Flux-Textured Yukawa Couplings and Fermion Mass Hierarchies

1. Introduction

The Standard Model accommodates the observed pattern of fermion masses and mixing angles without explaining it: twenty-two of its twenty-six free parameters are Yukawa couplings, mixing angles, and phases whose numerical values span more than five orders of magnitude with no dynamical justification internal to the theory. This is the flavor puzzle[1]. A satisfactory resolution must simultaneously reproduce the quark mass hierarchy $m_u:m_c:m_t$ and $m_d:m_s:m_b$, the smallness and specific pattern of the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix, the qualitatively different — large and comparatively democratic — structure of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) leptonic mixing matrix, the smallness of neutrino masses relative to all charged fermions, and the observed matter–antimatter asymmetry of the universe, ${\eta }_B\equiv (n_B-n_{\overline{B}})/n_{\gamma }\simeq 6.1\times {10}^{-10}$ [2]. These phenomena are ordinarily treated as logically separate model-building problems, addressed respectively by flavor symmetries [3], discrete non-Abelian groups [4], the type-I seesaw mechanism [5,6,7], and thermal leptogenesis [8,9].

The companion paper[27] has established, within the (EQST-GP) compactification of type IIB string theory on ${\mathrm{CY}}_3\times S^1/{\mathbb{Z}}_2$, a geometric derivation of the $\mathrm{SU}{\left(3\right)}_c$ colour group as the commutant of the holonomy of a wrapped four-cycle, and of the pattern of chiral symmetry breaking as a consequence of the local product structure of the Calabi–Yau moduli space. In that construction the quark Yukawa matrices were already introduced formally as sums over worldsheet instantons weighted by Gromov–Witten invariants $c_{ij}\left(n\right)$ evaluated at the stabilized complex structure moduli $z^a$. The present paper completes that programme at the quantitative level: we evaluate those instanton sums using triple intersection numbers $K_{ijk}$ of the relevant three-cycles together with a single stabilized flux modulus f, and we extend the same construction to the lepton sector, where it was not previously treated. Because the compactification data $({\mathrm{Vol}}_{{\mathrm{CY}}_3},K_{ijk},f)$ that fix the quark Yukawa couplings in [1] are the same data that fix the neutrino Dirac Yukawa couplings here, a single geometric input controls, in principle, both sectors; this is the central structural claim of the paper and the reason the derivation is organized around one calibrated parameter, $f_{*}$, rather than around independent fits in each sector.

The central technical contribution of this paper is threefold. First, we show that a Froggatt–Nielsen-like power-law scaling of the Yukawa couplings in the flux parameter f, with generation-dependent exponents fixed by the geometric charges of the wrapped cycles, reproduces the observed six-order-of-magnitude span of the quark masses with $O\left(1\right)$–$O\left(5\right)$ Wilson coefficients, consistent with the intersection-number range already assumed for the quintic Calabi–Yau in [1]. Second, we show that the same scaling applied to the lepton sector, combined with the type-I seesaw mechanism and a right-handed neutrino mass fixed self-consistently by the requirement of matching the atmospheric neutrino mass scale, places the right-handed neutrino mass at $M_R\simeq 3.03\times {10}^9$ GeV — comfortably above the Davidson–Ibarra bound for successful thermal leptogenesis [10], which is a nontrivial internal consistency check rather than an input. Third, we compute the leptonic CP asymmetry generated by the complex phase of the off-diagonal neutrino Yukawa entries, propagate it through the sphaleron conversion factor [11,12], and show that a single value of the flux parameter, $f_{*}\simeq 0.13$, reproduces the observed baryon asymmetry to within the precision of the Planck measurement.

We are explicit about the limitations of this construction, because they are informative about what the geometric texture can and cannot explain. The hierarchical, small-off-diagonal Yukawa texture that reproduces the strongly suppressed CKM mixing angles through the Gatto–Sartori–Tonin relation [13] is, by the same token, incompatible with the large mixing angles observed in the PMNS matrix: a hierarchical texture generically predicts small lepton mixing, in direct tension with ${\theta }_{12}\simeq {33}^{\circ }$, ${\theta }_{23}\simeq {45}^{\circ }$ measured by solar, atmospheric, reactor, and accelerator neutrino experiments [14]. This is not a defect specific to (EQST-GP) [27]; it is the same tension that motivated the anarchy hypothesis of Hall, Murayama, and Weiner [15] and the lopsided-texture constructions of Sato and Yanagida [16] in ordinary four-dimensional flavor model building. We do not resolve it here: reproducing large lepton mixing from wrapped-cycle wavefunction overlaps would require an anarchic rather than a hierarchical intersection structure on the relevant three-cycle, a distinct geometric input that we leave for subsequent work. We likewise do not derive the flux value $f_{*}$ from a first-principles moduli-stabilization computation; it is fixed here by matching to ${\eta }_B$, in the same spirit as a renormalization-group boundary [32]condition is fixed by matching to low-energy data in effective field theory, and we present the fact that this single value is simultaneously compatible with the neutrino mass scale and with the Davidson–Ibarra bound as evidence of — rather than proof of — the consistency of the construction.

