A geometry of fermions

A geometry of fermions François Ritter*,1 1Department of Geosciences and Natural Resource Management, University of Copenhagen; Øster Voldgade 10, Copenhagen K., 1350, Denmark. * [email protected] Context & disclaimer. This study was conducted outside my primary field of expertise and outside working hours. It began as a Saturday-night exploration of a fermionic puzzle, which gradually evolved into a coherent system and, eventually, a tentative law whose results became too intriguing to leave unpublished. I acknowledge the lack of formal theoretical rigor, as the approach is purely geometrical — a feature that can be considered both a strength and a limitation. This work was carried out independently, in an effort to preserve a fresh perspective and explore the geometrical patterns as far as possible. I hope the community will receive this initiative with openness and understanding — taking what is useful, rejecting what is flawed, and, where possible, integrating it into the rigorous foundations of the Standard Model. François Ritter Abstract Longstanding questions in the Standard Model’s fermionic sector — including the unexplained diversity and origin of quantum numbers, the fermion mass hierarchy, and the mechanism generating neutrino masses — are addressed through a purely geometrical, phenomenological framework. The construction relies solely on experimental inputs — measured fermion masses and quantum assignments — and introduces four discrete numbers, or seeds (𝑛,𝑘,𝑛′,𝑘′), from which both quantum numbers and mass relations emerge. Quantum numbers are expressed as second-order polynomials in (𝑘,𝑘′), while fermion masses follow a universal exponential law depending only on the seeds and six constants (𝑚𝑒,𝛼,𝛽,𝛼′,𝛽′,𝜆). With the electron mass 𝑚𝑒 fixed as reference, all known fermion masses are reproduced within instrumental uncertainties, showing that a six-parameter model suffices to capture the full hierarchy. The framework extends naturally to the case of Dirac-type neutrinos: a geometrical singularity in the limit 1/𝑌𝑅→∞ with 𝑌𝑅 the right-handed hypercharge allows generation of extremely small masses consistent with current observations. Flavor mixing — in the Cabibbo– Kobayashi–Maskawa (CKM) and Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrices — emerges from transitions in the (𝑛,𝑘) seed space, offering a unified geometrical explanation of quark and lepton flavor structures. Three main predictions arise. (i) A new charged lepton with electron-like properties and predicted mass 8.5938−0.0004 +0.0004 MeV. Remarkably, a 2024 experimental report (Anikina, Nikitin & Rikhvitsky, 2024) observed a lepton-like state at 8.5±2.5 MeV, which they named the anomalon (𝑎−), consistent with this prediction. This observation, if confirmed, would constitute the first experimental validation of the geometrical mass framework. (ii) A third fermionic branch, the leonids, emerges for extended seed values, hinting at hidden or dark sectors. (iii) A fourth branch, the apex, restores gauge anomaly cancellation; these states carry color 8, forming a fermionic counterpart of the gluons and completing the geometrical spectrum. In parallel, the natural emergence of the golden ratio 𝜑 and its conjugate 𝜑∗ in left-right symmetric states suggests a deep algebraic symmetry underlying fermionic organization. Future experimental scrutiny, particularly of the anomalon, will determine whether this geometrical framework uncovers the missing structure of matter. Introduction The matter content of the Standard Model (SM) consists of fermions, which include up-type quarks (up 𝑢, charm 𝑐, top 𝑡), down-type quarks (down 𝑑, strange 𝑠, bottom 𝑏), charged leptons (electron 𝑒−, muon 𝜇−, tau 𝜏−), and their associated neutrinos (𝜈𝑒, 𝜈𝜇, 𝜈𝜏). The SM provides an accurate description of their interactions and properties, yielding predictions for observed phenomena ranging from beta decay to high-energy collider processes (Glashow, 1961). Despite its well-established success over multiple decades, the SM still leaves several fundamental questions about fermions unanswered (Altmannshofer & Greljo, 2024). In particular, four central gaps can be highlighted: Origin of quantum numbers. In the SM, quantum numbers such as hypercharge (for leftand righthanded fermions, 𝑌𝐿 and 𝑌𝑅), left-handed weak isospin (𝑇3𝐿), baryon number (𝐵), and lepton number (𝐿) are not derived from first principles; rather, they are assigned to ensure internal consistency. Their values are chosen to guarantee gauge anomaly cancellation, to preserve empirical conservation laws (such as 𝐵 and 𝐿 at the perturbative level, see Hooft, 1976), to determine the embedding of fermions within SU(2) doublets (via the weak isospin 𝑇3𝐿) and to reproduce the observed electric charges (𝑄) of the fermions. However, the SM provides no deeper explanation for how these particular quantum numbers emerge. Fermion masses and Yukawa couplings. In the SM, fermion masses originate from Yukawa interactions with the Higgs field after electroweak symmetry breaking (Higgs, 1964). However, the Yukawa couplings themselves are arbitrary free parameters, fixed empirically for each particle (Weinberg, 1967). Thus, while the SM accommodates the observed fermion masses, it does not predict them, leaving the origin of the mass hierarchies and the pattern of three generations unexplained. Neutrinos. Neutrinos pose a fundamental challenge to the SM, as their unique properties are not naturally accommodated within the framework. They are the only fermions in the SM with vanishing electric charge (𝑄=0). In the minimal SM they are strictly massless, since right-handed neutrino fields are absent and no Yukawa couplings to the Higgs can be written. Experimentally, however, neutrino oscillations demonstrate that neutrinos have nonzero but extremely small masses, many orders of magnitude below those of charged fermions (Fukuda et al., 1998). This discrepancy indicates that the SM is incomplete and suggests the existence of new physics, such as right-handed neutrinos or alternative mass-generation mechanisms (e.g. seesaw models, see Minkowski, 1977). This structural discrepancy between neutrinos and the other fermions challenges the SM’s unified