NUMEN: Dynamics & Nuclear Experiments

• NUMEN is a dual framework that rigorously applies non-Archimedean analytic methods to arithmetic dynamics and experimental nuclear physics.
• It characterizes generalized Collatz maps using p-adic functions to map periodic and divergent orbits through explicit analytic and spectral techniques.
• The NUMEN collaboration integrates advanced detector technologies with quantum reaction models to extract 0νββ nuclear matrix elements from double charge exchange data.

NUMEN

NUMEN refers to a family of rigorous mathematical and experimental frameworks, as well as specific formal constructions, underpinning two domains: (1) non-Archimedean analytic approaches to Collatz-type (generalized $px+1$) dynamical problems, and (2) the international NUMEN collaboration and its experimental methods for extracting nuclear matrix elements (NMEs) of neutrinoless double beta decay ($0\nu\beta\beta$) from heavy-ion-induced double charge exchange (DCE) reaction cross sections. The term "numen" appears as a technical label for characteristic $p$-adic functions essential to the symbolic parameterization of orbit structures in arithmetic dynamics, as well as a brand for the integrated theory-experiment program targeting one of the most fundamental unresolved questions in neutrino physics: the quantitative mapping of DCE observables to $0\nu\beta\beta$ NMEs.

1. Mathematical Definition: The Numen of Generalized Collatz Maps

Let $p$ be an odd prime and consider the generalized Collatz map $H_p : \mathbb{Z} \to \mathbb{Z}$, defined by

$$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$

A core construction is the numen $\chi_p$, defined for $n \in \mathbb{N}_0$ by the recursion

$$\chi_p(0) = 0, \quad \chi_p(2n) = \chi_p(n), \quad \chi_p(2n+1) = p \chi_p(n) + 1$$

which extends uniquely to a "rising-continuous" function $0\nu\beta\beta$0 (Siegel, 2020). This function provides an explicit analytic parameterization of the orbits under $0\nu\beta\beta$1, yielding that the set of periodic points of $0\nu\beta\beta$2 coincides exactly with the set of rational integer values attained by $0\nu\beta\beta$3 on $0\nu\beta\beta$4 (Siegel, 2020). For arbitrary $0\nu\beta\beta$5-ary piecewise affine maps $0\nu\beta\beta$6 on rings of integers $0\nu\beta\beta$7 of a global field $0\nu\beta\beta$8 (the "Hydra map"), the numen generalizes to $0\nu\beta\beta$9 as the unique solution to the system

$p$0

with affine-linear branches $p$1 determined by $p$2, and operator-norm contraction at each place $p$3 ensuring analytic continuation (Siegel, 19 Jan 2026).

The numen encodes the dynamics of $p$4 in non-Archimedean terms, allowing arithmetic-dynamical properties (e.g., periodicity versus divergence) to be recast into the q-adic or adelic value-distribution of an analytic function on $p$5 (Siegel, 2020, Siegel, 19 Jan 2026).

2. Periodicity, Divergence, and the Correspondence Principle

For any $p$6, the following fundamental correspondences hold:

• $p$7 is a periodic point of $p$8 if and only if there exists $p$9 such that $0\nu\beta\beta$0.
• If $0\nu\beta\beta$1 and $0\nu\beta\beta$2, then the $0\nu\beta\beta$3-orbit of $0\nu\beta\beta$4 diverges (tends to $0\nu\beta\beta$5) (Siegel, 2020).

These relationships, collectively termed the "Correspondence Principle," demonstrate that periodic and divergent structures of all generalized $0\nu\beta\beta$6 maps are entirely determined by the numen's value-distribution: periodic orbits correspond to rational values achieved by $0\nu\beta\beta$7 on non-integer 2-adics, while certain irrational 2-adics correspond to divergent orbits. This principle generalizes to piecewise-affine Hydra maps on $0\nu\beta\beta$8, where all periodic points are achieved as images under $0\nu\beta\beta$9 of rational $p$0-adic points not in $p$1 (Siegel, 19 Jan 2026).