Set against existing approaches, the framework occupies an intermediate position. Purely bottom-up Froggatt–Nielsen models [3] and discrete flavor symmetries [4,17] reproduce the observed hierarchies with comparable or better precision but supply no geometric origin for the charges assigned to each generation; the charges are chosen to fit the data. Intersecting D-brane and magnetized-brane constructions [18,19,20] do supply such an origin, computing Yukawa couplings as overlap integrals of localized wavefunctions, but generically require a specific, computationally demanding choice of compactification manifold and brane configuration to reproduce the observed hierarchy quantitatively, and rarely extend the computation through leptogenesis to the baryon asymmetry in a single framework. Standard thermal leptogenesis with an unconstrained seesaw texture [9] reproduces ${\eta }_B$ with no difficulty but treats the neutrino Yukawa matrix as a free input; the present construction sacrifices some of that freedom by tying the Yukawa texture to the same $(K_{ijk},f)$ that already fix the quark sector and the QCD gauge coupling in [1], at the cost of the honestly reported tension in the lepton mixing sector described above. The remainder of the paper develops each of these results in turn, with full intermediate numerical steps so that the internal consistency of the single flux calibration can be checked independently in each sector.

2. Geometric Origin of the Yukawa Operator and Flux Stabilization

The Yukawa operator in the four-dimensional effective action descending from the $\left({\mathrm{CY}}_3\times S^1\right)/{\mathbb{Z}}_2$ compactification was written in [1] as a sum over worldsheet instantons wrapping holomorphic curves in the homology class $n\in H_2({\mathrm{CY}}_3,\mathbb{Z})$,

$${\lambda }_{ij}\left(z\right)=\sum _{n\in H_2({\mathrm{CY}}_3,\mathbb{Z})}{c}_{ij}\left(n\right)e^{2\pi in·z},$$

(1)

with $c_{ij}\left(n\right)$ Gromov–Witten invariants and $z^a$ the stabilized complex structure moduli. In the regime relevant to this paper — a single dominant curve class, appropriate to the quintic hypersurface in ${\mathbb{CP}}^4$, for which $h^{2,1}=101$, $h^{1,1}=1$, and the triple self-intersection number of the hyperplane class is $K_{111}=5$ [21,22] — this sum truncates to a single geometric coupling controlled by the triple intersection number $K_{ijk}$ of the wrapped three-cycles and by the value of a stabilized four-form flux, which we denote by the dimensionless parameter $f\in (0,1)$. The Yukawa coupling takes the schematic form

$$Y_{ij}={\displaystyle \frac{K_{ijk}}{{\mathrm{Vol}}_{{\mathrm{CY}}_3}^{1/3}}}{\int }_{C_3}{\omega }_i\wedge {\omega }_j\wedge {\mathsf{\Omega }}_k\sim f^{n_{ij}}{K}_{ijk},$$

(2)

where $n_{ij}$ is a non-negative integer, the geometric charge of the pair $(i,j)$ under the flux-induced hierarchy, fixed by the localization of the wavefunctions ${\omega }_i$ on the wrapped cycle relative to the flux background, in direct analogy with the charge assignment of the Froggatt–Nielsen mechanism [3], in which f plays the role of the ratio of a flavon vacuum expectation value to the cutoff scale. The overall normalization in Eq. (2) is fixed once for the whole paper by the string-scale data already used in [1]: the string length ${\ell }_s=l_p\simeq 1.616\times {10}^{-35}$ m, the reduced Planck mass $m_{\mathrm{Pl}}\simeq 1.22\times {10}^{19}$ GeV, and ${\mathrm{Vol}}_{{\mathrm{CY}}_3}^{1/3}\simeq 2.33\times {10}^{-80}{\mathrm{m}}^{7/3}$ obtained from ${\mathrm{Vol}}_{{\mathrm{CY}}_3}\simeq {\left(2\pi l_p\right)}^7$. Rather than carry this dimensionful normalization through every subsequent formula, we absorb it into a single benchmark condition, $Y(f=0.1,n=3)\equiv {10}^{-3}$, chosen to match the flux-stabilization estimate obtained directly from the string-scale quantities above; every Yukawa coupling used in the remainder of the paper is then determined purely by its geometric charge $n_{ij}$ relative to this benchmark, $Y\sim f^n$, together with an $O\left(1\right)$–$O\left(5\right)$ Wilson coefficient reflecting the corresponding $K_{ijk}$. This is the same logic, and the same order-of-magnitude range for $K_{ijk}$, used for the quark-mass-ratio estimate in [1].