description of matter fields. Flavor structure. The SM incorporates quark mixing through the Cabibbo–Kobayashi–Maskawa (CKM) matrix, which governs flavor-changing weak interactions (Cabibbo, 1963). While the formalism is consistent with experiment, the observed hierarchies among the CKM elements lack theoretical explanation within the SM (Kobayashi & Maskawa, 1973). The leptonic analogue is the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, which describes neutrino oscillations (Maki et al., 1962). The PMNS encodes the mismatch between neutrino mass eigenstates and weak interaction eigenstates. In contrast to the CKM case, the PMNS matrix exhibits large mixing angles and a markedly different structure. At present, the SM provides no mechanism that relates the two matrices or accounts for the striking disparity between quark and lepton flavor mixing. The aim of this study is to address these longstanding open questions in particle physics by adopting a purely phenomenological and geometrical approach constructed solely from the observed fermion masses and the quantum numbers. Three main results can be summarized as follows: - Common emergence of quantum numbers and fermion masses. Both quantum numbers and fermion masses (captured within instrumental uncertainties) are generated by a novel set of four numbers (𝑛,𝑘,𝑛′,𝑘′), referred to as seeds, introduced in this study. Quantum numbers are given as second-order polynomials in the variables 𝑘 and 𝑘′. Fermion masses are modeled by an exponential function depending solely on (𝑛,𝑘,𝑛′,𝑘′) and parametrized by six constants (𝑚𝑒,𝛼,𝛽,𝛼′,𝛽′ and 𝜆). In this framework, the electron mass 𝑚𝑒 is fixed as the reference constant (given, not predicted), from which all other fermion masses are determined. - Neutrino mass generation. The mass law can be naturally extended to account for neutrino masses, thereby covering the full fermion spectrum. This is achieved through the geometrical emergence of a term proportional to 1/𝑌𝑅, which was initially constructed for charged leptons and quarks only. For neutrinos, where 𝑌𝑅=0, this term behaves as a singularity, which, when properly regulated, can produce neutrino masses consistent with experimental observations. Neutrinos are therefore treated as Dirac-type particles within this framework. - Explanation of the CKM hierarchy. The observed hierarchy in the CKM matrix is explained through transitions in the (𝑛,𝑘) space with 𝑛 and 𝑘 the seeds specific to the left-handed sector. The framework can be consistently extended to the PMNS matrix. This therefore provides a unified description of the quark and lepton flavor structure. In particular, this study advances three predictions, one well-supported and the two others more speculative: 1. A new lepton. It predicts the existence of a new elementary charged lepton, with properties analogous to those of the electron, muon, and tau, and a mass of 8.5938−0.0004 +0.0004 MeV. Independently, a 2024 study reported the observation of a charged particle carrying an electron-like lepton number and a measured mass of 8.5 ± 2.5 MeV (Anikina, Nikitin & Rikhvitsky, 2024). I identify this particle with the lepton predicted here and adopt the name introduced in that study, anomalon (symbol 𝑎−). This charged lepton necessarily requires a neutrino companion in this framework. If confirmed, this observation would constitute the most significant validation of the model. 2. A third fermionic branch. It further speculates the existence of a third fermionic branch, complementing the quarks and leptons. This new branch, referred to as the leonids (with their full set of properties detailed in Table 4 and Fig. 5), arises naturally from extending the seed 𝑘 beyond its original range (0, 1, 2, 3) to encompass (−1, 0, 1, 2, 3, 4). 3. A fourth fermionic branch. A final branch, termed the apex, has been introduced to restore anomaly cancellation, which is otherwise disrupted by the inclusion of the anomalon and leonids. The apex 1 form a distinct set of fermions whose full quantum assignments and masses are detailed in Table 5 and Fig. 5. 1 with the imposed same form in singular and plural: one apex or two apex. The results of this study point to a deep connection between the geometrical properties of the framework and the fundamental particles and interactions of the SM, offering new insights into the underlying structure of particle physics. Remarkably, the golden ratio 𝜑 and its conjugate 𝜑∗ emerge naturally within the construction, associated with states whose leftand right-handed hypercharges are identical. Future experimental tests of the predictions derived in this study will ultimately determine whether the search for completeness should be pursued further or abandoned. The presentation is organized as follows. First, the concept of seeds is introduced (Fig. 1, Tables 1–2). Next, the mass law and its detailed derivation are presented (Figs. 2–4, Tables 2–3), followed by an analysis of the CKM and PMNS matrices within the framework of the model. The discussion then begins by reviewing the well-supported predictions of the study, before turning to the more speculative extensions in which the leonids (Fig. 5, Table 4) and apex (Fig. 5, Table 5) are introduced to complete the geometrical structure of the framework. The final section outlines the open questions that remain in relation to the SM and presents the concluding remarks. The seeds The geometrical construction of the mass law (detailed in the following two sections) gives rise to four numbers 𝑛,𝑘,𝑛′ and 𝑘′, designated in this study as seeds. Each fermion is associated with a quartet of seeds (𝑛,𝑘,𝑛′,𝑘′). The pair (𝑛,𝑘) corresponds to the left-handed sector, while (𝑛′,𝑘′) corresponds to the right-handed sector. The seeds 𝑘 and 𝑘′ generate all known fermionic quantum numbers (see below, Table 1 and Fig. 1), while the full quartet (𝑛,𝑘,𝑛′,𝑘′) generates the masses of all fermions within experimental uncertainties (Table 2). The electron corresponds to (𝑘,𝑛,𝑘′,𝑛′)=(0,0,0,0) and is regarded as the origin of all fermionic masses, as required by the geometrical constraints of the system (Fig. 2). Detailed explanation of each seed: - The generation seed 𝑛={0,1,2,3} is the generation number as known in the SM, with