Higgs Mass as Coherence Stiffness: A Categorical GL(N) Framework without an arbitrary $\mu^{2}$

Higgs Mass as Coherence Stiffness: A Categorical GL(N) Framework without an arbitrary µ2 Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We develop a coherence–first formulation of the Higgs mechanism in which vector boson masses and the Higgs mass arise as energetic costs of maintaining higher categorical coherence, rather than from an arbitrary scalar mass parameter. The construction is based on a non-abelian coherence lift of the electroweak gauge structure, formulated as a crossed-module–type theory with a two-form compensator field and a fundamental scalar that parametrizes admissible grammar morphisms. The scalar field does not enter through a free polynomial potential. Instead, it appears only through a constrained coherence invariant built from a fixed coherence tensor, yielding a GL ( N )-covariant but categorically restricted moduli space of grammars. As a result, there exists no independent local relevant counterterm corresponding to an arbitrary Higgs mass parameter µ2 ; this is shown by a cohomological classification of admissible counterterms consistent with locality, BRST invariance, and coherence constraints. Integrating out the two-form coherence field produces an effective mass operator for the gauge fields proportional to the square of the grammar morphism, reproducing the expected pattern of massive and massless electroweak gauge bosons. The Higgs mass is generated radiatively as the curvature of the one-loop effective potential on the constrained grammar moduli space and is therefore finite and controlled, rather than an independent input parameter. This framework preserves a fundamental scalar degree of freedom while providing a structural improvement of the Higgs naturalness problem rooted in categorical coherence rather than symmetry breaking. I. INTRODUCTION AND MOTIVATION A. The Higgs mechanism and the naturalness problem The Standard Model (SM) Higgs sector provides a minimal and remarkably successful mechanism for electroweak symmetry breaking and mass generation for the W± and Z bosons and the charged fermions. At the same time, it exposes one of the most persistent structural questions in quantum field theory: why is the Higgs mass parametrically small compared to ultraviolet scales at which the SM is expected to be modified? This question is usually called the naturalness or hierarchy problem. In the present work we will ultimately reframe this problem in coherence–first terms, but it is essential to review the conventional Higgs mechanism carefully and to isolate what is truly nontrivial. The SM Higgs Lagrangian and electroweak gauge structure. The electroweak gauge group is GEW =SU(2)L×U(1)Y,(1) with gauge fields Wa µ(a= 1,2,3) for SU(2)Land Bµfor U(1)Y. The corresponding field strengths are Wa µν =∂µWa ν−∂νWa µ+g abcWb µWc ν,(2) Bµν =∂µBν−∂νBµ,(3) with gauge couplings g and g0 (we include g0 in the covariant derivative below). The Higgs field is a complex scalar doublet φof hypercharge Y= 1/2, φ(x) = φ+(x) φ0(x), φ 7→ expiαa(x)σa 2expiβ(x)1 2φ, (4) where σa are Pauli matrices and αa ( x ) , β ( x )are gauge parameters. The covariant derivative acting on φ is Dµφ=∂µ−ig Wa µ σa 2−ig0Bµ 1 2φ. (5) The Higgs-sector Lagrangian is LH= (Dµφ)†(Dµφ)−V(φ), V (φ) = −µ2φ†φ+λ(φ†φ)2,(6) with λ > 0for stability. The parameter µ2 has mass dimension 2and is the key object in the naturalness discussion. 2 Classical vacuum and symmetry breaking pattern. At the classical level, the potential (6) is minimized for φ†φ=µ2 2λ≡v2 2,so that v=rµ2 λ.(7) Choosing unitary gauge, one may parametrize fluctuations around the vacuum by φ(x) = 1 √20 v+h(x),(8) where h ( x )is the physical Higgs excitation. Substituting (8) into the kinetic term yields the gauge boson mass terms. To display them, define W± µ:= 1 √2W1 µ∓iW2 µ.(9) The neutral gauge bosons mix through the Weinberg angle θWdefined by sin θW=g0 pg2+g02,cos θW=g pg2+g02,(10) with mass eigenstates Zµ Aµ=cos θW−sin θW sin θWcos θWW3 µ Bµ.(11) Derivation of W and Z masses. Insert (8) into (5) . Since the upper component is zero, only the action of σa/2on the lower component contributes. One finds (Dµφ)†(Dµφ) = 1 2∂µh ∂µh+(v+h)2 8hg2W1 µW1µ+W2 µW2µ+gW 3 µ−g0Bµ2i+··· ,(12) where ··· denotes interaction terms involving derivatives and higher powers of fields not needed for the mass extraction. Using (9), W1 µW1µ+W2 µW2µ= 2W+ µW−µ, so (v+h)2 8g2W1 µW1µ+W2 µW2µ=g2(v+h)2 4W+ µW−µ.(13) In the neutral sector, the mass matrix in the ( W3 µ, Bµ )basis is proportional to ( gW 3−g0B ) 2 , and diagonalized by (11), giving (v+h)2 8gW 3 µ−g0Bµ2=(g2+g02)(v+h)2 8ZµZµ.(14) Therefore the tree-level gauge boson masses are m2 W=g2v2 4, m2 Z=(g2+g02)v2 4, m2 γ= 0.(15) The photon remains massless because the vacuum leaves unbroken the U (1) em generator Q = T3 + Y [1, 2]. Derivation of the Higgs mass. Expanding the potential in (6) around the minimum, V(φ) = −µ2(v+h)2 2+λ(v+h)4 4= const + 1 2(2λv2)h2+λvh3+λ 4h4,(16) where the linear term vanishes by the definition of vin (7). Hence the tree-level Higgs mass is m2 h= 2λv2= 2µ2.(17) Equations (15) and (17) show that the electroweak scale and the Higgs mass are set by the single dimensionful parameter µ (together with dimensionless couplings). This is precisely why the parameter µ2 is central: it is a relevant deformation in the renormalization group sense, and nothing in the SM forbids it. 3 Naturalness as the arbitrariness of a relevant deformation. At the quantum level, the parameters in (6) are renormalized. A convenient way to organize this is via the effective action Γ[ φ, W, B, . . . ], obtained by integrating out quantum fluctuations. The naturalness problem appears when one asks how the renormalized Higgs mass depends on ultraviolet scales. In a cutoff-based estimate, the Higgs mass parameter receives additive radiative corrections of order δm2 h∼Λ2 16π26λ+9 4g2+3 4g02−6y2 t+···,(18) where yt is the top Yukawa coupling and Λis a UV cutoff representing the scale where new physics enters. Equation (18) is a schematic one-loop estimate; its detailed coefficients depend on conventions, but its structural content is that a gauge-invariant scalar mass term is renormalized additively and is sensitive to high scales [ 1 , 3 ]. If Λ v , the observed mh≃ 125 GeV then requires cancellations between the bare parameter and radiative corrections, i.e. tuning. It is important to state precisely what is at stake. Gauge symmetry does not forbid the operator φ†φ ; indeed, it is the unique dimension-two gauge-invariant scalar operator in the SM Higgs sector. Therefore the corresponding coefficient µ2 is an allowed, a priori independent relevant deformation of the theory. Naturalness concerns the absence of a structural reason why this allowed relevant deformation should take a value close to the electroweak scale rather than the ultraviolet scale. Put differently: the problem is not “symmetry breaking” as a phenomenon (which is a choice of vacuum in a gauge-redundant description), but the presence of an arbitrary relevant operator whose coefficient is not tied to other protected data. This observation motivates the coherence-first mechanism developed in the remainder of this paper. Our goal is to construct a gauge-consistent mass generation mechanism that retains a fundamental scalar degree of freedom but eliminates the arbitrariness of an independent µ2|φ|2 deformation by restricting admissible scalar counterterms to constrained coherence invariants. In that setting, the Higgs mass will arise as a controlled stiffness of categorical coherence rather than as an arbitrary input parameter. B. Beyond symmetry-first reasoning The preceding subsection emphasized that the naturalness issue is not “symmetry breaking” as such, but the existence of an allowed, independent relevant deformation—the gauge-invariant scalar operator φ†φ and its coefficient µ2 in (6) . A natural next question is whether modern generalizations of symmetry might address this arbitrariness. Over the last decade, the language of higher symmetries and non-invertible symmetries has substantially reshaped how symmetries are understood in QFT. These developments are deep and powerful, but their primary role is to classify and constrain infrared physics given a consistent QFT. They do not, by themselves, remove the freedom to write down the relevant scalar deformation that drives the Higgs naturalness problem. This subsection reviews these generalized symmetry notions sufficiently for our purposes and isolates their limitation in the present context. Higher-form and generalized global symmetries. The modern “generalized symmetry” viewpoint can be summarized as follows. A p -form global symmetry is one whose charged operators are p -dimensional extended objects, and whose symmetry generators are topological operators supported on codimension- ( p + 1) manifolds. The canonical example is a 1-form symmetry in a gauge theory, acting on Wilson lines. Mathematically, a p -form symmetry in d dimensions is generated by topological operators Ug (Σ d−p−1 ) satisfying fusion rules Ug(Σ) Ug0(Σ) = Ugg0(Σ),(19) and deformability (topological invariance) properties so long as Σdoes not cross charged operators. This framework organizes selection rules, superselection sectors, and anomaly constraints in a way that generalizes ordinary global symmetries [5–7]. A key structural consequence is that “symmetry” becomes a statement about the existence of certain topological defect operators and their fusion category, rather than merely about a Lie group acting on pointlike fields. It is also natural in this language to consider higher-group structures (e.g. 2-groups) in which ordinary (0-form) symmetries and higher-form symmetries mix via Postnikov classes. Such structures are relevant to anomaly inflow and to gauge theories coupled to topological sectors, and they provide a sharp way to encode generalized Ward identities. 4 Non-invertible symmetries and categorical defects. Non-invertible symmetries extend the generalized symmetry program beyond group-like fusion rules. Instead of (19) with inverses, one allows topological defects Dawhose fusion is of the form Da×Db=X c Nc ab Dc,(20) with nonnegative integer coefficients Nc ab . This is the algebraic structure of a fusion category rather than a group. Such non-invertible symmetries appear ubiquitously in two-dimensional QFT and have higher-dimensional avatars in the presence of topological sectors and defects; they impose generalized selection rules and constrain RG flows and phase structure [ 7 , 8 ]. Conceptually, non-invertible symmetries demonstrate that “symmetry” in QFT is not exhausted by group actions: the appropriate organizing structure may be genuinely categorical. Why generalized symmetries do not remove the Higgs relevant deformation. Despite their conceptual richness, higher and non-invertible symmetries address a different problem from Higgs naturalness. They classify and constrain consistent QFTs by identifying topological operators, defect fusion rules, and associated anomalies. They do not automatically eliminate the existence of gauge-invariant relevant operators in the microscopic Lagrangian. In particular, the operator φ†φ is compatible with all SM gauge symmetries and with the generalized symmetry structures typically present in the electroweak sector. Thus the freedom to add ∆Lrel =−µ2φ†φ(21) remains: generalized symmetry considerations do not forbid it. This can be seen in a standard BRST/cohomological characterization of admissible counterterms. In gauge theory, allowed local deformations (and counterterms) are classified by local BRST cohomology H0(s|d): a local functional ∆S=R∆Lis admissible if s∆L+d(···)=0,(22) modulo trivial deformations s ( ··· ) + d ( ··· )[ 9 , 10 ]. For an ordinary Higgs doublet, φ†φ is gauge invariant and therefore BRST closed, hence it represents an allowed ghost-number-zero local class. The point is structural: (21) is allowed because the scalar representation admits a dimension-two gauge invariant. Generalized symmetries may constrain IR phases or defect spectra, but they do not change this ultraviolet cohomological fact unless they enforce additional structure that removes φ†φ from the space of admissible local deformations. One can phrase the limitation in yet another way. Higher and non-invertible symmetries are generalizations of the symmetry principle: they enlarge the class of symmetry objects from groups to higher groups and fusion categories. But the naturalness problem is not, at root, a problem about what kind of symmetry object acts on fields. It is a problem about why a particular relevant deformation is present and arbitrary. To eliminate or constrain µ2 one needs a principle that restricts the space of allowed relevant operators more strongly than ordinary gauge invariance and the generalized symmetry constraints typically considered. What is needed: a restriction on admissible local deformations. The coherence-first approach proposed in this paper targets exactly this point. Instead of enlarging the taxonomy of symmetries, we enlarge (and constrain) the underlying notion of consistency by treating the local “grammar” of gauge structure as part of a higher coherence datum. In that setting, scalar degrees of freedom arise as coordinates on constrained moduli of admissible coherence morphisms, and admissible counterterms are functions on that constrained moduli stack rather than arbitrary polynomials in a Higgs field. The central mechanism explored below is that, once the scalar is restricted to enter only through coherence invariants built from constrained categorical data, an independent relevant deformation analogous to (21) can be absent, while a nonzero Higgs mass can still be generated radiatively as a controlled stiffness of coherence. C. Coherence-first perspective We now state the guiding principle of this paper and explain, in a way that can be made fully operational, how it leads to a different viewpoint on mass generation and naturalness. 