3. Analytic and Spectral Properties

The numen is uniquely defined by its base-$p$2 digit functional recursions, and under explicit operator-norm contraction can be extended to a measurable or continuous function on $p$3 with values in completions $p$4 (Siegel, 19 Jan 2026, Siegel, 2020). Analytically, $p$5 is often locally constant, and its pushforward of Haar measure yields a self-similar probability measure $p$6 on $p$7: $p$8 The $p$9-adic or $H_p : \mathbb{Z} \to \mathbb{Z}$0-adic Fourier transform (characteristic function) of $H_p : \mathbb{Z} \to \mathbb{Z}$1 satisfies a functional equation of iterated function-system (IFS) type, with implications for spectral gaps and the distribution of periodic values (Siegel, 19 Jan 2026). In Collatz-type cases, the numen further admits Dirichlet- and Mellin-transform techniques, enabling contour-integral reformulations of periodic point counting and the analysis of analytic continuation and pole structures (Siegel, 2021).

A notable technical point is that the distribution of periodic points translates into the zeros of explicit contour integrals involving Dirichlet-GF (generating function) transforms of $H_p : \mathbb{Z} \to \mathbb{Z}$2 and associated $H_p : \mathbb{Z} \to \mathbb{Z}$3 functions, whose poles and growth in left half-planes elucidate critical obstructions for closed-form summation (Siegel, 2021).

4. NUMEN Collaboration: Experimental Determination of NMEs via DCE

NUMEN also denotes the international collaboration led by INFN-LNS, focused on experimentally extracting the nuclear matrix elements $H_p : \mathbb{Z} \to \mathbb{Z}$4 relevant to $H_p : \mathbb{Z} \to \mathbb{Z}$5 decay. The program is founded on the formal analogy between heavy-ion double charge-exchange (DCE) reactions of the type

$H_p : \mathbb{Z} \to \mathbb{Z}$6

and the double beta decay process in candidate isotopes. In both cases:

• Initial and final many-body wavefunctions are identical.
• Transition operators are closely related, sharing Fermi, Gamow–Teller, and tensor structure (Cappuzzello et al., 2018, Bellone et al., 2019, Cavallaro et al., 2020).

The NUMEN methodology involves measuring absolute DCE cross sections in reactions such as $H_p : \mathbb{Z} \to \mathbb{Z}$7O, $H_p : \mathbb{Z} \to \mathbb{Z}$8Ne projectiles on $H_p : \mathbb{Z} \to \mathbb{Z}$9Ca, $$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$0Cd, etc., at forward angles and elevated intensity, relating these cross sections to $$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$1 via

$$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$2

where $$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$3 and $$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$4 encode reaction dynamics and operator structure, and $$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$5 is the sought-after NME (Cappuzzello et al., 2016).

The collaboration operates the upgraded MAGNEX spectrometer to achieve the required energy, angular, and mass resolution at high rates, employing advanced gas tracker detectors (THGEM/M-THGEM) and $$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$6–$$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$7 particle identification walls based on large-area SiC–CsI(Tl) telescopes, capable of operation under fluences up to $$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$8 ions/cm$$H_p(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ (px+1)/2 & \text{if } x \text{ is odd} \end{cases}$$9 and track rates up to tens of MHz (Lombardo et al., 29 Apr 2025, Carbone et al., 2024, Spatafora et al., 19 Jan 2026, Ciraldo et al., 2023, Carbone et al., 2017).

5. Reaction and Structure Theory Integration

NUMEN’s extraction of NMEs relies on combining DCE measurements with fully quantum-mechanical reaction theory. For most systems, the DCE process is modeled as a sequential two-step (DSCE) process, with the DCE amplitude given by

$\chi_p$0

where $\chi_p$1 is the charge-exchange interaction, $\chi_p$2 is the intermediate-state propagator, and the $\chi_p$3 are distorted waves. At forward angles, the cross section factorizes, and measured yields are directly sensitive to double-Fermi and double-Gamow–Teller NMEs (Bellone et al., 2019).