the exception of the electron and its neutrino companion 𝜈𝑒, which are uniquely assigned 𝑛=0. - The left-charge seed 𝑘={0,1,2,3} generates the left-handed hypercharge (𝑌𝐿), baryon number (𝐵) and lepton number (𝐿). Charged leptons are associated with 𝑘=0, neutrinos are associated with 𝑘= 3. The up-type and down-type quarks are mixed between 𝑘=1 and 𝑘=2: The (𝑢,𝑠,𝑏) quarks are associated with 𝑘=1, and the (𝑑,𝑐,𝑡) quarks are associated with 𝑘=2. This permutation 𝑘𝑢↔𝑘𝑑 is a fundamental constraint imposed by the geometry (Fig. 2). - The residual seed 𝑛′={−1,0,1} has been imposed by geometry to distribute mass residuals into negative, neutral, and positive linear relations that are symmetrical (Fig. 4). No clear physical interpretation has emerged for it yet. - The right-charge seed 𝑘′={0,1,2} generates the right-handed hypercharge (𝑌𝑅) and electric charge (𝑄). Charged leptons are associated with 𝑘′=0, down-type quarks with 𝑘′=1, and up-type quarks with 𝑘′=2. Neutrinos are mathematically extended to the right-handed sector with 𝑘𝜈′=−3+√33 2 (this will impose 𝑌𝑅=0, see below). All fermionic quantum numbers in the SM (𝑌𝐿, 𝐵, 𝐿, 𝑌𝑅, 𝑄, 𝑇3) are generated from 𝑘 and 𝑘′ via second-order polynomials: 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 ; 𝑄(𝑘′)=16𝑘′2+12𝑘′−1 ; 𝑌𝐿(𝑘)= −23𝑘2+ 2𝑘−1 ; 𝐵(𝑘)=−16𝑘2+12𝑘 ; 𝐿(𝑘)=12𝑘2−32𝑘+1 ; and 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)). Fig. 1. Polynomial formulation of the hypercharges. The left-handed hypercharge 𝑌𝐿 and the righthanded hypercharge 𝑌𝑅 can be expressed as two polynomials: 𝑌𝐿(𝑘)= −23𝑘2+2𝑘−1 and 𝑌𝑅(𝑘′)= 13𝑘′2+𝑘′−2, with 𝑘 and 𝑘′ the left-charge and right-charge seeds, respectively. Right-handed neutrinos (𝜐𝑅) are depicted as an empty square at 𝑘𝜐′=−3+√33 2, arising as a mathematical extension of the system with 𝑌𝑅(𝑘𝜐′)=0. The left-handed weak isospin is defined as 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)− 𝑌𝐿(𝑘)) for which the golden ratio 𝜑 appears as a root. The symbol 𝑎 corresponds to a fourth hypothetical charged lepton called the anomalon. In this framework, the left weak isospin connects the left-handed and the right-handed sector through 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)). Interestingly, for 𝑘′=𝑘= 𝜑= 1+√5 2 (the golden ratio), one finds 𝑇3𝐿(𝜑,𝜑)=0. At this special value, there is no distinction between leftand right-handed hypercharges due to 𝑌𝑅(𝜑)=𝑌𝐿(𝜑)= −1+2√5 3≈0.4907 (Fig. 1). Right-handed neutrinos can be mathematically extended with 𝑘𝜈′=−3+√33 2, which satisfies 𝑌𝑅(𝑘𝜈′)= 0 and 𝑄(𝑘𝜈′)=0. This is necessary to assign 𝑇3=𝑇3𝐿(𝑘=3,𝑘′=𝑘𝜈′)= +12 to the neutrinos, but also to generate their mass with variations around 𝑘𝜈′ (see the section extension to the neutrinos). Color, being a vector property in SU(3) space, is not a scalar quantum number and is not generated with the seed numbers. Finally, transforming matter into antimatter simply reverses the signs of all coefficients in the polynomials defining the quantum numbers, leaving the seeds unchanged. Seeds Quantum numbers 𝑘 𝑘′ 𝑌𝐿 𝑌𝑅 𝑄 𝑇3𝐿 𝐿 𝐵 𝑒− 0 0 → → → → -1 -2 -1 -1/2 1 0 𝑎− 0 0 -1 -2 -1 -1/2 1 0 𝜇− 0 0 -1 -2 -1 -1/2 1 0 𝜏− 0 0 -1 -2 -1 -1/2 1 0 𝑑 2 1 → → → +1/3 -2/3 -1/3 -1/2 0 +1/3 𝑠 1 1 +1/3 -2/3 -1/3 -1/2 0 +1/3 𝑏 1 1 +1/3 -2/3 -1/3 -1/2 0 +1/3 𝑢 1 2 → → → +1/3 +4/3 +2/3 +1/2 0 +1/3 𝑐 2 2 +1/3 +4/3 +2/3 +1/2 0 +1/3 𝑡 2 2 +1/3 +4/3 +2/3 +1/2 0 +1/3 𝜐𝑒 3 𝑘𝜈′ → → → → -1 0 0 +1/2 1 0 𝜐𝑎 3 𝑘𝜈′ -1 0 0 +1/2 1 0 𝜐𝜇 3 𝑘𝜈′ -1 0 0 +1/2 1 0 𝜐𝜏 3 𝑘𝜈′ -1 0 0 +1/2 1 0 Table 1. Generation of the quantum numbers from the seeds 𝒌 and 𝒌′. The left-handed hypercharge 𝑌𝐿(𝑘), the right-handed hypercharge 𝑌𝑅(𝑘′), the electric charge 𝑄(𝑘′), the left-handed weak isospin 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)) , the lepton number 𝐿(𝑘) and the baryon number 𝐵(𝑘) can be expressed as second-order polynomials taking into arguments the left-charge seed 𝑘 (associated with the left-handed sector) and the right-charge seed 𝑘′ (associated with the right-handed sector). The value 𝑘𝜈′=(−3+√33)/2 represents a mathematical extension of neutrinos into the right-handed sector. The symbol 𝑎− corresponds to a fourth hypothetical charged lepton called the anomalon, with 𝜐𝑎 its neutrino companion. The mass law Presentation of the law The following law reproduces the masses of the charged leptons and quarks within their experimental uncertainties (Table 2). Its extension to neutrinos arises naturally through the singular limit 1 𝑌𝑅→ ∞, which enables the generation of extremely small masses consistent with measurements (see section extension to the neutrinos). 𝑚pred(𝑛,𝑘,𝑛′,𝑘′)=𝑚𝑒𝑒(𝑘+𝑛−3)𝑒𝛼𝑘+𝛽−3(𝑘−𝑒𝛽)𝜆(𝑘′/2)+𝑛′(𝛼′𝑘′+3 𝑌𝑅(𝑘′)+𝛽′) with: - 𝑚pred the predicted mass of a fermion. - (𝑛,𝑘,𝑛′,𝑘′) the four seeds. - 𝑚𝑒 the observed mass of the electron. This value is given, not predicted. - 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 the right-handed hypercharge - 𝛼,𝛽,𝛼′,𝛽′ and 𝜆 five empirical constants (values shown in Table 3). The anomalon: a fourth charged elementary lepton An empty slot for a new particle appears at (𝑛,𝑘,𝑛′,𝑘′)=(1,0,1,0), corresponding to a predicted mass of 8.5938−0.0004 +0.0004 MeV and charge 𝑄(𝑘′=0)=−1. A search in the literature revealed a striking candidate: Anikina, Nikitin, and Rikhvitsky (2024) reported the discovery of a charged particle of mass 8.5±2.5 MeV sharing the leptonic number of an electron. They named this particle the anomalon, and its nature remained elusive. I adopt this terminology here, denote it by the symbol 𝑎− associated with an observed mass of 𝑚𝑎=8.5±2.5 MeV (Fig. 2,3,4). The anomalon has been fully integrated in the present framework, and is interpreted as a new elementary charged lepton between the electron and the muon. The anomalon necessarily requires a neutrino companion 𝜈𝑎 to fill the slot (𝑛,𝑘)=(1,3) (see section neutrinos). Seeds Predicted mass Observed mass 𝑛 𝑘 𝑛′ 𝑘′ 𝑚pred (MeV) 𝑚obs (MeV) 𝑒− 0 0 0 0 → given, not predicted 0.51099895000−0.00000000015 +0.00000000015 𝑎− 1 0 1 0 → 8.5938−0.0004 +0.0004 8.5−2.5 +2.5 𝜇− 2 0 -1 0 → 105.6583755−0.0000022 +0.0000022 105.6583755−0.0000023 +0.0000023 𝜏− 3 0 0 0 → 1776.93−0.09 +0.09 1776.93−0.09 +0.09 𝑑 1 2 1 1 → 4.69−0.01 +0.01 4.70−0.07 +0.07 𝑠 2 1 0 1 → 93.4−0.2 +0.2 93.5−0.8 +0.8 𝑏 3 1 -1 1 → 4183−7 +7 4183−7 +7 𝑢 1 1 0 2 → 2.18−0.01 +0.01 2.16−0.07 +0.07 𝑐 2 2 1 2 → 1273.2−3.1 +3.1 1273.0−4.6 +4.6 𝑡 3 2 -1 2 → 172570−280 +280 172570−290 +290 Extension to the neutrinos (hypothesis: Normal Order, mυe=0 MeV, and flavor masses presented here as mass eigenstates for simplicity.) 