5 Core conceptual shift: physics requires coherent gluing, not symmetry. Much of modern field theory is built from local data: local coordinate charts, local frames, local gauge choices, local effective descriptions, and local operator algebras. The existence of a globally meaningful theory is therefore a gluing problem: local descriptions must be compatible on overlaps and under composition. In differential geometry this is encoded by transition functions and cocycle conditions; in gauge theory it is encoded by local gauge transformations and curvature; and in categorical language it is encoded by coherence of compositions of morphisms [11–13]. The standard symmetry-first viewpoint begins by postulating a symmetry group and building a theory invariant under it. This strategy is powerful because group structure guarantees compositional consistency: associativity, inverses, and identities ensure that local transition data glue without contradiction. However, this guarantee is stronger than the minimal requirement of consistency. From the gluing standpoint, what is fundamentally required is that the compatibility relations close coherently; symmetry is one particularly rigid and elegant mechanism that enforces this closure. We therefore adopt the following guiding statement: Symmetry is a sufficient but not necessary condition for coherent gluing of local descriptions. The minimal requirement of consistency is the exact closure of the appropriate coherence relations. Symmetry emerges as a limiting case in which coherence defects are dynamically suppressed. This is not a rhetorical point. It becomes operational once we represent gluing relations as defect variables and require exact closure identities among them. In particular, higher gauge theory provides a natural language for gluing beyond group-valued cocycles: one distinguishes 1-morphisms (comparisons of local frames) and 2-morphisms (comparisons of comparisons), and coherence becomes the statement that the resulting higher diagrams commute up to specified higher data [14–16]. Coherence complexes and defects. A coherence-first field theory is organized not primarily by “symmetry generators” but by a hierarchy of compatibility conditions. At the first level, an ordinary connection A mediates comparisons between local descriptions; its curvature F ( A )measures the failure of those comparisons to close around loops. At the next level, a higher field B mediates comparisons between comparisons. The key structural idea is that a portion of the first-level defect may be repaired or absorbed by higher data through a structure map t:g1→g0,(23) as in crossed modules (strict 2-groups). One then defines the fake curvature (first-level coherence defect) Ffake := F(A)−t(B),(24) and a higher curvature (second-level coherence defect) H:= DAB+··· ,(25) with DA the appropriate covariant derivative and “ ··· ” indicating the standard higher-gauge corrections depending on the chosen 2-group structure [ 13 – 15 ]. Exact closure identities (higher Bianchi relations) link these defects. In abelian prototypes these reduce to identities of the schematic form DAFfake +t(H)≡0,(26) while in non-abelian settings additional terms express the coherence of the crossed-module data. The important point is that these are structural identities: they hold off-shell and constitute the minimal consistency backbone. From coherence to mass: energetic cost of maintaining closure. Once defects are identified, the most natural action principle is to penalize incoherence by assigning an energy cost to defects. For example, a minimal coherence functional has the schematic form S[A, B;grammar]∼1 2g2kF(A)−t(B)k2+1 2h2kHk2+··· ,(27) where k·k is defined by the spacetime metric (and, in discrete form, by a DEC Hodge star), and the ellipsis includes additional kinetic and topological terms. Crucially, (27) is not a “symmetry breaking” functional: it does not penalize violation of an assumed symmetry, but rather penalizes the failure of compatibility to be repaired. 6 To see how mass arises, expand around a background in which the grammar is constant (so t is constant) and B is stiff. At quadratic order in fluctuations, the B -field appears as a Gaussian compensator in the term kF−t ( B ) k2 . Integrating out B ( equivalently completing the square at the level of the quadratic functional integral ) produces an effective action for Acontaining a mass operator proportional to t t†: Seff[A]⊃1 2hA, M2 AAi, M2 A∝t t†,(28) up to convention-dependent factors involving ( g, h )and the choice of gauge for isolating transverse modes. The interpretation is direct: directions of the gauge field that require higher-level repair (i.e. lie in the image of t ) acquire an energetic penalty and therefore behave as massive vector modes. Massless directions are those annihilated by t† and correspond to unrepaired, freely propagating coherence modes (the analogue of an unbroken gauge direction). Thus vector boson masses arise as the energetic cost of maintaining higher coherence closure. The Higgs mass as coherence stiffness on grammar moduli. So far, t has been treated as fixed. The key additional step in this paper is that the “grammar” of repair is itself not taken as a fixed choice, but as part of the coherence datum. In symmetry-first language, this corresponds to allowing the structure responsible for mass generation to vary; in coherence-first language, it is allowing the repair morphism t to be selected from a constrained space of admissible coherence structures. The central proposal is that the Higgs field is a fundamental scalar coordinate on the moduli of admissible grammar morphisms, rather than a free scalar admitting an arbitrary polynomial potential. Concretely, we consider grammar morphisms tΦ depending on a fundamental scalar Φbut constrained categorically so that only a restricted set of invariants exists. In the ambitious framework developed below, Φis fundamental, yet the analogue of the arbitrary relevant deformation µ2φ†φ is absent because Φenters only through a coherence invariant built from a constrained coherence tensor Gij . The Higgs mass then arises not from an arbitrary input coefficient but as the curvature (stiffness) of an effective potential on the constrained grammar moduli space, m2 H= Hess Veff |Φ=Φ?(on the constrained grammar manifold),(29) where Veff is generated radiatively by integrating out gauge and coherence-compensator fluctuations. This gives the intended naturalness improvement: the dangerous independent relevant parameter is eliminated by categorical coherence constraints, while a finite nonzero Higgs mass is generated and controlled by the coherence sector. Preview of the mechanism. The remainder of the paper makes this precise in a concrete non-abelian setting relevant to electroweak physics. We construct a coherence-lifted gauge theory with a two-form compensator valued in a multiplicity-enhanced module, equipped with a GL ( N )grammar structure and an additional coherence tensor Gij whose admissible deformations are constrained by coherence (e.g. fixed determinant or fixed spectrum). The scalar field Φis fundamental but can enter the action only through the invariant Φ iGij Φ j , so there is no independent analogue of µ2|φ|2 . We then show: (i) integrating out the two-form field generates vector boson masses proportional to tΦt† Φ , and (ii) the Higgs mass is radiatively generated as the stiffness of Veff on the constrained grammar moduli. In this way, the Higgs mass is reinterpreted as an energetic cost of coherence, rooted in categorical constraints on admissible local deformations. II. COHERENCE LIFT: GENERAL FRAMEWORK A. Coherence complexes and defect fields This subsection formalizes the basic objects underlying a coherence-first formulation of gauge theory. The central point is that one should distinguish (i) local comparison data that relate descriptions between neighboring patches (or cells), (ii) defect data that measure failure of strict composition, and (iii) closure identities expressing coherent gluing at the next level. In ordinary gauge theory these structures are present but hidden by symmetry-first packaging. In higher gauge theory they become explicit: a 1-form connection captures comparisons, a 2-form field captures comparison-of-comparisons, and curvature-like objects encode coherence defects [13–15]. From gluing to curvature: ordinary gauge theory as first-level coherence. Let {Ui} be a good open cover of a manifold M . In principal-bundle language, local trivializations are glued on double overlaps 7 Uij =Ui∩Ujby transition functions gij :Uij →G0satisfying the cocycle condition gij gjk gki = 1 on Uijk.(30) The condition (30) is the algebraic form of strict gluing: comparisons compose to the identity around a triple overlap. A connection 1-form A∈ Ω 1 ( M, g0 )can be viewed as infinitesimal comparison data; its curvature F(A) := dA +A∧A∈Ω2(M, g0)(31) measures the failure of those comparisons to close around infinitesimal loops. The Bianchi identity DAF:= dF + [A, F]≡0(32) is then an exact closure relation expressing compatibility of the defect Facross higher overlaps. In this way, ordinary gauge theory is already a “coherence theory”: it replaces the strict cocycle condition (30) by a controlled defect F subject to an exact closure identity (32) . The coherence-first perspective emphasizes this gluing logic rather than taking invariance under G0as the primitive axiom. Crossed modules and 2-connections: adding the next coherence level. Higher gauge theory makes the next level of gluing explicit by introducing 2-morphisms (“comparisons of comparisons”). The algebraic backbone of a strict 2-group is a crossed module of Lie groups G1 t −→ G0, . :G0yG1,(33) with Peiffer identities ensuring coherent interaction between t and the action . . At the infinitesimal level this becomes a differential crossed module g1 t −→ g0, . :g0yg1,(34) with t equivariant and compatible with commutators [ 13 – 15 ]. A (strict) 2-connection then consists of a pair A∈Ω1(M, g0), B ∈Ω2(M, g1),(35) where Ais the usual 1-form connection and Bis the 2-form coherence field. Fake curvature as a coherence defect. The first coherence defect in a crossed-module 2-connection is the fake curvature FΦ:= F(A)−tΦ(B),(36) where tΦ is the structure map t (possibly dependent on a grammar field Φ, as developed later). If FΦ = 0, the 2-form field B repairs the failure of first-level closure exactly: the curvature F ( A )is purely “explained” by a higher morphism through tΦ ( B ). If FΦ6 = 0, the residual mismatch is a genuine coherence defect that cannot be repaired by the available higher data. In the coherence-first interpretation, (36) is the precise mathematical object measuring failure of strict gluing at the first level after allowing higher repair. Higher curvature as failure of second-level gluing. The next defect field is the 3-form curvature of the 2-form data. In the simplest (strict) case it is given by H:= DAB≡dB +A . B, (37) with A . B the action of g0 on g1 extended to forms. In semistrict or fully nonabelian settings, additional terms may appear (reflecting the full coherence structure), but (37) already captures the essential point: H measures failure of B -data to glue consistently across 3-cells (or triple overlaps), just as F measures failure of A-data to glue around loops [14, 16]. Exact closure identities: coherent gluing. The defining structural advantage of encoding gluing as a coherence complex is that defects satisfy exact closure identities. For a strict crossed-module 2-connection, one has generalized Bianchi identities of the schematic form DAFΦ+tΦ(H)≡0,(38) DAH+ (FΦ). B ≡0,(39) with the precise correction terms determined by the crossed-module relations. Equations (38) – (39) express that although gluing may fail at a given level (nonzero defect), the failures themselves are compatible 8 and do not generate contradictions at the next level. This is exactly what is meant by coherent gluing: higher diagrams close once one includes the next layer of morphisms. Clarifying terminology (“grammar”). In this paper we use the term grammar to denote the structural data that govern admissible composition and repair of local gauge-theoretic descriptions. Concretely, grammar is encoded by the structure morphism tΦ : g1→g0 (and its coherence constraints), which specifies how higher coherence data B may compensate first-level curvature defects through the combination FΦ = F ( A ) −tΦ ( B ). Grammar is therefore not a spacetime gauge symmetry and is not synonymous with symmetry breaking: it is a rule for consistent gluing/repair in the coherence complex. The internal GL ( N )covariance introduced later acts on this grammar data (multiplicity indices) and restricts the ring of admissible local deformations, which is the mechanism behind the absence of an independent relevant Higgs mass deformation. Coherence-first viewpoint and the role of the grammar map. The key conceptual point for this paper is that the structure map is not merely a fixed background choice. In a symmetry-first approach, one fixes the gauge structure (and therefore the allowed repair map) once and for all. In the coherence-first approach, the grammar of repair is itself treated as part of the physical data: we allow a family of admissible structure maps tΦ constrained by higher categorical coherence, and we regard Φas a fundamental coordinate on the corresponding moduli stack. The dynamics then penalizes coherence defects ( FΦ and H ), and the energetic cost of maintaining coherent gluing under a given grammar produces mass terms for gauge modes (via tΦt† Φ ) and, ultimately, a Higgs mass interpreted as stiffness on the constrained grammar moduli space. In summary, the coherence complex is the coupled structure (A, B;tΦ, .