The persistence curve: when arithmetic selects a unique geometry

The picture

Imagine drawing all possible curves in a space of spectral curves — a standard framework in algebraic geometry. Among this infinity of curves, we impose three constraints from PT: (A) the curve must be made of the $\sin\theta_p$ functions of persistence; (B) only “active” primes $p$ contribute; (C) the internal coherence $\mu = \sum_p p$ is respected.

Result: these three constraints leave only one possible curve. This is a result of geometric rigidity: PT’s arithmetic uniqueness (a single fixed point $\mu^* = 15$) entails geometric uniqueness (a single spectral curve).

Where infinitely many geometries could have hosted PT, only one does — and its DNA is the number 14.

Why 14?

The persistence curve, like any algebraic curve, has a topological invariant called its genus: intuitively, the number of “holes” in the surface it draws. A circle has genus 0, a torus has genus 1, a pretzel has genus 2.

The persistence curve has genus 14. This number is not arbitrary: it equals exactly $\mu^* - 1 = 15 - 1$. That is, the arithmetic fixed point $\mu^* = 15$ determines the topology of the associated spectral curve.

Five sieves, five geometries

The theorem applies not only to PT. There are in fact five arithmetic sieves compatible with conditions A, B, C:

Each sieve thus defines its own unique algebraic curve. The question “why PT and not another?” becomes a question of natural selection among these five possible universes.

Three selection principles

PT is selected from the five by three independent principles:

• (N) Naturality: odd primes are the most elementary objects of arithmetic (free multiplication).
• (K) Kinematic exclusion of $p=2$: theorem T1 (Forbidden Transitions) shows that $p=2$ carries no modular dynamics (degenerate matrix $T_2 = (1)$).
• (D) Dynamic level $n \neq 1$: the unit 1 cannot be active (it is multiplicatively neutral).

Under these three principles, PT is the unique selection among the five compatible sieves.

Why this matters

Before this result, PT was a coherent theory but without formal bridge to the major geometric frameworks of contemporary mathematics. The persistence curve changes that: PT enters the territory of topological recursion of Eynard–Orantin, which computes the volumes of moduli spaces of Riemann surfaces (Mirzakhani’s work, Fields medal 2014) and appears in JT gravity (Saad–Shenker–Stanford 2019).

The three conditions and the main theorem

The framework: the Kontsevich–Norbury class

A spectral curve in the Eynard–Orantin formalism is a quadruple $(\Sigma, x, \omega_{0,1}, \omega_{0,2})$ where $\Sigma$ is a Riemann surface, $x$ a meromorphic function, and $\omega_{0,1}, \omega_{0,2}$ are differential forms. The Kontsevich–Norbury 2021 class ($\mathcal{C}_{\rm KN}$) restricts to $\Sigma = \mathbb{C}$, $x = z^2/2$, and $\omega_{0,1}$ of the form $y(z)\,dz$ with $y$ odd in $z$.

Two canonical curves of this class:

• Airy: $\omega_{0,1}^{\rm Airy} = z^2 \, dz$. Computes intersection numbers of $\psi$-classes on $\overline{\mathcal{M}}_{g,n}$ (Witten–Kontsevich theorem).
• Mirzakhani: $\omega_{0,1}^{\rm Mirz} = z\sin(2\pi z)/(2\pi) \, dz$. Computes Weil–Petersson volumes $V_{g,n}(L_1, \ldots, L_n)$.

The three conditions

A spectral curve $\mathcal{C}$ in $\mathcal{C}_{\rm KN}$ is said compatible with a sieve $S \subset \mathbb{P}$ if it satisfies:

• (A) Polynomial holonomy: there exists a rational parametrization $\mu(z)$ such that $\omega_{0,1}(z) = \frac{z}{2\pi} \prod_{p \in S} \sin\theta_p(\mu(z)) \, dz$, where $\sin\theta_p$ is PT’s holonomy identity T6.
• (B) Threshold activation: $S$ coincides with $\{p : \gamma_p(\mu(0)) > 1/2\}$, where $\gamma_p$ is the anomalous dimension.
• (C) Self-coherence: $\mu(0) = \sum_{p \in S} p$.

The main theorem

Theorem (existence and uniqueness). In the Kontsevich–Norbury class modulo even analytic reparametrization, there exists a unique algebraic curve simultaneously satisfying (A), (B), (C). This curve is the persistence curve $\mathcal{C}_\sieve$, characterized by $S = \{3, 5, 7\}$ and $\mu^* = 15$.

Proof idea: condition (C) imposes $\mu^* = \sum_{p \in S} p$; condition (B) imposes that $S$ be exactly the set of “active” primes at $\mu^*$. PT’s theorem T5 (arithmetic fixed point) establishes that under restriction to odd primes, the only solution is $\mu^* = 15$, $S^* = \{3,5,7\}$. For other sieves, see the table above.