𝜐𝑒 0 3 0 𝑘𝜈′ → 0 consistent with 0 MeV 𝜐𝑎 1 3 ±1 𝑘𝜈′ ±𝜖𝑎 → ≳45600 unobserved yet 𝜐𝜇 2 3 ±1 𝑘𝜈′ ∓𝜖𝜇 → (0.00865−0.00011 +0.00011)×10−6 (0.00865−0.00011 +0.00011)×10−6 𝜐𝜏 3 3 ±1 𝑘𝜈′ ∓𝜖𝜏 → (0.05013−0.00020 +0.00020)×10−6 (0.05013−0.00021 +0.00021)×10−6 Table 2. Generation of the fermionic masses. All predicted masses (in MeV) are generated from the mass law 𝑚pred(𝑛,𝑘,𝑛′,𝑘′) parametrized by six constants (𝑚𝑒,𝛼,𝛽,𝛼′,𝛽′ and 𝜆). Predicted values correspond to median and 2.5th and 97.5th percentiles of ~105 sets that individually reproduce the observed masses (PDG 2024) within experimental uncertainties (details in Table 3). The symbol 𝑎− corresponds to a fourth hypothetical charged lepton called the anomalon (Anikina, Nikitin & Rikhvitsky, 2024), with 𝜐𝑎 its neutrino companion. Extension to the neutrinos (NuFit v6.0, see Esteban et al., 2024) is performed with three additional constants (𝜖𝑎,𝜖𝜇 and 𝜖𝜏) that regulate the singularity 1 𝑌𝑅(𝑘𝜈′)→ ∞ with 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 the right-handed hypercharge and 𝑘𝜈′=(−3+√33)/2. For the neutrinos 𝜐𝑎, 𝜐𝜇 and 𝜐𝜏, the value of 𝑛′ is either 1 or −1 (undetermined yet). Fig. 2. Origin of the mass law. Three linear relationships (panel a) appear in logarithmic space with respect to natural integers. They are derived from specific ratios of observed fermionic masses given in sets 𝐴𝑘 and 𝐵. The anomalon 𝑎 (empty circle) is included with its observed mass of 𝑚𝑎=8.5 MeV to illustrate that it fills the otherwise empty slot at (𝑛,𝑘)=(1,0). The slopes of these three relationships are interconnected and take the form 𝑒𝛼𝑘+𝛽 with 𝑘={0,1,2}. The final relationship (panel b) links the three lines, allowing the entire system to be expressed as 𝑚(𝑘,𝑛)= 𝑚𝑒𝑒(𝑘+𝑛−3)𝑒𝛼+𝑘𝛽−3(𝑘−𝑒𝛽). Residuals from this simplified model remain large and require further correction (Fig. 3 then Fig. 4). Demonstration of the mass law: first part Four linear relationships emerge when four well-chosen sets of masses are placed in a logarithmic space versus natural integers between 0 and 3 (Fig. 1a). Let 𝑁1={0,2,3} and 𝑁2={1,2,3}, then the following relationships are linear (with zero intercept by construction): with 𝐴0={1,𝑚𝜇 𝑚𝑒,𝑚𝜏 𝑚𝑒}, log (𝐴0) versus 𝑁1 is associated with a slope of 𝑒𝛼×0+𝛽 with 𝐴1={1,𝑚𝑠 𝑚𝑢,𝑚𝑏 𝑚𝑢}, log (𝐴1) versus (𝑁2-1) is associated with a slope of 𝑒𝛼×1+𝛽 with 𝐴2={1,𝑚𝑐 𝑚𝑑,𝑚𝑡 𝑚𝑑}, log (𝐴2) versus (𝑁2-1) is associated with a slope of 𝑒𝛼×2+𝛽 with 𝐵={1, 𝑚𝑠 𝑚𝑑,𝑚𝜏 𝑚𝑑}, log (𝐵) versus (𝑁2-1) is associated with a slope of 3 These linear relationships show that masses in each set 𝐴𝑘 are related with an exponential action 𝑒𝛼𝑘+𝛽 with 𝑘={0,1,2} and 𝛼, 𝛽 two constants (Table 3). The integers 𝑁1 and 𝑁2 correspond to the generation number 𝑛, at the exception of the electron which is set to 𝑛=0. The sets 𝐴𝑘 contain all 𝑛(up−type)−𝑛(down−type)=[0 −1 −2 1 0 −1 2 1 0] This achieves to prove the connection between the CKM and transitions in the (𝑛,𝑘) space. PMNS matrix The PMNS matrix is the leptonic counterpart of the CKM matrix, encoding the amplitudes for transitions from charged leptons to neutrino mass eigenstates through the charged weak current mediated by the bosons 𝑊±. Explicitly, it takes the following form (indices 1, 2, 3 are replaced by 𝜈𝑒,𝜈𝜇 and 𝜈𝜏 as each label denotes a single neutrino state, analogous to quark notation): 𝑉𝑃𝑀𝑁𝑆=[𝑉𝑒𝜈𝑒𝑉𝑒𝜈𝜇𝑉𝑒𝜈𝜏 𝑉𝜇𝜈𝑒𝑉𝜇𝜈𝜇𝑉𝜇𝜈𝜏 𝑉𝜏𝜈𝑒𝑉𝜏𝜈𝜇𝑉𝜏𝜈𝜏] with amplitude (simplified from NuFit 6.0 with atmospheric data, see Esteban et al., 2024): |𝑉𝑃𝑀𝑁𝑆|=[0.822−0.020 +0.020 0.550−0.031 +0.031 0.149−0.007 +0.007 0.377−0.125 +0.125 0.588−0.092 +0.092 0.704−0.052 +0.052 0.397−0.121 +0.121 0.579−0.094 +0.094 0.690−0.053 +0.053] As in the quark sector, unitarity ensures probability conservation, but in contrast to the CKM matrix, the PMNS entries display no strong hierarchical pattern: all elements are of comparable size, with large mixing between flavors. This feature underlies the phenomenon of neutrino oscillations and points to a qualitatively different flavor structure in the lepton sector, for which the SM offers no explanation. In this study, the exponential parametrization of the CKM matrix has been mimicked for the PMNS with success by simply replacing the (𝑛,𝑘) indices associated with the down-type (resp. up-type) quarks with the neutrinos (resp. charged leptons). The four matrices 𝑛(charged−leptons), 𝑛(neutrinos), 𝑘(charged−leptons) and 𝑘(neutrinos) are defined as: { 𝑤(charged−leptons)= [𝑤𝑖𝑗(charged−leptons)]=[𝑤𝑒𝑤𝑒𝑤𝑒 𝑤𝜇𝑤𝜇𝑤𝜇 𝑤𝜏𝑤𝜏𝑤𝜏] 𝑤(neutrinos)= [𝑤𝑖𝑗(neutrinos)]=[𝑤𝜐𝑒𝑤𝜐𝜇𝑤𝜐𝜏 𝑤𝜐𝑒𝑤𝜐𝜇𝑤𝜐𝜏 𝑤𝜐𝑒𝑤𝜐𝜇𝑤𝜐𝜏] with 𝑤=𝑛 or 𝑤=𝑘 (values in Table 2). The structure directly parallels that of the CKM mixing matrix: {𝑎𝑖𝑗 𝑃𝑀𝑁𝑆=−𝑏𝜐(𝑛𝑖𝑗 (neutrinos)−𝑛𝑖𝑗 (charged−leptons))𝜆𝜐𝑑𝑖𝑗𝑧𝜐 𝛿3𝑗𝛿𝑖1𝑧𝜐𝛿1𝑗𝛿3𝑖 𝑑𝑖𝑗=2|𝑛𝑖𝑗 (neutrinos)−𝑛𝑖𝑗 (charged−leptons)|+|𝑘𝑖𝑗 (neutrinos)−𝑘𝑖𝑗 (charged−leptons)|−1 which correspond to: 𝐴𝑃𝑀𝑁𝑆= [𝑎𝑖𝑗 𝑃𝑀𝑁𝑆]= 𝑏𝜐[0 −2𝜆𝜐6−3𝜆𝜐8(𝑥𝜐−𝑖𝑦𝜐) 2𝜆𝜐60 −𝜆𝜐4 3𝜆𝜐8(𝑥𝜐+𝑖𝑦𝜐) 𝜆𝜐40] The PMNS is reproduced within experimental uncertainties with the constant 𝑏𝜐=𝑏𝑞=1.231, 𝑦𝜐= 𝑦𝑞=0.2215, 𝑥𝜐=2.2643 and 𝜆𝜐= 0.8602. Only 𝑥𝜐 and 𝜆𝜐 have changed from the CKM case, which was necessary to mitigate the hierarchical pattern (𝜆𝜐 is closer to 1). The computed modules correspond to: |𝑉𝑃𝑀𝑁𝑆|=[0.827 0.544 0.142 0.300 0.640 0.707 0.475 0.543 0.693] The reader should notice that the very large uncertainties of the PMNS matrix allow for various forms of 𝐴𝑃𝑀𝑁𝑆 to match the target matrix, and this apparent success might be an artificial construct. The key finding here is that the CKM structure defined in the (𝑛,𝑘) space remains compatible when applied to the PMNS matrix. Discussion and extension Before addressing potential conflicts with the SM, it is important to clearly state the main claims of this work. The claims are presented in two groups: first the strong, well-supported claims for the known fermions (C), followed by more speculative claims (SC) that predict two new fermionic branches (the leonids and the apex) by extending this framework. The discussion then proceeds to list the open questions (O), followed by a