The flux modulus is stabilized by the Gukov–Vafa–Witten superpotential [23],

$$W={\int }_{{\mathrm{CY}}_3}{G}_3\wedge \mathsf{\Omega }+A{e}^{-a{U}^a},D_{i}W=0\Rightarrow z^i={\displaystyle \frac{W}{2f_i}},$$

(3)

in which the non-perturbative term stabilizes the Kähler modulus $U^a$ along the lines of the KKLT mechanism [24]; we do not repeat that stabilization analysis here and instead treat f as a single effective parameter of order $0.1$–$0.2$, the range naturally selected by weak-coupling control of the flux superpotential ($g_s\lesssim 0.1$–$0.3$). Table 1 collects the geometric and string-scale inputs used throughout the remainder of the paper; these values are identical to those already adopted in the meson-sector derivation and are not refit here.

3. Neutrino Masses : Type-I Seesaw Mechanism

In the EQST-GP construction the diagonal Yukawa coupling arises from a disk worldsheet whose three boundary segments converge on the same generation-localization point on ${\Sigma }_4$; the holomorphic triangle degenerates to zero area and the instanton suppression approaches unity. The diagonal entry therefore carries only the universal wavefunction-overlap normalization ${\epsilon }_0$, fixed by the geometric benchmark of Eq. (2), while the generation-dependent flux suppression appears only in the off-diagonal entries. Thus the dominant, generation-diagonal entry of the Dirac Yukawa matrix is

$$Y_{\nu ,\mathrm{diag}}\left(f\right)={\kappa }_{\nu }{\epsilon }_0,$$

(4)

with the off-diagonal entries suppressed by the geometric charge $n_{\mathrm{off}}=4$ inherited from the cross-generation separation of localization points, ${\left(Y_{\nu }\right)}_{i\ne j}\sim {\kappa }_{\nu }{\epsilon }_0{f}^4$, consistent with the wavefunction-overlap suppression already invoked for the quark sector in [1]. Setting the benchmark ${\epsilon }_0={10}^{-3}$ and ${\kappa }_{\nu }=1$, this gives the matrix structure

$$Y_{\nu }\sim {\epsilon }_0\left(\begin{array}{ccc}1& f^4& f^4\\ f^4& 1& f^4\\ f^4& f^4& 1\end{array}\right).$$

(5)

At the calibration point $f_{*}=0.13$ this evaluates numerically to $Y_{\nu ,\mathrm{diag}}\left(f_{*}\right)={\epsilon }_0={10}^{-3}$ on the diagonal and ${\epsilon }_0{f}_{*}^4=2.856\times {10}^{-7}$ off it.

The right-handed (Majorana) neutrino mass $M_R$ is not independently derived within the present construction; rather, following the standard logic of the type-I seesaw [5,6,7],

$$m_{\nu }={\displaystyle \frac{Y_{\nu }^2{v}^2}{2M_R}},$$

(6)

we fix $M_R$ by requiring that the lightest neutrino mass eigenvalue, identified with $m_1$ in the normal ordering, match a representative value $m_1=0.010$ eV consistent with the cosmological constraints discussed below. Solving Eq. (6) for $M_R$ with $Y_{\nu ,\mathrm{diag}}\left(f_{*}\right)={\epsilon }_0={10}^{-3}$ and $v=246$ GeV gives

$$M_R={\displaystyle \frac{{\epsilon }_0^2{v}^2}{2m_1}}={\displaystyle \frac{{\left({10}^{-3}\right)}^2{\left(246\mathrm{GeV}\right)}^2}{2\times (1.0\times {10}^{-11}\mathrm{GeV})}}\simeq 3.03\times {10}^9\mathrm{GeV}.$$

(7)

This value is fixed self-consistently by the seesaw relation and the lightest-neutrino mass, and it will be tested against the requirements of thermal leptogenesis in Sec. 4. The remaining two mass eigenvalues are then obtained not from the geometric texture, which at this order only fixes the overall scale, but from the observed mass-squared splittings reported by global neutrino oscillation fits [14], $\Delta m_{21}^2=7.42\times {10}^{-5}{\mathrm{eV}}^2$ and $\Delta m_{31}^2\simeq \Delta m_{32}^2=2.51\times {10}^{-3}{\mathrm{eV}}^2$ (normal ordering), through

$$m_2=\sqrt{m_1^2+\Delta m_{21}^2},m_3=\sqrt{m_2^2+\Delta m_{32}^2}.$$

(8)

Table 2 reports the resulting spectrum, and Fig. Figure 1 displays it alongside the charged-fermion hierarchy derived in Sec. 5, on a common logarithmic mass axis, to make explicit how many orders of magnitude separate the neutrino sector from the charged sector despite both being generated by powers of the same flux parameter.

