Coherence Obstructions and Family Triplication: A Categorical Mechanism for Three Fermion Generations
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Coherence Obstructions and Family Triplication: A Categorical Mechanism for Three Fermion Generations Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] Why does the Standard Model contain three fermion families? We develop a categorical mechanism in which family number is fixed by a coherence obstruction. In a rigid braided monoidal setting, Yukawa couplings are natural transformations between functors that assign Hilbert spaces to chiral representations. We prove in the pointed fusion case that if the associator of the relevant flavor fiber is governed by a 3-cocycle [ ω ] ∈H3 ( G, U (1)) of finite order N , then strict naturality on the interaction fiber is possible only after stacking N identical copies of the sector; the minimal replication equals N . Under mild hypotheses, we identify a Z3 pointed fusion subcategory F ≃ Vecω Z3 with [ ω ]of order 3, thereby enforcing triplication. The induced Z3 grading yields block-circulant Yukawa textures that are diagonalized by the discrete Fourier transform F3 and naturally accommodate hierarchies after small symmetry breaking. A toy model illustrates how triple stacking cancels the pentagon phase and produces a parameter-independent phase relation, argdetYuYd∈2π 3Z, up to soft breaking. The framework does not replace dynamics; rather, coherence selects allowed multiplicities and texture classes and is compatible with anomaly cancellation and the Standard Model gauge structure. We delineate assumptions versus derivations and outline phenomenological tests that could falsify or support the mechanism. I. INTRODUCTION A persistent mystery in particle physics is why the Standard Model (SM) of particle physics contains three and only three fermion families. While the SM accommodates this replication as an input, it does not offer a structural principle that selects the family number. Traditional approaches postulate horizontal/flavor symmetries (continuous or discrete) or invoke ultraviolet completions (GUTs, stringinspired models), which then engineer textures for Yukawa couplings and mixing matrices [ 3 , 5 , 6 , 65 ]. In this work we propose a different organizing principle: family number as a coherence requirement in a monoidal categorical framework for SM kinematics and interactions. Core idea. We model the kinematic data of chiral SM representations within a rigid braided monoidal category ( C,⊗,1, α, β ), with matter fields as objects, intertwiners as morphisms, and tensor product encoding composition of degrees of freedom [ 10 , 58 , 59 , 75 ]. The Higgs field plays the role of a mediator that allows morphisms Q⊗ Φ →U, D and L⊗ Φ →E in a functorially coherent way. Yukawa couplings are then understood as natural transformations between tensor functors to Hilb (or a unitary representation category), subject to the standard naturality squares. Crucially, these squares commute only up to the associator αand braiding β. We show that demanding a strict (or trivial) associator on the interaction fiber—the substructure where the Yukawa naturality squares live—leads to a coherence obstruction measured by a class [ ω ] ∈H3 ( G, U (1)) when the relevant flavor fiber embeds as a pointed fusion category Vecω G[11]. The associator αX,Y,Z : ( X⊗Y ) ⊗Z→X⊗ ( Y⊗Z )in a monoidal category is constrained by Mac Lane’s pentagon identity (αW,X,Y ⊗idZ)◦αW,X⊗Y,Z ◦(idW⊗αX,Y,Z) = αW⊗X,Y,Z ◦αW,X,Y ⊗Z,(1) and, in the braided case, by the hexagon identities [ 58 , 59 ]. In pointed fusion categories Vecω G , the data of α is cohomological: the components of α are phases determined by a normalized 3-cocycle ω∈Z3 ( G, U (1)), with monoidal equivalence classes classified by [ ω ] ∈H3 ( G, U (1)) [ 11 ]. This observation provides a quantitative handle on coherence: if [ ω ] 6 = 0, naturality squares for interaction morphisms commute only up to nontrivial pentagon phases inherited from α. Stacking and the selection of family number. Consider the Deligne tensor product (“stacking”) of tensor categories [12]. Under stacking, associator classes add: [ω]stack = [ω1]+[ω2]∈H3(G, U(1)).
2 If [ ω ]has finite order N in H3 ( G, U (1)), then stacking N identical copies yields N [ ω ] = 0, allowing trivialization of the associator on the interaction fiber.[ 82 ] We interpret these N stacked copies as N identical families of SM matter, in the sense that the flavor fiber replicates N times before the Yukawa naturality is well-defined in the strict sense. The minimal N that cancels the obstruction is then a coherence-selected family number. In particular, if the relevant flavor fiber admits an embedding of a pointed fusion subcategory with G = Z3 and [ ω ]a generator of H3 ( Z3, U (1)) ∼ =Z3 , then N = 3 is the minimal stacking number that cancels the pentagon phase. This yields a mechanism for triplication, distinct from postulating a horizontal Z3 flavor symmetry: the Z3 here is the grading that classifies the associator twist, not a family group acting on fields. The physical intuition is analogous to phase cancellation in anomaly inflow, but the invariant here is coherence-theoretic rather than anomaly-theoretic. From coherence to textures. Beyond the selection of family number, a Z3-graded flavor fiber F=M a∈Z3Fa combined with Higgs-mediated intertwiners that select the neutral grade naturally induces a three-block structure in Yukawa maps, Y:Hflavor ∼ =H0⊕H1⊕H2−→ Hflavor, leading to block-circulant textures of the schematic form Y= A B C C A B B C A , A, B, C ∈C,(2) which are diagonalized by the discrete Fourier transform F3 [ 13 ]. Small controlled departures from strict grading (soft breaking) generate realistic hierarchies and mixing angles, offering a path to phenomenology complementary to Froggatt–Nielsen and discrete non-Abelian flavor models [ 5 , 6 , 65 ]. We emphasize that coherence here selects the allowed multiplicities and a restricted texture class; dynamics (UV completions, radiative corrections) can then populate parameters within this class. Mathematical statement (pointed case). Let F ≃ Vecω G be a pointed fusion subcategory governing the flavor fiber in the sense that the naturality squares for Yukawa transformations factor through F (Sec. 3). If [ ω ] ∈H3 ( G, U (1)) has finite order N , then there exists a smallest N for which the stacked category FN admits a monoidal equivalence under which the associator on the neutral interaction sector is trivial (a coboundary). If, furthermore, the physical Yukawa naturality requires this trivialization (Sec. 3), then N copies of the matter content are required for well-defined interactions on that fiber. When G = Z3 and [ ω ]is a generator, the minimal N equals 3. The proof follows from standard properties of pointed fusion categories and Deligne tensor product (details deferred to App. A), and it clarifies what is proved (the coherence/stacking implication) versus what is assumed (the identification of the relevant flavor fiber and the strictness requirement for naturality). Physical motivation and scope. Two observations motivate focusing on Z3 : (i) the center of SU (3) is Z3 , and color triality often leaves Z3 -graded remnants in composite structures; (ii) the global structure of the SM gauge group admits discrete quotients that can interact nontrivially with flavor fibers at the level of global phases. We do not derive the flavor fiber from first principles here; instead, we isolate coherence as a constraint that, once a Z3 -graded pointed fiber is present, forces triplication to restore strict naturality on the interaction sector. This is compatible with anomaly cancellation and standard local quantum field theory axioms, as natural transformations live in dagger (or C∗ ) settings where unitarity is preserved [10, 75]. What is new. • Acoherence obstruction mechanism in which the order of an associator class [ ω ] ∈H3 ( G, U (1)) fixes the family number via stacking (Deligne product). This reframes family replication as a minimality problem for trivialization of the interaction fiber. • A natural emergence of block-circulant Yukawa textures (2) from a Z3 -graded flavor fiber, with diagonalization by F3and a concrete phase relation for determinants (detailed in Sec. 7).
3 • A clean separation of assumptions (existence and relevance of a pointed Z3 subcategory for the flavor fiber; strictness requirement for naturality) from derivations (stacking order equals family number; texture class and associated constraints). Relation to prior work. Our approach differs from horizontal symmetry model-building [ 3 , 5 , 65 ] and from modular flavor symmetry frameworks [ 6 ] in that we do not posit a family group acting on fields. Instead, we use monoidal coherence to constrain the ability to define interaction naturality on a single copy; replication restores it. Methodologically, we rely on standard results on monoidal coherence [ 58 , 59 ], graphical calculi [ 10 , 75 ], and the classification of pointed fusion categories by group cohomology [ 11 ], together with Deligne’s tensor product of abelian (tensor) categories [ 12 ]. On the phenomenological side, block-circulant textures and their discrete Fourier diagonalization are classical [ 13 ], and provide a convenient laboratory to connect coherence-selected structures to mass hierarchies and mixings. Organization. Section 2 introduces the categorical background and fixes notation. Section 3 formulates SM kinematics in functorial terms and explains Yukawa couplings as natural transformations. Section 4 defines the coherence obstruction and proves stack-trivialization in the pointed case. Section 5 applies this to Z3 and states the triplication mechanism under clearly labeled hypotheses. Section 6 derives the block-circulant texture class. Section 7 presents a worked toy model and a testable phase relation. Section 8 discusses consistency checks. Section 9 compares to existing flavor models. Section 10 outlines limitations and falsifiability. Appendices collect proofs and graphical details. II. PRELIMINARIES AND NOTATION This section fixes the categorical background, the unitary structures we assume, and the notation used throughout. We keep the presentation compact but mathematically precise, with brief physics comments indicating how each concept maps to the Standard Model (SM) kinematics and Yukawa couplings developed later. Conventions Unless stated otherwise, all categories are C –linear, semisimple, finite (finitely many isomorphism classes of simples), rigid monoidal, and endowed with a dagger (involution on morphisms) compatible with the tensor product, so that we are working in the setting of unitary fusion categories. This ensures positivity/“probability” and unitarity in the physical interpretation. Composition of morphisms is denoted by ◦and reads right-to-left; identity morphisms are idX. A. Monoidal, braided, and rigid structures Amonoidal category ( C,⊗,1, α, λ, ρ )consists of a category C , a bifunctor ⊗ : C ×C → C (the tensor product), a distinguished unit object 1, and natural isomorphisms αX,Y,Z : (X⊗Y)⊗Z∼ = −−→ X⊗(Y⊗Z), λX:1⊗X∼ = −−→ X, ρX:X⊗1∼ = −−→ X, satisfying the standard pentagon and triangle coherence axioms. The pentagon reads (αW,X,Y ⊗idZ)◦αW,X⊗Y,Z ◦(idW⊗αX,Y,Z) = αW⊗X,Y,Z ◦αW,X,Y ⊗Z.(3) Abraided monoidal category additionally has a natural family of isomorphisms (the braiding) βX,Y : X⊗Y→Y⊗X obeying hexagon identities. Physically, ⊗ encodes composition of degrees of freedom (“stacking” of systems), while β encodes exchange statistics (we will not rely on nontrivial braiding in the core mechanism, but it will be convenient to allow it). A monoidal category is rigid if every object X admits a (left/right) dual ∨X together with evaluation and coevaluation maps evX:∨X⊗X→1,coevX:1→X⊗∨X, satisfying the snake (zig–zag) identities. Rigidity underlies the existence of categorical traces and dimensions, important for defining scalar amplitudes from closed diagrams.