Φ)7−→ (FΦ, H)subject to exact closure identities,(40) and its defects encode the failure of strict gluing in a controlled, consistency-preserving manner. The remainder of this paper builds a mass-generation mechanism by combining defect-penalizing dynamics with a categorically constrained space of admissible grammars. B. Energy cost of coherence Having identified the coherence defects FΦ and H in Sec. II A, we now explain why a defect-based action of the form kFΦk2 + kHk2 is the most natural energetic principle in a coherence-first setting, and why it leads to mass generation for gauge modes once higher repair is present. The argument has two parts: (i) the geometric meaning of defect norms as a measure of non-glueability, and (ii) a concrete quadratic-field calculation showing how an effective mass operator emerges by integrating out the higher coherence field. Defect norms as quantitative incoherence. The coherence complex replaces strict gluing by controlled defects subject to exact closure identities. In this framework, “coherence” is not a binary property but a quantitative one: a configuration can be more or less coherent depending on the magnitude of its defects. Given a background metric on spacetime and invariant bilinear forms on the relevant Lie algebras, there is a canonical way to measure this magnitude: the L2norms of the defect forms. Let h·,·ig0 and h·,·ig1 be invariant inner products on g0 and g1 (e.g. minus the trace in a faithful representation, or the Killing form where appropriate). Using the spacetime Hodge star ?, define kFΦk2:= ZMhFΦ∧?FΦig0,kHk2:= ZMhH∧?Hig1.(41) The functional kFΦk2 penalizes failure of first-level closure after allowing higher repair through tΦ ( B ), while kHk2 penalizes failure of second-level closure (inconsistency of the repair data itself). These are the direct analogues of Yang–Mills energy kFk2 and higher-form kinetic energy kHk2 in higher gauge theory [14, 15]. From the gluing perspective, kFΦk2 and kHk2 are the most conservative energetic penalties one can write: they are local, gauge covariant, positive (for Euclidean signature), and directly measure the obstruction to strict gluing at the first two levels. Any other reasonable local measure of incoherence is equivalent to these norms at leading order in a derivative expansion, up to field redefinitions and higher-derivative corrections. 9 Coherence-first action functional. The minimal coherence energy is therefore taken to be S[A, B; Φ] = 1 2g2kFΦk2+1 2h2kHk2+SΦ[Φ] + Stop,(42) where g and h are stiffness parameters, SΦ governs the intrinsic dynamics (kinetic geometry and constraints) of Φ, and Stop denotes optional topological terms compatible with the coherence complex. Crucially, (42) is not a “symmetry breaking” functional: it does not penalize violation of an assumed symmetry; it penalizes incoherence, i.e. residual defects in the gluing hierarchy. Why this leads naturally to mass generation: completing the square. The appearance of a mass operator can be demonstrated explicitly by analyzing the quadratic fluctuations around a coherent background with constant grammar. To isolate the mechanism cleanly, we consider the following controlled approximation, which we state explicitly: Approximation (quadratic, stiff coherence sector). We linearize around a background with constant Φ = Φ ? (so tΦ = t? is constant), and we keep only quadratic terms in fluctuations of ( A, B ). We also assume the higher coherence sector is stiff enough that integrating out B at quadratic order is a good low-energy approximation (precise below). Let A be a g0 -valued 1-form and B a g1 -valued 2-form. Around the background, write tΦ→t? and expand F(A) = dA +O(A2), H =dB +O(AB),(43) so at quadratic order the action reduces to S(2)[A, B] = 1 2g2kdA −t?(B)k2+1 2h2kdBk2+··· ,(44) where ··· denotes gauge-fixing terms and possible small regulators (e.g. kBkBk2 ) that do not change the conceptual mechanism. The key observation is that (44) is Gaussian in B. To make the algebra transparent, let h·,·i denote the L2 inner product induced by the Hodge star and the invariant forms. Then kdA −t?(B)k2=hdA, dAi−2hdA, t?(B)i+ht?(B), t?(B)i.(45) Using the adjoint t† ? defined by hX, t? ( Y ) i = ht† ? ( X ) , Y i , we can rewrite the B -dependent part as a quadratic form: S(2)[A, B] = 1 2g2hdA, dAi+1 2hB, MBi−hB, J[A]i,(46) with M=1 g2t† ?t?+1 h2d†don Ω2(g1),J[A] = 1 g2t† ?(dA).(47) Here d† is the adjoint of d with respect to the chosen inner product. The functional integral over B is therefore a Gaussian: ZDBexp−1 2hB, MBi+hB, J[A]i∝(det M)−1/2exp1 2hJ[A],M−1J[A]i,(48) and the effective quadratic action for Abecomes S(2) eff [A] = 1 2g2kdAk2−1 21 g2t† ?dA, M−11 g2t† ?dA+1 2log det M.(49) Equation (49) is exact at quadratic order. The second term is the coherence-induced correction from integrating out the compensator. 16 Projector onto massive electroweak directions. To reproduce the qualitative electroweak pattern— massive W± and Z but a massless photon—the structure map must have a nontrivial kernel. In the coherence-first mass mechanism, massless gauge directions correspond to ker ( t† Φ ). Thus we require that the image of tΦlie in the “massive” subspace of g0and vanish on the electromagnetic direction. Let Q = T3 + Y denote the unbroken electromagnetic generator in the SM. At the Lie algebra level we implement the corresponding splitting by choosing a projector Pmassive :g0→g0, Pmassive(Q)=0, P2 massive =Pmassive,(82) whose image is a complement of u (1) em = span{Q} . One convenient choice is to work in the basis (T1, T2, Z, Q)where Z:= cos θWT3−sin θWY, Q := sin θWT3+ cos θWY, (83) with θW the Weinberg angle, and set Pmassive to be the identity on ( T1, T2, Z )and zero on Q . More invariantly, one may define Pmassive by orthogonal projection with respect to κAB once a normalized Q is chosen. The specific choice of projector does not affect the conceptual mechanism; it simply encodes the desired pattern of massive vs. massless gauge directions. Explicit definition of the grammar morphism tΦ .We are now ready to define the structure map. For U∈g0and v∈RN, set tΦ(U⊗v) := λ0ΦiGijvjPmassive(U),(84) where λ0>0is a fixed coherence scale. Several immediate properties follow. (i) G0 -equivariance. The map (84) is equivariant with respect to the electroweak gauge action because Pmassive is a g0 -module endomorphism (it commutes with the adjoint action on the massive subspace), and the contraction ΦiGijvjis a scalar: tΦX . (U⊗v)=tΦ([X, U]⊗v) = λ0(ΦTGv)Pmassive([X, U]) = [X, tΦ(U⊗v)],(85) for all X, U ∈g0 and v∈RN . This is precisely the crossed-module compatibility condition required for the coherence complex to close (Sec. II A) [14, 15]. (ii) GL(N)covariance and restricted invariants. Under a grammar transformation M∈GL(N), Φ7→ MΦ, v 7→ Mv, G 7→ M−TGM−1,(86) and therefore ΦTGv is invariant: (MΦ)T(M−TGM−1)(Mv)=ΦTGv. (87) Hence the map tΦ is well defined on the constrained grammar moduli and depends on Φonly through the coherence-covariant tensor G . This is the categorical restriction that prevents an arbitrary polynomial dependence on Φ. (iii) Kernel and the photon. Because Pmassive(Q)=0, we have tΦ(Q⊗v)=0 ∀v∈RN.(88) Consequently, the electromagnetic direction lies in the kernel of the induced mass operator (derived in Sec. V A): it is not repaired by coherence and therefore remains massless. This is the coherence-first analogue of the SM statement that the vacuum leaves U(1)em unbroken. Adjoint map and the mass operator tΦt† Φ .The appearance of vector masses depends on the positive operator tΦt† Φ . To compute it, we fix inner products on g0 and g1 . Let κAB be the invariant form on g0 and use Gij to pair the RNindices. Then the natural pairing on g1is hU⊗v, U0⊗v0ig1=hU, U0ig0(viGijv0j),(89) which makes tΦadmit an adjoint t† Φ:g0→g1defined by hX, tΦ(Y)ig0=ht† Φ(X), Y ig1.(90) A direct computation using (84)–(89) yields t† Φ(X) = λ0Pmassive(X)⊗Φ,(91) 17 where Φis viewed as an element of RN and the contraction uses Gij to identify covectors and vectors. Applying tΦagain gives (tΦt† Φ)(X) = λ2 0(ΦiGijΦj)Pmassive(X) = λ2 0I(Φ) Pmassive(X).(92) This identity is the key algebraic input for mass generation: the gauge boson mass operator is proportional to tΦt† Φ, and therefore proportional to the constrained grammar invariant I(Φ). Relation to the W/Z/γ pattern. Equation (92) shows that, once the higher coherence field B is integrated out (Sec. V A), the gauge boson mass operator takes the form M2 A(Φ) ∝tΦt† Φ∝I(Φ) Pmassive.(93) Thus: •modes in Im(Pmassive)acquire a mass proportional to I(Φ); •modes in ker(Pmassive)remain massless (the photon direction). This reproduces the qualitative electroweak structure of massive W/Z and massless γ at the level of the coherence-lift mass mechanism. A more detailed matching to the SM mass ratios can be obtained by refining Pmassive and the normalization of κAB (and by including additional electroweak-specific couplings if desired), but the present construction already captures the structural point relevant to naturalness: masses are controlled by a coherence invariant I(Φ) rather than by an arbitrary µ2. Why this removes an arbitrary µ2 while retaining a fundamental scalar. In the SM, the dimension-two gauge-invariant operator φ†φ allows an arbitrary relevant deformation −µ2φ†φ . Here, the scalar Φ can enter the gauge/coherence sector only through the invariant I (Φ) and only via the constrained coherence datum Gij . Since admissible deformations are functions on the constrained grammar moduli (not arbitrary polynomials in Φ), an independent µ2 -type parameter does not exist. Nevertheless, Φ remains a fundamental field: it appears in the definition of the structure map (84) and its fluctuations change I (Φ) and therefore change both gauge boson masses and the radiatively generated Higgs stiffness on grammar moduli (Secs. V A and VII B). D. Coherence constraint on Gij We now state the categorical/coherence assumption that is responsible for the naturalness improvement in the present framework. In the preceding subsections, the scalar Φ i was introduced as a fundamental grammar coordinate and the structure map tΦ was defined in terms of the bilinear invariant I (Φ) = Φ iGij Φ j . The tensor Gij is the crucial additional datum: it is the object that makes nontrivial GL ( N )- invariant scalar dependence possible, and it is also the locus where categorical coherence constrains admissible counterterms. Key assumption: Gij is coherence data, not a free coupling. In conventional effective field theory, specifying a symmetric tensor Gij on a multiplicity space would be equivalent to choosing an arbitrary set of couplings (a “metric” in field space) that can be deformed freely and renormalized arbitrarily. That is not the intended interpretation here. Instead, we assume: Coherence constraint. Gij is part of the categorical coherence datum of the theory: it is fixed by (or restricted to) a constrained class of admissible structures determined by coherence/normalization conditions, and it is not an independent local parameter that can be freely varied as an arbitrary relevant deformation. This is the structural replacement, in our setting, for the symmetry-based notion of “forbidden operators.” The point is not that gauge symmetry forbids a scalar mass term (it does not), but that categorical coherence restricts the space of admissible local deformations by restricting the admissible grammar tensors Gij. Mathematically, we may regard G as specifying a point (or orbit) in a constrained moduli space MG of coherence tensors, and the field theory is defined only for G∈ MG . Equivalently, admissible counterterms must descend to functionals on the constrained moduli, rather than being arbitrary polynomials in unconstrained scalar fields. 18 Why the constraint is necessary under GL ( N ).Recall that Φtransforms as a vector under GL ( N ), Φ 7→ M Φ, and that for a single vector there are no nontrivial GL ( N )invariants. Introducing G with the transformation G7→ M−TGM−1 produces the invariant I (Φ) = Φ TG Φand allows scalar dependence consistent with grammar covariance. However, if G were arbitrary and freely deformable, then the invariant I (Φ) would reintroduce precisely the same structural issue as the SM: a dimension-two invariant multiplied by an arbitrary coefficient. In other words, without a coherence constraint on G , one merely relocates the problem rather than solving it. The coherence-first mechanism therefore requires that the “radial” freedom associated with an arbitrary quadratic invariant is removed at the level of admissible coherence data. This is achieved by restricting G so that it cannot be varied to generate an arbitrary relevant deformation. Examples of coherence constraints on G .There are several natural and mathematically sharp ways to restrict G while retaining GL ( N )covariance and allowing nontrivial dynamics. We list three representative classes of constraints; in the remainder of the paper we keep the discussion general but assume that one such constraint is imposed. (i) Fixed determinant (volume-form constraint). Impose det G= det G0=constant.