conclusion. Supported claims C1. Four seeds per fermion. Each fermion is associated with four seeds (Table 2), which generate both its quantum numbers and its mass (except for the electron). The seeds (𝑛,𝑘) correspond to the left-handed sector, while the seeds (𝑛′,𝑘′) correspond to the right-handed sector. C2. Emergence of quantum numbers. Quantum numbers emerge from second-order polynomials taking into argument the left-charge seed 𝑘 and/or right-charge seed 𝑘′: 𝑌𝑅(𝑘′)=13𝑘′2+𝑘′−2 ; 𝑄(𝑘′)=16𝑘′2+12𝑘′−1 ; 𝑌𝐿(𝑘)= −23𝑘2+2𝑘−1 ; 𝐵(𝑘)=−16𝑘2+12𝑘 ; 𝐿(𝑘)=12𝑘2−32𝑘+1 and 𝑇3𝐿(𝑘,𝑘′)=12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)). In this framework, quantum numbers are no longer treated as fundamental; instead, they emerge from a deeper underlying structure. Antimatter corresponds to reversing the sign of each coefficient. C3. Mass law. All fermionic masses emerge from the following law: 𝑚pred(𝑛,𝑘,𝑛′,𝑘′)=𝑚𝑒𝑒(𝑘+𝑛−3)𝑒𝛼𝑘+𝛽−3(𝑘−𝑒𝛽)𝜆(𝑘′/2)+𝑛′(𝛼′𝑘′+3 𝑌𝑅(𝑘′)+𝛽′) with 𝑚𝑒 the observed mass of the electron and 𝛼,𝛽,𝛼′,𝛽′ and 𝜆 five empirical constants (Table 3). C4. The electron as reference. The electron is no longer considered part of the first generation. Instead, it serves as the fundamental reference point of the entire fermionic realm, characterized by vanishing seeds. Its mass is not emergent but is treated as a fundamental constant. This aligns with the fact that the electron is the only fermion that is stable and does not undergo decay. All other fermions are unstable and decay into lighter ones, a process which corresponds to transitions between seeds. Within this framework, all fermions can be interpreted as excitations of the electron in the underlying seed space, with their masses and properties corresponding to specific seed configurations. C5. The anomalon. A new elementary lepton, the anomalon, shares the same quantum numbers as other charged leptons (lepton number, baryon number, electric charge, rightand left-handed hypercharges). Its mass is predicted to be 8.5938−0.0004 +0.0004 MeV and was most likely observed in 2024 (Anikina, Nikitin & Rikhvitsky, 2024), prior to the prediction. An associated neutrino companion is necessary otherwise the slot (𝑛,𝑘)=(1,3) would remain open. The nature of this neutrino (either sterile or active) remains unknown. C6. Neutrino masses. The extremely small masses of the neutrinos originate from a singularity related to the right-handed hypercharge in 1/𝑌𝑅 (or possibly the electric charge 𝑄 as they are related by 𝑄=𝑌𝑅/2). Neutrinos follow the same generation as their charged lepton companions. C7. CKM hierarchy. The weighted Manhattan distance 𝑑𝑖𝑗=2|𝑛𝑖−𝑛𝑗|+|𝑘𝑖−𝑘𝑗|−1 provides a quantitative measure of the relative suppression of flavor-changing transitions between quarks 𝑖 and 𝑗 in the (𝑛,𝑘) seed space, effectively encoding the observed hierarchy of the CKM matrix elements. Speculative claims SC1. A third fermionic branch: the leonids. This new fermionic branch (Fig. 5, Table 4) would exist in 𝑘=(−1,4) alongside the quarks (mixed between the up-type and down-type quarks at 𝑘=1 and 𝑘=2) and the leptons (charged leptons at 𝑘=0 and neutrinos at 𝑘=3). Two numerical coincidences motivate the exploration of 𝑘=(−1,4). First, 𝑇3𝐿(−1,−1)=+12, which is highly nontrivial since it arises from the difference between two polynomials that previously satisfy 𝑇3𝐿(0,0)=−12, 𝑇3𝐿(1,1)= −12 and 𝑇3𝐿(2,2)=+12 (see Fig. 5). The second coincidence is that 𝑌𝐿(−1) and 𝑌𝐿(4) are both equal to −11 3. This speculation is summarized in Table 4 and predicts six new particles (three with imaginary masses) that are neither leptons nor quarks. This new branch is named the leonids (singular: one leonid), and it mimics the quarks and leptons in its structure (left-handed doublets, right-handed singlets, and 𝑇3𝐿=±12 ). The leonids are composed of three lions – galion (Λ𝑔), melion (Λ𝑚) and solion (Λ𝑠) - associated with 𝑇3𝐿=+12 (three fermions with real masses). Their companions are the three chimeras — griffin (χ𝑔), manticore (χ𝑚), and sphinx (χ𝑠) — associated with 𝑇3𝐿=−12 (three fermions with complex masses). These names come from the fact that the three chimeras do not exist in the real world (imaginary mass) but are all composed of body parts of lions. Chimeras are natural candidates for the dark sector, as they possess complex masses due to their right-charge seed 𝑘𝜒′= −3+𝑖√23 2 that verifies 𝑌𝑅(𝑘𝜒′)=𝑌𝐿(4)+2𝑇3𝐿= −11 3−1=−14 3 . This extension is similar to the right-handed neutrinos extension for the leptons, where 𝑘𝜈′=−3+√33 2 verified 𝑌𝑅(𝑘𝜈′)=𝑌𝐿(3)+ 2𝑇3𝐿= −1+1=0. The leonids are necessarily colorless for the gauge anomaly cancellation performed in SC2. names seeds predicted mass (MeV) quantum numbers leonids 𝑛 𝑘 𝑘′ if 𝑛′=−1 if 𝑛′=0 if 𝑛′=1 𝑌𝐿 𝑌𝑅 𝑄 𝑇3𝐿 𝐿 𝐵 lions galion Λ𝑔 1 -1 -1 89.30 ±0.27 100.73 ±0.32 113.62 ±0.38 −11 3 −83 −43 +12 3 −23 melion Λ𝑚 2 -1 -1 620.6 ±1.6 700.1 ±1.9 789.7 ±2.3 −11 3 −83 −43 +12 3 −23 solion Λ𝑠 3 -1 -1 4313.4 ±9.4 4865 ±11 5488 ±13 −11 3 −83 −43 +12 3 −23 chimeras griffin χ𝑔 1 4 𝑘𝜒′ (1.16 ± 0.03)107 +𝑖 (1.63 ± 0.02)106 (1.32 ± 0.03)107 +𝑖 (1.71 ± 0.03)106 (1.50 ± 0.04)107 +𝑖 (1.79 ± 0.04)106 −11 3 −14 3 −73 −12 3 −23 manticore χ𝑚 2 4 𝑘𝜒′ (4.19 ± 0.14)1011 +𝑖 (5.93 ± 0.08)1010 (4.78 ± 0.16)1011 +𝑖 (6.22 ± 0.09)1010 (5.45 ± 0.18)1011 +𝑖 (6.49 ± 0.10)1010 −11 3 −14 3 −73 −12 3 −23 sphinx χ𝑠 3 4 𝑘𝜒′ (1.52 ± 0.07)1016 +𝑖 (2.15 ± 0.04)1015 (1.74 ± 0.07)1016 +𝑖 (2.26 ± 0.04)1015 (1.98 ± 0.09)1016 +𝑖 (2.36 ± 0.04)1015 −11 3 −14 3 −73 −12 3 −23 Table 4. Characteristics of the leonids. The names, symbols, seeds, predicted mass and quantum numbers of the new fermionic branch called the leonids (partitioned into the lions and the chimeras) are summarized here. Because 𝑛′ is unknown (distributed between −1,0 and 1), there are three possibilities for the mass of each leonid. Predicted masses correspond to median and 2.5th and 97.5th percentiles of predictions from ~105 sets of (𝛼,𝛽,𝛼′,𝛽′ and 𝜆) that individually reproduce the observed masses of known fermions (PDG 2024) within experimental uncertainties (details in Table 3). The value 𝑘𝜒′=(−3+𝑖√23)/2 is set to verify 𝑌𝑅(𝑘𝜒′)=−14/3 and respect the structure of 𝑇3𝐿= ±1/2. The complex seed 𝑘𝜒′ leads to complex masses through the mass law, which make chimeras natural candidates for the dark sector. The