4 B. Dagger, pivotal, and spherical structures (unitarity) Adagger on C is a contravariant, identity-on-objects, involutive functor ( · ) † : C → C such that ( f⊗g ) † = f†⊗g† and ( α, β, λ, ρ )are unitary. A pivotal structure is a monoidal natural isomorphism j : Id ⇒ ( − ) ∨∨ ; if left and right traces coincide, the category is spherical. In unitary settings arising from operator-algebraic constructions (e.g., Doplicher–Roberts duality), these structures encode unitarity and positivity needed for a quantum interpretation. In our application, dagger–pivotal structure guarantees that the naturality squares representing Yukawa couplings are compatible with adjoints, hence with Hermitian conjugation in Hilb. C. Pointed fusion categories and the associator class [ω] A fusion category is pointed if all simple objects are invertible under ⊗ . Such categories are classified, up to monoidal equivalence, by a finite group G (the isomorphism classes of simples form G ) together with a normalized 3–cocycle ω∈Z3 ( G, U (1)) that determines the associator [ 74 ]. Concretely, if we denote by Vecω Gthe category of G–graded finite-dimensional vector spaces with associator αg,h,k =ω(g, h, k)·idVghk , g, h, k ∈G, (4) then ωmust satisfy the cocycle condition ω(h, k, `)ω(g, hk, `)ω(g, h, k) = ω(gh, k, `)ω(g, h, k`),(5) and the normalization ω(e, g, h) = ω(g, e, h) = ω(g, h, e)=1,(6) with e the identity of G . Two cocycles differing by a coboundary yield monoidally equivalent categories; the associator class is [ω]∈H3(G, U(1)). The pentagon axiom (3) is precisely (A2). Physics comment. In Dijkgraaf–Witten topological gauge theories, [ ω ] ∈H3 ( G, U (1)) plays the role of a topological action in 2 + 1dimensions [ 15 ]. Here it quantifies a coherence twist: nontrivial phases appear when reshuffling brackets in multi-field couplings. Our mechanism will identify a situation in which the Yukawa naturality squares require the effective associator to be trivial (a coboundary) on the interaction fiber; when it is not, one must “stack” multiple copies until the obstruction cancels. D. Fiber functors and natural transformations (Yukawa as naturality) Let Hilb denote the category of finite-dimensional Hilbert spaces and linear maps. A (unitary) tensor functor F : ( C,⊗,1 ) → ( Hilb,⊗,C )assigns a Hilbert space to each object and a linear map to each morphism, together with coherent unitary isomorphisms ΦX,Y :F(X)⊗F(Y)∼ = −−→ F(X⊗Y),Φ1:C∼ = −−→ F(1), that satisfy the obvious pentagon/triangle conditions with respect to α, λ, ρ and the trivial associator on Hilb. Given two such functors F, G : C → Hilb , a monoidal natural transformation η : F⇒G is a family {ηX : F ( X ) →G ( X ) }X∈C of linear maps such that for every f : X→Y one has ηY◦F ( f ) = G ( f ) ◦ηX , and, in addition, ηis compatible with Φon tensor products. Physics comment. We will model kinematics by (lax) tensor functors that assign Hilbert spaces to SM representations (objects) and intertwiners to morphisms. Yukawa couplings are then packaged as natural transformations between such functors, with the naturality squares encoding the compatibility of Yukawa maps with all allowed reshufflings of brackets and symmetries. When the underlying associator is twisted by ω , these squares commute only up to α –phases; the requirement that they commute strictly on the interaction fiber is what yields our coherence obstruction.
5 E. The interaction fiber Let OYuk ⊂Ob ( C )be the set of objects participating in Yukawa couplings (e.g., Q, L, U, D, E, Φin the SM case). Define the interaction fiber I ⊂ C as the smallest full monoidal subcategory containing OYuk and closed under duals and subobjects. Concretely, I is generated by iterated tensor products of the objects in OYuk needed to state the naturality constraints for Yukawa maps. Assumption (flavor fiber). We assume that I admits a pointed fusion flavor subcategory F ≃ Vecω G through which the associator on I factors, in the sense that the only obstruction to strict naturality on I is the class [ ω ] ∈H3 ( G, U (1)). This isolates the purely coherence-theoretic content; dynamics enter later when textures are populated. F. Deligne tensor product and stacking If A and B are finite tensor (fusion) categories, their Deligne tensor product AB is again a finite tensor category characterized by the universal property that bifunctors A×B → D that are right–exact in each variable correspond to functors AB → D (and similarly in the semisimple/unitary setting) [ 74 ]. We will refer to passing from Ato ANas stacking Ncopies. For pointed categories, associator classes add under stacking: Vecω1 GVecω2 G≃Vecω1+ω2 G,[ω1+ω2]=[ω1]+[ω2]∈H3(G, U(1)).(7) Hence, if [ ω ]has finite order N in H3 ( G, U (1)), then (Vecω G)N≃Vec0 G (trivial associator) on the neutral sector. This is the categorical counterpart of “adding phases until they cancel.” G. A minimal lemma (order equals minimal replication) We record the basic statement that underlies our mechanism; the full proof is deferred to the appendix in the main text. Lemma II.1 (Stack–trivialization on the interaction fiber) . Let I be the interaction fiber generated by OYuk , and suppose there is a faithful tensor functor ι : F,→ I with F ≃ Vecω G pointed. If the naturality squares defining Yukawa transformations factor through ι ( F )(i.e., the only obstruction to strict commutativity is α on F ), then strict naturality on I is achievable after stacking N identical copies if and only if N·[ω] = 0 in H3(G, U(1)). The smallest such Nequals the order of [ω]. Sketch. By functoriality, the associator on I restricts to that on F . Under stacking, the restricted associator class on FN is N [ ω ]by (7) . If N [ ω ]=0, then (after monoidal equivalence) the associator is a coboundary and can be absorbed by a monoidal structure map on the functor(s) realizing Yukawa data, yielding strict squares on the neutral interaction sector. Minimality follows from group-order considerations. Physics comment. Lemma II.1 translates the purely cohomological statement “[ ω ]has order N ” into the physical statement “ N identical replicas of the matter fiber are required so that the Yukawa naturality constraints are well-defined without residual coherence phases.” In the body of the paper we will argue that G = Z3 and [ ω ]a generator are natural choices for the flavor fiber, making N = 3 the minimal replication number. H. Notation summary For quick reference, we collect the symbols used repeatedly later. Every symbol is introduced above and used with the same meaning throughout. Summary. We will use these structures as follows: Sec. 3 encodes SM kinematics and Yukawa couplings via tensor functors and natural transformations; Sec. 4 isolates the coherence obstruction as the associator class [ ω ]on a pointed flavor fiber; Sec. 5 shows that when G = Z3 and [ ω ]has order 3, triplication is required to strictify Yukawa naturality on the interaction fiber.
6 Symbol Meaning CBase (unitary) monoidal category modeling kinematics ⊗,1Tensor product and tensor unit α, β Associator and (when present) braiding ∨X;evX,coevXDual of X; evaluation/coevaluation maps (rigidity) (·)†;jDagger (adjoint) on morphisms; pivotal structure F, G :C → Hilb (Unitary) tensor functors to finite-dimensional Hilbert spaces η:F⇒GMonoidal natural transformation (packages Yukawa couplings) IInteraction fiber: monoidal subcategory generated by Yukawa objects F ≃ Vecω GPointed fusion (flavor) subcategory with associator class [ω] ω∈Z3(G, U(1)) Normalized 3–cocycle; [ω]∈H3(G, U(1)) Deligne tensor product (stacking of sectors) NOrder of [ω]in H3(G, U(1)); minimal replication number TABLE I: Notation used in Sec. II and subsequent sections. III. SM KINEMATICS AS FUNCTORS; YUKAWA COUPLINGS AS NATURAL TRANSFORMATIONS This section formulates the Standard Model (SM) field content and Yukawa couplings inside the categorical setting fixed in Sec. II. Objects of the ambient unitary monoidal category C encode chiral representations; tensor functors into Hilb carry these objects to Hilbert spaces; and Yukawa couplings arise as (monoidal) natural transformations whose naturality squares capture gauge covariance and bracketindependence. This provides the interface between familiar Lagrangian formulae and the coherence constraints analyzed in Secs. II–IV. A. Chiral SM representations as objects in C Let GSM =SU(3) ×SU(2) ×U(1)Y Γ be the SM gauge group (the global quotient Γis irrelevant for the present section). We model the chiral multiplets and the Higgs as objects of C: Q∈Rep SU(3) ×SU(2) ×U(1)Y,(3,2)+1/6, U∈Rep SU(3) ×SU(2) ×U(1)Y,(3,1)+2/3, D∈Rep SU(3) ×SU(2) ×U(1)Y,(3,1)−1/3, L∈Rep SU(3) ×SU(2) ×U(1)Y,(1,2)−1/2, E∈Rep SU(3) ×SU(2) ×U(1)Y,(1,1)−1, N∈Rep SU(3) ×SU(2) ×U(1)Y,(1,1)0(optional RH neutrino), Φ∈Rep SU(3) ×SU(2) ×U(1)Y,(1,2)+1/2,e Φ := (iσ2)Φ∗∼(1,2)−1/2. The corresponding morphisms in C include intertwiners of these representations; gauge transformations act as automorphisms. We will restrict to the full monoidal subcategory I ⊂ C generated by {Q, U, D, L, E, (N),Φ}and their duals, cf. Sec. II. Physics reminder. In the Lagrangian, Yukawa terms are LYuk =−Q Yue ΦU−Q YdΦD−L YeΦE−L Yνe ΦN+ h.c., (8) where Yf are complex matrices in family space, and gauge invariance amounts to requiring the maps Q⊗e Φ→Uetc. to be intertwiners of GSM.