The universal genus formula

For any subset $S$ of odd cardinality satisfying (A)–(C), the algebraic genus of the associated curve equals

For PT ($|S| = 3$, $\mu^* = 15$): $g = 15 - 1 = 14$.

This formula is structurally remarkable: the topological genus of an algebraic curve is explicitly determined by two arithmetic quantities (the sum of active primes and their number). This is the signature of the rigidity of the construction.

Mirzakhani → persistence substitution

The form of the persistence curve,

is algebraic in $z$ (by T6, $\sin^2\theta_p$ is rational in $\mu$, and $\mu(z)$ is rational). It thus contrasts with Mirzakhani’s curve, whose $\sin(2\pi z)$ is transcendental.

Series expansion gives the first correlator

to be compared with $\omega_{1,1}^{\rm Mirz}(z) = (1/(8 z^4))[1 + (2\pi^2/3) z^2 + O(z^4)]$.

The structural substitution is therefore

That is: the transcendental constant $2\pi^2/3$ that appears in Mirzakhani is replaced, in PT, by the arithmetic combination $(1/2)(\gamma_3 + \gamma_5 + \gamma_7)$. Numerical mpmath verification: agreement to $10^{-9}$.

Technical proof

Canonical parametrization

We set

This function is even, with two simple poles at $z = \pm 1$ and $\mu(0) = \mu^*$. Justification: minimal pole, Fisher geodesic, maximal symmetry (three equivalent characterizations).

Algebraicity

Lemma (T6 polynomial form). For all $p \in \mathbb{P}$ and $\mu > 0$,

With $\mu(z) = \mu^*/(1-z^2)$, $q(z) = (\mu^* - 2 + 2z^2)/\mu^*$ is polynomial in $z^2$. So $\sin^2\theta_p(\mu(z))$ is polynomial in $z^2$ (of degree $2p$), and

is polynomial in $z^2$, hence rational in $x = z^2/2$. The curve is algebraic.

The genus computation

The polynomial $Y(x) := y(z)^2$ writes

where $D$ is an integer and $F_d(x)$ an irreducible polynomial of degree $d$ over $\mathbb{Q}$.

• Raw degree: $\deg N = 31 = 2\mu^* + 1$.
• Multiplicity of $(2x-1)$: $|S^*| = 3$ (triple coincidence $q = 1$ for $p = 3, 5, 7$).
• Squarefree degree: $31 - 2 = 29 = 2\mu^* - 1$.
• For $y^2 = N_{\rm sqf}(x)$ with $\deg N_{\rm sqf}$ odd squarefree, the genus equals $g = (\deg N_{\rm sqf} - 1)/2$ (Hartshorne IV §1, Thm 1.3).
• So $g = (29 - 1)/2 = 14 = \mu^* - 1$.

Arithmetically structured invariants

Non-existence theorem for $\mu > 17$

Theorem: for the odd primes sieve $\mathbb{P}_{\rm odd}$, there is no self-coherence solution with $\mu^* > 17$.

Proof (sketch):

1. Uniform asymptotic: $\gamma_n(\mu) = g(2n/\mu) + O(1/\mu)$ with $g(x) = x/(e^x - 1)$.
2. Threshold: $g(x) = 1/2 \iff e^x = 1 + 2x \iff x_0 \approx 1.2564$.
3. Activation: $\gamma_p(\mu) > 1/2$ asymptotically $\iff p < (x_0/2) \cdot \mu \approx 0.6282 \mu$.
4. Exhaustive verification: for $\mu \in [18, 1000]$, $T(\mu) := \sum_{p \in S(\mu)} p > \mu$ (mpmath 30 digits computation).
5. Analytic bound: for $\mu > 1000$, we use Dusart’s 2010 theorem (Thm 5.2): $\sum_{p \leq N} p \geq N^2/(2 \ln N)$ valid for $N \geq 563$. At $\mu = 1000$, $N = 600 \geq 563$, bound valid. Combined with super-linear growth of $T(\mu)/\mu$, we conclude $T(\mu) > \mu$ for all $\mu > 17$.

Combined with exhaustive search for $\mu \leq 17$: the only solutions are $\mu^* = 15$ ($S^* = \{3,5,7\}$ on $\mathbb{P}_{\rm odd}$) and $\mu^* = 17$ ($S = \{2,3,5,7\}$ if we allow $p = 2$). Under restriction to $\mathbb{P}_{\rm odd}$, uniqueness is strict.

Bridge with the quasi-soliton

The persistence curve and the asymptotic expanding quasi-soliton (companion paper ricci_soliton_fr.tex) are two faces of the same geometric object. A quantitative bridge connects the geometric residue $\lambda(\mu)$ to the leading coefficient $B_0(\mu) = 1/(8 a_1(\mu))$ of the correlator $\omega_{1,1}^{\sieve}$:

The exact identity (explicit constant $C$) remains an open program.