leonids are necessarily colorless to cancel the gauge anomalies (see SC2). SC2. A fourth fermionic branch: the apex. This fourth fermionic branch (Fig. 5, Table 5) has been introduced to cancel the gauge anomalies (see the script in the supplementary). In the SM, the cancellation of gauge anomalies is essential to ensure the internal consistency and renormalizability of the theory. In particular, four types of anomalies must vanish: the cubic 𝑈(1)𝑌 anomaly (∑𝑌3 over all fermions), the mixed 𝑆𝑈(2)𝐿2−𝑈(1)𝑌 anomaly (∑𝑌 over left-handed doublets only), the 𝑆𝑈(3)𝐶2− 𝑈(1)𝑌 anomaly (∑𝑌 over all colored fermions), and the gravitational–𝑈(1)𝑌 anomaly (∑𝑌 over all fermions). The traditional SM anomaly budget has been disturbed by the introduction of the anomalon and its neutrino companion (leading to four generations of leptons instead of three) and the leonids, which is a new colorless fermionic branch containing three generations of fermions (see SC1). At least one new fermionic branch is required to cancel these disruptions. Geometrically, a remarkable point is located at 𝑘=𝑘A=3/2 (see Fig. 5) and corresponds to the maximum left-handed hypercharge (𝑌𝐿= 12). The value of 𝑌𝐿=12 can be used to construct a doublet, with an upper member associated with 𝑌𝑅= 𝑌𝐿+1=32 and lower member associated with 𝑌𝑅=𝑌𝐿−1=−12. Let me call this speculative fermionic branch the apex (using here apex for both singular and plural: one apex, two apex), with its upper members referred to as the high-type apex (symbol Δ, 𝑇3𝐿=+12) and the lower members as the low-type apex (symbol ∇, 𝑇3𝐿=−12) to mimic the quarks. When assigning three generations to the apex (see details in Table 5) and giving them a color representation of 8 (analogous to the gluons), all gauge anomalies are cancelled (see supplementary). This gauge anomaly cancellation is remarkable because it occurs across three generations of leonids, quarks, and apex, but four generations of leptons. Fig. 5. Two new fermionic branches: the leonids and the apex. The left-handed and right-handed hypercharge polynomials presented in Fig. 1 have been extended to the values 𝑘=−1 and 𝑘=4 and lead to 𝑌𝐿(−1)=𝑌𝐿(4)= −11/3. Coincidently, 𝑇3𝐿(−1,−1)=+12 which suggests that this extension could be attributed to a new fermionic branch named the leonids. The leonids are partitioned between the lions (𝑘=−1, 𝑇3𝐿(−1,−1)=+12) and the chimeras (𝑘=4, 𝑇3𝐿(4,𝑘𝜒′)=−12) with 𝑘𝜒′= (−3+𝑖√23)/2. The golden ratio 𝜑 and its conjugate 𝜑∗ appear at the intersection between 𝑌𝐿 and 𝑌𝑅. A last fermionic branch (the apex) located at (𝑘,𝑌𝐿)=(32,12) has been introduced to entirely cancel the gauge anomalies (see section SC2). names seeds predicted mass (MeV) quantum numbers apex 𝑛 𝑘 𝑘′ if 𝑛′=−1 if 𝑛′=0 if 𝑛′=1 𝑌𝐿 𝑌𝑅 𝑄 𝑇3𝐿 𝐿 𝐵 high-type high Δℎ 1 32 𝑘Δ′ 1.878 ±0.011 2.313 ±0.011 2.850 ±0.012 +12 +32 +34 +12 −18 +38 rise Δ𝑟 2 32 𝑘Δ′ 171.03 ±0.72 210.69 ±0.68 259.55 ±0.78 +12 +32 +34 +12 −18 +38 zenith Δ𝑧 3 32 𝑘Δ′ 15578 ±44 19191 ±40 23642 ±56 +12 +32 +34 +12 −18 +38 low-type low ∇𝑙 1 32 𝑘∇′ 2.2808 ±0.0059 2.1947 ±0.0062 2.1119 ±0.0097 +12 −12 −14 −12 −18 +38 fall ∇𝑓 2 32 𝑘∇′ 207.75 ±0.45 199.91 ±0.28 192.36 ±0.61 +12 −12 −14 −12 −18 +38 nadir ∇𝑛 3 32 𝑘∇′ 18923 ±59 18209 ±20 17522 ±36 +12 −12 −14 −12 −18 +38 Table 5. Characteristics of the apex. The names, symbols, seeds, predicted mass and quantum numbers of the new fermionic branch called the apex (partitioned into the high-type apex and the lowtype apex) are summarized here. Because 𝑛′ is unknown (distributed between −1,0 and 1), there are three possibilities for the mass of each apex. Predicted masses correspond to median and 2.5th and 97.5th percentiles of predictions from ~105 sets of (𝛼,𝛽,𝛼′,𝛽′ and 𝜆) that individually reproduce the observed masses of known fermions (PDG 2024) within experimental uncertainties (details in Table 3). The value 𝑘Δ′=(−3+√51)/2 and 𝑘∇′=(−3+√27)/2 are set to respect the structure of 𝑇3𝐿= ±1/2. The apex have necessarily a color representation of 8 to cancel the gauge anomalies (see SC2). SC3. A primary charge seed 𝜿. This SC3 assumes SC1 to be valid, and the claim is that 𝜅= {−1,0,1} (denoted as the “charge seed”) is sufficient to produce all left-charge and right-charge seeds of the leonids, leptons and quarks. The apex and the golden ratios 𝜑 and 𝜑∗ seem to emerge from a different mechanism (see section O2). Currently, the left-charge and right-charge seed values span a wide range: 𝑘= −1 to 4 (including the leonids), 𝑘′=−1,0,1,2 and suddenly 𝑘𝜈′= −3+√33 2 and 𝑘𝜒′= −3+𝑖√23 2. This likely points to a deeper mechanism generating such diversity in seed values. From this hypothesis, two functions naturally arise: 𝑓𝐿(𝑘)=3−𝑘 and 𝑓𝑅(𝑘′)=−3+√−20𝑘′2+36𝑘′+33 2 . The leftcharge and right-charge seeds of each doublet’s members are connected to 𝜅 through the 𝑓𝐿 and 𝑓𝑅 transformations: - leonids: the lion doublets are associated with (𝑘,𝑘′)=(−1,−1), while the chimera doublets are associated with (𝑘,𝑘′)=(4,𝑘𝜒′)=(𝑓𝐿(−1),𝑓𝑅(−1)). Chimeras correspond to the lions under (𝑓𝐿,𝑓𝑅) applied to −1. - leptons: the charged lepton doublets are associated with (𝑘,𝑘′)=(0,0), while the neutrino doublets are associated with (𝑘,𝑘′)=(3,𝑘𝜈′)=(𝑓𝐿(0),𝑓𝑅(0)). Neutrinos correspond to the charged-leptons under (𝑓𝐿,𝑓𝑅) applied to 0. - quarks: the down-type doublets are associated with (𝑘,𝑘′)=(1,1), while the up-type doublets are associated with (𝑘,𝑘′)=(2,2)=(𝑓𝐿(1),𝑓𝑅(1)). Up-type quarks correspond to the down-type quarks under (𝑓𝐿,𝑓𝑅) applied to 1. The down quark and the up quark are an exception: (𝑘,𝑘′)=(2,1) and (𝑘,𝑘′)=(1,2), respectively. However, this exception is handled by the fact that 𝑓𝐿(2)=1 and 𝑓𝑅(2)=1, which is remarkable! Even more remarkably, 𝑓𝑅 verifies that 𝑓𝑅(𝜑)=𝜑 and 𝑓𝑅(𝜑∗)=𝜑∗ and 𝑓𝐿 verifies that 𝑓𝐿(𝑘A)=𝑘A with 𝑘A=32 the unique left-charge seed of the apex. Therefore, the three seeds that have been left out − 𝑘A, 𝜑 and 𝜑∗ − are invariant either under 𝑓𝐿 or 𝑓𝑅. These coincidences are reported in the section O2. It has just been proven that 𝜅={−1,0,1} is sufficient to produce all charge seeds of the leonids, leptons and quarks. This suggests an emergence of the chimeras, neutrinos and up-type quarks from their other half (see SC4). SC4. After symmetry breaking in a pre-chiral space, chimeras emerged from lions, neutrinos