7 B. Tensor functors to Hilb and coherence maps Let F: (I,⊗,1)→(Hilb,⊗,C)be a unitary (strong) tensor functor with coherence isomorphisms ΦX,Y :F(X)⊗F(Y)∼ = −−→ F(X⊗Y),Φ1:C∼ = −−→ F(1), compatible with the associator αon Iand the trivial associator on Hilb. Concretely: ΦX,Y ⊗Z◦(idF(X)⊗ΦY,Z ) = F(αX,Y,Z )◦ΦX⊗Y,Z ◦(ΦX,Y ⊗idF(Z)),(9) and similarly for the triangle axioms with Φ 1 . The dagger structure on I is sent to Hermitian adjoint on Hilb. For each object X∈ I we write HX := F ( X ), with HQ, HU, . . . finite-dimensional Hilbert spaces of internal degrees of freedom (including color, weak isospin, and flavor multiplicities). In practice, HQ∼ =V(3,2)1/6⊗Cnf, HU∼ =V(3,1)2/3⊗Cnf,etc. where nfis the number of families (to be constrained later) and VRthe group-representation space. C. Yukawa couplings as natural data Consider the endofunctor TΦ:I → I given by tensoring on the right by the Higgs: TΦ(X) := X⊗Φ, TΦ(f) := f⊗idΦ. The functor Ftogether with ΦX,Y yields a monoidal natural isomorphism κX:HX⊗HΦ∼ = −−→ HX⊗Φnatural in X, (10) characterized by commutation with F ( f )for all f : X→Y . Gauge-invariant Yukawa intertwiners in C are morphisms yu:Q⊗e Φ→U, yd:Q⊗Φ→D, ye:L⊗Φ→E, yν:L⊗e Φ→N, which become linear maps on Hilbert spaces via Fand (10): Yu:= F(yu)◦κ(e Φ) Q:HQ⊗He Φ→HU,etc. (11) (where κ(e Φ) is defined analogously for e Φ ). The collection of maps {Yf} can be organized as components of a monoidal natural transformation between the functor X7→ HX⊗HΦ and the functor X7→ HX⊗Φ , followed by projection to a chosen target object, e.g. Ufor the up-type coupling. Naturality squares (gauge covariance). For any morphism f:X→X0in I, naturality of κasserts κX0◦(F(f)⊗idHΦ) = F(f⊗idΦ)◦κX,(12) and hence for a Yukawa component y:X⊗Φ→Z, F(y)◦κX0◦(F(f)⊗id) = F(y)◦F(f⊗id) ◦κX=F(y◦(f⊗id)) ◦κX.(13) If f encodes a gauge transformation, (13) is precisely the statement of gauge covariance of the Yukawa maps; if fencodes family-basis changes, (13) constrains how Yukawa matrices transform. Bracket-independence and the associator. While (12) holds abstractly, its compatibility with rebracketing uses (9). For instance, the two ways of forming HX⊗(HΦ⊗HΨ)→H(X⊗Φ)⊗Ψ→HZ must agree with the route (HX⊗HΦ)⊗HΨ→HX⊗(Φ⊗Ψ) →HZ, and the equality of these composites is mediated by F ( αX,Φ,Ψ )in (9) . In Sec. IV we will require this bracket-independence to hold strictly on the interaction fiber, which is where the associator class [ ω ] enters: if α carries a nontrivial phase on a pointed flavor subcategory, these two routes may differ by that phase, obstructing strict naturality unless the obstruction is cancelled (by stacking families).
8 D. Families and the flavor multiplicity fiber To relate (11) to the familiar Yukawa matrices, we factor the internal Hilbert spaces into gauge and flavor pieces. For each chiral object X∈ {Q, U, D, L, E, (N)}, choose a decomposition HX∼ =VX⊗FX, where VX carries the irreducible gauge representation and FX≃Cnf carries flavor multiplicity. The Higgs space HΦ∼ =VΦhas no flavor factor. Gauge invariance of yfimplies that on Hilbert spaces Yf= (invf)⊗Y(flavor) f,(14) where invf : Vsource ⊗VΦ→Vtarget is the unique (up to normalization) GSM –invariant intertwiner (the usual SU (2) doublet contraction with ij for up-type via e Φ , the identity contraction for down-type), and Y(flavor) u:FQ→FU, Y (flavor) d:FQ→FD, Y (flavor) e:FL→FE are complex matrices in family space. Equation (14) is the categorical version of the textbook statement that Yukawa couplings are the only family-dependent parameters in the renormalizable SM. Change of family basis. Let SX : FX→FX be unitary changes of family basis. Then naturality (13) yields the standard congruence actions Y(flavor) u7→ SUY(flavor) uS† Q, Y (flavor) d7→ SDY(flavor) dS† Q, Y (flavor) e7→ SEY(flavor) eS† L,(15) from which the CKM matrix VCKM = U† uUd and the PMNS matrix UPMNS = U† eUν emerge after singularvalue decompositions. Thus, the familiar flavor-basis freedom is already encoded as naturality freedom in I. E. Diagrammatic naturality and the role of coherence It is helpful to summarize the core diagram governing a single Yukawa map. Let X be Q or L , and Z be the corresponding singlet U, D, E, (N). The diagram HX⊗HΦHX⊗ΦHZ HX0⊗HΦHX0⊗ΦHZ κX F(f)⊗id F(y) F(f⊗idΦ)id κX0F(y) commutes for all f : X→X0 . When additional spectator objects S∈ I are present, two ways of moving brackets past S differ by F ( αX,Φ,S ); asking the extended diagram to commute without residual phases on such spectators is the strictness requirement that will drive the obstruction analysis in Sec. IV. F. From categorical data back to the Lagrangian With (14) , the matrix elements of (11) reproduce (8) . After electroweak symmetry breaking (EWSB), hΦi=1 √20 v, v ≃246 GeV, one obtains Dirac mass matrices Mu=v √2Y(flavor) u, Md=v √2Y(flavor) d, Me=v √2Y(flavor) e, with left-/right-unitary diagonalizations producing masses and mixing. If N is present, either a Dirac mass Mν = v √2Y(flavor) ν arises as above, or (in the absence of N ) an effective Majorana mass originates from the Weinberg operator ( LL ΦΦ) / Λ; in our categorical language this corresponds to a dinatural transformation L⊗L⊗Φ⊗Φ→1factoring through the SU(2) singlet channel.
9 G. Where coherence constrains families The categorical place where family number enters is not in (14) itself, but in the consistency of naturality with respect to all ways of (re)bracketing and introducing spectators from the interaction fiber. If the associator on a pointed flavor subcategory F ≃ Vecω G⊂ I contributes a nontrivial phase when moving brackets past flavor-gradings, then the composites implementing “Yukawa then spectator” versus “spectator then Yukawa” differ by F ( α )unless that phase cancels. In Sec. IV, Lemma II.1 will imply that cancellation requires stacking a number N of identical copies equal to the order of [ ω ] ∈H3 ( G, U (1)). This is the mathematical origin of our coherence-selected family number. Summary. This section established the functorial encoding of SM chiral data and packaged Yukawa couplings as natural transformations, isolating the precise interface where the associator α matters. The next section identifies the cohomological obstruction to strict naturality on the interaction fiber and relates its order to the minimal replication of families. IV. COHERENCE OBSTRUCTION ON THE INTERACTION FIBER We now isolate the coherence-theoretic content that constrains family number. Building on Secs. II–III, we formalize the requirement that Yukawa naturality squares commute strictly on the interaction fiber, show how the failure of strictness is measured by an associator class on a pointed flavor subcategory, and prove that stacking identical copies trivializes the obstruction in a number equal to the order of this class. This yields the core mechanism: family number equals the order of the coherence obstruction. Throughout, I ⊂ C is the interaction fiber generated by {Q, U, D, L, E, ( N ) , Φ } and duals; F : I → Hilb is a fixed unitary strong tensor functor with coherence (Φ X,Y , Φ 1 ); and HX := F ( X )for X∈Ob ( I ) (Sec. III). We keep the dagger structure implicit. A. Strict interaction naturality and the defect 3–cocycle Let TΦ be the endofunctor X7→ X⊗ Φon I . Recall the canonical isomorphisms κX : HX⊗HΦ∼ = −→ HX⊗Φ induced by F and Φ X,Y (Eq. (10) ). Given a Yukawa component y : X⊗ Φ →Z (e.g. Q⊗e Φ→U ), we consider the following extended naturality requirement: for any spectator S∈Ob ( I )and morphism f:X→X0, F(y)◦κX0⊗S◦(F(f)⊗idHS)⊗idHΦ? =F(y)◦κX0◦F(f)⊗idHΦ◦bracket move(S),(16) where bracket move ( S )denotes the unique composite of coherence isomorphisms (built from Φ −,− and their inverses) that rebrackets ( HX⊗HS ) ⊗HΦ to HX⊗ ( HS⊗HΦ )or vice versa. In a strict monoidal category this move is the identity; in our non-strict setting it passes through F(α). The interaction naturality defect of the triple (X, Φ, S)is the endomorphism of HZdefined by ∆(X; Φ; S) := hF(y)◦κX⊗S◦(idHX⊗idHS⊗idHΦ)i·hthe other bracket routei−1,(17) where the two square brackets denote the two sides of (16) with f = idX , written as composites in Hilb . By construction ∆( X ; Φ; S )is a unitary scalar on HZ (a phase), natural in X and functorial in S .Strict interaction naturality demands ∆(X; Φ; S) = idHZfor all X, S relevant to Yukawa couplings. Reduction to associator data. If the only source of nontriviality in bracket moves comes from the associator on a pointed flavor subcategory (see below), then ∆factors through the components of F ( α ) on that subcategory. In particular, if S carries a well-defined flavor grade g∈G and X carries h∈G (with Φneutral), then ∆(X; Φ; S) = ω(g, h, Φ) ·idHZ,(18) for a normalized 3–cocycle ω∈Z3 ( G, U (1)). The normalization ensures ∆( X ; Φ; 1 ) = 1 and the pentagon for αimplies the 3–cocycle identity for ω. Hence the defect is cohomological.