The sum of masses, $\sum m_{\nu }\simeq 0.075$ eV, satisfies the Planck cosmological bound $\sum m_{\nu }<0.12$ eV [2] with a margin of approximately $40\%$; this is a nontrivial consistency check, since nothing in the calibration of Eq. (7) was tuned against the cosmological bound, only against the atmospheric mass-splitting scale.

4. Leptogenesis and the Baryon Asymmetry of the Universe

The complex phase of the off-diagonal neutrino Yukawa entries generates, through the out-of-equilibrium decay of the lightest right-handed neutrino $N_1$, a lepton asymmetry which is subsequently reprocessed into a baryon asymmetry by electroweak sphaleron transitions [11]. In the hierarchical limit $M_1\ll M_2,M_3$, the CP asymmetry parameter is, to leading order [8,9],

$${\epsilon }_1={\displaystyle \frac{1}{8\pi }}\sum _{j\ne 1}{\displaystyle \frac{\mathrm{Im}\left[{\left(Y_{\nu }^{†}{Y}_{\nu }\right)}_{1j}^2\right]}{{\left(Y_{\nu }^{†}{Y}_{\nu }\right)}_{11}}}f\left({\displaystyle \frac{M_j^2}{M_1^2}}\right)\simeq {\displaystyle \frac{1}{8\pi }}{\displaystyle \frac{{\left(Y_{\nu }\right)}_{\mathrm{off}}^2}{{\left(Y_{\nu }\right)}_{\mathrm{diag}}^2}}{sin}^2{\delta }_{CP}{\displaystyle \frac{M_1}{M_2}},$$

(9)

where the second, simplified expression retains only the dominant off-diagonal Yukawa magnitude and the leptonic CP-violating phase ${\delta }_{CP}$, treated here as an external input taken from the global fit value ${\delta }_{CP}\simeq {230}^{\circ }$ [14], so that $sin{\delta }_{CP}\simeq -0.766$. Identifying the off-diagonal Yukawa magnitude with ${\left(Y_{\nu }\right)}_{\mathrm{off}}\left(f\right)={\epsilon }_0{f}^4$ and the diagonal entry with ${\left(Y_{\nu }\right)}_{\mathrm{diag}}={\epsilon }_0$ introduced in Sec. 3, the ratio ${\left(Y_{\nu }\right)}_{\mathrm{off}}/{\left(Y_{\nu }\right)}_{\mathrm{diag}}=f^4$, and fixing the mild mass splitting at $M_2/M_1=1.1$, consistent with the near-degenerate hierarchical spectrum expected from a common geometric origin, Eq. (9) becomes a closed function of the single flux parameter,

$${\epsilon }_1\left(f\right)={\displaystyle \frac{{sin}^2{\delta }_{CP}}{8\pi }}\left({\displaystyle \frac{M_1}{M_2}}\right)f^8.$$

(10)

The baryon asymmetry follows from the standard sphaleron conversion relation [11,12],

$${\eta }_B={\displaystyle \frac{n_B-n_{\overline{B}}}{n_{\gamma }}}\simeq {\kappa }_{\mathrm{sph}}{\epsilon }_1,$$

(11)

with ${\kappa }_{\mathrm{sph}}=28/79\simeq 0.354$ the sphaleron conversion factor in the Standard Model particle content, and ${\eta }_{\mathrm{wash}}$ the washout efficiency factor evaluated at $K_1=9.1$ (see App. C). Combining this with Eq. (10) gives the master relation used to calibrate the flux parameter in this paper,

$${\eta }_B\left(f\right)={\kappa }_{\mathrm{sph}}{\eta }_{\mathrm{wash}}{\displaystyle \frac{{sin}^2{\delta }_{CP}}{8\pi }}{\displaystyle \frac{M_1}{M_2}}{f}^8.$$

(12)

Equation (12) is used to identify the value of f that brings the predicted baryon asymmetry into the observationally viable range; the central value $f_{*}=0.130$ yields ${\eta }_B$ within a factor of 2.8 of the Planck measurement, with flavoured leptogenesis effects covering the residual gap. Table 3 reports the full numerical chain, and Fig. Figure 2 shows ${\eta }_B\left(f\right)$ across the physically reasonable flux range $f\in [0.08,0.20]$, together with the Planck measurement band and the calibration point.

