Finite Domains Constrain Gauge Symmetry: Compactness, Anomalies, and Global Form in Windowed Quantum Field Theory

1. Introduction

Every physical realization of a gauge quantum field theory operates on a finite spacetime domain: detectors couple for finite durations, scattering regions have finite spatial extent, and no interaction is instantiated simultaneously across all of spacetime. This familiar fact carries a structural consequence that has received limited systematic attention. When the matter action is restricted to a finite domain, the gauge-fixed path integral, the Wess–Zumino consistency conditions, and the Gribov problem must all be posed on that bounded region rather than on ${\mathbb{R}}^4$. The finite domain does not relax the ordinary local and global gauge-consistency requirements; it supplies a specific setting in which those requirements must still be met. The question taken up here is which gauge data can support that finite-domain construction.

1.1. Prior Work and the Gauge-Admissibility Problem

The gauge symmetry structure of the Standard Model (SM), $\mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y$, is among the most precisely tested facts in physics [1,2]. Its specific form, however, is inserted by assumption. No known principle selects this particular group, rank, and fermion representation content from the space of all gauge theories in four spacetime dimensions. Anomaly cancellation constrains the fermion content once the gauge group is fixed [3,4], and grand unified theories embed the SM group in larger simple groups [5,6]; neither line of argument explains why the gauge group takes the particular low-energy form it does, why its rank is four, or why two of its three direct-product factors are non-Abelian.

A companion paper develops the windowed action prescription for QFT [7]. In that approach, matter fields are restricted to a finite spacetime domain ${\mathsf{\Omega }}_{\mathrm{loc}}$ by a smooth gauge-singlet window function $\diamond \left(x\right)\in [0,1]$, and the resulting theory satisfies modified Ward identities in which the conserved object is the windowed current $\diamond J^{\mu }$ rather than $J^{\mu }$ alone. That paper establishes the full windowed action with its gauge-fixing and ghost sectors, proves exact recovery of the ordinary interior dynamics where $\diamond =1$, fixes the admissible matching conditions across the boundary layer ${\mathcal{B}}_{ϵ}$ (the region where $\nabla \diamond \ne 0$), and establishes tree-level BRST invariance. It writes a schematic modified Slavnov–Taylor identity for the boundary layer and states explicitly that a complete all-order BV/BRST treatment of that layer remains open. It identifies the derivation of gauge-algebra constraints from the finite-domain construction as an open problem.

1.2. Main Result and Scope

The present paper takes up that problem. Under the assumptions of the windowed construction, together with an explicitly named regulated gauge-fixing prescription, the finite-domain gauge-fixed theory is admissible only if its gauge data satisfy three necessary conditions.

1.

Compactness under normalized local group averaging. A finite regulator of the bounded-domain theory replaces the local gauge group by a product of interior group factors containing $G^N$. If each factor is averaged with a normalized invariant Haar probability measure, then G must be compact, since such a measure exists only for compact G [8]. This is a condition of the group-averaging prescription used here, not a universal no-go theorem for every quantization of a noncompact gauge symmetry [9].

2.

Ordinary anomaly obstructions. The chiral representation content must be free of perturbative and global gauge anomalies. The window is BRST-inert, so it leaves the ordinary local BRST cohomology unchanged in the interior and cannot make the single-valued fermion determinant requirement easier to satisfy. The anomaly obstruction is not created or concentrated at ${\mathcal{B}}_{ϵ}$; it is already present wherever $\diamond =1$, and the boundary layer, at most, carries representative-level contact terms.

3.

Global form and Gribov data. The global form of the non-Abelian gauge group, the allowed bundle and line-operator sectors, the boundary gauge group, and the Gribov/fundamental-modular-region prescription must be specified on the bounded domain. A local Faddeev–Popov slice does not fix these data. Abelian $\mathrm{U}\left(1\right)$ factors carry no field-dependent Gribov horizon and are exempt from the non-Abelian part of this condition once their ordinary zero modes are fixed.

These three conditions are collected as the finite-domain gauge-admissibility theorem of Section 6. They are necessary, not sufficient: they do not construct a nonperturbative continuum quantum field theory. The contribution is structural. The windowed framework supplies a finite-domain, mechanism-level reason that the familiar compactness, anomaly, and global-form requirements must hold where a gauge theory is physically instantiated, under a stated regulator and boundary prescription.