(94) This constraint is GL ( N )covariant in the sense that det ( M−TGM−1 ) = ( det M ) −2det G , so fixing det G restricts GL ( N )to the subgroup preserving the volume form (or, equivalently, it selects a constrained moduli within the full GL ( N )orbit). Operationally, (94) removes one would-be scaling degree of freedom of G, preventing an arbitrary rescaling of I(Φ) from being introduced as a free relevant parameter. (ii) Fixed eigenvalue spectrum (isomorphism-class constraint). Assume G is symmetric positive definite (or of fixed signature) and restrict it to a fixed isospectral class, spec(G) = {λ1, . . . , λN}fixed,(95) so that admissible G differ only by change of basis G = M−TG0M−1 with M in the stabilizer of the spectrum. This removes all continuous scaling freedom of quadratic invariants, leaving only orientationtype moduli. In categorical language, this is the statement that the coherence tensor defines an isomorphism class of bilinear forms, fixed by normalization conditions rather than by local couplings. (iii) Fixed categorical normalization. In many higher-categorical settings, structure morphisms are defined only up to equivalence and must satisfy normalization conditions expressing unit compatibility (for example, choices of inner products on multiplicity spaces may be fixed by requiring functorial compatibility with composition, or by specifying a canonical Frobenius/trace structure on the relevant category). In such cases Gis not a freely deformable field but part of a chosen categorical datum, fixed up to discrete choices. While the precise categorical origin depends on the chosen coherence 2-category, the operational consequence is the same: G belongs to a constrained moduli MG and cannot absorb an arbitrary relevant scalar deformation. Interpretation: removal of an arbitrary µ2 .We can now state precisely how the coherence constraint eliminates an arbitrary Higgs mass parameter. In the SM, the dimension-two gauge-invariant operator φ†φ exists and its coefficient µ2 is an independent relevant deformation. In the present framework, the only admissible scalar dependence enters through the grammar invariant I(Φ) = ΦiGijΦj,(96) but G itself is constrained and not freely deformable. Consequently, there is no independent analogue of a freely adjustable µ2 coefficient: any would-be quadratic term is either a constant (if I is fixed by the constraint) or is tied to constrained coherence data whose admissible deformations are not local relevant parameters. This can be phrased in BRST/cohomological terms. Admissible counterterms are elements of the local BRST cohomology H0 ( s|d )[ 9 , 10 ]. In the SM, φ†φ defines a nontrivial class at ghost number zero, so its coefficient is an admissible counterterm. In our setting, admissible counterterms must be invariant not only under gauge-of-description redundancies but also under the categorical restrictions defining MG . Equivalently, admissible local deformations must be functions on the constrained grammar moduli; a free µ2 deformation is absent because G cannot be freely scaled to generate it. The Higgs mass will instead arise radiatively as the curvature of the effective potential on the constrained moduli, and will therefore be generated and controlled rather than arbitrarily inserted. 19 Practical remark. In the remainder of the paper, we will keep the constraint class general and denote by MG the admissible moduli of coherence tensors. All subsequent statements about counterterms and radiative mass generation should be understood as statements about deformations within MG . When we compute the one-loop effective potential, the dependence on Φappears only through I (Φ), and the physical Higgs mass is the Hessian of Veff ( I )restricted to the allowed moduli directions. This is the precise operational meaning of “coherence constrains counterterms.” IV. FULL LAGRANGIAN AND SYMMETRIES A. Explicit Lagrangian with indices We now assemble the full local Lagrangian density implementing the coherence-first Higgs mechanism in the GL ( N )grammar setting. The construction combines: (i) the ordinary electroweak gauge kinetic term, (ii) the coherence-lift (fake-curvature) term in which the 2-form coherence field repairs gauge curvature through the grammar map tΦ , (iii) a kinetic term for the B -sector ensuring the higher coherence constraint is dynamical, and (iv) a kinetic term for the fundamental grammar scalar Φexpressed through the constrained coherence tensor Gij . All terms are local, gauge covariant, and power-counting renormalizable in four dimensions. Field content and index conventions. We work on a four-dimensional spacetime manifold M with metric gµν and volume element p|g|d4x . The electroweak gauge algebra is g0 = su (2) ⊕u (1) Y , with a basis {TA} , where the index A runs over the three SU (2) generators and one U (1) generator. Structure constants are denoted [TA, TB] = fCABTC,(97) with the understanding that fCAB = 0 whenever any index corresponds to the abelian U (1) direction. We fix an invariant bilinear form κAB on g0 , used to contract gauge indices in kinetic terms. The multiplicity indices i, j = 1, . . . , N are contracted by the constrained coherence tensor Gij introduced in Sec. III D. The dynamical fields are: •Ag0-valued gauge connection Aµ=AA µTA. •Ag1-valued 2-form Bµν =BAi µν TA⊗eiwith g1=g0⊗RN. • A fundamental real scalar Φ i ( x )(grammar coordinate), transforming under GL ( N )as described in Sec. III B. Gauge curvature and B-sector covariant derivative. The field strength of Ais FA µν =∂µAA ν−∂νAA µ+fABC AB µAC ν,(98) and the g0 -covariant derivative acting on g1 -valued fields is induced by the adjoint action on the g0 factor. For BAi µν, this gives (DρBµν)Ai =∂ρBAi µν +fABC AB ρBCi µν .(99) The corresponding 3-form curvature (higher coherence defect) is the covariant exterior derivative HAi µνρ = (D[µBνρ])Ai =∂[µBAi νρ]+fABC AB [µBCi νρ].(100) In the present core construction we take g1 to be abelian as a Lie algebra (Sec. III A), so there is no additional [ B, B ]term in H . This is sufficient to realize coherence lift and the naturalness mechanism; semistrict corrections can be added later. Grammar map and fake curvature. The grammar morphism tΦ:g1→g0is defined in Sec. III C by tΦ(U⊗v) = λ0ΦiGijvjPmassive(U),(101) with fixed λ0> 0and a projector Pmassive satisfying Pmassive ( Q ) = 0 for the electromagnetic generator Q . At the level of components, the induced map acting on BAi µν is (tΦ(Bµν))A=λ0ΦiGij BBj µν (Pmassive)AB,(102) 20 where (Pmassive)ABdenotes the matrix of the projector in the chosen basis. The fake curvature (first-level coherence defect after repair) is then (FΦ)A µν := FA µν −(tΦ(Bµν))A.(103) This is the central object in the coherence-lift term: it measures the residual non-glueability of the gauge curvature after allowing the higher coherence field Bto contribute according to the grammar tΦ. Full Lagrangian. The full Lagrangian density is L=1 4g2κAB FA µνFB µν | {z } (i) gauge kinetic +1 4g2κAB h(FΦ)A µν(FΦ)B µν −FA µνFB µνi | {z } (ii) coherence lift +1 12h2κAB Gij HAi µνρHBj µνρ | {z } (iii) B-sector kinetic +κΦ 2∂µΦiGij ∂µΦj | {z } (iv) Φ-sector kinetic +Lconstr[G, Φ]. (104) Here Lconstr [ G, Φ] denotes the implementation of the coherence constraint on Gij (Sec. III D) and, if desired, any constraint on Φcompatible with the chosen grammar moduli (e.g. constraints defining a submanifold of RN ). For example, a fixed-determinant constraint det G = det G0 may be imposed by a Lagrange multiplier field, or treated as a restriction on admissible background data in the path integral, depending on the chosen categorical interpretation. The decomposition shown in (104) is arranged to emphasize the physical logic: the gauge kinetic term is standard; the coherence-lift term replaces F by FΦ = F−tΦ ( B )and is responsible for mass generation; the B -sector kinetic term endows the higher coherence field with stiffness; and the Φkinetic term defines dynamics on grammar moduli through the coherence tensor Gij. Simplified expression and interpretation of the coherence-lift term. Expanding the coherence-lift term explicitly, 1 4g2κAB(FΦ)A µν(FΦ)B µν =1 4g2κAB FA µνFB µν −2FA µν(tΦ(Bµν))B+ (tΦ(Bµν))A(tΦ(Bµν))B.(105) The middle term couples the gauge curvature to the coherence field through the grammar map, and the last term is an algebraic stiffness for the repair contribution. This is the precise higher-coherence analogue of the gauge-boson mass generation mechanism: when B is integrated out (or when it is stiff), these terms produce an effective quadratic operator for A proportional to tΦt† Φ (Sec. V A). In particular, because tΦ contains Pmassive , the electromagnetic direction is in the kernel of the induced mass operator, reproducing the presence of a massless photon without imposing symmetry breaking by hand. Power counting and locality. The Lagrangian (104) is local and, at the level of naive power counting in four dimensions, renormalizable in the following sense. The field dimensions are [Aµ]=1,[Bµν] = 1,[Φ] = 0,[G]=0,[λ0]=1,[κΦ]=2.(106) so that Fµν and tΦ ( Bµν )both have dimension 2. The kinetic terms kFΦk2 and kHk2 therefore have the usual dimension 4. The dimensionful parameter κΦ sets the stiffness of the grammar moduli kinetic term and plays the role of a coherence-scale normalization. The scalar kinetic term has dimension 4as well because Gij is dimensionless. The crucial departure from the SM is that there is no independent dimension-two scalar operator with arbitrary coefficient: scalar dependence enters through coherence invariants built from constrained Gij and through derivatives. The counterterm analysis in Sec. VI C will make this precise in BRST terms. Relation to standard electroweak Lagrangians. If one were to set B = 0 and treat Φas a conventional Higgs doublet with a polynomial potential, (104) would reduce to the symmetry-first SM Higgs sector. Here we do not take that route. The coherence field B and the grammar map tΦ replace the usual role of a vacuum expectation value in generating vector boson masses: mass generation arises from the energetic cost of maintaining coherence, encoded by the defect norm kFΦk2 , rather than from inserting an arbitrary relevant deformation µ2|φ|2. The next subsection formulates the symmetry principles of (104) explicitly (gauge, 2-gauge, and grammar covariance), and later sections show by explicit Gaussian functional integration that integrating out B produces a gauge-boson mass operator proportional to tΦt† Φ , with coefficient controlled by the constrained grammar invariant I(Φ). 21 B. Gauge, 2-gauge, and GL(N)symmetries The Lagrangian (104) is built to encode three distinct layers of invariance: (i) the usual electroweak gauge symmetry G0 = SU (2) ×U (1) Y acting on the g0 indices, (ii) a 2-gauge (one-form) redundancy acting on the coherence 2-form field B and shifting A through the grammar map tΦ , and (iii) an internal GL ( N ) grammar covariance acting on the multiplicity indices. The first is the familiar local gauge redundancy of Yang–Mills theory; the second is the higher-gauge redundancy characteristic of crossed-module (2-group) structures; the third is not a spacetime gauge symmetry at all but an automorphism symmetry of the coherence data (the “grammar group”). Keeping these roles sharply separated is essential: the naturalness mechanism depends precisely on the fact that admissible local deformations must respect the coherence (grammar) structure rather than only the spacetime gauge symmetry. G0 gauge symmetry. Let g ( x ) ∈G0 be a spacetime-dependent gauge transformation. The g0 -valued connection transforms as usual, Aµ7→ Ag µ:= gAµg−1+ (∂µg)g−1,(107) and the field strength transforms covariantly, Fµν 7→ Fg µν := gFµν g−1.(108) The coherence field BAi µν takes values in g1 = g0⊗RN and transforms in the induced representation, i.e. by the adjoint action on the g0factor and trivially on the multiplicity factor: Bµν 7→ Bg µν := g . Bµν ≡(Adg⊗Id) Bµν .(109) In components this reads BAi µν 7→ (Bg µν)Ai = (Adg)ABBBi µν .(110) The grammar scalar Φiis taken to be a G0singlet in this construction, so Φ7→ Φunder G0.(111) (We stress that this is a deliberate departure from the SM Higgs doublet: Φis not a G0 -charged order parameter, but a grammar coordinate.) Equivariance of tΦ and covariance of the fake curvature. Because tΦ acts as Pmassive on g0 and as scalar contraction on RN, it is equivariant under G0: tΦ(g . U) = Adg(tΦ(U)) for U∈g1.(112) Consequently, tΦ(Bµν)7→ tΦ(Bg µν) = g tΦ(Bµν)g−1.(113) Hence the fake curvature FΦ,µν := Fµν −tΦ(Bµν)(114) transforms covariantly: FΦ,µν 7→ Fg Φ,µν =gFΦ,µν g−1.(115) The gauge and coherence-lift kinetic term κABFA ΦFB Φ is therefore invariant, as is the B -sector term built from HAi µνρ, since His a covariant 3-form in the same representation: Hµνρ 7→ Hg µνρ = (Adg⊗Id)Hµνρ.(116) This establishes invariance of (104) under local G0gauge transformations. 22 2-gauge (one-form) redundancy. The second invariance is the higher-gauge (2-group) redundancy associated with the presence of the 2-form B. Let Λµ(x)be a g1-valued 1-form parameter, Λµ= ΛAi µTA⊗ei∈Ω1(M, g1).(117) The 2-gauge transformation acts as Aµ7→ A0 µ=Aµ−(tΦ(Λµ)),(118) Bµν 7→ B0 µν =Bµν + (D[µΛν]),(119) where Dµ is the covariant derivative induced by A in the g1 representation. In components, using (102) , (118) becomes AA µ7→ AA µ−λ0ΦiGij ΛBj µ(Pmassive)AB.(120) To understand why this is the correct redundancy, recall that FΦ is the defect after allowing repair by tΦ ( B ). The transformation (118) – (119) shifts the presentation of repair data: Λencodes a reparameterization of the 2-level comparison, accompanied by a compensating shift of the 1-level comparison through tΦ . This is the continuum analogue of the 2-gauge redundancy familiar in crossed-module gauge theory [14, 15]. Invariance of FΦ at constant grammar and the Leibniz correction. If Φ(hence tΦ ) is constant, one checks directly that FΦis invariant to leading order: F0 Φ=F(A−tΦ(Λ)) −tΦ(B+DΛ) = F(A)−tΦ(B) + O(Λ2),(121) with the cancellation governed by the identity DAtΦ (Λ) = tΦ ( DA Λ) when tΦ is constant and equivariant. When Φvaries in spacetime, tΦ does not commute with DA and a Leibniz-type correction appears, exactly as discussed in the coherence-complex identities (Sec. II A). The correct statement is that the 2-gauge transformations are defined together with the coherence complex so that the structural closure identities remain exact; the explicit form of the correction is fixed once a precise grammar moduli prescription is chosen. For the perturbative computations of masses and counterterms we will work around constant backgrounds Φ=Φ?, where this subtlety disappears and the standard 2-group redundancy applies. GL ( N )grammar covariance. Finally, we describe the internal grammar covariance. Let M∈GL ( N ) act on the multiplicity space RNby v7→ Mv. This induces the transformations Φ7→ Φ0=MΦ, BAi µν 7→ (B0 µν)Ai =MijBAj µν ,(122) while leaving AA µ unchanged (since GL ( N )acts only on multiplicity indices). The coherence tensor transforms covariantly as G7→ G0=M−TGM−1,(123) so that the invariant I(Φ) = ΦTGΦis unchanged: (Φ0)TG0Φ0= ΦTGΦ.