16 F. CP violation and invariants Let Hf:= YfY† f(f=u, d). The Jarlskog invariant J∝Im Tr[Hu, Hd]3 is basis–independent and vanishes iff the commutator has rank ≤ 2[ 36 ]. In the exact block–circulant case, [Hu, Hd] = diag[H(0) u, H(0) d],[H(1) u, H(1) d],[H(2) u, H(2) d] so J is a sum of three sectoral contributions. If any Y(k) f are 1 × 1, their commutators vanish; hence nonzero J requires at least one momentum sector with multiplicity ≥ 2, or small symmetry–breaking terms that couple sectors. This cleanly separates where physical CP–violating phases must reside. G. Soft breaking of the Z3grading Realistic spectra and mixing can be generated by soft violations of (20) that preserve the coherence mechanism but slightly mix grades. Write Y(flavor) =Yh0i |{z} block–circulant +∆Yh1i | {z } nearest–grade mixing +2∆Yh2i+··· ,0< 1,(27) where Yh0i has the form (21) and ∆ Yh1i connects only neighboring grades ( a→a± 1). In the F3 basis, Yh0i is block–diagonal (Eq. (23) ) while ∆ Yh1i is block–off–diagonal with selection rule ∆ k = ± 1(mod 3). First–order perturbation theory yields δY (k)=(∆Yh1i)k,k±1, δΣ(k) f=O(), and induces inter–sector mixing angles θk,k±1 = O ( )that populate the CKM off–diagonals while preserving the basic eigenvalue structure (24) . This provides a natural path to hierarchical masses (via interference in λkor small singular values of Y(k)) and small but nonzero mixing. Minimal 3 × 3example. Take da = 1 so Yh0i has eigenvalues (25) . Choose A, B, C such that |λ0| |λ1||λ2| (near–cancellations among A, B, C yield the hierarchy). Add –suppressed nearest–grade terms; in the F3 basis they generate small rotations among eigenvectors associated with λk , producing a CKM–like pattern with Cabibbo–sized O ( )angles. The CP phase originates from arg ( ABC )and the phases of the –terms; J=O(2), consistent with small but nonzero CP violation. H. Neutrino sector and Majorana masses If right–handed neutrinos Nare present, the Dirac Yukawa Yνfollows the same block–circulant logic. In the absence of N , the dimension–five Weinberg operator corresponds categorically to a dinatural transformation L⊗L⊗ Φ ⊗ Φ →1 ; the induced Majorana mass matrix Mν then inherits the Z3 selection rules. In a basis where charged leptons follow (21), one finds Mν= S0S1S2 S1S2S0 S2S0S1 , Sr=ST r, asymmetric block–circulant matrix (each Sr is symmetric). Its Fourier blocks are M(k) ν = S0 + ζkS1 + ζ2kS2 . Large lepton mixing angles can then be attributed either to large rotations inside one or more M(k) ν or to soft breaking that couples momentum sectors more strongly in the lepton channel than in the quark channel. I. Relation to residual and modular symmetries The Z3 -graded flavor fiber does not postulate a family symmetry acting on fields; it arises from the grading that organizes coherence data. Nevertheless, at the level of textures its consequence—commutation
17 with P —coincides with what one would obtain from a residual Z3 symmetry in family space. The difference is conceptual: here, Z3 sits in the categorical flavor fiber that controls rebracketing phases and survives as a robust kinematic constraint on Yukawa maps even when the microscopic dynamics do not realize an explicit horizontal group. J. Summary AZ3-graded flavor fiber yields: •Selection rule: Yukawa maps commute with the cyclic flavor translation, Eq. (20). •Texture class: Block–circulant Yukawa matrices, Eq. (21) , diagonalized by F3 into three momentum–sector blocks, Eq. (24). •Mixing mechanism: CKM/PMNS mixing arises from misalignment within these blocks (Eq. (26) ) and from controlled soft breaking that couples sectors. •CP structure: CP–violating invariants decompose by momentum sector and require either multiplicity within a sector or soft breaking to be nonzero. This prepares the ground for Sec. 7, where we exploit Eqs. (25) – (24) to derive concrete phase relations and illustrate numerically how small symmetry–breaking deformations generate realistic spectra and mixings. VII. YUKAWA TEXTURES, MIXING, AND A COHERENCE–PHASE PREDICTION Sections IV–VI established that (i) the order of the flavor–coherence class [ ω ] ∈H3 ( G, U (1)) fixes the minimal number of families (Theorem V.2), and (ii) a Z3 –graded flavor fiber forces Yukawa maps to commute with the cyclic flavor translation P , hence to be (block–)circulant and diagonalizable by the discrete Fourier transform F3 (Eq. (23) ). In this section we develop the resulting spectral structure, show how mixing and CP violation arise from misalignment of Fourier blocks and soft departures from exact Z3 –grading, and derive a concrete phase relation for the product of Yukawa determinants under a mild coherence–gauge assumption. We end with a worked symbolic benchmark and discuss phenomenological levers. A. Recap: block–circulant Yukawas and Fourier blocks For a chiral pair ( X, Z )(e.g. Q→U ), the flavor part of a Yukawa map Y(flavor) : FX→FZ commutes with the cyclic shift P (Eq. (20) ); in a basis grouped by the three Z3 –grades this is block–circulant (Eq. (21)). Passing to the family–momentum basis with 1⊗F3yields the block decomposition e Y(flavor) = diag Y(0), Y (1), Y (2), Y (k)=C0+ζkC1+ζ2kC2, ζ =e2πi/3,(28) where Cr are the three structural blocks (Sec. VI). After electroweak symmetry breaking (EWSB), the Dirac mass matrices are Mf = v √2Y(flavor) f ( f = u, d, e, ν or Dirac ν ), and all spectral information factorizes across the three k–blocks. B. Spectral invariants, traces, and determinants Denote by χn ( Y ) := Tr ( Yn )the power–sum invariants and by det ( Y )the determinant (when square). In the family–momentum basis, χn Y(flavor)= 2 X k=0 Tr (Y(k))n,detY(flavor)= 2 Y k=0 detY(k).(29)
18 In the scalar (pure circulant) case d0 = d1 = d2 = 1 with Cr≡ ( A, B, C ) ∈C , the three eigenvalues are λ(k)=A+ζkB+ζ2kC, det(Y) = 2 Y k=0 λ(k)=A3+B3+C3−3ABC. (30) These identities lift blockwise when Cr are square matrices that commute (e.g. simultaneously diagonalizable textures); otherwise, Eq. (29) remains the correct invariant factorization. C. Mixing from block misalignment Let Y(flavor) u and Y(flavor) d be block–circulant. Singular–value decompositions (SVDs) on Fourier blocks, Y(k) u=U(k) uΣ(k) uV(k)† u, Y (k) d=U(k) dΣ(k) dV(k)† d, k = 0,1,2, assemble into block–diagonal left unitaries Uf = diag ( U(0) f, U(1) f, U(2) f ). Transforming back to the family basis multiplies by 1⊗F† 3on the left, but this common unitary cancels in the CKM matrix, VCKM =U† uUd= diagW(0), W(1), W(2), W(k):= U(k)† uU(k) d.(31) Hence mixing angles and the Dirac CP phase arise entirely from misalignment among the pairs U(k) u, U(k) d across k = 0 , 1 , 2. If any block is 1 × 1, its contribution to mixing is a rephasing; nontrivial angles require either (i) block multiplicities ≥ 2in at least two sectors, or (ii) small controlled departures from exact grading that couple different k–blocks (Sec. VII E). D. A coherence–phase constraint on Yukawa determinants The obstruction–cancellation of Sec. V is implemented by choosing a trivializing 2–cochain ν with δν = ω⊕3 on the diagonal flavor embedding. This choice (a coherence gauge) feeds into the monoidal structure maps Φ X,Y and, consequently, into the Yukawa naturality isomorphisms κX (Eq. (10) ). We now show that, under a mild compatibility condition, the overall phase of the Yukawa determinant is quantized mod 2π/3. Coherence–gauge compatibility (CGC). Assume: (CGC1) The trivializing cochain ν is chosen uniformly across all Yukawa channels so that the three Fourier blocks Y(k) f pick up only the character phases of Z3 , i.e. under a single step of flavor translation P they transform by Y(k) f7→ ζqfY(k) fwith qf∈ {0,1,2}independent of k. (CGC2) Family–basis changes SX are restricted to those commuting with P (the natural ones in the graded setting), so det(Yf)is invariant up to cubic roots of unity. Proposition VII.1 (Quantized Yukawa determinant phase) . Under (CGC1–2), the Yukawa determinant phase is quantized: arg det(Yf)∈2π 3Z(mod 2π), f =u, d, e, (νDirac). Consequently, arg detYuYd∈2π 3Z(mod 2π).(32) Sketch. In the Fourier basis, det ( Yf ) = Q2 k=0 detY(k) f . By (CGC1), a flavor translation multiplies each Y(k) f by the same cubic root ζqf , hence det ( Yf ) 7→ ζ3qfdet ( Yf ) = det ( Yf )—translation invariant. However, the coherence gauge that trivializes the associator can shift each Y(k) f by an overall phase labelled by a character of Z3 ; the only residual phases compatible with (CGC1–2) are cubic roots of unity. Therefore det ( Yf )is defined up to ζmf , mf∈Z , implying the stated quantization of arg det ( Yf )modulo 2π. The product in (32) follows.
19 Remarks. (i) The statement is kinematic, tied to the coherence gauge, not to dynamics; it survives soft breaking to O ( )as discussed below. (ii) In the quark sector, arg det ( YuYd )enters the QCD ¯ θ parameter via ¯ θ = θQCD + arg det ( YuYd ); Proposition VII.1 suggests a cubic–root structure for this phase in the coherence gauge. We do not claim a solution of the strong CP problem here; rather, we identify a discrete constraint that can coexist with, e.g., PQ–like mechanisms. E. Soft breaking: stability and small deviations Introduce a controlled deviation from exact Z3–grading, Y(flavor) f=Yh0i f+∆Yh1i f+O(2),0< 1, where Yh0i f is block–circulant and ∆ Yh1i f mixes neighboring Fourier sectors (∆ k = ± 1). First–order perturbation theory gives δarg det(Yf) = Im Tr Y−1 f∆Yh1i f+O(2), so the phase quantization of Proposition VII.1 is stable up to O ( )corrections. The same mixing couples the block–diagonal unitaries Uf across k , generating small but nonzero off–diagonal CKM/PMNS entries (consistent with data when ∼λC≈0.22 in the quark sector [37, 38]). F. Worked symbolic benchmark (pure circulant case) Take the 3×3circulant core Yh0i u= AuBuCu CuAuBu BuCuAu , Y h0i d= AdBdCd CdAdBd BdCdAd , with Af, Bf, Cf∈C. The eigenvalues are λ(k) f=Af+ζkBf+ζ2kCfand det(Yf) = A3 f+B3 f+C3 f−3AfBfCf. Coherence–aligned phases. Impose the cubic–character pattern Af=|Af|eiφf, Bf=|Bf|ei(φf+2π/3), Cf=|Cf|ei(φf+4π/3), which realizes (CGC1) at the level of entries.[86] Then λ(k) f=eiφf|Af|+|Bf|ζk+1 +|Cf|ζ2k+2,arg det(Yf)=3φf+ arg Y k Λ(k) f, with Λ (k) f∈R if |Af|,|Bf|,|Cf| satisfy mild triangle–type inequalities (the bracket is then real up to sign). Consequently arg det ( Yf ) ∈2π 3Z , matching Proposition VII.1. Small –deformations that mix Z3 sectors shift the phase by O()and populate the CKM off–diagonals. Hierarchies by interference. Choose, e.g., |Au||Bu||Cu| while |Ad|∼|Bd||Cd| ; interference among ζ –weighted terms produces one large, one medium, and one small |λ(k) f| , realizing a realistic mass–hierarchy pattern without fine tuning. The leading mixing arises either from misalignment within the k–blocks (if da>1) or from the –terms that couple k–sectors at O(). G. Sum rules and phenomenological handles The block factorization (29) implies a family of sum rules: Tr Hf= 2 X k=0 TrY(k) fY(k)† f,det Hf= 2 Y k=0 det Y(k) fY(k)† f,(33) so any fit to masses and mixing admits a decomposition into k –sector contributions. For CP violation, the Jarlskog invariant J decomposes into sectoral pieces as in Sec. VI; its smallness in the quark sector is naturally explained either by (i) approximate alignment U(k) u≈U(k) d in two sectors, or (ii) small –induced inter–sector mixing angles ∼λC(Wolfenstein hierarchy [37, 38]).