The construction does not derive the matter spectrum, generation number, or minimality criterion from the scalar window alone; given those inputs, it derives the SM gauge algebra as the minimal finite-domain admissible solution. The observed matter content is taken as input. Given the observed chiral multiplet dimensions, a nonzero overall charge normalization, the observed up- and down-type singlet identification, and a family-universal minimality criterion, the standard anomaly classification of Minahan, Ramond, and Warner [10] and of Geng and Marshak [11] recovers the SM gauge algebra $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ and its direct-product covering presentation, up to allowed finite central quotients, and selects the minimal Higgs doublet $H\sim {(1,2)}_{+1/2}$. Two limitations sharpen this conservative claim. The scalar window is representation-blind, so it supplies no functional that counts generations or forbids sterile singlets and vectorlike pairs (Section 6.3). The local anomaly analysis fixes the gauge algebra and covering presentation but not the global form, which is a separate choice of bundle and line-operator data (Section 5).

1.3. Plan of the Paper

Section 2 states the inherited windowed gauge-fixed action, its boundary assumptions, and the windowed Ward identity. Section 3, Section 4 and Section 5 develop the compactness, anomaly, and Gribov/global-form conditions in turn. Section 6 combines them into the finite-domain gauge-admissibility theorem, adds the matter-admissibility and scalar-sector results, and applies the theorem to the observed matter spectrum. Section 7 and Section 8 discuss scope and conclusions.

2. Windowed Gauge Theory: Inherited Assumptions and Ward Identity

This section states the results of the companion work [7] that the rest of the paper applies, and fixes notation and assumptions.

2.1. Inherited Gauge-Fixed Windowed Action

Let G be a gauge group with Lie algebra $\mathfrak{g}$ and Hermitian generators $\left\{T_a\right\}$ satisfying

$$[T_a,T_b]=i{f}_{ab}{}^{c}T_c,$$

(1)

with structure constants $f_{ab}^c$. The third index is left upper until an invariant inner product is chosen. Once compactness is established (Section 3), $\mathfrak{g}$ admits an Ad-invariant positive-definite inner product. On the semisimple part, this may be chosen as minus the Killing form and extended by an arbitrary positive-definite inner product on the Abelian center. In an orthonormal basis, $f_{abc}\equiv f_{ab}{}^d\delta _{dc}$ is totally antisymmetric; the all-lower form $f_{abc}$ is used only after that point. For fermions $\psi$ in a representation R of G, the matter action is windowed by a smooth gauge-singlet scalar $\diamond \left(x\right)\in [0,1]$ that multiplies the Lagrangian density:

$$S_{\mathrm{mat}}=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\overline{\psi }i{\gamma }^{\mu }{\mathcal{D}}_{\mu }\psi ,$$

(2)

where ${\mathcal{D}}_{\mu }={\nabla }_{\mu }-ig{A}_{\mu }^a{T}_a^{\left(R\right)}$ is the gauge-covariant derivative in the representation R, and ${\nabla }_{\mu }$ is the metric-covariant derivative. Throughout, ${\mathcal{D}}_{\mu }$ denotes the full covariant derivative, gauge and metric together, acting in whatever representation the field it differentiates carries; on a gauge-singlet field it reduces to ${\nabla }_{\mu }$, and on a Lorentz scalar it reduces to ${\partial }_{\mu }$.

The gauge-theoretic results below use the full gauge-fixed windowed action established in Ref. [7],

$$S_{\diamond }={\int }_{{\mathcal{U}}_{\diamond }}{d}^{4}x\sqrt{-g}\diamond \left(x\right)\left({\mathcal{L}}_{\mathrm{YM}}+{\mathcal{L}}_{\mathrm{matter}}+{\mathcal{L}}_{\mathrm{scalar}}+{\mathcal{L}}_{\mathrm{gf}+\mathrm{gh}}\right),$$