(124) It follows that tΦis GL(N)covariant in the precise sense that tΦ0(U⊗v0) = tΦ(U⊗v)whenever Φ0=MΦ, v0=Mv, G0=M−TGM−1,(125) and therefore the Lagrangian (104) is invariant under the combined GL ( N )transformation of (Φ , B, G ). Why GL ( N )is not a spacetime gauge symmetry. It is essential to emphasize the conceptual status of GL ( N ). The G0 gauge symmetry is a local redundancy of spacetime description: it acts pointwise on fields in spacetime and generates the usual Ward identities. The 2-gauge redundancy is likewise a local redundancy associated with the higher coherence level. By contrast, GL ( N )acts on the internal grammar (multiplicity) indices and should be viewed as an automorphism symmetry of the coherence data rather than as an additional spacetime gauge symmetry. In particular: •GL ( N )does not act on spacetime indices and does not introduce additional gauge constraints on physical observables; •GL ( N )organizes which scalar invariants are admissible: the scalar can enter only through GL ( N )- invariant coherence combinations such as I(Φ); 23 • categorical coherence constraints restrict admissible deformations of Gij (Sec. III D), thereby restricting the space of admissible counterterms more strongly than gauge symmetry alone. This is precisely how the naturalness mechanism operates: it is not a symmetry that forbids µ2 ; it is the restricted ring of admissible grammar invariants, together with coherence constraints on G , that removes an arbitrary independent relevant deformation. Summary of the symmetry structure. The combined invariances can be summarized as: spacetime gauge redundancies: G0and 2-gauge,grammar covariance: GL(N).(126) The first two ensure the usual consistency of the gauge and coherence complex; the third restricts admissible scalar dependence to coherence invariants. In later sections we will exploit this structure in two ways: (i) to compute the coherence-induced gauge boson mass operator (by integrating out B ), and (ii) to classify admissible counterterms and show the absence of an arbitrary Higgs µ2deformation. C. BRST structure (overview) To formulate the counterterm theorem and the one-loop effective potential in a way that is both unambiguous and robust under quantum corrections, we organize the gauge and higher-gauge redundancies of Sec. IV B in BRST form. This provides a canonical bookkeeping device for (i) gauge fixing, (ii) defining the quantum functional integral without double counting redundant configurations, and (iii) classifying admissible local counterterms. In particular, the admissible counterterms in a local gauge theory are controlled by the local BRST cohomology H0 ( s|d ): a local deformation is admissible if it is BRST-closed modulo total derivatives, and nontrivial if it is not BRST-exact [9, 10]. The present model has three layers of redundancy: 1. ordinary G0gauge symmetry with a 0-form parameter g(x); 2. 2-gauge symmetry with a 1-form parameter Λ(x)valued in g1(and reducibility); 3. internal GL ( N )grammar covariance acting on multiplicity indices, which we treat as an internal (non-spacetime) redundancy constraining admissible invariants. For counterterm classification it is sufficient to treat GL ( N )covariance as a rigid internal symmetry constraining local invariants. If one wishes to gauge a subgroup, the corresponding BRST sector can be added without changing the logic. We now list the corresponding ghost fields and the BRST transformations of the dynamical variables at the level needed for counterterm analysis. Ghost fields and grading. We introduce the following ghost multiplet: •c ( x ) ∈ Ω 0 ( M, g0 ): the usual Grassmann-odd 0-form ghost for G0 gauge transformations (ghost number +1). •ρ(x)∈Ω1(M, g1): a Grassmann-odd 1-form ghost for the 2-gauge symmetry (ghost number +1). •η ( x ) ∈ Ω 0 ( M, g1 ): a Grassmann-even 0-form ghost-for-ghost encoding the reducibility of the 2-gauge symmetry (ghost number +2). •χ ( x ) ∈ Ω 0 ( M, gl ( N )): a Grassmann-odd 0-form ghost encoding infinitesimal GL ( N )transformations on the multiplicity indices (ghost number +1). In components, χij ( x )acts on vectors by ( χ Φ) i = χijΦj. We denote the BRST differential by s; it raises ghost number by +1 and is nilpotent, s2= 0. BRST transformations: gauge and grammar sectors. We write the BRST transformations of the dynamical fields ( A, B, Φ , G )in a form consistent with the symmetries of Sec. IV B. For the g0 -valued connection A , the standard non-abelian BRST variation is modified by the 2-gauge ghost through the grammar map: sAµ= (Dµc)−(tΦ(ρµ)),(127) where ( Dµc ) A = ∂µcA + fABC AB µcC . Here tΦ ( ρµ )is the g0 -valued 1-form obtained by applying the grammar map to the g1-valued ghost ρµ(explicitly as in (102) with Bµν →ρµ). 24 For the 2-form field BAi µν , BRST covariance includes three contributions: a 2-gauge shift, an adjoint G0 action, and a grammar GL(N)action: sBµν = (D[µρν]) + c.Bµν + (Id ⊗χ)Bµν.(128) In components, (c.Bµν)Ai =fABC cBBCi µν and ((Id ⊗χ)Bµν)Ai =χijBAj µν . For the grammar scalar Φi, we treat GL(N)as the relevant internal covariance and take sΦ = χΦ,(129) i.e. ( s Φ) i = χij Φ j . (Recall Φis a G0 singlet by construction.) For the coherence tensor Gij , the covariance (80) implies sG =−χTG−Gχ, (130) or in components, (sG)ij =−χkiGkj −χkjGik.(131) This ensures that the scalar invariant I(Φ) = ΦTGΦis BRST invariant: s(ΦTGΦ) = 0,(132) as expected. Ghost sector and reducibility. The ghost variations encode the algebra of the symmetries. For the non-abelian G0ghost c, sc =−1 2[c, c],(133) the standard BRST rule. For the GL(N)ghost χ, sχ =−1 2[χ, χ],(134) reflecting the Lie algebra gl ( N ). For the 2-gauge ghost ρ and the ghost-for-ghost η , the standard reducible 2-form structure gives sρ =Dη +c.ρ+ (Id ⊗χ)ρ, sη =c.η+ (Id ⊗χ)η, (135) where D is the covariant derivative induced by A on g1 . The inclusion of η encodes the gauge-for-gauge redundancy ρ∼ρ+Dσ characteristic of 2-form gauge fields. Nilpotency and compatibility conditions. Nilpotency s2 = 0 holds provided: (i) the G0 and GL ( N ) algebras close (standard), (ii) the action of g0 on g1 is a representation (Sec. III A), and (iii) the grammar map tΦ is equivariant with respect to G0 and compatible with the GL ( N )covariance (Sec. III C). These are precisely the algebraic coherence conditions encoded by the crossed-module structure. We defer the explicit nilpotency checks and the BV master-action formulation (including antifields) to Appendix A, since the present subsection aims only to record the transformations needed for the counterterm classification. Purpose: counterterms and the role of coherence constraints. With the BRST differential defined, admissible local counterterms ∆Lmust satisfy the cohomological condition s∆L+d(···)=0,(136) modulo trivial deformations s ( ··· ) + d ( ··· )[ 9 , 10 ]. In the SM, the gauge-invariant operator φ†φ defines a nontrivial element of H0 ( s|d ), allowing an independent relevant deformation. In our coherence-first setting, the scalar enters only through the constrained grammar invariant I (Φ) = Φ TG Φand admissible deformations of G are restricted by coherence constraints (Sec. III D). The BRST structure above makes this precise: the allowed scalar counterterms are not arbitrary polynomials in Φ, but functions of BRST-invariant grammar combinations consistent with the constrained moduli of G . This is the technical backbone for the counterterm theorem proved in Sec. VI C. All mass operators and one-loop quantities in this paper are computed around constant grammar backgrounds Φ = Φ ? , where the standard crossed-module gauge algebra closes strictly and no additional correction terms are required. 25 V. INTEGRATING OUT THE COHERENCE FIELD B(MAIN RESULT I) A. Gaussian functional integral and completion of the square In this section we derive the first main result: integrating out the higher-coherence 2-form field B generates an effective mass operator for the gauge field A proportional to tΦt† Φ , with coefficient controlled by the constrained grammar invariant I (Φ) = Φ TG Φ. This is the precise coherence-first analogue of gauge-boson mass generation: vector masses arise as an energetic cost of maintaining coherence once higher repair is allowed and carries stiffness. The derivation is completely explicit because the B -dependence of the Lagrangian is Gaussian at quadratic order. Setup and controlled approximations. We work in a background-field expansion around a configuration with constant grammar data: Φ(x)=Φ?, Gij =G(?) ij (constrained),so that tΦ→t?.(137) This is the correct setting for extracting mass operators and for perturbative renormalization: the mass matrix is defined at a stationary point of the effective potential and depends on constant background values. We will also make the standard (controlled) quadratic approximation: •Keep terms quadratic in (A, B)about the background. • Treat the B sector as stiff so that integrating out B gives a meaningful low-energy effective description for A. The precise stiffness scale will be tracked by the parameter h in the B kinetic term; physically, this corresponds to the higher coherence field being heavy compared to the external momenta of interest. Quadratic B -dependence of the action. Start from the Lagrangian (104) . For clarity, split the fakecurvature term as LFΦ=1 4g2κAB (FA µν −(t?(Bµν))A)(FB µν −(t?(Bµν ))B).(138) Expand (138): LFΦ=1 4g2κAB FA µνFB µν −1 2g2κAB FA µν(t?(Bµν))B+1 4g2κAB (t?(Bµν))A(t?(Bµν))B.(139) The B-sector kinetic term is LH=1 12h2κAB Gij HAi µνρHBj µνρ, HAi µνρ = (D[µBνρ])Ai.(140) To isolate the Gaussian structure, it is convenient to work at quadratic order around a background with small A fluctuations (or in background-field gauge). At that order one may replace D by ∂ in the H term, since the additional AB couplings only affect higher-order vertices and do not change the quadratic Gaussian integration of B that produces the leading mass operator. Keeping full covariance is straightforward but clutters notation. The final mass operator is gauge covariant; we will state the result in invariant form. Thus, at the level needed for the mass operator, the action is a quadratic functional of Bof the form S[A, B; Φ?] = SA[A] + 1 2hB, MBBi−hB, J[A]i,(141) where: •SA[A]contains the pure gauge kinetic term 1 4g2RκABFA µνFB µν (and gauge fixing); • MB is the positive operator on 2-forms induced by the combination of the algebraic t† ?t? term and the differential operator from the Hkinetic term; • J[A]is the linear source for Binduced by the cross term F·t?(B). 32 1. as a restriction on the admissible configuration space and hence on the admissible local functionals (the simplest viewpoint adopted here); 2. by introducing Lagrange multiplier fields enforcing the constraint, enlarging the BRST complex accordingly, and then computing H0(s|d)in the enlarged system. Either implementation leads to the same physical conclusion: admissible counterterms must descend to the constrained grammar moduli, and the independent relevant deformation responsible for the SM Higgs naturalness problem is absent. B. Scalar sector with GL(N)+ coherence constraint We now prove the central structural statement behind Main Result II: in the GL ( N )grammar setting with coherence tensor Gij, the admissible scalar dependence is generated by a single invariant I(Φ) = ΦiGijΦj,(178) and, because Gij is constrained coherence data (Sec. III D) and Φis a dimensionless grammar coordinate, there is no independent relevant deformation analogous to µ2|φ|2 in the SM. We emphasize that the point is not merely that I (Φ) is invariant; rather, the grammar symmetry and coherence constraint restrict the entire ring of admissible local scalar counterterms. Transformation law and the invariant. Recall the grammar covariance: Φ7→ Φ0=MΦ, G 7→ G0=M−TGM−1, M ∈GL(N).(179) A direct computation shows that I(Φ) is invariant: I(Φ0) = (Φ0)TG0Φ0= ΦTMT(M−TGM−1)MΦ=ΦTGΦ = I(Φ).(180) Statement to be proved (invariant ring in the scalar sector). Let A be the algebra of local scalar densities built from Φand G without derivatives. Derivative terms are treated separately and include ( ∂ Φ) TG ( ∂ Φ) and higher-derivative operators; they do not affect the existence of a relevant µ2 -type deformation. We seek the GL ( N )-invariant subalgebra AGL(N) , i.e. scalar functions f (Φ , G )satisfying f(MΦ, M−TGM−1) = f(Φ, G). We claim: Proposition. On the open subset where G is nondegenerate (in particular for symmetric positive-definite G ), the GL ( N )-invariant scalar functions of the pair (Φ , G )are generated by I (Φ) = Φ TG Φand invariants of Galone (such as det Gand Tr Gk). Since Gis constrained coherence data, invariants of Galone are fixed (or restricted to a discrete set), and the only nontrivial Φ-dependence is through I(Φ). Proof via reduction to a normal form. Assume G is symmetric positive-definite (the case relevant for the kinetic pairing and for positivity of the quadratic form). Then there exists a (non-unique) M∈GL ( N ) such that G=M−TM−1.(181) Equivalently, writing S:= G−1/2, we have G=S−TS−1with S∈GL(N). Define the GL(N)-equivariant change of variables ψ:= S−1Φ∈RN.(182) Under (179) , S transforms by left multiplication and one checks that ψ transforms only under the stabilizer of the canonical form, i.e. under O(N): G7→ I⇒ψ7→ Rψ, R ∈O(N).