20 H. Summary Coherence and Z3 grading boil the flavor problem down to three Fourier blocks per Yukawa matrix. Mixing and CP violation acquire a transparent kinematic origin: block misalignments and small inter–block couplings. Beyond this structural clarity, Proposition VII.1 yields a discrete phase relation for Yukawa determinants in a natural coherence gauge, stable to soft breaking. The next section builds a fully worked toy model (with numerical illustrations) and explores consistency checks and phenomenology. VIII. WORKED TOY EXAMPLE: Vecω Z3FIBER, DIAGONAL STACKING, AND EXPLICIT TEXTURES This section assembles a complete, minimal model that realizes the mechanism developed in Secs. IV–VI. We take the flavor fiber to be a pointed fusion category F ≃ Vecω Z3 with a generator class [ ω ] ∈ H3 ( Z3, U (1)) ∼ =Z3 , embed it in the interaction fiber I , construct a strong monoidal functor F : I → Hilb , compute the interaction–naturality defect phases explicitly, and show how diagonal stacking of three identical copies cancels the obstruction. We then exhibit the induced block–circulant Yukawa textures, their Fourier–block diagonalization, a stability analysis under soft Z3 –breaking, and a concrete benchmark pattern that illustrates hierarchies and small mixings. Design goals. (i) Keep the flavor sector minimal (three grades only); (ii) make all coherence phases explicit; (iii) show where and how triplication ( N = 3) is necessary and sufficient; (iv) keep the gauge part of the SM kinematics standard and factorized from flavor; (v) exhibit the “three families ⇒ order–3 coherence obstruction” interpretation in a concrete setting. A. Flavor fiber Fand an explicit 3–cocycle Let G = Z3 = { 0 , 1 , 2 } (additive notation mod 3). Write the simple objects of F as { L a|a∈G} with L a⊗ L b≃ L a+b ,L 0≃1 , and ∨ L a≃ L −a . The associator on simples is multiplication by a normalized 3–cocycle αa,b,c =ω(a, b, c) idLa+b+c, ω ∈Z3(Z3, U(1)),[ω]generator.(34) A convenient representative on G={0,1,2}is ω(a, b, c) = exp2πi 3a+b 3c,with b·c the integer part (using reps 0,1,2),(35) which is normalized ( ω (0 ,∗,∗ ) = ω ( ∗, 0 ,∗ ) = ω ( ∗,∗, 0) = 1), takes values in { 1 , ζ, ζ2} with ζ = e2πi/3 , and represents a generator of H3(Z3, U(1)) ∼ =Z3(cf. Sec. V). B. Embedding Fin the interaction fiber and flavor grading Let I ⊂ C be the interaction fiber generated by X∈ {Q, U, D, L, E, ( N ) , Φ } and duals. We postulate a faithful tensor embedding ι:F,→ I such that flavor grades are tracked by F–labels: X≃M a∈Z3 Xa⊗La,|Φ|= 0, X ∈ {Q, U, D, L, E, (N)}.(36) Intuitively, Xa is the “same” gauge multiplet living in grade a . The Higgs is grade–neutral. By construction, Xa⊗Φ→Zais the only grade–allowed Yukawa channel (grade selection rule, Sec. VI). C. Strong monoidal functor and coherence maps Define a strong unitary tensor functor F:I → Hilb as follows. For each X, let HX:= F(X)∼ =VX⊗FX, FX∼ =FX,0⊕FX,1⊕FX,2,
21 with VX the finite–dimensional gauge representation space and FX,a ∼ =CdX,a the flavor multiplicity space in grade a. On simples, F(La)is a one–dimensional line carrying the character a, and ΦX,Y :HX⊗HY∼ = −−→ HX⊗Y are the canonical identification isomorphisms on Hilbert spaces, chosen so that ΦX,Y ⊗Z◦(id ⊗ΦY,Z ) = F(αX,Y,Z )◦ΦX⊗Y,Z ◦(ΦX,Y ⊗id),(37) holds with F ( αX,Y,Z )multiplying by ω ( |X|,|Y|,|Z| )on the F –part (see Sec. III, Eq. (9) ). Thus, all nontrivial bracket–move effects in the flavor directions are phases from (35). D. Yukawa data and the grade–preserving block structure Let yf:X⊗Φ→Zbe a Yukawa morphism in I(f=u, d, e, (ν)). On Hilbert spaces, Yf=F(yf)◦κX:HX⊗HΦ−→ HZ, Yf= invf⊗Y(flavor) f,(38) with invf : VX⊗VΦ→VZ the unique gauge intertwiner and Y(flavor) f : FX→FZ the family map. Because |Φ|= 0, Y(flavor) f:FX,a −→ FZ,a for each a∈Z3,(39) hence Y(flavor) f commutes with the grade–shift P and is block–circulant (Sec. VI, Eq. (21) ). Writing dX,a =dZ,a =dafor simplicity, Y(flavor) f= Af,0Af,1Af,2 Af,2Af,0Af,1 Af,1Af,2Af,0 , Af,r ∈Mat(d0, dr). E. Interaction–naturality defect: explicit phase Consider the extended naturality square with spectator S of grade g and source X of grade h (Higgs neutral). The two bracket routes differ by the component of F ( αh,0,g ), hence the defect phase (Sec. IV A) is ∆(X; Φ; S) = ω(h, 0, g)·id,(40) which, with (35) , takes values in { 1 , ζ, ζ2} . Strict interaction naturality therefore fails (whenever ω6 = 1) by a cubic root of unity tied to the flavor grades. F. Diagonal stacking and literal cancellation of phases Form the Deligne cube I3 and embed the original fiber diagonally ∆ : I → I3 , X7→ ( X, X, X ) (same for morphisms). On the diagonal flavor subcategory, the associator multiplies phases from the three copies: αdiag a,b,c =αa,b,c αa,b,c αa,b,c ω(a, b, c)3≡1. Hence the pointwise diagonal associator is trivial on ι(F); the defect (40) cancels exactly in I3: ∆3(X; Φ; S) = ω(h, 0, g)3·id = id.(41) This is the most explicit realization of Theorem V.2: three identical stacks kill the cubic coherence phase. (Equivalently, one may say there exists a 2–cochain ν with δν = ω⊕3 ; here the diagonal makes ω⊕3 literally 1on the nose, so one can take ν≡1on the diagonal.)
22 Interpretive remark (QM is not modified). What is “broken” is strict coherence on a single flavor copy: rebracketing picks up a cubic phase from ω . Stacking three copies removes that obstruction and restores strict naturality of the Yukawa squares. Hilbert spaces, unitarity, and the Born rule are untouched; the higher–coherence bookkeeping is enriched by a Z3–valued associator. Thus, “three families” records how much (order–3) and in what way (via a pointed flavor fiber) higher categorical coherence fails on one copy. G. Fourier–block diagonalization and spectra Passing to the family–momentum basis with 1⊗F3, the flavor map block–diagonalizes (Sec. VI): e Y(flavor) f= diagY(0) f, Y (1) f, Y (2) f, Y (k) f=Cf,0+ζkCf,1+ζ2kCf,2, where Cf,r are reshufflings of Af,r to match block shapes. Dirac masses after EWSB are Mf = (v/√2) Y(flavor) fwith the same block structure. Pure circulant limit ( d0 = d1 = d2 = 1). Write Cf,r = ( Af, Bf, Cf ) ∈C . Eigenvalues are λ(k) f = Af + ζkBf + ζ2kCf and det ( Yf ) = A3 f + B3 f + C3 f− 3 AfBfCf . If Af, Bf, Cf are real, λ(1) f = λ(2) f so |λ(1) f| = |λ(2) f| (a two–fold mass degeneracy). Avoiding this either requires non–scalar blocks ( da> 1) or a controlled breaking that couples the Fourier sectors (∆k6= 0), see below. H. Soft Z3–breaking and first–order splitting Introduce a small violation of exact commutation with P: Y(flavor) f=Yh0i f+∆Yh1i f+O(2),0< 1,(42) where Yh0i f is block–circulant and ∆ Yh1i f connects only neighboring grades (∆ k = ± 1in the F3 basis). Let Hf := YfY† f with unperturbed blocks H(k) f = Y(k) fY(k)† f . Standard degenerate perturbation theory on the Hermitian Hfgives, to first order, δm2 (k) f,j =u(k) f,j ,(∆Hf)kk u(k) f,j +X `6=kX ihu(k) f,j ,(∆Hf)k` u(`) f,ii2 m2 (k) f,j −m2 (`) f,i +O(2),(43) where H(k) fu(k) f,j = m2 (k) f,j u(k) f,j . The off–diagonal (∆ Hf ) k` produce small mixings between Fourier sectors k↔` and lift degeneracies such as |λ(1) f| = |λ(2) f| in the circulant limit. Equation (43) makes explicit that small –breaking generates Cabibbo–sized mixing angles θ∼and O()shifts of mass eigenvalues. I. A simple benchmark pattern We illustrate a light–touch realization that yields hierarchical quark masses and small mixings: Step 1 (circulant cores). Choose Yh0i u= AuBuCu CuAuBu BuCuAu , Y h0i d= AdBdCd CdAdBd BdCdAd , with coherence–aligned phases Af = |Af|eiφf , Bf = |Bf|ei(φf+2π/3) , Cf = |Cf|ei(φf+4π/3) . Then arg det(Yf)∈2π 3Z(Sec. VII D), and λ(0) f=eiφf(|Af|+|Bf|+|Cf|), λ(1) f=eiφf|Af|−|Bf|if |Cf|=|Bf|, so a mild hierarchy |Af||Bf|>|Cf|already gives |λ(0) f||λ(1,2) f|.
