Title: The Innocent Lepton: How to Be in the Emergence Tree Authors: Kibrom Kidane Date: 2026-04-01 DOI: 10.5281/zenodo.19932394 URL: https://xmoro.com/research/the-innocent-lepton-how-to-be-in-the-emergence-tree Abstract -------- When compact G2 symmetry breaks to SU(3), one confined branch closes on an SU(3)3 topological phase. In this framework the charged-lepton mass amplitude is represented by a Z3-equivariant Bogoliubov–de Gennes (BdG) operator on the three family labels; the physical masses are mk = ∆2 k. The direct finite G2 →SU(3) path retains the center harmonic v= [1,− 1 1 2 ,− 2 ] and therefore keeps a microscopic sector label. The charged-lepton branch used here is the cyclic closure: [∆,S] = 0, so the mass matrix is circulant and has eigenvalues ∆k = A+ Bcos θ+ 2πk 3. On this branch the chiral SU(3)3 modular tensor category and the octonionic G2/SU(3) Clebsch–Gordan data fix the amplitude ratio B/A= √2; the fundamental conformal weight gives the phase θ= h(3) = 2/9. The same branch defines the UV coupling αG2 (MPl) = |Z3|/(2πD2 tot) = 1/(24π), and the projected pair two-point norm gives the trace-lift normalization ceff = 1/2. One-loop dimensional transmutation then gives mk = 1 2 MPl e−9π 2 /2 1 + √2 cos 2 9 + 2πk 3 , k= 0,1,2, which reproduces the electron, muon, and tau masses with a common 0.12% scale residual, of the same order as α(0)/(2π), from the branch constants above. The Koide relation Q= 2/3 follows as an algebraic consequence of |B/A|= √2 on the positive branch. The mass formula depends only on the charged-lepton-layer closure algebra: the cyclic BdG form and the chiral data shared by the three SU(3)3 modular invariants. The D(6) modular invariant is the full-RCFT certificate of this condition, with M1,J = 1 and M8,8 = 3. Calculation-level details are collected in the Supplemental Material Introduction This paper studies the charged-lepton sector at the effective layer where its mass observable closes. The starting point is the compact exceptional Lie group , the automorphism group of the octonions . Among simple Lie groups, compact is the unique one that contains as a maximal subgroup while itself having trivial center. When this symmetry breaks, , the child center emerges as a center label: its elements lie in the embedded , but they are not central in the parent and do not define parent superselection sectors. The three center sectors are therefore an emergent partition of the confined Hilbert space. This emergent is the origin of three family labels. For an independent algebraic route from the octonions to three Standard Model generations, see Furey and Hughes . After breaking, the child confines. The Chern–Simons/Wess–Zumino–Witten correspondence gives the confined topological sector an description, and the requirement that the center current be a dimension-one boson ( ) uniquely selects . The resulting modular tensor category is the algebraic closure data at the charged-lepton layer. The organizing viewpoint is emergence by observable closure: a mass spectrum is closed at a given effective layer when it is fully determined by the layer's algebraic data, with no free parameters inherited from the parent theory. The direct finite path retains the center harmonic and does not close. The path that closes imposes cyclicity ( ) on the BdG mass-amplitude operator, so every admissible mass matrix is circulant and the non-circulant term is excluded. The charged-lepton spectrum is unusually rigid because the cyclic BdG identity and the chiral data already close the mass observable; quark and gravitational observables require additional layers. A branch that is complete at the charged-lepton layer may still fail a later interaction test, so the branch constructed here is a candidate solution that subsequent layers can refine or reject. Throughout the paper, is the internal family-space mass-amplitude operator, not the Dirac mass matrix itself. The physical charged-lepton mass matrix is , so the masses are . Lorentz spin and electroweak charges live on the ordinary charged-lepton fields; after electroweak symmetry breaking appears in the scalar Dirac mass term . The charged-lepton pole masses also satisfy Koide's empirical relation to high precision . In this paper Koide's relation is the algebraic verification supplied by the prior BdG identity: once the effective branch gives , the Koide value follows. Theoretical attempts to derive this relation include the geometric interpretation of Foot and Esposito–Santorelli , the family-gauge model of Sumino , the discrete-flavor proposals of Ma and Koide , the empirical -symmetric parametrization of \.Zenczykowski , the trigonometric extension of Brannen , and the exceptional-Jordan-algebra ladder of Singh . The present paper differs from these in placing the derivation at the level of the chiral modular tensor category. To our knowledge, this is the first derivation of from the chiral data together with octonionic channel normalization, and it further identifies the phase and absolute scale from the same branch data. The following purely algebraic result was proved in . [BdG/Koide identity ] Let be a real -equivariant triplet with , and set . Then Consequently, on the positive branch, if and only if . Given Theorem , the present paper determines the branch data: The ratio and phase are determined by the chiral modular tensor category (MTC), together with the octonionic Clebsch–Gordan data that fix the relative diagonal and off-diagonal BdG channels. The absolute scale follows from and the trace-lift normalization . The calculation-level details are collected in the Supplemental Material . Structure. Section defines the effective charged-lepton branch. Section traces the two paths through the emergence tree: the failed finite- path and the closed cyclic BdG branch. Section determines the data kept by that branch: the shared MTC data ( ), the amplitude ratio , and the phase . Section fixes the absolute scale. Section gives the mass spectrum and Koide verification. Section discusses the label-free condition at the full-RCFT