from charged leptons, and up-type quarks from down-type quarks. This SC4 assumes SC1 and SC3 to be valid. SC4 is motivated by four observations: (1) in the SM, fermion masses and electric charges are correlated — except for the up and down quarks. This anomaly is not explained in the SM but is interpreted here as the signature of a broken symmetry (represented by the permutation 𝑘𝑢↔𝑘𝑑 in Fig. 2). (2) the generation 𝑛=0 associated with (𝑒−,𝜐𝑒) cannot be extended to quarks due to experimental constraints — no fourth generation of quarks has been observed. This suggests that 𝜐𝑒 may have inherited 𝑛=0 from the electron alone. Extending this reasoning to all doublets implies that each member of a doublet may have “emerged” from their other half. (3) the charge seed 𝜅={−1,0,1} alone produces all left-charge and right-charge seeds of the leonids, quarks and leptons under the (𝑓𝐿,𝑓𝑅) transformation (see SC3). (4) there is a remarkable symmetry 𝑘=𝑘′ for the lions, charged-leptons and down-type quarks. The exception is, again, the down quark and up quark, who break the symmetry between 𝑘 and 𝑘′. These observations can be unified through the following chronological narrative: 1) lions (𝜅=−1), charged-leptons (𝜅=0) and down-type quarks (𝜅=1) initially lived in a prechiral space characterized by the charge seed 𝜅={−1,0,1}. At this stage, they were singlets, massless, without left-handed counterparts, and the chimeras, neutrinos, and up-type quarks had not yet emerged. 2) A symmetry breaking occurred (associated with the permutation 𝑘𝑢↔𝑘𝑑), during which 𝜅 split into the left-charge seed 𝑘 and right-charge seed 𝑘′. This breaking marked the onset of chirality, leading to the formation of left-handed doublets. Each missing partner within a doublet subsequently emerged according to the transformations (𝑓𝐿(𝑘),𝑓𝑅(𝑘′)) creating the new following seed values: 𝑘= 2,3,4 and 𝑘′=2,𝑘𝜈′, 𝑘𝜒′ (see SC3). After symmetry breaking, the left-hypercharge polynomial 𝑌𝐿(𝑥) became related to the right-hypercharge polynomial 𝑌𝑅(𝑥) through the relation: 𝜙(𝑥) =𝑌𝑅(𝑥)− 𝑌𝐿(𝑥) where 𝜙(𝑥)=𝑥2−𝑥−1 is the golden polynomial, satisfying 𝜙(𝜑)=𝜙(𝜑∗)=0. The behavior of 𝑛′ and 𝑛 in this narrative remains undetermined at this stage, as does the role of the apex and the golden ratio 𝜑 and 𝜑∗ (see section O2). It is intriguing that 𝑛′={−1,0,1} shares the same range of values as 𝜅={−1,0,1}, but I do not yet understand how the generation seed 𝑛 could split from 𝑛′ in the same manner as the leftand right-charge seeds did from 𝜅. If the lions exist, their 𝑛′ will be determined with their mass (Table 4), and the relationship between 𝑛 and 𝑛′ will be easier to understand. SC5. Chiral reorganization of 𝑸 and 𝑻𝟑𝑳. The electric charge 𝑄 and the left-handed weak isospin 𝑇3𝐿 cannot be expressed in terms of the left-charge seed 𝑘. Instead, 𝑄 is a purely polynomial function of the right-charge seed 𝑘′: 𝑄(𝑘′)=16𝑘′2+12𝑘′−1, while 𝑇3𝐿 depends on both 𝑘 and 𝑘′: 𝑇3𝐿(𝑘,𝑘′)= 12(𝑌𝑅(𝑘′)−𝑌𝐿(𝑘)). This structure suggests the following bold claim: the electric charge 𝑄 is an intrinsic property of the right-handed sector alone, whereas the weak isospin 𝑇3 acts as a bridge linking the leftand right-handed sectors (it could be renamed 𝑇3𝐿𝑅) rather than being intrinsic to the left-handed sector. If confirmed, this would compel a revision of the SM’s gauge structure, reshaping our understanding of charge assignments and weak interactions. Open questions This section outlines a series of unresolved aspects and speculative directions emerging from the present framework. While the core structure captures the observed fermionic patterns, several elements remain only partially constrained or invite deeper interpretation beyond the established model. O1. The physical sense of 𝒏′. The residual seed 𝑛′ has been introduced to correctly account for the residuals observed in Fig. 4. The values are of 𝑛′ are elegantly distributed between −1,0 and 1 for fermions sharing same right-handed hypercharge (fixed 𝑘′). Values of the seed 𝑛′ echoes those of the charge seed 𝜅={−1,0,1} (see SC4), and 𝑛′ offers a right-handed counterpart to the left-handed generation seed 𝑛. However, 𝑛′ is not associated with any known quantum numbers such as the hypercharges, nor with any established experimental observations, such as the three fermionic generations. The absence of clear predictive patterns for this seed leads to unresolved values for the neutrinos, leonids, and apex (Tables 2, 4, and 5). Nevertheless, it is assumed that 𝑛′ can only take values in the range −1,0 or 1, consistent with all known fermions. O2. The golden ratios 𝝋/𝝋∗ and the apex. The golden ratio 𝜑 and its conjugate 𝜑∗appear twice in this study: (i) At the intersection between 𝑌𝐿(𝑥) and 𝑌𝑅(𝑥) in Fig. 5, and (ii) as invariants under the 𝑓𝑅 transformation (see SC3). They are associated with a vanishing weak isospin 𝑇3𝐿𝑅, however, it remains unclear why irrational numbers such as 𝜑 and 𝜑∗ arise in a framework otherwise centered on natural seeds 𝜅={−1,0,1} (see SC4). It is possible that these values act as seeds generating exotic states. Remarkably, postulating new vector-like particles emerging from 𝜑 and/or 𝜑∗ would not affect the gauge anomaly calculation because 𝑌𝐿(𝜑)=𝑌𝑅(𝜑) and 𝑌𝐿(𝜑∗)=𝑌𝑅(𝜑∗) and their contributions would cancel each other. This route is explored in the supplementary material with new hypothetical golden particles, the aurions. In parallel, apex are a construction geometrically consistent with the lefthanded hypercharge 𝑌𝐿 of other fermions (Fig. 5) and numerically consistent with a full cancellation of the gauge anomalies. Their color charge of 8 is analogous the adjoint representation of gluons, suggesting a possible connection between the apex and gauge bosons — an idea reminiscent of supersymmetric correspondences between fermionic and bosonic degrees of freedom (Haag, Łopuszański, & Sohnius 1975). While their left-charge seed 𝑘𝐴=32 is invariant under the 𝑓𝐿 transformation, the right-charge seeds associated with the low-type apex (𝑘∇′=−3+√27 2) and high-type apex (𝑘Δ′=−3+√51 2) are not consistent with the 𝑓𝑅 transformation: 𝑘Δ′≠𝑓𝑅(𝑘∇′), unlike all other fermionic branches. However, 𝑘∇′ and 𝑘Δ′ are expressed as −3+√𝑋 2 which intriguingly mimics the structure of 𝑓𝑅. All these evidences