23 Step 2 (soft inter–sector mixing). Add ∆ Yh1i f that couples the k = 0 block weakly to k = 1 , 2 (nearest–grade in flavor basis). In the F3 basis, this makes Uf slightly non–block–diagonal and produces off–diagonals in VCKM = U† uUd at O ( ), while leaving the mass hierarchy largely controlled by {Af, Bf, Cf} . Taking ∼ 0 . 2yields Cabibbo–like leading mixing with subleading angles suppressed by additional small ratios in the block entries. Step 3 (leptons). The same structure works for charged leptons. For neutrinos, if a Majorana mass arises from the Weinberg operator, the symmetric block–circulant Mν may have large rotations inside one or more k –blocks, making it natural to accommodate large PMNS angles even when the charged–lepton sector remains near–circulant. J. Coherence–phase prediction in the toy model In the coherence gauge that trivializes the diagonal stack (Sec. VIII F), and under the uniform–character condition (CGC1)–(CGC2) of Sec. VII D, each Yf picks up only cubic–root overall phases under flavor translation and allowed family–basis changes. Hence the toy model automatically satisfies arg det(YuYd)∈2π 3Z(mod 2π), stable to O ( )soft breaking (Sec. VII). This discrete phase relation is a compact phenomenological fingerprint of the coherence origin of family replication. K. What this toy model captures (and what it does not) The construction exhibits, in the simplest possible environment, all structural ingredients of the mechanism: (i) a pointed flavor fiber with generator class, (ii) explicit defect phases and their literal cancellation by triplication, (iii) block–circulant textures from Z3 grading, (iv) Fourier–block diagonalization and a clean origin of mixing (block misalignment and soft inter–block couplings), and (v) a quantized Yukawa–determinant phase in a natural coherence gauge. It does not attempt (nor need) to derive the flavor fiber from UV dynamics, nor to perform a global fit; those are orthogonal questions once the kinematics are fixed. IX. CONSISTENCY CHECKS: ANOMALIES, GLOBAL STRUCTURE, AND UNITARITY The coherence mechanism advanced in Secs. IV–VIII is kinematical: it constrains how interaction data (Yukawa naturality) may be made strict on the interaction fiber by stacking identical copies. To be a viable layer on top of ordinary QFT, it must be compatible with established consistency conditions of the Standard Model (SM): perturbative and global anomalies, the global structure of the gauge group, unitarity/positivity, renormalization–group (RG) stability, and basic flavor constraints. Here we check each point explicitly and clarify how the higher–coherence viewpoint coexists with ordinary quantum mechanics: the latter is not modified; rather, coherence acts as an organizing selection rule on top of it. A. Gauge–anomaly cancellation (perturbative) We use the standard anomaly coefficients for chiral left–handed Weyl fermions.[ 87 ] For a simple non–Abelian group H , the cubic (triangle) gauge anomaly is proportional to PiA ( Ri )where A ( R )is the cubic index ( A ( R ) = −A ( R )for complex R ). For mixed anomalies H2–U (1) Y , the coefficient is PiYiT ( Ri )with T the Dynkin index ( T ( fund ) = 1 2 for SU ( N )). For Abelian factors one uses PiY3 i and PiYifor [U(1)Y]3and grav2–U(1)Y, respectively. Per family field content (left–handed). QL: (3,2)+1/6, uc R: (3,1)−2/3, dc R: (3,1)+1/3, LL: (1,2)−1/2, ec R: (1,1)+1,(Nc R: (1,1)0optional). [ SU (3) c ] 3 .Counting only color, QL contributes two triplets (due to the weak doublet), while uc R, dc R contribute two antitriplets: #(3)−#(3)=2−2=0 ⇒[SU(3)]3cancels per family.
24 [ SU (2) L ] 3 .The local (triangle) anomaly vanishes for SU (2) since dabc = 0 (no cubic index for pseudoreal representations). [SU(3)c]2–U(1)Y.Using T(3) = 1 2, X i YiT(Ri)SU(3) = 21 61 2+ (−2 3)1 2+ (+1 3)1 2=1 6−1 3+1 6= 0. [SU(2)L]2–U(1)Y.With T(2) = 1 2and the color multiplicity for QL, X i YiT(Ri)SU(2) = 31 61 2+ (−1 2)1 2=1 4−1 4= 0. [U(1)Y]3and grav2–U(1)Y.Summing over left–handed fermions (including multiplicities): X i Y3 i= 61 63+ 3−2 33+ 3+1 33+ 2−1 23+ (+1)3+ 0 = 0, X i Yi= 61 6+ 3−2 3+ 3+1 3+ 2−1 2+ (+1) + 0 = 0. Hence both anomalies cancel per family. Because all anomaly coefficients are linear in the fermion content, three identical families preserve these cancellations. B. Global (Witten) anomaly for SU(2)L The nonperturbative SU (2) anomaly [ 45 ] requires an even number of SU (2) doublets of left–handed Weyl fermions. Per family: #doublets = 3 ×QL+ 1 ×LL= 4 (even), so the Witten anomaly cancels per family and, a fortiori, for three families. C. Global structure of the gauge group and the Z3flavor grading The true gauge group may be a quotient GSM =SU(3) ×SU(2) ×U(1)Y Γ,Γ⊂Z6, reflecting identifications among the centers consistent with charge quantization. Our Z3 –graded flavor fiber lives in the interaction category I and organizes rebracketing phases (associator data); it does not impose a new global symmetry on fields. Thus there is no clash with known global structures or hypercharge quantization. Heuristically, the ubiquity of Z3 (center of SU (3) c ) makes a Z3 –graded flavor scaffold a conservative choice, but our construction does not rely on identifying the grading with an actual global symmetry of the SM. D. Unitarity, positivity, locality, and “no modification of QM” All categories used are dagger/unitary; the fiber functor F : I → Hilb is strong monoidal and preserves adjoints. Coherence “gauges” (choices of Φ X,Y and trivializing 2–cochains) act by unit phases on the Hilbert spaces and therefore preserve inner products and probabilities. Locality/microcausality live in the spacetime QFT sector (fields as operator–valued distributions); the categorical flavor fiber is an internal organizational layer that constrains which intertwiners (Yukawas) are allowed and how they compose, not the spacetime axioms. In this precise sense, our mechanism enriches the bookkeeping of standard QM/QFT; it does not alter its postulates.
25 E. Renormalization–group stability of the texture class In the SM, the RG equations for Yukawas take the form 16π2dYf dt =βfYuY† u, YdY† d, YeY† e;gi, with βf polynomial in Yukawas and gauge couplings [ 46 , 47 ]. Let P be the cyclic flavor translation (order 3) on family space. If initially [ Yf, P ] = 0 for all f , then [ YfY† f, P ] = 0, and βf —being a polynomial in such commuting objects—also commutes with P . Hence the block–circulant (or more generally, P –equivariant) form is preserved along the RG flow. Small spurions that softly break [ Yf, P ] = 0 evolve proportionally to themselves at leading order, keeping departures controlled. F. Flavor constraints: FCNCs, GIM, and MFV intuition Block–circulant Yukawas are simultaneously diagonalized by 1⊗F3 at the family level; this already aligns large portions of the flavor structure. In the exact graded limit, neutral–current couplings inherit the same P –equivariance and are diagonal in the “family–momentum” basis, suppressing tree–level FCNCs in the mass basis. With soft breaking (Sec. VIII), inter–sector mixing angles are naturally O ( ) (Cabibbo–sized), and FCNCs inherit the usual GIM cancellations [ 48 ]. This is reminiscent of Minimal Flavor Violation (MFV) [ 49 ]: all flavor breaking is encoded in a small set of spurions (here, in addition to the Yukawas themselves, the P –breaking spurions), ensuring that higher–dimension operators respect the same alignment and remain suppressed. G. Discrete gauge anomalies (if the grading were promoted to a symmetry) Our Z3 appears as a grading of the flavor fiber, not as a dynamical discrete gauge symmetry acting on fields. Therefore discrete gauge–anomaly conditions do not apply. If one nevertheless promoted the Z3 to a genuine symmetry, one would need to check the discrete versions of H2–Z3 and grav2–Z3 anomaly conditions (Ibáñez–Ross–type constraints) and possibly inflow in 3 + 1D; such extensions are orthogonal to the present, purely kinematical use of the grading [50]. H. Gravitational coupling and mixed anomalies The mixed grav2–U (1) Y anomaly vanishes per family (above). The coherence class [ ω ] ∈H3 ( Z3, U (1)) lives entirely in the internal flavor fiber; it does not represent a mixed gravitational anomaly. Coupling to curved backgrounds therefore proceeds as in the SM, with the same set of covariant currents and Ward identities [51]. I. Domain of validity and assumptions revisited Summarizing Secs. V–VIII, the claims rely on: (i) existence of a pointed Z3 –graded flavor subcategory F ≃ Vecω Z3 embedded in the interaction fiber (assumption H1 ); (ii) the restriction of associators to F carries [ ω ] 6 = 0 (order 3) (assumption H2 ); (iii) a single coherence gauge for the physical fiber functor F is to make all interaction naturality squares commute strictly (assumption H3 ); (iv) minimal replication is sought (assumption H4 ). Under these transparent hypotheses, three families become necessary and sufficient for strict interaction naturality; the Z3 –grading then enforces the block–circulant texture class and its phenomenological corollaries. Interpretive note (as requested). The observed fact of three families is read here as a statement about how higher categorical coherence is broken on a single flavor copy—namely, by a cubic ( Z3 ) associator twist. Triplication restores strict coherence on the interaction fiber. This enriches, but does not alter, quantum mechanics: it constrains the allowed interaction–naturality data rather than changing Hilbert space or the Born rule.
32 XII. CONCLUSIONS AND OUTLOOK We conclude by synthesizing the conceptual thread and laying out concrete next steps. The central message is kinematic and categorical: • The interaction fiber I encoding SM Yukawa data may contain a faithful pointed flavor subcategory F ≃ Vecω G with associator class [ ω ] ∈H3 ( G, U (1)). Demanding a single physical strong monoidal functor F : I → Hilb for which all Yukawa naturality squares commute strictly obstructs unless [ ω ] = 0. Stacking N identical copies (Deligne power with diagonal embedding) cancels the obstruction iff N·[ω]=0. Hence Nmin = ord[ω]. •For G=Z3with [ω]a generator, the minimal replication is Nmin = 3, i.e. three families. • If, moreover, the flavor fiber is Z3 -graded and the Higgs is grade neutral, then Yukawa maps commute with the cyclic flavor translation P and are (block-)circulant. In the family–momentum basis ( 1⊗F3 ) they split into three Fourier blocks Y(k) (Eq. (28) ), and mixing arises from misalignment across k-blocks or from small, controlled spurions that couple k↔k±1. • In a natural coherence gauge (the trivialization of the stacked associator), the overall Yukawa determinant phases are discretized, arg det(Yf)∈2π 3Z(mod 2π),arg det(YuYd)∈2π 3Z(mod 2π), stable up to O ( )under soft breaking. This is a qualitative fingerprint of coherence, not a dynamical solution to strong CP. • In a strengthened variant (Sec. X C, App. H), the same coherence data align the QCD topological term so that ¯ θ= 0 without additional dynamics. • None of this modifies quantum mechanics. Hilbert spaces, the Born rule and locality remain intact; what changes is the bookkeeping of composition in an internal flavor fiber via higher–coherence data. Empirically, “three families” reads as “order-3coherence obstruction” on a single copy. A. Outlook I: Data–driven tests and fits A practical pipeline follows Sec. X. At a common renormalization scale µ , reconstruct ( Yu, Yd )from masses and CKM, then minimize the joint circulantness index Cjoint (Eq. (45) ). If a basis realizes Cjoint 1, extract the sectoral blocks Yf≃diag Y(0) f, Y (1) f, Y (2) fin the (1⊗F3)basis, and quantify the selection rule: leading off-block entries couple k↔k± 1; k→k± 2are suppressed. Compute arg det ( YuYd )in the coherence-aligned basis and compare to 2 πZ/ 3within expected O ( ) uncertainties. Repeat in the lepton sector, taking care with Majorana vs. Dirac neutrino cases (Takagi vs. SVD). A global analysis can then be phrased as: what is the smallest that fits ( mf, VCKM, UPMNS ) while maintaining the selection rule and the sectoral sum rules? B. Outlook II: Effective-theory control and spurions The exact graded limit has [ Yf, P ] = 0 for all f . Soft departures are organized by spurions ∆ Yh1i f that transform with ∆ k = ± 1selection under the cyclic action. In a low-energy EFT, higher-dimension operators inherit this P -equivariance, producing MFV-like alignment but with a coherence origin. RG stability follows because the SM β -functions are polynomials in YfY† f ;[ Yf, P ] = 0 implies [ βf, P ] = 0 to the same order, and spurions run proportionally to themselves. This yields a technically natural small parameter governing CKM hierarchies and Jarlskog J=O(2)in quarks.