level, and Section the scope and falsifiability. The emergent charged-lepton branch By the emergent charged-lepton branch we mean the following closed set of data: together with the scale normalizations defined in Eqs. and , and the remaining discrete conditions ( cyclicity, the local-OPE phase convention, and unit-adjoint normalization). Within this branch, the logic runs in one direction. The chiral MTC and octonionic channel data give , which by Theorem is equivalent to on the positive branch. The phase is the minimum of the center-symmetry effective potential, dominated by the lightest charged primaries with conformal weight . The absolute scale is determined by the MTC data through . Thus Koide follows as a verification of the branch identification. The chiral data that close the mass observable are These are common to every consistency condition at the charged-lepton layer: every realization of the label-free property presents the same cyclic algebra to the BdG mass operator. The Supplemental Material keeps the dependency ledger and arithmetic checks. Paths through the emergence tree The direct microscopic path asks whether finite dynamics already erases the center label. It does not. The broken channels retain the center harmonic The Supplemental Material gives the character restriction and finite-temperature determinant calculation. Its result is that the finite broken-vector profile has the form at every finite mass scale. Thus parent locality and charge conjugation identify the two nontrivial center labels with each other, while the identity label remains distinguished. The mass operator on this path is not circulant: it retains and therefore does not close on cyclic data alone. Since the obstacle is the residual label that distinguishes the identity sector, the natural next step is to ask what branch erases exactly that distinction while keeping the three-sector structure. The branch that closes keeps only the cyclic three-sector data. The physical content is that the confined sector does not spontaneously break its own center symmetry: it treats the three center labels democratically, just as the center-symmetric confined regime of pure Yang–Mills preserves the center rather than selecting a vacuum. The admissible mass operator must respect the cyclic action that permutes the three sectors. With the cyclic shift, the closed mass algebra is the -equivariant algebra Any operator satisfying Eq. is circulant: it preserves the cyclic relation among the three labels while excluding the non-circulant term . The Supplemental Material gives the commutator check explicitly. Diagonalizing the cyclic mass operator in the Fourier basis gives The rest of the paper determines the remaining branch data and the absolute scale. The branch data: k=3, B/A=sqrt(2), theta=2/9 The closed cyclic branch carries three pieces of data that determine the mass observable: the level , the amplitude ratio , and the phase . Each is fixed by the chiral MTC together with the octonionic channel structure. The shared SU(3)3 MTC data After breaking, the child confines. The Chern–Simons/WZW correspondence implies that the confined topological sector carries chiral data. The center current generates the emergent ; requiring it to be a dimension-one boson, , gives , whose unique positive solution is . The Hilbert space factorizes as , so massive matter affects only while the topological sector carries the mass-matrix phase information . This branch therefore keeps the following chiral data: The Supplemental Material carries the level-selection calculation, the ten integrable weights, and the quantum-dimension table. These chiral MTC data are shared by every charged-lepton consistency condition. The amplitude ratio B/A=sqrt(2) The branch sees two octonionic channels: the diagonal singlet channel (Clebsch–Gordan coefficient ) and the off-diagonal antisymmetric channel (Clebsch–Gordan coefficient ). For a simple intermediate line in a one-dimensional fusion channel, the unit-quantum-trace projector evaluates to : the endomorphism algebra is with quantum trace , so the projector with unit trace is . The charged intermediate line therefore carries weight . If is the diagonal coefficient and the off-diagonal circulant coefficient, then . The circulant eigenvalues contain the pair , so the cosine amplitude is while . Hence By Theorem , this is exactly the Koide point on the positive branch. The Brannen phase theta=2/9 The Koide ratio is independent of (Eq. ); the phase is determined by the chiral data through the center-symmetry effective potential. Here is the dimensionless local Fourier phase of the BdG insertion, not a closed-loop holonomy angle. Each charged primary with triality contributes to the effective potential with statistical weight and phase lag . The lightest charged primaries are and , both with and ; minimizing gives The mass insertion is a same-point two-point coupling, so it reads the local OPE exponent rather than the closed-loop monodromy phase . The Supplemental Material gives the full effective-potential calculation and verifies that produces a negative mass amplitude, falling outside the positive branch. The absolute scale The amplitude ratio and phase determine only mass ratios. The absolute scale is set by the total chiral dimension and the center order of the category. [Scale normalization] For the charged-lepton branch, the UV coupling at the Planck scale is Equivalently, in the adjoint-trace convention, . In the total-category form, Equivalently, dividing by the center order gives Below , the coupling runs by the one-loop equation used below. The Supplemental Material records the equivalent chiral-dimension, adjoint-trace, and full-RCFT readings of the same normalization. The branch-normalized one-loop coefficient is The subtraction is the effective scalar sector from the six real broken-channel directions : one complex fundamental, contributing at one loop. The Supplemental Material gives the normalization table, including the standard convention where the same content gives ; the invariant quantity