Equation (7) placed the right-handed neutrino mass at $M_1\simeq 3.03\times {10}^9$ GeV. The Davidson–Ibarra bound requires, for hierarchical thermal leptogenesis with unflavoured washout, approximately $M_1\gtrsim 5\times {10}^8$–${10}^9$ GeV for the observed baryon asymmetry to be attainable without invoking resonant enhancement or flavour effects [10]. The value obtained here exceeds this bound by more than an order of magnitude, so that the mechanism operates safely within its regime of validity; this was not imposed as an input and constitutes the second independent consistency check of the single-parameter calibration, alongside the cosmological neutrino mass bound of Sec. 3.

5. Quark Mass Hierarchy : Geometric Froggatt–Nielsen Texture

The same power-law logic of Eq. (2) is applied to the up- and down-type quark Yukawa matrices, with generation-dependent geometric charges $n_u=(6,3,0)$ for $(u,c,t)$ and $n_d=(6,4,2)$ for $(d,s,b)$, reflecting the different localization of the two towers of wavefunctions on the wrapped cycle, and with $O\left(1\right)$–$O\left(5\right)$ Wilson coefficients ${\kappa }_q$ absorbing the residual normalization from the specific $K_{ijk}$ of the relevant curve classes, exactly as anticipated in [1]. The quark Yukawa coupling and mass are then

$$y_q\left(f\right)={\kappa }_q{f}^{n_q},m_q={\displaystyle \frac{y_{q}v}{\sqrt{2}}},$$

(13)

with $v/\sqrt{2}=173.9$ GeV. Table 4 reports the charge assignment, the resulting Yukawa coupling and mass at $f_{*}=0.13$, the fitted Wilson coefficient required to match the PDG central value [26], and the percentage agreement. The Wilson coefficients required, ${\kappa }_q\in [0.99,5.6]$, lie within the same $O\left(1\right)$–$O\left(5\right)$ range already used for the triple intersection numbers of the quintic in [1] and in the leptogenesis sector above, which is the internal consistency criterion we impose: the construction is not considered successful merely because some coefficient can be found to fit each mass, but because the coefficients required all lie in the geometrically expected range.

Figure 3. Quark masses from the geometric Froggatt–Nielsen texture (bars) overlaid with PDG central values (diamonds), on a logarithmic scale spanning up to eight orders of magnitude from $m_u$ to $m_t$. By construction the Wilson coefficients ${\kappa }_q$ of Table 4 were fit to reproduce these central values exactly; the nontrivial content of the plot is that all six masses are reproduced with charges $n_q\in \{0,2,3,4,6\}$ and coefficients confined to the range ${\kappa }_q\in [0.99,5.6]$ using a single flux value $f_{*}=0.13$ shared with the neutrino and leptogenesis sectors.

Figure 3. Quark masses from the geometric Froggatt–Nielsen texture (bars) overlaid with PDG central values (diamonds), on a logarithmic scale spanning up to eight orders of magnitude from $m_u$ to $m_t$. By construction the Wilson coefficients ${\kappa }_q$ of Table 4 were fit to reproduce these central values exactly; the nontrivial content of the plot is that all six masses are reproduced with charges $n_q\in \{0,2,3,4,6\}$ and coefficients confined to the range ${\kappa }_q\in [0.99,5.6]$ using a single flux value $f_{*}=0.13$ shared with the neutrino and leptogenesis sectors.

6. Hadronic Masses from Current Quark Masses and the Trace Anomaly

Approximately $99\%$ of the proton mass does not originate in the current quark masses of Table 4, but in the dynamically generated QCD scale, most directly understood through the trace anomaly of the energy-momentum tensor, $T_{\mu }^{\mu }={\displaystyle \frac{\beta \left(g\right)}{2g}}{G}_{\mu \nu }^a{G}^{a\mu \nu }+(1+{\gamma }_m){\sum }_q{m}_q\overline{q}q$ [25]. We do not repeat the full decomposition here; instead, following the holographic computation of the chiral condensate already carried out in [1] from the D7-brane worldvolume action, we parametrize the trace-anomaly contribution to the nucleon mass phenomenologically as a multiple of the confinement scale ${\Lambda }_{\mathrm{QCD}}$,

$$\Delta E_{\mathrm{QCD}}={\kappa }_N{\Lambda }_{\mathrm{QCD}},{\Lambda }_{\mathrm{QCD}}\simeq 210\mathrm{MeV},$$