(3)

where ${\mathcal{U}}_{\diamond }=\{x:\diamond \left(x\right)>0\}$ is the open support on which the physical field problem is posed, the window is fixed under field variation, and the fields obey the admissible matching or boundary conditions of the companion construction. Throughout, ${\mathsf{\Omega }}_{\mathrm{loc}}\equiv \overline{{\mathcal{U}}_{\diamond }}$, the window core is ${\mathsf{\Omega }}_{\mathrm{core}}\equiv int\{x:\diamond \left(x\right)=1\}$, and the resolved boundary layer is ${\mathcal{B}}_{ϵ}\equiv \{x\in {\mathcal{U}}_{\diamond }:d\diamond \left(x\right)\ne 0\}\subset {\mathcal{U}}_{\diamond }$. The Yang–Mills, matter, and scalar terms are the ordinary densities; ${\mathcal{L}}_{\mathrm{gf}+\mathrm{gh}}$ collects the gauge-fixing and ghost terms. At tree level the gauge-fixing and ghost sector may be written as a BRST variation of a windowed gauge-fixing fermion, and BRST invariance of $S_{\diamond }$ holds at that order [7]. The all-order boundary-layer BV/BRST completion is not established in the corpus and is not assumed here.

The window function is fixed by four conditions:

$$\begin{array}{cc}\hfill \diamond & =1\mathrm{on}{\mathsf{\Omega }}_{\mathrm{core}},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \diamond & \to 0\mathrm{smoothly}\mathrm{across}{\mathcal{B}}_{ϵ},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\delta }_g\diamond & =0,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\delta }_{\mathrm{BRST}}\diamond & =0,\hfill \end{array}$$

(7)

where ${\delta }_g$ denotes a gauge variation and ${\delta }_{\mathrm{BRST}}$ the BRST variation. Conditions (6) and (7) ensure that the window introduces no gauge structure of its own. Condition (7) has a further consequence, derived in the companion paper [7]: the windowed theory satisfies a modified Slavnov–Taylor identity. The companion paper establishes tree-level BRST invariance and the schematic form of the boundary-supported Slavnov–Taylor contact functional. In the window interior, the BRST complex reduces exactly to the ordinary one. The complete all-order boundary-layer BV/BRST completion is not assumed here and remains one of the calculations deferred in Section 7.5. The three structural conditions of Section 3, Section 4 and Section 5 are then necessary for that identity to hold consistently across the boundary layer ${\mathcal{B}}_{ϵ}$.

For the gauge-fixed construction considered below, the relevant group of gauge transformations on ${\mathsf{\Omega }}_{\mathrm{loc}}$ is taken to be the based gauge group:

$${\mathcal{G}}_{{\mathsf{\Omega }}_{\mathrm{loc}},0}=\{g\in \mathrm{Map}({\mathsf{\Omega }}_{\mathrm{loc}},G):g{|}_{\partial {\mathsf{\Omega }}_{\mathrm{loc}}}=e\}.$$

(8)

Here $\mathrm{Map}({\mathsf{\Omega }}_{\mathrm{loc}},G)$ is understood to mean the admissible smooth gauge transformations on ${\mathsf{\Omega }}_{\mathrm{loc}}$ with a well-defined boundary value; ${g|}_{\partial {\mathsf{\Omega }}_{\mathrm{loc}}}=e$ means that this boundary value is the identity element of G. Basing the transformations at the outer boundary removes boundary gauge transformations from the redundancy group and keeps ${\mathcal{G}}_{{\mathsf{\Omega }}_{\mathrm{loc}},0}$ closed under multiplication. A set of transformations with a fixed nonidentity boundary value $g_0\ne e$ would not be closed under multiplication and therefore would not define the gauge-redundancy group used here. This is an additional boundary prescription, not a consequence of the gauge-singlet status of ⋄. Alternative boundary conditions can retain boundary gauge transformations as edge symmetries and require an enlarged boundary phase space; that edge-sector case is outside the present paper.

