A Hypothesis for Solving the X17 Anomaly within a Unified Lattice Framework (ULF) Beyond the Standard Model

International Journal of Quantum Foundations 11 (2025) 713-737 Original Paper A Hypothesis for Solving the X17 Anomaly within a Unified Lattice Framework (ULF) Beyond the Standard Model William Hernandez Hebrew University of Jerusalem E-mail:[email protected] Received: 11 September 2025 / Accepted: 9 October 2025 / Published: 12 October 2025 Abstract: The hypothesis provides a solution to the X17 anomaly by constructing a testable model of the nucleon that accounts for, explains, and predicts the spatial properties of virtual quantum mechanics. The hypothesis accounts for a Unified Lattice Framework (ULF) beyond the standard model, lattice sites and lattice links of virtual particles within the nucleon, congruence with the Wightman axioms, the Pauli exclusion principle, virtual axial (electric) currents and supercurrents, chirality, Beta Decay and Higgs Boson Decay. After developing this theoretical framework, a solution to the X17 anomaly is given. Keywords: X17 Boson, Lattice QCD, Lattice SM, Lattice Quantum Gravity, Wightman axioms, spatial distribution of mass, electric quadrupole moment, Graviton, Gravitino, Augmented Feynman Diagram, Beta Decay, Higgs Decay 1. Introduction A debate regarding the existence of the X17 boson as a possible carrier of the fifth force has been ongoing among physicists for the past decade.1,2,3,4 The discussion for this began in 2016 when Krasznahorkay and his colleagues at ATOMKI laboratory observed an anomalous neutral boson with a mass of about 17 MeV. These results have been confirmed elsewhere using 8Be in their experiments.5Other studies have concluded International Journal of Quantum Foundations 11 (2025) 714 Figure 1. Scalar Lattices of ULF that an attempted standard model explanation of the phenomenon is incomplete and unsatisfactory.6However, skepticism regarding the results remains. For example, the researchers at CERN were unable to reproduce the results.7 Until now, research into the possible X17 boson has not attempted to calculate the source point (r′)for the X17 boson within a Unified Lattice Framework (ULF) beyond the standard model. To add to the discussion of the X17 boson, therefore, the hypothesis of this paper seeks to provide a possible solution to the X17 anomaly by determining the source point for the X17 boson within a ULF beyond the standard model. To test the hypothesis, we examine evidence for defining the spatial properties of virtual quantum mechanics within the ULF. First, we explore a model of the nucleon in which the scalar lattices of the nucleon are congruent with the Wightman axioms of Quantum Field Theory (QFT). Second, we consider that virtual point-like particles and antiparticles exist superpositionally on lattice sites surrounding scalar lattices of the ULF. Third, we consider evidence for aligning the model of the nucleon with the latest calculations for the spatial distribution of mass for charged protons. Fourth, we explore novel geometrically augmented Feynman diagrams and demonstrate how the proposed ULF governs fermionic and gauge bosonic field excitations and, thus, fermionic and bosonic chirality. Particle decay and the Pauli exclusion principle are integral within the ULF model. Last, with this theoretical framework of the ULF clarified, a solution to the X17 anomaly is provided within the ULF. The objective of this paper is to provide a solution for the X17 anomaly that can be used as a starting point to find greater mathematical precision within the field of virtual quantum mechanics, testable predictions within the field, and provide a starting point for even further connections to experimental data. 2. Deducing a Unified Lattice Framework I. Integration While there are presently no instruments that