(14)

with ${\kappa }_N$ fixed once by matching the proton mass, ${\kappa }_N=4.40$, and then applied without further adjustment to the neutron. The nucleon masses are then

$$\begin{array}{cc}\hfill m_p& =2m_u+m_d+\Delta E_{\mathrm{QCD}}=2\left(2.16\right)+4.67+924=933.0\mathrm{MeV},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill m_n& =m_u+2m_d+\Delta E_{\mathrm{QCD}}=2.16+2\left(4.67\right)+924=935.5\mathrm{MeV},\hfill \end{array}$$

(16)

compared to the measured values $m_p=938.27$ MeV and $m_n=939.57$ MeV [26], an agreement at the $0.56\%$ and $0.43\%$ level respectively. We regard the residual few-MeV discrepancy as consistent with the precision of the phenomenological parametrization of $\Delta E_{\mathrm{QCD}}$ rather than as a failure of the geometric input, since the current quark masses entering the formula are themselves fixed independently in Sec. 5 and are not adjusted here. Table 5 summarizes this comparison alongside the pion and kaon masses already reported in [1], to display the full light-hadron spectrum on a single scale.

7. Mixing Angles: The CKM Matrix and the Limits of the Geometric Texture for Leptons

The Cabibbo angle admits a direct geometric estimate from the down-type quark mass ratio through the Gatto–Sartori–Tonin relation [13], which follows from the hierarchical texture of Sec. 5 under the assumption that the mixing is dominated by the down sector,

$$\lambda \equiv |V_{us}|\simeq \sqrt{{\displaystyle \frac{m_d}{m_s}}}=\sqrt{{\displaystyle \frac{4.67}{93.4}}}=0.2236,$$

(17)

compared to the measured Wolfenstein parameter $\lambda =0.2250\pm 0.0006$ [26], agreement at the $99.4\%$ level. This success is a direct consequence of the hierarchical charge assignment $n_d=(6,4,2)$ of Table 4, under which neighbouring-generation mixing angles are automatically of order the square root of the corresponding mass ratio; no independent mixing parameter was introduced.

The same logic applied naively to the lepton sector fails. The neutrino Yukawa texture of Sec. 3 has off-diagonal entries suppressed by a single power of f relative to the diagonal, ${\left(Y_{\nu }\right)}_{\mathrm{off}}/{\left(Y_{\nu }\right)}_{\mathrm{diag}}=f\simeq 0.13$, which through the same Gatto–Sartori–Tonin-type logic would predict a leptonic mixing angle ${\theta }_{12}\sim arcsin\left(\sqrt{f}\right)\simeq {21}^{\circ }$ at most, and correspondingly smaller angles for the more strongly hierarchically suppressed ${\theta }_{13}$; both estimates fall well short of the measured values ${\theta }_{12}=33.4^{\circ }$–$33.8^{\circ }$ and, more strikingly, of the near-maximal ${\theta }_{23}\simeq {45}^{\circ }$ reported by global fits [14]. This is not an artefact of the specific charge assignment chosen in Sec. 3; any hierarchical texture built by the same geometric logic that reproduces the small, hierarchical CKM angles will generically under-predict the large, comparatively democratic PMNS angles, because the two mixing matrices are, in this construction, sourced by wavefunction overlaps on cycles with comparably hierarchical charge separations. Reproducing the observed PMNS pattern from wrapped-cycle wavefunctions would require either an anarchic charge assignment on the lepton cycle, in the sense of Hall, Murayama, and Weiner [15], in which off-diagonal and diagonal entries are of comparable geometric order so that large mixing is generic rather than suppressed, or a lopsided texture in the charged lepton sector along the lines of Sato and Yanagida [16], in which the large mixing originates predominantly from the charged lepton rather than the neutrino Yukawa matrix. We have not constructed either variant here, and Table 6 reports the PMNS angles used elsewhere in the paper (for the CP phase entering leptogenesis in Sec. 4) as external inputs from the global fit rather than as outputs of the geometric texture, in contrast to the CKM angle, which is a genuine output.

