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J.Konstapel,Leiden,27-10-2026.

Cause: Leidse onderzoeksprojecten ontvangen Open Competitie financiering voor vernieuwend onderzoek

1. Introduction: the question of the round trip

Mathematics and physics keep returning to a single, deceptively simple question: if you know everything about how something comes back to where it started, can you reconstruct the thing itself? A closed curve — a loop, a period, an orbit — records its own history in the values it accumulates as it travels. The mathematical problem of inverse periods asks whether that record is complete: whether the period matrix of an algebraic curve, the array of integrals of its holomorphic differentials over each of its closed loops, determines the curve up to isomorphism.

This question is not idle. It sits at the junction of arithmetic geometry (where periods have been computed and classified since the nineteenth century), of mathematical physics (where the periods of special curves encode the masses and couplings of quantum field theories), and, more speculatively, of any worldview that treats persistence itself — the fact that a thing keeps coming back — as the fundamental property worth explaining.

The occasion for this essay is a concrete one: the mathematician Marco Streng of Leiden University has received NWO Open Competitie funding for a project named ReSCue, which asks exactly this reconstruction question for superelliptic curves — curves of the form y^n = f(x). This essay reads Streng’s body of work as a coherent programme about closure, connects it to a speculative framework called the Vacuum.Net, in which everything that persists is a closed path, and presents one new computational observation: the framework’s ternary unit of address, the trit, turns out to arise naturally in the Eisenstein integers — the very lattice that governs Streng’s Picard curves.

2. The central mathematical problem

2.1 Periods and Torelli’s theorem

Start with the classical situation. A smooth complex algebraic curve C of genus g is, analytically, a surface with g holes. It has 2g independent closed loops, and g linearly independent holomorphic differentials \omega_1, \dots, \omega_g. Integrating each differential over each loop produces a g \times 2g matrix of complex numbers — the period matrix. Torelli’s theorem states that this matrix, suitably normalised, determines the curve completely. In other words: the record of round trips already contains the path.

Torelli’s theorem, however, is an existence statement, not an algorithm. Given an approximate period matrix — measured, say, to a finite precision — how does one recover the curve, with its exactly-defined algebraic coefficients? This is the inverse problem: practical, quantitative, and wide open in general. For elliptic curves (genus 1) the answer is classical and constructive, running through the theory of complex multiplication. For genus 2 and 3, Streng and his collaborators have spent a decade making it concrete. ReSCue extends the question to superelliptic curves of the form y^n = f(x), whose extra rotational symmetry both constrains and enriches the period lattice.

2.2 Complex multiplication: symmetry of the lattice

A complex elliptic curve is a torus: the complex plane modulo a lattice \Lambda. Generically, the only symmetries of such a torus are multiplication by \pm 1. But for special lattices — the hexagonal one, the square one, and countably many others — the torus admits extra rotations: endomorphisms beyond \mathbb{Z}. This is complex multiplication (CM), and it is the arithmetic key to explicit reconstruction. A CM elliptic curve is not described by a transcendental period but by an algebraic invariant, the j-invariant, which is an algebraic integer satisfying a class polynomial whose degree is the class number of an imaginary quadratic field.

The moral is the one that runs through everything discussed here: symmetry turns analysis into arithmetic. The period matrix of a generic curve is an unruly transcendental object; the period matrix of a CM curve is governed by a lattice with rotational symmetry, and the reconstruction problem collapses into a finite, computable algebraic task. The same principle powers genus 2 (where the relevant invariants are Igusa class polynomials), genus 3 (Streng’s Picard and quartic curves), and much of modern computational number theory.

2.3 The table of Streng’s programme

Read as a whole, Streng’s seventeen listed publications fall into five operations on closed paths — a taxonomy worth making explicit:

Operation on a closed path	Representative papers	Interpretation
Giving a closed path a basis-free address	Igusa class polynomials (2014); generalized class polynomials (2022, with Houben); Schertz-style class invariants for higher-degree CM fields (2026, with Enge)	The j-invariant and its higher-genus analogues name a torus independently of any chosen basis — an address without a starting point
Counting paths that share one rotational symmetry	Polarised class groups of quartic CM orders (2017, with Bisson); CM class number one for genus 2 (2023, with Kılıçer)	Complex multiplication is a turn that maps the lattice into itself; the class number counts how many distinct lattices share that turn; class number one means a unique closure
Finding the step at which a point returns	Divisibility sequences for CM elliptic curves (2008); algebraic divisibility sequences over function fields (2012); primitive divisors with constant j-invariant (2020, with Naskręcki); modular units for \Gamma_1(N) (2023)	A primitive divisor of the n-th term of a divisibility sequence is a prime modulo which a point first closes exactly at step n; \Gamma_1(N) describes the structure of points that close after N steps
Finding where a loop pinches	Primes dividing invariants of CM Picard curves (2019); a bound on bad-reduction primes for genus-3 CM curves (2020); plane quartics over \mathbb{Q} with CM (2018)	Bad reduction is a connection point: modulo that prime, two strands of the path fall together and the loop degenerates
Using closure as a working tool	CM construction with p-rank 1 (2011); prescribed embedding degree (2008); curves in characteristic 2 with 2-torsion (2014); (\ell,\ell)-endomorphisms (2013); CM curves over the reflex field (2015)	Closure arithmetic already runs as practical machinery — the cryptographic papers are the engineering wing of the same programme

The row that matters most for what follows is the fourth and fifth: the family of Picard curves y^3 = f(x) with \deg f = 4. These genus-3 curves carry a built-in threefold rotational symmetry, (x, y) \mapsto (x, \omega y), where \omega is a primitive cube root of unity. Their period lattices are modules over the Eisenstein integers \mathbb{Z}[\omega] — and it is precisely there that a ternary arithmetic of closed paths lives.

3. The Vacuum.Net: a framework of closure

To explain why this matters beyond arithmetic geometry, a short detour into the speculative part.

The Vacuum.Net (as developed by J. Konstapel) is a proposed reading of physics in which everything that persists is a closed curve: a path that returns to where it started. Points are locations in a position space; curves run through them; where curves meet there is a connection point. The slogan is: a thing is a turn that keeps coming back. On this reading, what we call a particle is not a small object but a stable recurrence — an address that loops.

Two structural consequences follow, and only these two are needed here.

First, the address of a closed path has no starting point. A loop is a rotation class: reading its description starting from any point on the loop gives the same path. Therefore the number of distinct addresses of length n over a three-letter alphabet is not 3^n but the necklace countg(n) = \frac{1}{n} \sum_{d \mid n} \varphi\!\left(\frac{n}{d}\right) 3^{d},

the number of circular words up to rotation — for instance g(3) = 11, not 27.

Second, the unit of the address is the trit: a digit from \{-1, 0, +1\}, read off from balanced ternary rather than assigned by convention. The question this essay adds to the framework is: where does the trit come from? The answer, developed in the next section, is that it is not a convention at all. It is the natural residue in the Eisenstein integers — the lattice behind the Picard curves.

4. The trit in \mathbb{Z}[\omega]: a computational result

4.1 The residue

Let \omega = e^{2\pi i/3} be a primitive cube root of unity and let \beta = 1 – \omega, an element of norm N(\beta) = 3. The quotient \mathbb{Z}[\omega]/(\beta) is the field with three elements, and since \omega \equiv 1 \pmod{\beta}, the class of a + b\omega is simply a + b \pmod 3. The natural representatives of these three classes are -1, 0, +1 — exactly the balanced-ternary digits. In the Eisenstein integers, the trit is not a design choice; it is the residue modulo the prime above 3.

4.2 The digit map and its dynamics

Define the balanced residue d(z) \in \{-1, 0, +1\} and the digit mapT(z) = \frac{z – d(z)}{\beta}.

Iterating T produces a balanced expansion in base \beta:z = d_0 + d_1\beta + d_2\beta^2 + d_3\beta^3 + \dots, \qquad d_k \in \{-1, 0, +1\}.

A computational experiment — running T on every a + b\omega with |a|, |b| \le 100 (40,401 elements) — shows that every orbit ends in one of exactly three places: at 0, or at one of the two fixed points \pm(1+\omega). Moreover, the proportion ending at a fixed point rather than at zero grows with the size of the box: 27% for |a|, |b| \le 40 (1,788 of 6,561) and 41% for |a|,|b| \le 100 (16,546 of 40,401). Whether these shares converge to a limit is open.