A critical component is the deployment of structure models (large-scale shell model, QRPA, IBM-2, Energy Density Functional) to provide one- and two-body transition densities. Recent work has connected single- and double-nucleon transfer cross sections (measured in the same campaign) with shell-model and Interacting Boson–Fermion–Fermion Model (IBFFM) spectroscopic amplitudes, which are also used in the computation of $\chi_p$4 NMEs (Sgouros et al., 2021, Sgouros et al., 2023, Vsevolodovna et al., 2021).

The cross-validation of reaction theory (via DWBA/CCBA) and empirically tuned structure inputs is systematically benchmarked on transfer reactions and is key to reducing uncertainty in $\chi_p$5 down to 15–20% in target nuclei (Cappuzzello et al., 2018, Cappuzzello et al., 2016, Bellone et al., 2019).

6. Technical Advances in Detector and Data Acquisition Systems

To achieve the necessary background rejection, sensitivity ($\chi_p$6 nb), and particle discrimination ($\chi_p$7 for $\chi_p$8), NUMEN has advanced detector technologies:

• Development of $\chi_p$9 mm$n \in \mathbb{N}_0$0 fully depleted SiC sensors (100 $n \in \mathbb{N}_0$1m active layer) with energy resolution $n \in \mathbb{N}_0$2 FWHM at MeV-scale $n \in \mathbb{N}_0$3 energies, validated for depletion uniformity and high radiation tolerance at up to $n \in \mathbb{N}_0$4 ions/cm$n \in \mathbb{N}_0$5 fluence (Carbone et al., 2024, Spatafora et al., 19 Jan 2026).
• Construction and beamline validation of 720 SiC–CsI(Tl) $n \in \mathbb{N}_0$6–$n \in \mathbb{N}_0$7 telescopes forming a high-granularity PID wall for the MAGNEX focal plane (Lombardo et al., 29 Apr 2025).
• Implementation of micro-pattern gas detectors (THGEM/M-THGEM) operated at low-pressure isobutane for high-rate heavy-ion tracking (Ciraldo et al., 2023).
• Integration of fast, low-noise, multichannel front-end electronics and FPGA/data-driven DAQ to support event rates approaching $n \in \mathbb{N}_0$8/s per tower.

These systems attain the sub-nanobarn DCE cross-section sensitivity and PID purity required to extract ground-state matrix elements in $n \in \mathbb{N}_0$9 candidates (Lombardo et al., 29 Apr 2025, Carbone et al., 2024).

7. Broader Implications and Future Prospects

The numen formalism in arithmetic dynamics—non-Archimedean analytic encoding of symbolic orbits—provides a new lens for the study of Collatz-type problems, relating dynamical periodicity/divergence questions to analytic, spectral, and measure-theoretic properties (e.g., through $$\chi_p(0) = 0, \quad \chi_p(2n) = \chi_p(n), \quad \chi_p(2n+1) = p \chi_p(n) + 1$$0-adic Fourier transforms, Dirichlet series, and self-similar measures) (Siegel, 2020, Siegel, 2021, Siegel, 19 Jan 2026). Ongoing research emphasizes global-field generalization, adelic spectral methods, and rigorous mapping between the analytic structure of numens and the completeness properties of dynamical orbits.

On the experimental and nuclear theory front, the NUMEN collaboration represents a major effort to connect nuclear reaction data to fundamental lepton-number-violating processes. By providing high-precision DCE data, establishing formal factorization between DCE cross sections and $$\chi_p(0) = 0, \quad \chi_p(2n) = \chi_p(n), \quad \chi_p(2n+1) = p \chi_p(n) + 1$$1 NMEs, and benchmarking nuclear structure models, NUMEN aims to substantially reduce matrix-element uncertainties, thus sharpening the physics potential of future $$\chi_p(0) = 0, \quad \chi_p(2n) = \chi_p(n), \quad \chi_p(2n+1) = p \chi_p(n) + 1$$2 searches (Cappuzzello et al., 2018, Cappuzzello et al., 2016, Cavallaro et al., 2020).

The unifying theme is that numens—in both their analytic and experimental incarnations—serve as the bridge between discrete dynamical structure and observable or theoretically relevant quantities: be it the arithmetic intractability of orbit classification, or the quantum matrix elements that define the neutrino mass scale.