(183) Therefore any GL ( N )-invariant scalar f (Φ , G )may be rewritten as an O ( N )-invariant scalar ˜ f ( ψ )(up to invariants of G alone). But the O ( N )-invariant polynomials (and smooth invariants) of a single vector ψ are functions of ψTψonly [17]. Thus f(Φ, G) = FψTψ=FΦTGΦ,(184) for some scalar function F , which proves that the only nontrivial Φ-dependence is through I (Φ) = Φ TG Φ. This completes the proof. More generally, for nondegenerate G of fixed signature one reduces to a standard diagonal form and obtains invariants generated by Φ TG Φas well. For systematic treatments of invariant theory for classical groups and related settings see [17, 18]. 33 Consequences for admissible local potentials and the absence of an arbitrary µ2 .The proposition implies that, in the scalar sector, any admissible local (derivative-free) contribution to the effective action must be of the form ∆Lpot =p|g| VI(Φ),(185) up to constants determined by invariants of G alone. Because Gij is constrained coherence data (Sec. III D), those G-only invariants are fixed and do not introduce new freely tunable relevant parameters. Now note the crucial dimensional point. In our grammar construction, Φis a dimensionless coordinate on grammar moduli (it is not a canonical dimension-one scalar like the SM Higgs doublet). Consequently, [Φ] = 0,[G]=0,[I(Φ)] = 0,(186) and any local potential term V(I)must carry mass dimension four through an overall scale: V(I)=Λ4f(I),(187) for some dimensionless function f . There is therefore no dimension-two operator of the form µ2I (Φ) analogous to the SM term −µ2φ†φ ; in particular, there is no independent relevant deformation parametrized by µ2 in the grammar sector. This is the precise sense in which the “arbitrary µ2 ” is removed: the scalar coordinate is fundamental but dimensionless, its admissible local dependence is restricted to I (Φ), and the only scales enter through coherence-sector stiffness parameters and radiatively generated effective potentials. Compatibility with BRST cohomology. The result above can be restated cohomologically: the scalar contribution to H0 ( s|d )is generated (up to G -only invariants) by functions of the single GL ( N )-invariant I (Φ). Because G is constrained and I is dimensionless, the relevant part of H0 ( s|d )in the scalar sector is trivial: there is no independent relevant µ2 -type counterterm. Quantum corrections can and will generate an effective potential Veff ( I )(Sec. VII), but its curvature at the minimum yields a Higgs mass controlled by coherence stiffness rather than by an arbitrary bare relevant parameter. Interpretation. The SM naturalness problem is fundamentally the freedom to write a gauge-invariant dimension-two local deformation with an arbitrary coefficient. In the coherence-first GL ( N )grammar model, the scalar is not a free polynomial field: it is a grammar coordinate whose admissible dependence is constrained by the categorical coherence datum Gij . The invariant ring collapses to I (Φ), and the relevant deformation disappears. This is the structural input that enables a naturalness improvement while retaining a fundamental scalar degree of freedom. C. Counterterm theorem (formal statement) We now state the counterterm theorem in a form suitable for use in renormalization arguments and for comparison with the Standard Model. The theorem makes precise the naturalness claim of this paper: the analogue of an independent relevant Higgs mass deformation is absent. The underlying reason is structural: admissible scalar dependence is restricted to a constrained grammar invariant, and the coherence constraints on Gij eliminate the freedom to introduce an arbitrary dimension-two mass parameter. Setting. Consider the coherence-first model defined by: 1. The gauge and coherence algebras (g0,g1)with g0=su(2) ⊕u(1)Yand g1=g0⊗RN(Sec. III A). 2. A fundamental scalar grammar coordinate Φ i and a symmetric coherence tensor Gij transforming covariantly under GL ( N ), together with a coherence constraint restricting G to a constrained moduli MG(Sec. III D). 3. A GL(N)-covariant grammar morphism tΦof the form (84) and the local Lagrangian (104). 4. A BRST differential s encoding the G0 gauge symmetry, the 2-gauge redundancy, and the grammar covariance (Sec. IV C), with nilpotency s2= 0 on the constrained configuration space. Admissible local counterterms are classified by local BRST cohomology H0(s|d)(Sec. VI A). 34 Relevant scalar mass deformation in the Standard Model. For comparison, recall that in the SM Higgs sector the gauge-invariant operator φ†φ has mass dimension 2and therefore defines a relevant local deformation ∆LSM,rel =−µ2φ†φ, (188) which is BRST closed and represents a nontrivial element of H0 ( s|d )for the SM BRST differential. The coefficient µ2 is therefore an independent relevant coupling; its additive renormalization is the origin of the naturalness problem. Statement of the theorem. In the present model, the scalar sector differs in two crucial respects: (i) the scalar Φis a grammar coordinate and enters only through the GL ( N )-invariant I (Φ) = Φ TG Φ(Sec. VI B), and (ii) G is constrained coherence data rather than a freely tunable parameter (Sec. III D). These facts restrict the relevant part of the BRST cohomology and eliminate an independent dimension-two scalar deformation. Theorem (no independent relevant Higgs mass deformation). In the GL ( N )coherence-first grammar model with coherence constraint G∈ MG , there exists no nontrivial local counterterm in H0 ( s|d )corresponding to an independent relevant scalar mass deformation. More precisely: there is no ghost-number-zero local functional of canonical dimension 2whose coefficient is a free parameter and whose BRST class is independent of the constrained coherence data. Proof sketch and logic. The proof is a direct consequence of the invariant-ring result of Sec. VI B combined with dimensional analysis. We outline it in three steps. Step 1: scalar dependence is generated by I (Φ) . By Sec. VI B, any derivative-free GL ( N )- invariant scalar functional must be a function of I(Φ) = ΦTGΦ,(189) up to invariants of G alone. Since the BRST differential acts by grammar covariance and G0 gauge covariance, I (Φ) is BRST invariant, and no other independent scalar invariant exists. Thus the scalar sector of H0 ( s|d )is generated by functions of I (Φ) (modulo constants and G -only invariants fixed by the coherence constraint). Step 2: no dimension-two invariant deformation exists. In our grammar construction, Φand G are dimensionless, hence I (Φ) is dimensionless. Therefore no local term of the form µ2I (Φ) has the correct mass dimension to appear as a dimension-two deformation in the four-dimensional Lagrangian. Any potential term must have mass dimension four, and hence must be of the schematic form ∆Lpot =p|g|Λ4fI(Φ),(190) with Λa scale determined by coherence-sector stiffnesses or radiative effects. There is therefore no analogue of (188) : the independent relevant deformation is absent already by dimensional and invariant-theoretic constraints. Step 3: coherence constraint prevents reintroducing a free relevant parameter through G. One might worry that an effective µ2 could be reintroduced by allowing arbitrary deformations of G . This is precisely what the coherence constraint excludes: G is restricted to MG and does not carry an arbitrary local scaling mode that would function as an independent relevant coupling. Equivalently, invariants of G alone are fixed (or restricted to a discrete set) and do not provide a tunable relevant deformation. Together, these steps imply that the relevant scalar-mass counterterm of the SM does not exist in the admissible deformation space of the coherence-first grammar model. This is the structural naturalness improvement: the dangerous relevant coupling is absent from the cohomology of admissible local deformations. What replaces µ2 : controlled radiative stiffness. The absence of an independent relevant deformation does not imply the absence of a Higgs mass. Instead, it implies that the Higgs mass must be generated as a controlled stiffness of the effective potential on grammar moduli. In Sec. VII we compute the one-loop effective potential Veff ( I )obtained by integrating out gauge and coherence fields in a background Φ configuration. The physical Higgs mass is then m2 H= HessMgrammar VeffΦ=Φ?,(191) 35 a curvature on constrained moduli rather than a free input. Radiative corrections renormalize the coherence stiffness data and the functional form of Veff ( I ), but cannot generate an independent dimension-two scalar mass term because such a term is not an admissible local deformation. Interpretation (naturalness result). The Higgs naturalness problem can be summarized as: “a gaugeinvariant relevant deformation exists and is independent.” The coherence-first grammar model eliminates precisely this structural ingredient. In that sense, the theorem above is the naturalness result: the dangerous relevant parameter is absent, while a nonzero Higgs mass arises as a controlled radiative stiffness on the constrained grammar moduli space. VII. ONE-LOOP EFFECTIVE POTENTIAL AND HIGGS MASS (MAIN RESULT III) A. Background field method We now compute the one-loop effective potential for the grammar scalar and extract the Higgs mass as the curvature (Hessian) of the effective potential on constrained grammar moduli. The key structural input is that the scalar enters the gauge/coherence sector only through the invariant I (Φ) = Φ TG Φand through constrained coherence data. Consequently, the one-loop potential is a function Veff ( I ), and the Higgs mass is determined by derivatives of Veff evaluated at a stationary point I? . We begin by setting up the background field method and identifying the spectrum of massive modes that contribute to the functional determinants. Background-field setup. We work in Euclidean signature for the effective action calculation (standard for functional determinants), and analytically continue to Lorentzian signature at the end if desired. Let Φ?be a constant background value and define the invariant I?:= I(Φ?)=ΦT ?GΦ?.(192) We expand all fields as background plus fluctuations: Aµ=¯ Aµ+aµ, Bµν =¯ Bµν +bµν,Φ=Φ?+ϕ, (193) with Φ ? constant and ϕ a small fluctuation. For the effective potential we may take background gauge fields ¯ Aµ = 0 and ¯ Bµν = 0, since Veff is defined by evaluating the effective action on constant scalar backgrounds and vanishing gauge backgrounds: Γ[Φ?] = Zd4x Veff(I?).(194) The nontrivial dependence arises through the background value of tΦ? and hence through the induced mass operator for gauge bosons and for the coherence field. Quadratic fluctuation action and gauge fixing. At one loop, only the quadratic action in the fluctuations (a, b)contributes to Veff . We choose background-field gauge for the G0gauge symmetry, Lgf =1 2ξκAB (∂µaA µ)(∂νaB ν),(195) and include the corresponding Faddeev–Popov ghost action (details standard and omitted here; see [ 1 , 2 ]). For the 2-gauge symmetry associated with B , one similarly introduces a gauge-fixing condition for bµν (e.g. Lorenz-type condition ∂µbµν = 0) and the corresponding ghost-for-ghost sector [ 9 , 10 ]. For the present subsection, the essential point is that the resulting one-loop determinants can be organized into determinants of Laplace-type operators shifted by background-dependent masses. Quadratic operators and background-dependent masses. The coherence-lifted kinetic term contains the combination Fµν −tΦ? ( Bµν ), so even at ¯ A = ¯ B = 0 the quadratic part couples aµ and bµν . However, because the b -dependence is Gaussian, the coupled quadratic system can be diagonalized by the same completion-of-the-square logic used in Sec. V. Concretely, at quadratic order one finds a block structure S(2)[a, b; Φ?] = 1 2a bOaa(Φ?)Oab(Φ?) Oba(Φ?)Obb(Φ?)a b,(196) where Obb is a Laplace-type operator on 2-forms in the g1 representation (with stiffness scale h ), and Oab is controlled by the background grammar map tΦ?. 36 Integrating out b (or equivalently taking the Schur complement of the block operator) produces an effective quadratic operator for the gauge fluctuation aof the form Oeff(Φ?) = −∆11g0+M2 A(Φ?) + ··· ,(197) where ∆ 1 is the vector Laplacian (including gauge-fixing terms), and M2 A (Φ ? )is the coherence-induced mass operator derived in Sec. V B: M2 A(Φ?) = h2 g2tΦ?t† Φ?=h2 g2λ2 0I?Pmassive.(198) Thus the spectrum of gauge fluctuations splits into: •massless modes in ker(Pmassive)(photon direction and any additional kernel directions), • massive modes in Im ( Pmassive )with common mass-squared proportional to I? (up to coupling normalizations). This is the background-dependent mass spectrum that feeds into the one-loop potential. Spectrum counting and degrees of freedom. To write the one-loop potential in the familiar form, one counts physical degrees of freedom in the massive sector. In four dimensions, a massive vector field carries three physical polarizations, while a massless vector carries two. In background-field gauge the one-loop determinants are computed with gauge-fixing and ghosts, but the net result can be expressed in terms of physical polarizations and ghost contributions as usual [ 1 , 2 ]. Denote by nmass the dimension of Im ( Pmassive ) as a real vector space. In the electroweak-inspired choice Pmassive projects onto span{T1, T2, Z} and annihilates Q , so nmass = 3. Then the gauge contribution to the one-loop effective potential has the schematic form V(1) eff (I?) = 1 2nmass (d−1) Zd4p (2π)4logp2+m2 A(I?)+(ghost and b-sector determinants),(199) where (d−1) = 3 is the number of massive polarizations and m2 A(I?) = h2 g2λ2 0I?.(200) The remaining determinants (from the b -sector and its ghosts) depend on I? through the same grammar map tΦ? , and therefore also reduce to functions of I? alone. This establishes the key structural statement: Veff(I)depends on the scalar background only through the invariant I(Φ) = ΦTGΦ.