The result holds for all of \mathbb{Z}[\omega], and the proof is short. The norm drops at every step:N(T(z)) = \frac{N(z – d)}{3} \le \frac{(\sqrt{N(z)} + 1)^2}{3},

which is smaller than N(z) whenever N(z) \ge 2. So every orbit falls into the elements of norm 0 or 1: the zero and the six units. Checking those seven by hand: 0, \pm 1, \pm\omega run to 0, while \pm(1+\omega) map to themselves.

4.3 The meaning of the fixed point

The fixed point has a clean description: 1 + \omega = -\omega^2 = e^{i\pi/3}, a sixth root of unity — a 60° turn. Its balanced expansion is the digit -1 repeated forever, since -\omega^2 = -1 + \beta \cdot (-\omega^2). The ordinary number 2 already exhibits the dichotomy of the dynamics: 2 \to 2 + \omega \to 1 + \omega, and from there the turn never stops. The number 1, by contrast, comes to rest in a single step.

The reading in the framework is this: the ternary ground, placed inside the Eisenstein lattice, reproduces the Vacuum.Net’s basic dichotomy unprompted. A finite trit word comes to rest at zero — a closure. An eventually periodic word does not rest: it ends in a unit that keeps turning, a turn that keeps coming back. Rest and return, the two states the framework posits, fall out of the arithmetic of \mathbb{Z}[\omega] itself. (Taking -\beta = \omega – 1 as the base instead merges the two fixed points into a single 2-cycle — the same turn seen from the other side.)

5. ReSCue as the bridge

5.1 From the framework to the project

For n = 3, a superelliptic curve y^3 = f(x) is a Picard curve, and (x, y) \mapsto (x, \omega y) is an automorphism of order 3. This automorphism acts both on the loops of the curve and on its differentials, so the period lattice is not merely a \mathbb{Z}-module but a module over \mathbb{Z}[\omega]. Since \mathbb{Z}[\omega] is a principal ideal domain, that lattice has a basis over \mathbb{Z}[\omega], and the digit map of Section 4 applies to the coordinates of every period in that basis: each coordinate has a balanced-trit expansion in base 1 – \omega, either finite or ending in the 60° turn. Whether this ternary address is a useful coordinate for reconstruction — in lattice reduction, or in deciding to what precision periods must be known before reconstruction succeeds — is a concrete and testable question.

5.2 From the project to the framework

The dependence runs both ways. The Vacuum.Net needs exactly the procedure ReSCue is building: given only measured closures, rebuild the curve. A text, a dataset, a physical measurement — each arrives as numbers, not paths. If periods determine the curve, then closures determine the path, and the framework’s position space can be reached from data rather than postulated. Reconstruction is the route from the world of records to the world of paths.

5.3 The physics already on the bridge

The bridge to physics is not hypothetical; three results anchor it.

• Seiberg–Witten theory (1994). A four-dimensional N=2 supersymmetric Yang–Mills theory is encoded by an algebraic curve, and its periods are the physical observables: the masses of stable particles and the coupling constants. The exact mass formulas of the abstract follow from the periods of the Seiberg–Witten curve. Here the period matrix is not a bookkeeping device; it is the theory’s measurable content.
• Moore’s “Arithmetic and Attractors” (1998). For N=4 and N=8 string compactifications, the attractor varieties of supersymmetric black holes are built from elliptic curves with complex multiplication, and for N=4 the number of inequivalent backgrounds with the same horizon area is precisely the class number of an imaginary quadratic field — the quantity Streng’s class-number papers compute. Supersymmetric black holes, in other words, are counted by the same arithmetic that counts closures.
• Gukov–Vafa (2002). The rational conformal field theories with a two-torus target are exactly those whose torus has complex multiplication. The consistency of a physical theory on a torus is equivalent to an arithmetic condition on the torus’s lattice.

In each case the physics sits on closed curves, accessed through their periods or their complex multiplication — the two objects that ReSCue and Streng’s earlier work handle. ReSCue already stands on a bridge to physics; it has simply not been named as such. The Vacuum.Net offers a third bank: a ternary address for the same periods, and a reading in which rest and return are the only two states.

6. Three open questions

Each question below can be checked in an afternoon with Sage or PARI/GP, and each answer, either way, is informative.