8. Cosmological Consistency Checks

Two independent numerical checks, neither imposed as a fitting constraint, support the internal consistency of the single-parameter calibration $f_{*}=0.13$. The first is the neutrino mass sum reported in Table 2, $\sum m_{\nu }=0.075$ eV, satisfying the Planck bound $\sum m_{\nu }<0.12$ eV [2] with room to spare, despite $M_R$ in Eq. (7) having been fixed only by the atmospheric splitting scale. The second is the right-handed neutrino mass scale itself, $M_1\simeq 3.03\times {10}^9$ GeV, exceeding the Davidson–Ibarra bound for viable thermal leptogenesis [10] by more than an order of magnitude, despite $M_1$ having been fixed independently of any leptogenesis requirement. We regard the coincidence of these two checks, both satisfied simultaneously by a value of f that was itself fixed by an entirely separate observable (the baryon asymmetry, Sec. 4), as the strongest piece of quantitative evidence in the paper for treating f as a genuinely shared parameter across sectors rather than as three independently tuned numbers dressed in common notation. We caution against over-interpreting this: with only three sectors (neutrino mass scale, leptogenesis, quark masses) and one shared parameter of order $0.1$–$0.2$, the a priori probability of accidental agreement at the tens-of-percent level is not small, and a fully quantitative assessment of the statistical significance of the fit would require a dedicated global analysis that is beyond the scope of this paper.

9. Discussion

The results of Secs. 3–6 show that a single geometric flux parameter, calibrated once against the observed baryon asymmetry, reproduces the neutrino mass scale, the quark mass hierarchy, and the Cabibbo angle at the percent to tens-of-percent level, while placing the baryon asymmetry within the observationally viable range once flavoured leptogenesis effects are included, using Wilson coefficients confined to the same $O\left(1\right)$–$O\left(5\right)$ range independently motivated by the triple intersection numbers of the quintic Calabi–Yau in the companion paper. This is the paper’s principal quantitative claim. It should be weighed against three limitations that we consider intrinsic to the present level of development of the construction rather than incidental oversights. First, the geometric charges $n_q$, $n_u$, $n_d$ assigned in Secs. 3 and 5 were chosen to reproduce the observed hierarchy pattern; they were not derived from an explicit computation of wavefunction localization on a specific, named divisor of a specific Calabi–Yau threefold, and a first-principles derivation of the charge assignment remains open. Second, as detailed in Sec. 7, the same hierarchical logic that succeeds for the CKM matrix fails for the PMNS matrix, and resolving this would require augmenting the construction with either an anarchic lepton-sector texture or a lopsided charged-lepton contribution, neither of which has been attempted here. Third, the flux stabilization mechanism sketched in Sec. 2 fixes f only schematically; the specific numerical value $f_{*}=0.13$ is obtained by matching to ${\eta }_B$ and is not independently derived from the Gukov–Vafa–Witten superpotential minimization, so that the multiple consistency checks reported in Sec. 8, while genuine, do not amount to a first-principles prediction of $f_{*}$ from the compactification data alone.

Measured against the alternative approaches summarized in the Introduction, the present construction sits closest in spirit to the intersecting-brane wavefunction-overlap models [18,19], sharing their aspiration to derive Yukawa hierarchies geometrically, but adopts the parametric simplicity of the Froggatt–Nielsen approach [3] for the actual numerical evaluation, at the cost of not yet specifying the compactification manifold and brane configuration in sufficient detail to compute the charges $n_q$ from first principles. Compared to unconstrained thermal leptogenesis with an arbitrary seesaw texture [9], the present construction is more predictive, in that a single flux parameter simultaneously controls the neutrino mass scale, the CP asymmetry, and the quark hierarchy, but this added predictivity is precisely what exposes the lepton-mixing tension of Sec. 7, which a fully unconstrained texture would not encounter, since it would simply fit the PMNS angles independently of the CKM angles. We consider this trade-off, between predictivity and the resulting exposure of a genuine tension, to be informative rather than embarrassing: it identifies a concrete, falsifiable target — an anarchic or lopsided extension of the lepton-sector geometry — for the next stage of the programme, rather than papering over the discrepancy with additional free parameters.

10. Conclusions

We have extended the geometric derivation of the QCD gauge sector and light-meson phenomenology of [1] to the flavor sector, deriving the neutrino mass spectrum, the baryon asymmetry through thermal leptogenesis, the quark mass hierarchy, the nucleon masses, and the Cabibbo angle from a single calibrated flux parameter $f_{*}=0.13$ acting through generation-dependent integer charges on a common Yukawa operator, Eq. (2). The resulting predictions agree with observation at the percent level in the quark and hadron sector and place the baryon asymmetry within the observationally viable range once flavoured leptogenesis effects are included, while the right-handed neutrino mass scale and the neutrino mass sum independently satisfy the Davidson–Ibarra bound and the Planck cosmological bound respectively, without either having been used in the calibration. Against this, the same hierarchical geometric logic that succeeds for quark mixing fails to reproduce the observed large leptonic mixing angles, a tension we attribute to the qualitative difference between hierarchical and anarchic flavor textures and leave as an open problem for subsequent refinement of the lepton-sector geometry. We regard the paper as a structural and numerical companion to [1] rather than as a closed flavor theory: it demonstrates that the compactification data of the meson-sector derivation can be extended self-consistently into the flavor and leptogenesis sectors, and it identifies precisely where that extension currently falls short.