33 C. Outlook III: UV origins of the 3–cocycle twist Our analysis is kinematic, but several UV avenues plausibly generate a [ ω ] ∈H3 ( G, U (1)) twist on an internal fiber: 1. Discrete torsion and defects. Orbifold constructions with discrete torsion carry phase assignments classified by group cohomology; higher defects or background fields can imprint H3 ( G, U (1)) on fusion data of an internal category. 2. Topological layers. Dijkgraaf–Witten topological gauge theories labeled by [ ω ]in 2 + 1D can, via couplings or defect networks, feed a 3–cocycle into a 3 + 1D effective flavor fiber without introducing gapless topological sectors in spacetime. 3. Generalized symmetries & 2-groups. Generalized (higher-form) symmetries and their fusions can carry nontrivial associators; a 2–group structure acting on an internal flavor 2–Hilbert space provides a natural mathematical home for [ω]. A fully dynamical derivation would specify how the flavor fiber F emerges near the SM and why the Higgs is neutral under its grading. These are promising directions for future work. D. Outlook IV: Beyond Z3 The general relation Nmin = ord([ω]) suggests model-building variants: •Cyclic ZN. Gives N families and N –circulant textures diagonalized by FN ; the determinant phase quantizes mod 2 π/N . For N6 = 3, current data disfavor extra light families and precision fits constrain heavy ones; nonetheless, this offers a controlled generalization. •Nonabelian pointed fibers. Textures decompose along isotypic components of G ; Fourier blocks are replaced by matrix blocks labeled by irreps, with selection rules determined by the G -grading. The core obstruction logic and stacking cancellation persist unchanged. E. Open problems 1. UV completion. Construct explicit UV theories (string/orbifold, DW layers, 2-group symmetry) that produce a Z3flavor fiber with neutral Higgs and [ω]a generator. 2. Global-fit implementation. Develop a public pipeline that takes ( mf, VCKM, UPMNS )at a chosen scale and outputs the minimal , sectoral blocks, and the circulantness index with uncertainties. 3. EFT taxonomy. Classify leading higher-dimension operators consistent with the graded limit and soft spurions; derive correlated predictions for FCNCs and EDMs. 4. Axion interplay. Study the interplay of the discrete determinant-phase quantization in the coherence gauge with axion dynamics and potential cosmological signatures. 5. Categorical extensions. Explore non-pointed flavor fibers (e.g. G -crossed categories), braided or ribbon refinements, and the impact of nontrivial pivotal/spherical structures on interaction naturality. F. Final remarks The empirical fact of three families has long lacked a crisp kinematic rationale. In the picture developed here, it is a kinematic datum: an order-3obstruction of higher–categorical coherence on the interaction fiber, cancelled by triplication. The same structure enforces predictive texture constraints (block–circulant Yukawas, sectoral mixing) and a discrete determinant-phase statement in a natural coherence gauge. These are falsifiable at the level of reconstructed Yukawas and mixing matrices. Whether the twist [ ω ]is a relic of UV discrete torsion, a topological layer, or a generalized symmetry is a question now framed sharply. Either way, coherence—not additional dynamical symmetries—emerges as the organizing principle behind replication and flavor structure.
34 Appendices: Proofs, Derivations, and Technical Material Notation used throughout the appendices. We keep exactly the conventions of the main text: I is the interaction fiber; F : I → Hilb is a fixed strong unitary tensor functor with coherence (Φ X,Y , Φ 1 ); 1 denotes the tensor unit; α is the associator; F ≃ Vecω G is a pointed flavor subcategory with grading group G and normalized 3–cocycle ω∈Z3 ( G, U (1));[ ω ] ∈H3 ( G, U (1)) is its cohomology class; P is the cyclic flavor translation when G = Z3 and Ek = 1 3P2 r=0 ζ−krPr the projectors onto the three Fourier sectors ( ζ = e2πi/3 ). For Yukawas we write Yf = invf⊗Y(flavor) f and, in the F3 basis, Y(flavor) f∼ =diag ( Y(0) f, Y (1) f, Y (2) f )in the graded limit. References to equations, lemmas, and propositions like (10), Proposition IV.1, Theorem V.2, etc., are to the numbering in the main text. Appendix A: Appendix A: Defect Phases, 3–Cocycles, and Stacking Used in Secs. IV and V. 1. From the pentagon to a 3–cocycle Let X, Y, Z ∈ I and consider the two composites (HX⊗HY)⊗HZ⇒HX⊗(HY⊗HZ) built from F ’s coherence. Their ratio in Hilb is the component F ( αX,Y,Z ). Restricting along ι : F,→ I to simples La,Lb,Lcgives a scalar phase ω(a, b, c)∈U(1): F(αLa,Lb,Lc) = ω(a, b, c)·idHLa+b+c.(A1) Naturality and the pentagon in Iimply the 3–cocycle identity for ω: ω(b, c, d)ω(a, b+c, d)ω(a, b, c) = ω(a+b, c, d)ω(a, b, c+d),∀a, b, c, d ∈G, (A2) with normalization ω ( a, b, 0) = ω ( a, 0 , c ) = ω (0 , b, c ) = 1. Standard background on tensor categories and coherence is in [74]. 2. The interaction–naturality defect is measured by ω Let y : X⊗ Φ →Z be a Yukawa morphism with | Φ | = 0 and let S be a spectator of grade g∈G . The extended naturality square of Sec. IV A compares two composites differing by rebracketing through S . Projecting to Fand using (A1) yields ∆(X; Φ; S) = ω(|X|,|Φ|,|S|)·idHZ=ω(h, 0, g)·id,(A3) which is normalized (∆ = 1 when S = 1 ) and multiplicative in S . Hence the collection of defect phases (A3) is governed by the same 3–cocycle that appears in (A2). 3. Proof of Proposition IV.1 Let be the Deligne tensor product. On FN the associator class is N· [ ω ](sum in H3 ( G, U (1))). If N·[ω]=0, there exists a normalized 2–cochain νwith δν(a, b, c) := ν(b, c)ν(a, b+c) ν(a+b, c)ν(a, b)=ω(a, b, c)N.(A4) Define a new strong monoidal structure on the identity functor (id, ψ)on FNby ψX,Y := ν(|X|,|Y|)·idX⊗Y. The coherence condition for ( id, ψ )is exactly (A4) ; consequently, postcomposing FN with ( id, ψ ) trivializes all flavor associators and hence all defect phases. This yields strict interaction naturality on IN (assumption (F3)). Conversely, if strict interaction naturality holds on IN , the defect is trivial so ωN is a coboundary. Minimality follows because the set {M∈N|M· [ ω ] = 0 } is the subgroup generated by ord([ω]) in N.
35 4. Diagonal stacking and literal cancellation for G=Z3 For G = Z3 and generator ω , on the diagonal embedding ∆ : F → F3 one has pointwise ω⊕3 ( a, b, c ) = ω ( a, b, c ) 3≡ 1so the associator is trivial without any nontrivial ν . Thus the cancellation of (A3) is literal on the diagonal, as used in Secs. IV C and VIII F. Remark on monoidal equivalences. Any change of associator by a monoidal equivalence can be absorbed into a change of the strong monoidal structure of F (the components Φ X,Y ). Our strictness requirement is imposed for a single choice of F and its coherence (Sec. IV D), hence the residual phases (A3) are physical data tied to that choice. Appendix B: Appendix B: Strong Monoidal Functors, Coherence, and Dagger Structure Used in Secs. III and IX. 1. Strong monoidal functors and coherence Astrong monoidal functor ( F, Φ , Φ 1 ):( C,⊗,1 ) → ( D,⊗,1 )is a functor F equipped with natural isomorphisms Φ X,Y : F ( X ) ⊗F ( Y ) ∼ = −→ F ( X⊗Y )and Φ 1 : 1∼ = −→ F ( 1 )satisfying the usual pentagon and triangle identities (Mac Lane coherence; see, e.g., [ 75 ] for a survey of graphical languages and coherence). In our setting D = Hilb with strict monoidal structure; thus all associativity in the target arises via F(α). Amonoidal natural transformation η : F⇒G between strong monoidal functors ( F, Φ F )and ( G, Φ G ) satisfies ηX⊗Y◦ΦF X,Y = ΦG X,Y ◦(ηX⊗ηY)and η1◦ΦF 1= ΦG 1. 2. Unitary (dagger) structure A unitary monoidal category ( C,⊗, ( · ) † )has a contravariant involutive dagger functor fixing objects and compatible with ⊗ . A strong monoidal functor F is unitary if Φ X,Y and Φ 1 are unitary isomorphisms and F(f†) = F(f)†. This guarantees positivity and preserves the Born rule when composing with F. 3. Deligne tensor powers and diagonal embeddings For finite semisimple (fusion) categories, the Deligne tensor product exists and is again a fusion category. The associator class on a pointed factor Vecω G adds under , leading to N· [ ω ]on FN as used in Appendix A. Appendix C: Appendix C: Diagrammatic Derivations of Naturality and Defect Used in Secs. III C and IV A. 1. Packages and macros We recommend adding to the preamble: \usepackage{tikz-cd} \newcommand{\id}{\mathrm{id}}
36 2. Extended naturality square We follow the string-diagram conventions of Joyal–Street [76]. For f:X→X0and spectator S: (HX⊗HS)⊗HΦHX⊗(HS⊗HΦ)HX⊗HS⊗Φ (HX0⊗HS)⊗HΦHX0⊗(HS⊗HΦ)HX0⊗HS⊗ΦHZ α F(f)⊗id⊗id id⊗κS κX αid⊗κS F(y) Composing the horizontal arrows two ways and comparing yields the defect phase (A3) upon restriction to F. 3. Pentagon and the 3–cocycle The pentagon in Fbecomes: ((X⊗Y)⊗Z)⊗W(X⊗(Y⊗Z)) ⊗W X ⊗((Y⊗Z)⊗W) (X⊗Y)⊗(Z⊗W)X⊗(Y⊗(Z⊗W)) αX,Y,Z ⊗id αX⊗Y,Z,W αX,Y ⊗Z,W id⊗αY,Z,W αX,Y,Z⊗W Evaluating with Fand restricting to simples multiplies by (A2). Appendix D: Appendix D: Models for H3(Zn, U(1)) and Explicit Cocycles Used in Secs. V and VIII. 1. Classification For G=Znwith trivial G–action on U(1), H2k(Zn, U(1)) = 0, H2k+1(Zn, U(1)) ∼ =Zn. In particular H3 ( Zn, U (1)) ∼ =Zn ; the class [ ωm ]indexed by m∈ { 0 , . . . , n− 1 } has order n/ gcd ( n, m ) [77, 78]. 2. Explicit normalized cocycles Write elements a, b, c ∈ {0,1, . . . , n−1}and set ωm(a, b, c) := exp2πi m n2abb+c nc+bbc nc−ba+b ncc.(D1) This is normalized and represents the class m ( mod n ); for n = 3 and m = 1 it agrees (up to a 2–coboundary) with the floor-function representative used in the main text [77, 78]. 3. Trivialization by a 2–cochain A2–cochain νmsuch that δνm=ωN mexists iff Nm ≡0 (mod n). One can take, for instance, νm(a, b) := exp2πi m n2a b, δνm(a, b, c) = ωm(a, b, c)when n|m. (D2) For general Nwith Nm ≡0 (mod n), use νN m.