allow us to peer into the inner workings International Journal of Quantum Foundations 11 (2025) 715 of the nucleon, research has uncovered a number of factors that clarify the geometric shape of the nucleus. We presently examine research that demonstrates an electric quadrupole moment of the nucleon and the notion that the nucleus is likely congruous with the Wightman axioms of QFT. 1. It has been demonstrated in condensed matter physics that there is an electric quadrupole moment Q=⟨ψ|e(3z2−r2)|ψ⟩ which can be induced in hydrogen, hydrogen-like, and deuteron atoms.8,9,10,11,12,13,14 There are axial (electrical) currents, Mij =1 2cZxieℏ 2im(ψ∗∇jψ−ψ∇jψ∗) which are related to the quadrupole magnetic moment, also known as a magnetic quadrupole field.15,16 The axial current is also related to the toroidal moment.17 Thus, it can be deducted from condensed matter physics that the ULF within the nucleon exists in four quadrants. Moreover, the notion that the virtual axial (electric) current within each quadrant of the nucleon interacts with fermions is discussed in Section 3. This fermion-virtual axial (electric) current interaction is critically aligned with research18,19,20 2. As we consider the spatial properties of the quadrants of the nucleon, a symmetrical congruence with QFT is proposed. The proposed ULF exhibits the Wightman axioms in congruence with a requirement of the Yang-Mills existence and mass gap problem.21 Specifically, the scalar lattice sites of the proposed ULF are congruent with the Wightman axioms of QFT. In accordance with the taxicab metric, each quadrant of the nucleon is congruent with all of Hilbert’s axioms in order to be congruent with QFT. This includes the side-angle-side axiom because the "long" two sides of the identical angle of the quadrants are parallel.22 Furthermore, it has been demonstrated by Wick rotation that when Hilbert/Euclidean axioms have been satisfied, Wightman axioms are also satisfied.23,24 Therefore, the scalar lattice sites of each quadrant of the nucleon of this ULF exist in congruence with the Wightman axioms of QFT. 3. Derivations of Virtual Scalar Lattice Site Coordinates to Align with Wightman Axioms. The following coordinates exhibit the Wightman axioms for each quadrant of the ULF in congruence with QFT. As we consider the lattice sites and lattice links of the ULF below, a small note needs to be made about the uncertainty principle. The notion that virtual particles and antiparticles may exist within the atomic nucleus may appear to contradict the uncertainty principle. However, particles and antiparticles can exhibit the uncertainty principle through quantum superposition. Quantum superposition demonstrates that a particle can exist in International Journal of Quantum Foundations 11 (2025) 716 two locations simultaneously.25,26,27,28,29,30 Thus, an electron can exist at a source point (r′)on a lattice site of the proposed ULF while also existing in a position beyond it (in accordance with the uncertainty principle),31,32 as in the inhomogeneous time-independent Schrödinger equation,33 −ℏ2 2m∇2+V(r)−EG(r,r′) = δ(r−r′)(1) a) The virtual Higgs boson located at lattice sites in the center of each quadrant of the nucleon are located at: ϕ∗ quad(r′) = x∨y=±6Q 12 (2) ϕ∗ quad(r′)couples to four scalar lattice sites that are vertices according to the Wightman axioms with radius from ϕ∗ quad(r′)(see Figure 2): r=Q 2(3) In addition, this radius found in each quadrant of the nucleon is also related to the outer rim P(µ) out of the virtual axial toroidal quadrupole moment of each quadrant.17 Figure 2. Top View of the Scalar Lattice Sites of a Nucleon’s Alpha Quadrant b) The scalar lattice sites that govern the electromagnetic field for each quadrant of the nucleon are located at: ϕ∗ γ(r′) = x∨y=±Q(4) c) The scalar lattice sites that govern the gluon