(201) Grammar moduli and Higgs fluctuations. The Higgs/grammar fluctuation ϕ corresponds to fluctuations of Φon the constrained grammar moduli space (Sec. III D). Because Veff is a function of I (Φ), the Higgs mass is extracted by expanding I(Φ) around Φ?. To second order, I(Φ?+ϕ) = I?+ 2 ϕTGΦ?+ϕTGϕ. (202) The quadratic term in the effective potential then yields the mass matrix on the tangent space of the constrained moduli: (m2 H)ij =∂2Veff ∂Φi∂ΦjΦ=Φ? =dVeff dI I? 2Gij +d2Veff dI2I? 4(GΦ?)i(GΦ?)j,(203) with the understanding that physical Higgs fluctuations are restricted to the admissible grammar moduli directions (e.g. quotienting by any redundant directions and imposing constraints from MG ). At a stationary point of the effective potential, dVeff/dI|I? = 0, and the first term drops, so the Higgs mass is controlled by d2Veff/dI2|I? and the coherence tensor Gij . This is the precise sense in which the Higgs mass is a stiffness of coherence on grammar moduli. 37 Outlook to the explicit one-loop formula. In the next subsection we evaluate the one-loop determinant in a standard renormalization scheme and obtain an explicit expression Veff(I) = α I2logm2 0I µ2+β I2+··· ,(204) where α is determined by the spectrum of massive gauge/coherence modes and µ is the renormalization scale. We then extract m2 H from the Hessian (203) restricted to the physical grammar moduli. This completes Main Result III: a finite, controlled Higgs mass generated radiatively without an independent relevant µ2deformation. B. One-loop determinant We now evaluate the one-loop effective potential for a constant grammar background Φ=Φ ? , or equivalently for the constant invariant I?:= I(Φ?)=ΦT ?GΦ?,(205) using the background-field setup of Sec. VII A. The result is a Coleman–Weinberg type potential Veff(I) = α I2logm2 0I µ2+β I2+(subleading),(206) with calculable coefficient α determined by the spectrum of massive modes and the coherence-lift parameters. The crucial structural consequence is that there exists no independent relevant scalar mass deformation in the admissible counterterm space. In particular, the local BRST cohomology admits no operator playing the role of an independent Higgs mass term. Radiative corrections therefore renormalize only dimension–four coherence invariants and kinetic coefficients, rather than inducing an additive renormalization of a dimension–two scalar mass parameter. The effective potential is thus generated radiatively from the background–dependent masses of gauge and coherence modes, and depends on Φ only through the dimensionless grammar invariant I (Φ). In dimensional regularization, this absence manifests itself as the renormalization of dimension–four operators only; in cutoff schemes, any power–law sensitivity cannot be absorbed into a forbidden dimension–two counterterm. General form of the one-loop effective action. For a set of bosonic quadratic fluctuation operators Ok and fermionic operators D`around a background, the one-loop effective action is Γ(1) =1 2X k log det Ok−X ` log det D`,(207) with appropriate ghost subtractions in gauge theories [ 1 , 2 ]. For a constant background Φ ? , the effective potential is defined by Γ[Φ?] = Zd4x Veff(I?),(208) so that Veff is obtained by evaluating (207) per unit volume. Background-dependent masses. From Sec. VII A, the coherence-lift mechanism induces a gauge-boson mass operator M2 A(Φ?) = m2 0I?Pmassive, m2 0:= h2 g2λ2 0.(209) Thus, in a basis adapted to Pmassive, the gauge fluctuation spectrum splits into: •nmass massive gauge directions with mass-squared m2 A(I?) = m2 0I?; •dim ker(Pmassive)massless gauge directions (including the photon). In the electroweak-motivated projector choice, nmass = 3 (corresponding to W± and Z ) and dim ker ( Pmassive ) = 1(the photon). The B -sector and ghost determinants also depend on I? through the same algebraic operator t† Φ?tΦ?∝I? . Their detailed contribution affects the finite constant β and can contribute to α depending on gauge-fixing choices, but the central structural result is unchanged: all I -dependence enters through the background mass scale m2 A(I) = m2 0Iand similar I-dependent eigenvalues. 38 Coleman–Weinberg evaluation in dimensional regularization. We evaluate the determinant contribution of a bosonic degree of freedom with mass musing dimensional regularization in d= 4 −dimensions: V(1) bos (m2) = 1 2µ4−dZddp (2π)dlog(p2+m2),(210) where µis the renormalization scale. Differentiating with respect to m2gives a convergent integral: ∂V (1) bos ∂m2=1 2µ4−dZddp (2π)d 1 p2+m2=1 2µ4−d1 (4π)d/2Γ 1−d 2(m2)d 2−1.(211) Integrating back and expanding near d= 4 yields the standard result (in MS scheme): V(1) bos (m2) = m4 64π2log m2 µ2−3 2+(field-independent constant).(212) For a massive vector, the net contribution after gauge-fixing and ghost subtraction is equivalent to 3 physical polarizations times the scalar result (with a scheme-dependent constant shift), i.e. V(1) vec (m2) = 3m4 64π2log m2 µ2−cvec,(213) where cvec = 5 / 6in a common MS convention for gauge bosons [ 1 , 2 ]. (Any change in cvec corresponds to a finite renormalization and is absorbed into βbelow.) Effective potential as a function of I .Using m2 A ( I ) = m2 0I , the gauge contribution from the massive sector is V(1) gauge(I) = nmass 3 (m2 0I)2 64π2log m2 0I µ2−cvec+··· ,(214) where ··· denotes the I -dependent contributions from the b -sector and the ghost system (which are also functions of m2 0I ) and I -independent constants. Collecting all one-loop contributions, the effective potential takes the form Veff(I) = α I2log I µ2+β I2+γ I2log(m2 0) + (constants),(215) which may be re-expressed (absorbing log(m2 0)into βby a finite renormalization) as Veff(I) = α I2logm2 0I µ2+β I2,(216) where αand βare renormalized coefficients. In particular, the leading coefficient receives a transparent contribution from the massive gauge sector: αgauge =nmass 3m4 0 64π2=nmass 3 64π2h2λ2 0 g22 ,(217) with additional contributions from the B -sector determinants and associated ghosts of the same schematic scaling. A fully explicit accounting of the B -sector and ghost contributions requires specifying the 2-gauge fixing and is provided in Appendix C. For the present main-text argument, it suffices that these contributions depend on Φonly through I(Φ) and therefore preserve the functional form (216). No quadratic divergence in an independent mass parameter. The standard naturalness concern in the SM is the additive renormalization of a dimension-two parameter µ2 multiplying a gauge-invariant operator. In the present framework, there is no independent dimension-two scalar deformation in the admissible counterterm space (Main Result II). Accordingly, the one-loop potential is generated as a Coleman–Weinberg type function of the dimensionless invariant I , with an overall scale set by coherence-sector stiffnesses and by the renormalization scale µ . In dimensional regularization, the one-loop divergences appear as poles in  multiplying dimension-four operators (here I2 ), and are absorbed into β (and into wavefunction renormalizations). There is no place for a quadratically divergent additive shift of an independent µ2 parameter because such a parameter is not an admissible local deformation: the dangerous relevant coupling is absent by construction and by the cohomological classification. This is the precise sense in which the naturalness problem is structurally improved: radiative corrections renormalize a stiffness functional Veff ( I )on constrained grammar moduli rather than an arbitrary relevant mass parameter. In the next subsection, we use (216) to extract the Higgs mass as the Hessian of Veff on the constrained grammar manifold. 39 C. Higgs mass extraction We now extract the Higgs mass from the one-loop effective potential computed in Sec. VII B. The central point is that the scalar enters the effective action only through the constrained grammar invariant I(Φ) = ΦTGΦ,(218) so the Higgs/grammar mode is a fluctuation of Φon the constrained grammar moduli whose mass is determined by the curvature (Hessian) of Veff ( I )restricted to the physical moduli directions. This makes precise the slogan “Higgs mass as coherence stiffness”: the Higgs mass is the stiffness of the radiatively generated coherence potential along admissible grammar directions. Stationary points and the definition of I? .Let Φ ? be a constant background that extremizes the effective action. Since Γ[Φ] = Zd4x Veff(I(Φ)) (219) for constant backgrounds, stationarity implies ∂Veff ∂ΦiΦ=Φ? = 0.(220) By the chain rule, ∂Veff ∂Φi=dVeff dI ∂I ∂Φi=dVeff dI 2(GΦ)i,(221) where (GΦ)i:= GijΦj. Thus, for generic Φ?with GΦ?6= 0, stationarity implies dVeff dI I=I? = 0, I?:= I(Φ?)=ΦT ?GΦ?.(222) This is the standard statement that the effective potential is extremized as a function of the invariant I . Hessian in Φ-coordinates. The mass matrix for small scalar fluctuations ϕ = Φ − Φ ? is given by the second derivative of the effective potential, evaluated at the stationary point and projected onto physical grammar moduli directions: (M2 H)ij := ∂2Veff ∂Φi∂ΦjΦ=Φ? .(223) Differentiating (221) gives ∂2Veff ∂Φi∂Φj=d2Veff dI2 ∂I ∂Φi ∂I ∂Φj+dVeff dI ∂2I ∂Φi∂Φj.(224) Since ∂I ∂Φi= 2(GΦ)i,∂2I ∂Φi∂Φj= 2Gij,(225) we obtain (M2 H)ij = 4 d2Veff dI2I? (GΦ?)i(GΦ?)j+ 2 dVeff dI I? Gij.(226) At a stationary point (222) the second term vanishes, leaving the rank-one Hessian (M2 H)ij = 4 d2Veff dI2I? (GΦ?)i(GΦ?)j.(227) This makes the geometry transparent: only fluctuations of Φalong the direction G Φ ? acquire curvature from Veff ( I ). All orthogonal directions correspond to flat directions of Veff as a function of I alone. In the full theory these additional directions may be lifted by additional constrained invariants (or by constraints defining the grammar moduli), but (227) captures the universal contribution to the Higgs/grammar mass from Veff(I). 40 Projection to physical grammar moduli and the “geometry factor.” The physical Higgs mass is not simply an eigenvalue of (227) , because Φlives on constrained grammar moduli. There are two sources of projection: 1. Constraints defining the admissible Φ-manifold (e.g. a fixed-norm condition, or more generally a constraint induced by the coherence moduli stack). 2. Redundancies/quotients associated with grammar covariance (identifications on Φrelated to admissible transformations of Gwithin MG). Denote the physical tangent space at Φ ? by TΦ?Mgrammar . The physical mass squared is the Hessian restricted to this space and normalized by the kinetic metric. Since the kinetic term is LΦ,kin =κΦ 2∂µΦiGij ∂µΦj,(228) the canonically normalized fluctuation along a tangent vector v∈TΦ?Mgrammar has norm kvk2 kin := κΦviGijvj.(229) Therefore the physical squared mass for a mode vis m2 H(v) = vi(M2 H)ijvj κΦviGijvj.(230) Using (227), we obtain m2 H(v) = 4d2Veff dI2I? (vTGΦ?)2 κΦvTGv .(231) This is the desired “Hessian times geometry” statement: the mass is governed by the curvature of Veff with respect to the invariant I , times a geometric factor depending on how the physical mode projects onto the gradient direction GΦ?within the constrained grammar moduli. In particular, for the physical Higgs direction aligned with v∝ Φ ? (or more generally v∝G Φ ? after normalization), the geometry factor simplifies, yielding m2 H=4 κΦ d2Veff dI2I? I?,(232) where we used I? = Φ T ?G Φ ? and normalized the mode with respect to the kinetic metric. Equation (232) is the central quantitative statement of Main Result III: the Higgs mass is proportional to the curvature of the radiatively generated coherence potential and to the background grammar invariant I?. Explicit evaluation for Veff(I) = αI2log(m2 0I/µ2) + βI2.Using (216), compute derivatives: dVeff dI = 2αI logm2 0I µ2+αI + 2βI, (233) d2Veff dI2= 2αlogm2 0I µ2+ 3α+ 2β. (234) Stationarity (222) fixes I?implicitly by dV/dI|I?= 0, i.e. 2αlogm2 0I? µ2+α+ 2β= 0 ⇒logm2 0I? µ2=−1 2−β α.(235) Substituting (235) into (234) gives d2Veff dI2I? = 2α−1 2−β α+ 3α+ 2β= 2α, (236) a standard Coleman–Weinberg result: the curvature at the minimum is controlled by the logarithmic coefficient. Therefore m2 H=8α κΦ I?,(237) with α given (at leading order) by the determinant coefficients computed in Sec. VII B. This is a fully explicit expression for the Higgs mass in terms of coherence-sector couplings and the vacuum grammar invariant. 41 Interpretation: “mass generated and controlled.” Equations (232) – (237) complete Main Result III. The Higgs mass is not an arbitrary input parameter µ2 ; it is generated by radiative effects as the stiffness of the coherence potential on grammar moduli, and it is controlled by the same coherence data that determine gauge boson masses through tΦt† Φ . Radiative corrections renormalize α and β (dimension-four data), but cannot generate an independent relevant scalar mass deformation because such a deformation is absent from the admissible counterterm space (Main Result II). D. Interpretation We now interpret Main Result III in conceptual and physical terms, emphasizing what is new relative to the Standard Model and what is (and is not) being claimed. The short summary is: the Higgs mass is radiatively generated as a coherence stiffness on constrained grammar moduli, is tied to the gauge/coherence sector through tΦt† Φ , and does not arise from an independent relevant deformation. This provides a structural improvement of the Higgs naturalness problem. Radiatively generated mass as stiffness on grammar moduli. In the present framework, the fundamental scalar Φis not a conventional gauge-charged Higgs doublet with an arbitrary polynomial potential. Instead, it is a grammar coordinate whose admissible dependence is restricted to the invariant I(Φ) = ΦTGΦ,(238) with G constrained coherence data. The one-loop computation yields an effective potential Veff ( I )of Coleman–Weinberg form (216) , and the Higgs mass is the Hessian of Veff restricted to the physical grammar moduli: m2 H=4 κΦ d2Veff dI2I? I?