is the exponent . Dimensional transmutation gives Finally, the trace-lift normalization is The finite-size CFT coefficient from the local scalar primary is . The pair space has full Hilbert–Schmidt trace while the adjoint-screened projector has trace ; trace preservation gives the lift factor . Therefore . The Supplemental Material gives the complete projector calculation. The overall mass-amplitude scale is then where carries dimensions of mass, since and the mass amplitudes have dimensions of . Mass spectrum and Koide verification Combining Theorem with Eqs. , , and , the charged-lepton masses are The three values are assigned to in increasing order. The Koide value has already been fixed by Eq. and Theorem ; this section checks that the same branch data also reproduce the observed ordering, Brannen phase, and absolute scale. Using the unreduced Planck mass GeV (distinct from the reduced Planck mass GeV), the Supplemental Material gives the numerical substitution and the increasing mass order . The residual is nearly uniform across the three generations, so it is primarily an overall-scale residual rather than a failure of the mass-ratio structure. If the scale is floated while keeping and fixed, the largest ratio-level residual is about . The branch formula fixes the scale through Eq. , leaving the common overshoot as the leading discrepancy. The Supplemental Material shows that this residual is of the same order as ; deriving the pole-mass matching correction is left as a separate test. The label-free condition at the full-RCFT level The previous sections determine the mass formula from the closed cyclic BdG branch and the chiral MTC data. The same label-free condition extends to the full RCFT layer: modular invariants that erase the center label are compatible conditions; those that retain it are not. The full RCFT admits three modular invariants : They differ in their left/right pairing and boundary interpretation, but they share the chiral data entering the mass prediction. The Koide/Brannen ratio and phase depend only on this shared chiral data. The invariant is the full-RCFT certificate of the same label-free condition. Its partition-function witnesses are The first witness says that the vacuum row contains the full simple-current orbit ; the second is the fixed-point-resolution test for the object. The Supplemental Material expands the corresponding partition function. These modular-invariant data are the full-RCFT expression of the same label-free consistency condition used by the cyclic mass algebra. Scope, limitations, and falsifiability The BdG/Koide identity (Theorem , ) enters only through . The paper is an emergence-layer derivation from octonionic data to the MTC-level mass observable. The mass formula uses chiral MTC data shared by all three modular invariants. The modular invariant is a consistency condition for the same label-free property that defines the closed branch. It is independently testable through a direct partition-function calculation. The remaining common residual is an pole-mass matching target. The formula itself is the tree-level MTC/bootstrap result. The scale normalization Eq. is (Definition ), equivalently . The coefficient is the one-loop running with the effective scalar sector, and (Eq. ) is the projected pair two-point norm. Quark, gravitational, and full neutrino sectors belong to later layers. The quark masses and CKM mixing angles provide a concrete falsification target: any consistent extension must reproduce the observed six quark masses and the CKM matrix from the next-layer branch data, without contradicting the charged-lepton result here. The proposal is falsifiable. It would fail if the local mass operator coupled to rather than to while remaining on the positive branch; if an independent calculation gave a different value for or ; or if the observed residuals were not accountable as a common scale correction. At the tree level, the six quark masses, CKM mixing angles, and -violating phase must all follow from the same branch framework. The certificate is independently testable: a direct partition-function calculation can confirm or rule out , equivalently . Neutral-sector outlook. A natural next test is whether the same emergence tree accommodates the neutral sector. If the tree admits the branch then the two singlets give exactly two right-handed neutrinos. A rank-2 Type-I seesaw then forces one light neutrino to be massless. For normal ordering, The numerical uncertainty is then inherited from the oscillation splittings; the structural prediction is . Using NuFIT 6.0 central values , the sum lies just above the oscillation floor and close to current DESI DR2 cosmological sensitivity in CDM , with future reach from surveys such as Euclid . This conditional neutral prediction is not an input to the charged-lepton mass formula. Conclusion The direct finite- path keeps the center label: the broken-vector determinant carries the harmonic . The cyclic BdG branch erases this label: the mass matrix is circulant, and the non-circulant term is not an admissible mass term. On this positive branch, the earlier BdG identity reduces the Koide condition to the amplitude ratio , and the present paper supplies the MTC-level data These inputs assemble into the charged-lepton branch formula which reproduces the electron, muon, and tau masses to about . The ratio and phase are determined by the shared chiral data of the modular tensor category together with the octonionic channel normalization; the absolute scale is set by (Definition ), equivalently , and is separately falsifiable through the criteria in Section . At the full-RCFT layer, is the categorical expression of the same label-free condition ( , ); which closed branches persist through subsequent layers (in particular, whether the same framework reproduces the quark masses, CKM mixing, and gravitational sector) will narrow the set further. Acknowledgments The author thanks the open-source communities behind Python, mpmath, and \ for the tools used in the numerical checks accompanying this work.