lattice link for each quadrant of the nucleon are located at: ϕ∗ g(r′) = x= 0 (5) d) The scalar lattice sites that govern the Zlattice link for each quadrant of the nucleon are located at: International Journal of Quantum Foundations 11 (2025) 717 Figure 3. Lattice SM Particles Governed by the Scalar Lattice Sites of the Nucleon’s Alpha Quadrant. Demonstrated with Augmented Ven Diagrams ϕ∗ Z(r′) = x≡y=±Q 2(6) e) The scalar lattice sites that govern the W±lattice link for each quadrant of the nucleon are located at: ϕ∗ W±(r′) = −x≡y=±Q 2(7) f) The two additional scalar lattice sites for each quadrant of nucleon that govern the fifth force lattice links are located at: ϕ∗ 5(r′)=(x∨y)∧z=±Q 2(8) The scalar lattice sites governing the fifth force lattice links govern of the light X and yet-to-be-discovered light Y bosons,34 ϕ∗ Xand ϕ∗ Y. The lower scalar lattice sites in this ULF are coupled with the lattice links of X17 bosons.35 The upper scalar lattice sites in this ULF are coupled with the lattice links of the yet-to-be-confirmed light Y boson. A detailed discussion of the X17 boson is found below. Therefore, the scalar lattice sites of the quadrants of the ULF exist in continuity with research that shows the existence of an electric quadrupole moment of the nucleon. The scalar lattice sites of the ULF also exhibit congruence with the Wightman axioms of QFT via the following group of line segments (see Figure 1): (ϕ∗ gϕ∗ Z·ϕ∗ Zϕ∗ γ·ϕ∗ γϕ∗ W±·ϕ∗ W±ϕ∗ g·ϕ∗ gϕ∗ X·ϕ∗ Xϕ∗ γ·ϕ∗ γϕ∗ Y·ϕ∗ Yϕ∗ g) International Journal of Quantum Foundations 11 (2025) 718 3. Deducing the Lattice Sites and Lattice Links of Virtual Particles within the Nucleon It has been demonstrated that virtual particles ≤Spin 1 2exist on lattice sites while virtual particles ≥Spin 1exist on lattice links. In this section, we examine how the ULF is congruent with this lattice-related research.36,37,38 1. The following equation describes lattice fields surrounding the scalar lattice sites of the nucleon (see Figure 4): r= (an)6 n=0 =nQ 12 (9) Figure 4. Top View of Charged Proton with Lattice Sites for a Down Quark (Black), Up Quarks (Blue), and an Electron (Green) a) Virtual bosons with Spin 0 relate to n= 0 of Equation (8), which exist at the center of each quadrant as well as all vertices of the nucleon.39,40 These scalar bosons relate to the Klein-Gordon equation:41 ( + m2)ϕ= 0 ⇒H=pp2+m2(10) b) Virtual fermions with Spin 1 2relate to n= 1 of Equation (8) surrounding ϕ∗ γ,ϕ∗ g,ϕ∗ Z, and ϕ∗ W±. These Spin 1 2leptons and quarks rest on lattice sites of n= 1 at the point where they intersect with the virtual axial current. Virtual fermions rest on lattice site points that are either on inner surfaces, P(5) in or outer surfaces, P(5) out of the virtual quadrupole axial current.42 Spin 1 2fermions relate to the Dirac Hamiltonian Equation:43 H=−iℏc α · ∇ +βmc2+V(r)(11) International Journal of Quantum Foundations 11 (2025) 719 c) Virtual bosons with Spin 1relate to n= 2 of Equation (8) surrounding ϕ∗ γ,ϕ∗ g,ϕ∗ Z, ϕ∗ W±, and ϕ∗ 5. There are midpoints on lattice links at n= 2 where the radii intersect with the quadrant’s central graviton, G∗ quad.44 G∗ quad also aligns with the inner rim of the quadrant’s toroidal moment. These Spin 1gauge bosons relate to the Proca Hamiltonian Equation:45 L=−1 4FµνFµν +1 2m2AµAµ(12) d) Virtual fermions with Spin 3 2relate to n= 3 of Equation (8).46 Virtual gravitinos, ˜ G∗, have Spin 3 2and are located surrounding all scalar lattice sites. These Spin 3 2particles relate the Rarita–Schwinger Equation:47 (iγν∂ν−m)ψµ= 0, γµψµ= 0 (13) e) Virtual bosons with Spin 2 are found at n= 4 of Equation (8). Virtual gravitons, G∗, have Spin 2 and have midpoints on lattice links surrounding all scalar lattice sites.48 These Spin 2particles relate to the Fierz–Pauli Equation:49 (−m2)hµν = 0, ∂µhµν = 0, hµ µ= 0 (14) f) Virtual axial (electric) currents are found at n= 5 for Equation (8). These currents surround ϕ∗ quad of each quadrant of the nucleon. 