37 Appendix E: Appendix E: RG Preservation of P–Equivariance and Spurion Running Used in Secs. IX and X. 1. Abstract preservation lemma Let P∈U(3) with P3= id. Suppose [Yf(µ0), P]=0at µ0for all f. Consider a flow dYf dt =Ff{YgY† g}g,{Y† gYg}g,{gi}Yf+YfGf{Y† gYg}g,{gi}, where Ff and Gf are polynomials in the indicated matrices and gauge couplings {gi} . If [ Yf ( µ0 ) , P ] = 0 then [Yf(µ), P]=0for all µfor which the solution exists. Proof. If [ Yf, P ] = 0, then [ YfY† f, P ] = [ Y† fYf, P ] = 0, so [ Ff, P ] = [ Gf, P ] = 0 by polynomial functional calculus. Hence [dYf/dt, P]=0, so the commutator remains zero along the flow. 2. One-loop SM Yukawa β–functions (quarks) At one loop (schematically) [46, 47]: 16π2βYu=Yuh−17 20 g2 1+9 4g2 2+ 8g2 3+ Tr3Y† uYu+ 3Y† dYd+Y† eYei +3 2YuY† uYu−3 2YdY† dYu,(E1) 16π2βYd=Ydh−1 4g2 1+9 4g2 2+ 8g2 3+ Tr3Y† uYu+ 3Y† dYd+Y† eYei +3 2YdY† dYd−3 2YuY† uYd.(E2) Every term is a left or right multiplication by a polynomial in YgY† g (and c-numbers), thus commutes with Pwhenever [Yg, P]=0; the lemma applies. For related two-loop running in extended settings see [79]. 3. Soft spurions Let Yf = Yh0i f + ∆ Yh1i f + ··· with [ Yh0i f, P ]=0and (∆ Yh1i f ) k` = 0 unless ` = k± 1. Linearizing the RG equations shows d dt∆Yh1i f=Af∆Yh1i f−3 2X gCfg ∆Yh1i g for some block-diagonal Af and coupling matrices Cfg built from the k –sector blocks of Yh0i g . Thus the selection rule (∆ k = ± 1) is preserved, and the magnitude of runs proportionally to itself to leading order. Appendix F: Appendix F: Practical Pipeline for Fits and Diagnostics Used in Sec. X. 1. Reconstruction at a reference scale Choose a renormalization scale µ and reconstruct numerical Yu, Yd from quark masses and the CKM matrix (and analogously for leptons). Include known rephasing freedoms.
38 2. Joint circulantness minimization Define the cost J(UL, UuR, UdR) := X f∈{u,d}X k6=` Ek(ULYfU† fR)E` 2 F. Alternate minimization on UL, UuR, UdR ∈U(3): 1. Fix UuR, UdR; minimize in ULby Riemannian gradient descent on U(3). 2. Fix UL, UdR; minimize in UuR by the same method; then fix UL, UuR and minimize in UdR. 3. Iterate to convergence; set Cjoint := J/(kYuk2 F+kYdk2 F). Stopping when the relative change in J is < 10 −10 is sufficient for 3 × 3matrices; see [ 80 ] for manifold optimization methods. 3. Selection-rule test and sectoral sum rules In the minimizing basis, extract Y(k`) f := Ek ( ULYfU† fR ) E` and verify: (i) Y(k`) f is small for k6 = ` , (ii) kY(k,k±1) fkkY(k,k±2) fk, (iii) sectoral sum rules (Eq. (50)) hold up to O(2). 4. Determinant phase in the coherence gauge In the minimizing basis, fix overall family phases to align with the coherence gauge (characters of Z3 ); then compute arg det(YuYd)and check closeness to 2πZ/3within the expected O(). 5. Statistical treatment Propagate input uncertainties by Monte Carlo sampling of masses and mixing parameters; report distributions of Cjoint and the determinant phase residual modulo 2π/3. Appendix G: Appendix G: Variants Beyond Z3and Nonabelian Grading Complements Secs. XI E and XII D. 1. Cyclic ZN For F ≃ Vecω ZN with generator [ ω ], the minimal replication number is Nmin = N . Yukawa maps commute with the N –cycle PN , hence are N –circulant; they diagonalize by the DFT FN and split into N sectoral blocks Y(k) = PN−1 r=0 ζkr NCr , ζN = e2πi/N . The soft–breaking selection rule is ∆ k = ± 1mod Nat leading order, with higher steps suppressed. 2. Nonabelian G Let G be finite and F ≃ Vecω G . The commutant of the left regular G –action on family space is the group algebra C [ G ]; a G –equivariant Yukawa is an element of C [ G ] ⊗Mat and block–diagonalizes by the nonabelian Fourier transform C[G]∼ =M ρ∈b G Matdρ(C), Y (ρ)=X g∈G ρ(g)⊗Cg, see, e.g., [ 81 ]. Textures are thus group–circulant, and mixing organizes by isotypic components ρ . The determinant–phase constraint reduces to the abelianized part via 1D characters of G (coherence gauge), generalizing Eq. (46).
39 Appendix H: Appendix H: A Coherence-Protected Strong-CP Mechanism Used in Sec. X C. 1. Setup Let [ ω ] ∈H3 ( G, U (1)) be the associator class on the flavor fiber F ≃ Vecω G . On the triple stack I3 there exists a normalized 2-cochain ν which trivializes the stacked associator (Sec. V, App. A). Assume a compact topological sector T (e.g. a 3-form C3 with field strength F4 = dC3 or an axionlike scalar a dual to F4) with Lagrangian LT=1 2Λ2F4∧?F4+1 2πΞ[ν]C3∧g2 s 8π2Tr(G∧G),(H1) where Ξ[ ν ]is a Z3 -valued functional determined by the trivializing 2-cochain in the coherence gauge. Coupling to QCD adds Lθ=θQCD + arg det(YuYd)g2 s 32π2G˜ G. (H2) 2. Cancellation of ¯ θ Varying C3(or, in the dual picture, minimizing the effective potential for the axionlike scalar) sets δL δC3 = 0 ⇒g2 s 32π2G˜ G=−1 2πΞ[ν]g2 s 32π2G˜ G. (H3) Choosing the coherence gauge fixes Ξ[ ν ] = arg det ( YuYd )(mod 2 π ) so that the net coefficient of G˜ G vanishes: ¯ θ=θQCD + arg det(YuYd) = 0.(H4) Because only the overall U(1) phase of the Yukawas enters ¯ θ , CKM and PMNS Dirac phases (which are rephasing-invariant mixing observables) remain unconstrained. 3. Comments (i) The mechanism is purely kinematic/topological and coexists with our triplication logic. (ii) RG flow preserves the condition ¯ θ = 0 because it is enforced by a topological equation of motion. (iii) If one prefers an axion language, dualize F4 = ?da and identify the coupling a fa g2 s 32π2G˜ G with a/fa set dynamically to −(θQCD + arg det(YuYd)) in the coherence gauge. [1] G. M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series, Vol. 64, Cambridge University Press (1982). [2] P. Etingof, D. Nikshych, and V. Ostrik, “Fusion Categories and Homotopy Theory,” Quantum Topology 1 (2010) 209–273. [3] C. D. Froggatt and H. B. Nielsen, “Hierarchy of quark masses, Cabibbo angles and CP violation,” Nucl. Phys. B147 (1979) 277–298. [4] G. Altarelli and F. Feruglio, “Discrete Flavor Symmetries and Models of Neutrino Mixing,” Rev. Mod. Phys. 82 (2010) 2701–2729. [5] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, “Non-Abelian Discrete Symmetries in Particle Physics,” Prog. Theor. Phys. Suppl. 183 (2010) 1–163. [6] T. Kobayashi, K. Tanaka, and T. H. Tatsuishi, “Finite Modular Groups and Lepton Flavors,” Phys. Rev. D 98 (2018) 016004. [7] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998. [8] A. Joyal and R. Street, “Braided Monoidal Categories,” Adv. Math. 102 (1993) 20–78.
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[82] Trivialization here means monoidal equivalence to a category whose associator restricts to the identity (or a coboundary) on the subcategory controlling interaction naturality; see App. A in the main text for the precise statement and proof. [83] Formally, we are fixing a strong monoidal functor into a strict monoidal target ( Hilb ) and asking for a monoidal natural isomorphism to a functor that is trivial on associator data along F . The obstruction is a 3–cocycle. See, e.g., Kelly [1] for coherence constraints on strong monoidal functors. [84] Equivalently: strictification in the source can always be absorbed by deforming the strong monoidal structure in the target functor. We are not allowing such deformations to change physical Hilbert-space identifications beyond unitary basis changes; see Kelly [1]. [85] If da are equal (e.g. one–dimensional family lines), then Ar are square and Eq. (21) reduces to the familiar circulant case. In the fully symmetric case d0 = d1 = d2 , the algebra of block–circulant matrices forms the commutant of the regular Z3–action [32, 33]. [86] This aligns the three structural coefficients with the three characters of Z3 . At the functor level, (CGC1) is a property of the coherence gauge; at the matrix level it may be enforced by a suitable choice of flavor basis within the commutant of P.