(for the physical Higgs direction). (239) Thus the Higgs mass is the curvature (stiffness) of a radiatively generated potential on the constrained grammar manifold. In particular, for Veff(I) = αI2log(m2 0I/µ2) + βI2we found (237): m2 H=8α κΦ I?,(240) which exhibits the defining features: •it is generated at one loop (and corrected at higher loops); • it is controlled by the coefficient α , determined by the spectrum of massive gauge/coherence modes; •it scales with the vacuum grammar invariant I?, which also controls gauge boson masses. This is the precise mathematical sense in which “Higgs mass arises as an energetic cost of coherence”: it is the stiffness associated with varying the grammar invariant in a background where coherence lift gives gauge bosons mass. Tied to the gauge sector: common origin with vector masses. A defining structural advantage of the coherence-first formulation is that the scalar and vector masses originate from the same object: the grammar morphism tΦ . From Main Result I, integrating out the coherence field B produces the gauge boson mass operator M2 A(Φ) = h2 g2tΦt† Φ=h2 g2λ2 0I(Φ) Pmassive.(241) Thus the same invariant I (Φ) that controls the scalar effective potential also controls the vector mass scale: m2 A∝I?, m2 H∝I?,(242) with proportionality coefficients determined by stiffness and loop factors. This is conceptually similar to the SM relation m2 h∼λv2 and m2 W∼g2v2 , except that in the SM the common scale v is set by an arbitrary relevant deformation µ2 . Here the common scale is a grammar invariant constrained by coherence data and selected radiatively. 48 IX. DISCUSSION AND OUTLOOK We conclude by summarizing the main results, clarifying the precise sense in which the Higgs naturalness problem is structurally improved, and outlining the most important open directions. The central message is that mass generation can be understood as an energetic cost of maintaining higher categorical coherence, and that the arbitrariness of the Standard Model Higgs relevant deformation can be removed by restricting admissible scalar counterterms through constrained grammar data. Summary of the mechanism and results. The theory is organized around a coherence complex with fields (A, B)and grammar morphisms tΦ(Sec. II). The first-level coherence defect is the fake curvature FΦ=F(A)−tΦ(B),(255) and the second-level defect is H=DAB. Coherence is quantified by a defect-based action S∼1 2g2kFΦk2+1 2h2kHk2+··· ,(256) which penalizes incoherence rather than “breaking” a redundancy. Integrating out the coherence field B yields an effective mass operator for gauge fluctuations: M2 A(Φ) = h2 g2tΦt† Φ,(257) so gauge boson masses are eigenvalues of tΦt† Φ and massless directions are those in ker ( t† Φ )(Secs. V–V C). In the explicit electroweak-motivated construction, M2 A(Φ) = h2 g2λ2 0(ΦTGΦ) Pmassive,(258) so the photon remains massless because the electromagnetic generator lies in ker(Pmassive). The naturalness improvement (Main Result II) is a statement about admissible deformations. Counterterms are classified by local BRST cohomology H0 ( s|d )[ 9 , 10 ]. In the Standard Model, the dimension-two invariant φ†φ exists and supports an independent relevant deformation µ2φ†φ . In our coherence-first grammar model, the scalar sector is constrained by grammar covariance and coherence constraints on Gij , and the invariant ring collapses to the single grammar invariant I (Φ) = Φ TG Φ(Sec. VI B). Because G is constrained coherence data rather than a freely tunable coupling, there exists no independent relevant scalar mass deformation compatible with locality, BRST invariance, and coherence constraints (Sec. VI C). The Higgs mass is then generated radiatively as the stiffness of the one-loop effective potential on constrained grammar moduli: m2 H=4 κΦ d2Veff dI2I? I?,(259) and for the Coleman–Weinberg form Veff ( I ) = αI2log ( m2 0I/µ2 ) + βI2 one finds m2 H = (8 α/κΦ ) I? (Secs. VII B–VII C). In this sense, the Higgs mass is a coherence stiffness: it is the curvature of a radiatively generated coherence functional, not an arbitrary input parameter. What is (and is not) claimed. We emphasize the precise scope of the naturalness statement. We do not claim that ultraviolet physics is irrelevant or that all regulator-dependent effects vanish. We claim a structural fact: the dangerous independent relevant scalar deformation present in the SM is absent from the admissible counterterm space of the coherence-first grammar theory. Radiative corrections therefore renormalize dimension-four stiffness data (the effective potential on grammar invariants and kinetic coefficients) rather than introducing an independent dimension-two mass parameter. This is a well-defined and scheme-independent improvement of the naturalness logic. Open questions and next steps. Several extensions are important for turning this structural mechanism into a complete phenomenological framework. (i) Fermion masses and Yukawa structure. In the SM, fermion masses and flavor structure are tied to the SU (2) doublet nature of φ and its Yukawa couplings. In the present framework the grammar scalar Φis not a G0 -charged doublet; fermion masses must therefore be generated by additional coherence couplings (e.g. through tΦ -dependent operators, categorical selection rules, or a generalized coherence-Yukawa sector). A key question is whether categorical coherence constraints can also restrict flavor-relevant deformations, potentially providing a new viewpoint on flavor hierarchies. 49 (ii) Renormalization group flow of coherence data. While counterterms are restricted by H0 ( s|d ), the remaining couplings (stiffness parameters, effective potential coefficients, and admissible grammar invariants) will run under RG flow. A natural and potentially powerful direction is to formulate the RG flow directly on the constrained moduli of coherence data: how do ( λ0, h, g, κΦ )and the constrained class of Gij evolve, and can coherence constraints lead to attractive fixed structures? This would connect the present mechanism to modern perspectives on generalized symmetry and anomaly constraints, but at the deeper level of coherence closure rather than symmetry postulates. (iii) Phenomenology and precision constraints. A realistic electroweak phenomenology requires matching not only the mass pattern but also couplings (Higgs couplings, gauge self-interactions, precision electroweak parameters). In this paper we focused on the structural mechanism: coherence-lift mass generation and the absence of an independent µ2 deformation. Detailed matching will require specifying the electroweak projector embedding, the relation between ( g, g0 )and the coherence stiffness parameters, and the contributions of the B sector to higher-dimensional operators. The framework predicts correlated deviations, since both vector and scalar masses are controlled by the same grammar invariant I? and by the same coherence stiffness data. Broader message. The conventional Higgs narrative is frequently framed in terms of “spontaneous symmetry breaking,” but gauge symmetry is a redundancy, and the deeper question is why the theory is globally consistent and why certain deformations are admissible. The coherence-first perspective suggests a different organizing principle: the fundamental requirement is coherent gluing of local descriptions, encoded by closure of a coherence complex. Symmetry is a sufficient and often elegant way to guarantee coherence, but not a necessary one. Within this view, mass generation can arise from the energetic cost of maintaining coherence when higher repair data are stiff, and the naturalness problem is addressed by restricting admissible deformations through categorical grammar constraints rather than by postulating protective symmetries. In this sense, mass can arise from coherence, not symmetry breaking. Appendix A: Full BRST/BV complex This appendix records a concrete BRST/BV realization of the redundancies of the coherence-first grammar model introduced in Secs. IV B–IV C. The purpose is twofold: (i) to make the nilpotent structure underlying counterterm classification completely explicit, and (ii) to exhibit the reducibility (ghost-for-ghost) structure associated with the 2-form coherence gauge sector. 1. Fields, ghosts, and gradings We denote the dynamical fields collectively by ϕ={AA µ, BAi µν,Φi, Gij}.(A1) The gauge and grammar redundancies are encoded by the BRST differential s acting on fields and ghosts. We adopt the following ghost content: •cA(x)∈Ω0(M, g0): Grassmann odd, ghost number +1 (ordinary G0gauge ghost). •ρAi µ(x)∈Ω1(M, g1): Grassmann odd, ghost number +1 (2-gauge ghost). •ηAi(x)∈Ω0(M, g1): Grassmann even, ghost number +2 (ghost-for-ghost). •χij(x)∈Ω0(M, gl(N)): Grassmann odd, ghost number +1 (grammar covariance ghost). Explicitly, (sχ)ij=−χikχkj, with χGrassmann odd. When one treats GL ( N )as a rigid internal symmetry (as in the main text), χ may be taken spacetimeconstant; keeping it as a local ghost is a convenient bookkeeping device and does not change the classification of admissible local invariants. 50 2. BRST transformations and nilpotency We list the BRST transformations (cf. Sec. IV C) in component form. Let Dµ denote the g0 -covariant derivative on g0-valued fields: (Dµc)A=∂µcA+fABC AB µcC.(A2) Let the induced action on g1=g0⊗RNbe (Dµρν)Ai =∂µρAi ν+fABC AB µρCi ν,(Dµη)Ai =∂µηAi +fABC AB µηCi.(A3) The structure map tΦacts as in (102); in particular (tΦ(ρµ))A=λ0ΦiGij ρBj µ(Pmassive)AB.(A4) BRST on dynamical fields. sAA µ= (Dµc)A−(tΦ(ρµ))A,(A5) sBAi µν = (D[µρν])Ai +fABC cBBCi µν +χijBAj µν ,(A6) sΦi=χijΦj,(A7) sGij =−χkiGkj −χkjGik.(A8) One checks immediately that the invariant I(Φ) = ΦiGijΦjis BRST closed: s(ΦiGijΦj)=0.(A9) BRST on ghosts (closure and reducibility). scA=−1 2fABC cBcC,(A10) sχ =−χ2,i.e. (sχ)ij=−χikχkj,(A11) sρAi µ= (Dµη)Ai +fABC cBρCi µ+χijρAj µ,(A12) sηAi =fABC cBηCi +χijηAj.(A13) Nilpotency. Nilpotency s2 = 0 holds provided: (i) fABC satisfy the Jacobi identity, (ii) the action of g0 on g1 is a representation, and (iii) tΦ is g0 -equivariant (Sec. III C). The only nontrivial check in the present model is that the tΦ ( ρµ )term in sA is compatible with sρµ and sc ; this is exactly the equivariance relation (85). 3. BV master action (minimal sector) To make counterterm classification and gauge fixing fully systematic, one may pass to the BV formalism. Introduce antifields for each field/ghost: A∗µ A, B∗µν Ai ,Φ∗ i, G∗ij, c∗ A, ρ∗µ Ai, η∗ Ai, χ∗ji,(A14) with antifield ghost number gh(ϕ∗) = −1−gh(ϕ). The minimal BV action is SBV =S0[A, B, Φ, G]+Zd4xA∗µ AsAA µ+1 2B∗µν Ai sBAi µν + Φ∗ isΦi+G∗ij sGij +c∗ AscA+ρ∗µ Ai sρAi µ+η∗ Ai sηAi +χ∗jisχij, (A15) where S0 is the classical action from (104) (including constraint implementation). By construction, (SBV, SBV)=0(classical master equation) if and only if s2= 0. Gauge fixing is then implemented by a gauge-fixing fermion Ψgf, setting antifields to ϕ∗=δΨgf/δϕ. Appendix B: Detailed Gaussian integration over B This appendix provides a self-contained, detailed derivation of the Gaussian integration over B that yields Main Result I. We present both the block-operator (Schur complement) viewpoint and the completion-of-the-square viewpoint. 51 1. Quadratic action and operator notation Around a constant grammar background (Φ ?, G? ), the Lagrangian quadratic in ( a, b )takes the schematic form S(2)[a, b] = 1 2ha, Oaaai+1 2hb, Obbbi+ha, Oabbi,(B1) where a is a g0 -valued 1-form fluctuation and b a g1 -valued 2-form fluctuation. The operator Obb is Laplace type shifted by an algebraic term from t† ?t?: Obb =1 g2t† ?t?+1 h2d†d+··· ,(B2) and Oab comes from the cross-term F·t?(b). 2. Completion of the square Rewrite (B1) as S(2)[a, b] = 1 2ha, Oaaai+1 2hb, Obbbi−hb, J[a]i,(B3) with J[a] = −Obaa. Shift b=bcl[a] + ˜ bwith bcl[a] = O−1 bb J[a]. Then S(2)[a, b] = 1 2ha, Oaaai− 1 2hJ[a],O−1 bb J[a]i+1 2h˜ b, Obb˜ bi.(B4) Integrating out ˜ bgives the determinant factor and the Schur complement: Oeff =Oaa −Oab O−1 bb Oba.(B5) At low momentum, O−1 bb is dominated by (t† ?t?)−1and yields the local mass operator in Sec. V B. 3. Low-energy local limit Assuming p2 ( h2/g2 ) spec ( t† ?t? ), we approximate O−1 bb ≈g2 ( t† ?t? ) −1 on the relevant subspace, which gives Oeff ⊃h2 g2t?t† ?,(B6) hence M2 A(Φ?) = h2 g2t?t† ?. Appendix C: One-loop determinant calculation This appendix collects the determinant formulas used in Secs. VII B–VII C. We restrict to the standard background-field setting with ¯ A=¯ B= 0 and constant Φ=Φ?. 1. Basic integrals and renormalization For a bosonic degree of freedom with mass m, dimensional regularization gives V(1) bos (m2) = m4 64π2log m2 µ2−3 2(C1) (up to an additive constant). For a massive vector, the net contribution is V(1) vec (m2) = 3m4 64π2log m2 µ2−5 6,(C2) after gauge fixing and ghost subtraction in a common convention [2]. 52 2. Counting massive gauge modes Let nmass = dim Im(Pmassive). Then the gauge contribution to Veff is V(1) eff (I) = nmass V(1) vec (m2 0I) + ··· ,(C3) with m2 0 = ( h2/g2 ) λ2 0 . The ellipsis includes b -sector and ghost-for-ghost determinants which are also functions of m2 0Iand therefore preserve the functional dependence Veff (I). Appendix D: Electroweak projector construction Here we give an explicit construction of Pmassive used in Sec. III C and in the mass operator (165) . Let {T1, T2, T3, Y } be a basis of su (2) ⊕u (1) Y with invariant inner product κ . Define the Weinberg angle by (10) and the rotated generators Z:= cos θWT3−sin θWY, Q := sin θWT3+ cos θWY. (D1) Then one may define Pmassive(T1) = T1, Pmassive(T2) = T2, Pmassive(Z) = Z, Pmassive(Q)=0,(D2) and extend linearly. 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