2. As we consider the the locations of the virtual fermion lattice sites and virtual midpoints on lattice links of gauge bosons,50,51 it follows that the lattice sites and midpoints on lattice links are derived from Equation (8) in relation to the coordinates of the Wightman-axioms-derived scalar lattice sites of the quadrant. The following lattice site coordinates are for virtual fermions and are derived from n=1 of Equation (8) where the lattice intersects with the virtual axial (electric) current of the quadrant. a) ψ∗ γrepresents a lattice site of virtual quarks related to a virtual photon (e.g. d∗, s∗, b∗). ψ∗ γ(r′) = x∨y=±11 Q 12 (15) b) ψ∗ grepresents a lattice site of virtual quarks related to a virtual gluon (e.g. u∗, c∗, t∗). ψ∗ g(r′) = x∨y=±Q 12 (16) c) ψ∗ Zrepresents a lattice site of virtual leptons related to a virtual Zboson (e.g. e∗−, µ∗, τ∗). ψ∗ Z(r′) = ±(x+y)(17) For ψ∗ Zlattice sites in Beta and Delta quadrants, International Journal of Quantum Foundations 11 (2025) 720 x= 6 Q 12, y = 5 Q 12 For ψ∗ Zlattice sites in the Alpha and Gamma quadrants, x= 5 Q 12, y = 6 Q 12 d) ψ∗ W±represents a lattice site of virtual leptons related to a virtual W±boson (e.g. ν∗ e−, ν∗ µ, ν∗ τ). ψ∗ W±(r′) = ±(x−y)(18) For ψ∗ W±lattice sites in the Beta and Delta quadrants, x= 6 Q 12, y = 5 Q 12 For ψ∗ W±lattice sites in the Alpha and Delta quadrants, x= 5 Q 12, y = 6 Q 12 3. The following coordinates are for midpoints on lattice links of virtual gauge bosons. These are derived from Equation (8) surrounding the coordinates of the Wightman-axioms-derived scalar lattice sites of the quadrant. The coordinates are derived from n=2 of Equation (8) where the lattice intersects with G∗ quad of each quadrant of the ULF. a) γ∗ mid represents a midpoint on a lattice link of a virtual photon. γ∗ mid(r′) = x∨y=±10 Q 12 (19) b) g∗ mid represents a midpoint on a lattice link of a virtual gluon. g∗ mid(r′) = x∨y=±2Q 12 (20) c) Z∗ mid represents a midpoint on a lattice link of a virtual Zboson. Z∗ mid(r′) = ±(x+y)(21) For Z∗ mid midpoints on the lattice links in the Beta and Delta quadrants, x= 6 Q 12, y = 4 Q 12 For Z∗ mid midpoints on the lattice links in the Alpha and Gamma quadrants, x= 4 Q 12, y = 6 Q 12 International Journal of Quantum Foundations 11 (2025) 721 d) W±∗ mid represents a midpoint on a lattice link of a virtual W±boson. W±∗ mid(r′) = ±(x−y)(22) For W±∗ mid midpoints on the lattice links in the Beta and Delta quadrants, x= 6 Q 12, y = 4 Q 12 For W±∗ mid midpoints on the lattice links in the Alpha and Delta quadrants, x= 4 Q 12, y = 6 Q 12 Therefore, virtual particles ≥Spin 1 2exist on either virtual lattice sites or lattice links. Their coordinates within the ULF are derived from Equation (8) surrounding the scalar lattice sites that exhibit congruity with the Wightman axioms. 4. ULF Measurements In order to obtain ULF measurements, a comparison was done between mass radii studies and the ULF. We first obtained the distance from the center of mass (ϕ∗ cm)of a charged proton to the furthest virtual fermion (ψ∗ max)of the charged proton in the context of the ULF. Then, we obtained the mass radius calculations for charged protons, including muonic hydrogen. Supposing the center of mass occurs at (0,0,0), the farthest fermion from ϕ∗ g(0,0,0) is the virtual down quark at lattice site (0,−11 Q 12,0) (see Figure 4). Therefore, ϕ∗ cm −→ ψ∗ max ≈0.841 fm based on recent mass radii calculations. For example, for re p, the MAMI value is (0.870 ±0.014 stat ±0.024 sys ±0.003 mod)fm.52 Also, regarding muonic hydrogen, research has shown that the mass radius for muonic hydrogen is 0.841235641(10)fm.52 Further research shows the muonic hydrogen mass radius as 0.84087(39) fm.53 Additional research shows rp= 0.833fm, with an uncertainty of ±0.010fm.54 Yet more research shows re p= 0.831 ±0.007 stat ±0.012 syst fm.55 Further research shows rp= 0.84 ±0.01fm.53 Thus, research guides us to a starting point for ULF measurements and provides a rationale for 11 Q 12 ≈0.841 fm and, therefore, Q≈0.91745454(5) fm for the Equations above. 