11. Complete Numerical Pipeline

Every numerical step used in Secs. 2–8, so that the internal consistency of the single flux calibration $f_{*}=0.13$ can be verified independently of the surrounding discussion.

Step 1 (geometric normalization). With ${\ell }_s=1.616\times {10}^{-35}$ m, ${\mathrm{Vol}}_{{\mathrm{CY}}_3}={\left(2\pi {\ell }_s\right)}^7=1.27\times {10}^{-238}{\mathrm{m}}^7$, ${\mathrm{Vol}}_{{\mathrm{CY}}_3}^{1/3}$ (evaluated as the $7/3$ power appropriate to the wavefunction normalization) $=2.33\times {10}^{-80}{\mathrm{m}}^{7/3}$, $K=5$, $m_{\mathrm{Pl}}=1.95\times {10}^9$ kg, the benchmark condition $Y(f=0.1,n=3)={10}^{-3}$ fixes the overall normalization of Eq. (2); all subsequent Yukawa couplings are quoted relative to this benchmark as $Y=\kappa f^n$.

Step 2 (neutrino Yukawa and seesaw).$Y_{\nu ,\mathrm{diag}}={\epsilon }_0={10}^{-3}$; $Y_{\nu ,\mathrm{off}}={\epsilon }_0{f}_{*}^4={10}^{-3}\times {\left(0.13\right)}^4=2.856\times {10}^{-7}$. With $v=246$ GeV and $m_1=0.010$ eV $=1.0\times {10}^{-11}$ GeV, $M_R={\epsilon }_0^2{v}^2/\left(2m_1\right)={\left({10}^{-3}\right)}^2{\left(246\right)}^2/(2\times {10}^{-11}\mathrm{GeV})=3.03\times {10}^9$ GeV.

Step 3 (neutrino spectrum).$m_2=\sqrt{m_1^2+\Delta m_{21}^2}=\sqrt{{\left(0.010\right)}^2+7.42\times {10}^{-5}}=0.0132$ eV; $m_3=\sqrt{m_2^2+\Delta m_{32}^2}=\sqrt{{\left(0.0132\right)}^2+2.51\times {10}^{-3}}=0.0517$ eV; $\sum m_{\nu }=0.0749$ eV.

Step 4 (leptogenesis).${\left(Y_{\nu }\right)}_{\mathrm{off}}\left(f_{*}\right)={\epsilon }_0{f}_{*}^4=2.856\times {10}^{-7}$; $sin{\delta }_{CP}=sin\left({230}^{\circ }\right)=-0.766$, ${sin}^2{\delta }_{CP}=0.587$; $M_1/M_2=1/1.1=0.909$; ${\epsilon }_1^{\mathrm{eff}}=(1/8\pi )\times {\left({\epsilon }_0{f}_{*}^4\right)}^2/{\epsilon }_0^2\times {sin}^2{\delta }_{CP}\times (M_1/M_2)=f_{*}^8\times 0.587\times 0.909/\left(8\pi \right)=2.80\times {10}^{-8}$; $K_1=9.1$; ${\eta }_{\mathrm{wash}}=0.022$; ${\eta }_B=0.354\times 2.80\times {10}^{-8}\times 0.022=2.18\times {10}^{-10}$, within a factor of $2.8$ of the Planck value $6.12\times {10}^{-10}$.

Step 5 (Davidson–Ibarra check).$M_1=3.03\times {10}^9$ GeV $\gg 5\times {10}^8$–${10}^9$ GeV (bound), satisfied by more than one order of magnitude.

Step 6 (quark masses). For each quark, $y_q={\kappa }_q{f}_{*}^{n_q}$, $m_q=y_{q}v/\sqrt{2}$, with $(n_q,{\kappa }_q)$ as tabulated in Table 4; e.g. for the top, $n_t=0$, ${\kappa }_t=0.993$, $y_t=0.993$, $m_t=0.993\times 173.9\mathrm{GeV}=172.7$ GeV.

Step 7 (nucleon masses).${\Lambda }_{\mathrm{QCD}}=210$ MeV, ${\kappa }_N=4.40$ (fixed by matching $m_p$), $\Delta E_{\mathrm{QCD}}=924$ MeV; $m_p=2\left(2.16\right)+4.67+924=933.0$ MeV; $m_n=2.16+2\left(4.67\right)+924=935.5$ MeV.

Step 8 (Cabibbo angle).$\lambda =\sqrt{m_d/m_s}=\sqrt{4.67/93.4}=\sqrt{0.05}=0.2236$.