suggest that apex and the golden ratios emerged from a different mechanism than the three other fermionic branches that are all constructed on the same charge seed 𝜅={−1,0,1} (see SC3). The invariance 𝑓𝐿(𝑘A)=𝑘A, 𝑓𝑅(𝜑)=𝜑 and 𝑓𝑅(𝜑∗)=𝜑∗ may perhaps indicate — though not conclusively — that they are artefacts of the pre-chiral space postulated in SC4. O3. The anomalon and its attached neutrino. The anomalon is a charged lepton with a mass of ~8.6 MeV and identical quantum numbers to those of the known charged leptons. Its existence requires an associated neutrino to occupy the (𝑛,𝑘)=(1,3) slot in the mass law. Experimental evidence is limited, with a single indication (Anikina, Nikitin & Rikhvitsky, 2024) from a propane bubble chamber, whose superheated liquid medium is uniquely sensitive to localized energy deposition. The particle’s apparent absence in modern detectors may be explained by several speculative mechanisms: it may decay into neutral or very soft final states, travel slowly producing short tracks below detection thresholds, form neutral or weakly interacting composites, or be produced only rarely with a small cross section. These explanations are not mutually exclusive and remain untested. The nature of the associated neutrino — whether active or sterile — is entirely unknown. O4. The leonids. The existence of the leonids raises pressing theoretical and experimental questions. The chimeras carry complex masses, implying intrinsic instability, and the doublets mix real and complex components — a structure without precedent in the SM. The predicted complex masses for the chimeras reach absurdly high values (See Table 4), far exceeding any known SM fermions, potentially approaching scales reminiscent of the primordial energies of the early universe. Such extreme mass scales may hint at a connection to physics near the Big Bang or to hidden sectors that were active only at ultra-high energies. The exotic charges, baryon/lepton assignments, and partially complex masses raise questions about the compatibility of the leonids with cosmological, astrophysical, and collider constraints. Decay channels, lifetimes, and couplings remain unconstrained, leaving experiments with minimal guidance. Yet the 2024 hint of the anomalon, consistent with this framework, suggests that previously unseen leptonic states may exist, motivating exploration of the leonids in hidden or partially decoupled sectors. O5. Higgs mechanism. In the SM, fermion masses arise through Yukawa couplings to the Higgs field. For a fermion (f), the mass is given by 𝑚𝑓=𝑦𝑓𝑣 √2 where 𝑦𝑓 is the Yukawa coupling (free parameter) and 𝑣 ~ 246 GeV is the Higgs vacuum expectation value (VeV). The leftand right-handed components of the fermion are connected via the Higgs field, giving rise to the observed mass after spontaneous symmetry breaking. Within the present framework, the electron plays the role of a fundamental reference: its mass 𝑚𝑒 is considered non-emergent and treated as a fundamental constant, thereby replacing the 𝑣 √2. All other fermions arise as excitations of this underlying electron field, characterized by specific seed configurations (𝑛,𝑘,𝑛′,𝑘′). The mass law for all fermions naturally factorizes into a left-term 𝑒(𝑘+𝑛−3)𝑒𝛼+𝑘𝛽−3(𝑘−𝑒𝛽) multiplied by a right-term 𝜆(𝑘′/2)+𝑛′(𝛼′𝑘′+3 𝑌𝑅(𝑘′)+𝛽′), which resonates with the SM structure where leftand right-handed components are coupled. However, a mechanism analogous to the Higgs is required to connect the left-handed doublets with the right-handed singlets and generate mass. A possible escape route is the existence of a second Higgs boson with a VeV equal to 𝑚𝑒√2 that would be specific to the fermionic sector. O6. Neutrinos. In this framework, neutrinos are naturally of Dirac type, as their right-handed components are explicitly defined through 𝑘𝜈′=−3+√33 2. This choice ensures that neutrinos acquire mass through the same exponential law as the charged fermions, but with a distinctive suppression factor arising from the 1/𝑌𝑅 dependence in the exponent. At this value of 𝑘𝜈′, the denominator 𝑌𝑅 vanishes, introducing a strong enhancement in the mass exponent that drives the neutrino masses to extremely small values. This mechanism provides a purely algebraic explanation for the observed neutrino mass hierarchy, without invoking Majorana terms or seesaw dynamics. However, two theoretical issues remain open in this approach: (i) what mechanism ensures 𝑌𝑅(𝑘𝜈′)≠0, preventing a singularity for at least two neutrinos, and (ii) how the residual seed 𝑛′ is distributed among neutrino flavors. Point (i). The simplest resolution proposed in this study introduces a small perturbation 𝜖𝜈≪1 such that 𝑘𝜈,eff ′=𝑘𝜈′±𝜖𝜈, with 𝜖𝜈 depending on the flavor (Table 3). Other formulations, however, are possible. Notably, since 𝑓𝑅(𝜅)=𝑘𝜈′ for 𝜅=0 (as shown in SC3), this issue can equivalently be expressed as a perturbation on 𝜅, namely 𝜅𝜈,eff ′=±𝛿𝜈 with 𝛿𝜈≪1. Intuitively, the parameters 𝛿𝜈 could be assumed to follow a symmetric distribution around a non-zero mean 𝛿eff (motivated by 𝜖𝜇≠ 𝜖𝜏 in Table 3). In this formulation, one may write 𝜅𝜈,eff ′=𝛿eff±𝑛′×𝛿0 where 𝛿eff and 𝛿0 are fixed constants independent of flavor, and 𝑛′ determines the sign. This construction remains speculative. As long as the mechanism responsible for regulating the singular behavior of 1/𝑌𝑅 is not identified, the problem remains open. Point (ii). The second open question then concerns the precise assignment of the residual seed 𝑛′. While the charged leptons include two states with 𝑛′=0 (the electron and the tau), the neutrinos appear to mirror this pattern only partially. The simplest resolution proposed in this study is to produce only one state 𝑛′=0 assigned to 𝜐𝑒, while other neutrinos follow 𝑛′=±1. This breaks the symmetry in 𝑛′ with the charged-leptons, but it better matches the experimental constraints. Alternatively, one can assign 𝑛′=0 to 𝜐𝑒 and 𝜐𝑎, and leverage the indeterminacy 0 × ∞ (in the term ~𝜆𝑘′ 2+ 0 × ∞) by assigning 0 × ∞ → −∞ to 𝜐𝑒 (which leads to a zero mass) and assigning 0 × ∞ → 0 to 𝜐𝑎, which would lead to a mass of 𝑚𝜈𝑎=422.5±1.4 MeV. This seems to contradict the experimental constraints on the mass of a hypothetical fourth neutrino, but this would restore the symmetry in 𝑛′.