
Thomas Krause is an independent researcher in Lübeck, Germany. He writes the book series Geometry of Reality.