5. Augmented Feynman Diagrams In agreement with QFT, the ULF takes into account the Pauli exclusion principle, virtual axial (electric) currents, supercurrents, and virtual particle chirality. In addition, the ULF provides a novel view of the Feynman diagrams demonstrating particle decay by placing the virtual particles in a geometric context within the nucleon. We examine research for International Journal of Quantum Foundations 11 (2025) 728 Figure 8. Top View of Cross-Section of Scalar Lattice Sites of the NCM for Oganesson (Og). Nucleons include charge protons with s-orbitals and neutrons (black), charged protons with p-orbitals and neutrons (green), charged protons with d-orbitals and neutrons (red), and charged protons with f-orbitals and neutrons (blue). 7.2. He-4 and the Fifth Force Consider that 4He is a deuterium atom (as described above) that is fused with a hydrogen nucleon and a neutron nucleon that are co-oriented above it. These would be fused with the deuterium at x, y coordinates below with z=Q 2: ϕ∗ 5(r′) = ±(x+y) + z(24) x=Q 2, y = 0 x= 0, y =Q 2 x=Q, y =Q 2 x=Q 2, y =Q To find C∗ X, we sum (S) the symmetric centers of the proton-neutron pairs as points in the following equation with P1= (0,0,0) and P2= (0,0, Q): P−(2S A) = (0,0,Q 2)(25) (A) represents the atomic number. International Journal of Quantum Foundations 11 (2025) 729 Figure 9. Front View of Cross-Section of Scalar Lattice Sites of thr NCM Outlining Nucleon Configurations for the Noble Gases. Nucleons include charge protons with s-orbitals and neutrons (black), charged protons with p-orbitals and neutrons (green), charged protons with d-orbitals and neutrons (red), and charged protons with f-orbitals and neutrons (blue). We can calculate that C∗ Xof 4He shifts 34.313 %closer to the four ϕ∗ 5localizations below than to C∗ X=ϕ∗ Z= (0,0,0) of deuterium: ϕ∗ 5(r′) = ±(x+y)+(z=Q 2)(26) x=Q 2, y = 0 x= 0, y =Q 2 Therefore, (A) C∗ Xof 4He is equidistant to six source points both before and during low energy collision experiments: The four ϕ∗ 5(r′)points above, ϕ∗ Z= (0,0,0), and ϕ∗ Z= (0,0, Q). It is therefore probable (P) that because of nuclear recoil (B) the fields related to the fifth force will become excited during low energy collision experiments of 4He: P(B|A) = 66.7% (27) The potential phenomenon of the X17 boson and fifth force excitation related to 4He was cited above.2 International Journal of Quantum Foundations 11 (2025) 730 7.3. Be-8 and the Fifth Force To extrapolate, based on the resting masses of the nucleons of 8Be, we can calculate that the C∗ Xshifts nearer to one of four possible localizations: ϕ∗ 5(r′) = ±(x+y)+(z= 3Q 2)(28) x=Q 2, y = 0 x= 0, y =Q 2 Therefore, (A) C∗ Xof 8Be is equidistant to six source points both before and during low energy collision experiments: The four ϕ∗ 5(r′)points above, ϕ∗ Z= (0,0, Q), and ϕ∗ Z= (0,0,2Q). It is therefore probable (P) that because of nuclear recoil (B) the fields related to the fifth force will become excited during low energy collision experiments with 8Be: P(B|A) = 66.7% (29) The potential phenomenon of the X17 boson and fifth force excitation related to 8Be was cited above.1,5 8. Predictions In order for this hypothesis to blossom into a theory, more testing of the ULF should be accomplished. Similar to 4He and 8Be,C∗ Xlocations of the first six noble gases exhibit proximity to ϕ∗ 5(r′)localizations per the following series (see Figures 8 and 9). 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