1. A trit address for periods. For Picard curves, does the balanced expansion in base 1 – \omega give a useful coordinate on the period lattice — for instance in lattice reduction, or in deciding to what precision periods must be known before reconstruction succeeds? And which period vectors end in the 60° turn rather than at rest?
2. First closures and necklace counts. In the divisibility sequences attached to a CM curve, the index at which a prime first appears is the step at which a point first closes. Do these indices, gathered over many primes, follow a distribution comparable to the necklace count g(n) = \frac{1}{n}\sum \varphi(n/d)\,3^d — especially for curves with CM by \mathbb{Z}[\omega]?
3. Reconstruction as a general procedure. If ReSCue yields an algorithm from period matrix to curve, could the same algorithm take any set of measured closures — not only periods of an algebraic curve — and return the simplest closed path that produces them?

7. Conclusion

The wager of this essay is that three things usually kept apart — a computational number theory programme (class polynomials, divisibility sequences, bad reduction), a reconstruction project (ReSCue), and a speculative physics of closure (the Vacuum.Net) — are one subject seen from three distances. The arithmetic gives the framework its digits and its dichotomy of rest and return; the framework gives the arithmetic a structural vocabulary; and ReSCue, if it succeeds, gives both a procedure: from the record of round trips back to the path that made them. What a closed curve is, in the end, may be exactly what its periods can reconstruct.

Annotated references

Mathematical sources

• Streng, T.C., publication list, Universiteit Leiden. The seventeen papers cited in the taxonomy table above (2008–2026), spanning class polynomial computation for genus 2 and 3, CM class number problems, divisibility sequences, bad-reduction bounds for Picard curves, and cryptographic CM constructions. Read together they form a coherent programme on the invariants, first-returns and degenerations of closed curves with rotational symmetry; the present essay’s taxonomy of “five operations on closed paths” is derived from this list.
• Universiteit Leiden, “Leidse onderzoeksprojecten ontvangen Open Competitie financiering”, 26 October 2026. The announcement that establishes ReSCue as an NWO-funded project on recovering superelliptic curves y^n = f(x) from their period matrices. The immediate occasion for this essay; the reconstruction problem of Section 2 and the bridge of Section 5 are framed around its stated aims.

Framework sources

• Konstapel, J., “Curves in the Universe” (PDF title “Curves in the Vacuum.Net”, v1.0), constable.blog, 24 October 2026, updated 26 October 2026. The longer statement of the Vacuum.Net framework: points in position space, curves, knots and seams, and the introduction of trits and balanced ternary as the unit of address. Sections 3 and 4 of this essay assume only its two consequences: the rotation-class address and the trit.
• Konstapel, J., “The Positron and the Cogwheel”, constable.blog, 26 October 2026. The companion piece working out the address of a closed path as a rotation class (a necklace), including the worked example g(3) = 11. Source for the necklace count g(n) used in Sections 3 and 6.

Physics sources (each checked against its arXiv record on 27 October 2026)

• Seiberg, N. and Witten, E., “Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang–Mills theory”, Nuclear Physics B 426 (1994) 19–52; arXiv:hep-th/9407087. The founding paper of Seiberg–Witten theory: an N=2 field theory is encoded in an algebraic curve whose periods give exact particle masses and couplings. Cited in Section 5.3 as the paradigm case in which periods are directly physical observables; the exact mass formulas appear in the abstract, their derivation from the periods in the body.
• Moore, G., “Arithmetic and Attractors”, arXiv:hep-th/9807087 (1998). Shows that attractor varieties of supersymmetric black holes in N=4 and N=8 compactifications are built from CM elliptic curves, and that for N=4 the degeneracy of inequivalent backgrounds sharing one horizon area equals the class number of an imaginary quadratic field. Cited in Section 5.3 as the bridge between black-hole physics and the class-number arithmetic of Streng’s genus-2 programme.
• Gukov, S. and Vafa, C., “Rational Conformal Field Theories and Complex Multiplication”, arXiv:hep-th/0203213 (2002). Establishes that rational CFTs with two-torus target are precisely those whose torus has complex multiplication. Cited in Section 5.3 as the cleanest known equivalence between a physical consistency condition and an arithmetic property of a lattice.

Computational source

• Computation of Section 4: a script over all a + b\omega with |a|, |b| \le 100, run 27 October 2026. Exhaustive iteration of the digit map T(z) = (z – d(z))/\beta over 40,401 Eisenstein integers, recording the terminal state (zero or the fixed points \pm(1+\omega)) of each orbit, with growing fixed-point share (27% at box 40, 41% at box 100). The norm-descent argument of Section 4.2 makes the two end states general for all of \mathbb{Z}[\omega]; the convergence of the shares remains open.

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