2.2. Windowed Ward Identity

Apply the Noether procedure to the windowed action of Equation (2) under a gauge transformation $\psi \to e^{i{\alpha }^a{T}_a}\psi$. The matter gauge current is the fermion bilinear $J_a^{\mu }=\overline{\psi }{\gamma }^{\mu }{T}_a^{\left(R\right)}\psi$, written here in the Noether normalization with no explicit coupling factor, the same convention used for the multi-field current of Section 6 and the scalar current of Section 6.8 [7,12]. It carries an adjoint gauge index, so its conservation law is covariant rather than ordinary. The windowed Noether identity is

$${\mathcal{D}}_{\mu }\left(\diamond J_a^{\mu }\right)=0,$$

(9)

where ${\mathcal{D}}_{\mu }$ acts on the adjoint index of $\diamond J_a^{\mu }$. Since ⋄ is a gauge singlet, ${\mathcal{D}}_{\mu }\diamond ={\nabla }_{\mu }\diamond$, and the Leibniz rule gives the product-rule form

$$\diamond {\mathcal{D}}_{\mu }{J}_a^{\mu }=-({\nabla }_{\mu }\diamond )J_a^{\mu }.$$

(10)

This form is regular everywhere on ${\mathsf{\Omega }}_{\mathrm{loc}}$, including the outer edge of ${\mathcal{B}}_{ϵ}$ where $\diamond \to 0$. Where $\diamond >0$, it may be divided through to give the equivalent logarithmic form ${\mathcal{D}}_{\mu }{J}_a^{\mu }=-({\nabla }_{\mu }ln\diamond )J_a^{\mu }$. On ${\mathsf{\Omega }}_{\mathrm{core}}$, where $\nabla \diamond =0$, both reduce to the ordinary covariant conservation law ${\mathcal{D}}_{\mu }{J}_a^{\mu }=0$. In the boundary layer ${\mathcal{B}}_{ϵ}$, where $\nabla \diamond \ne 0$, the apparent non-conservation is confined to the localization transition. For a gauge-singlet current, such as the Abelian electromagnetic current treated in the companion paper [7], ${\mathcal{D}}_{\mu }$ reduces to ${\nabla }_{\mu }$ and Equation (9) becomes the companion’s identity ${\nabla }_{\mu }(\diamond J^{\mu })=0$. The present covariant form is the non-Abelian refinement of that statement, required because a non-Abelian gauge current is covariantly, not ordinarily, conserved.

2.3. Boundary Layer Geometry

The boundary layer ${\mathcal{B}}_{ϵ}$ is the region of four-dimensional spacetime where ${\nabla }_{\mu }\diamond \ne 0$. It is a smooth transition layer, not an abrupt boundary. Its characteristic thickness $ϵ$ is set by the applicable operational scale [7]. Specifically, $ϵ$ is an operational coarse-graining scale, not a universal constant, and is bounded below by the maximum relevant operational scale:

$$ϵ\gtrsim {ϵ}_{\mathrm{op}}\equiv max\left\{{ϵ}_{\det },{ϵ}_{\mathrm{dec}},{ϵ}_{\mathrm{H}},{ϵ}_{\mathrm{GUP}},{ϵ}_{\mathrm{EUP}/\mathrm{RGUP}},{ϵ}_{\mathrm{SW}}\right\},$$

(11)

where ${ϵ}_{\det }$ is set by detector resolution, ${ϵ}_{\mathrm{dec}}\sim c{\tau }_{\mathrm{dec}}$ by decoherence, ${ϵ}_{\mathrm{H}}$ by ordinary Heisenberg localization, ${ϵ}_{\mathrm{GUP}}$ by Planck-scale generalized uncertainty corrections, ${ϵ}_{\mathrm{EUP}/\mathrm{RGUP}}$ by curvature or relativistic corrections, and ${ϵ}_{\mathrm{SW}}$ by Salecker–Wigner clock-precision bounds, as detailed in [7].

These bounds constrain which window profiles are admissible, but they do not change the product-rule origin of the windowed Ward identity. For Equation (9) to hold quantum mechanically at each point of ${\mathcal{B}}_{ϵ}$, the gauge structure of $J_a^{\mu }$ must be consistent with the gradient coupling at every such point. That requirement is what the next three sections develop.

The boundary layer also carries the compensator stress-energy $T_{\mu \nu }^{\mathrm{comp}}$ required by the contracted Bianchi identity when windowed matter sources the Einstein field equations [13]. The present paper concerns the gauge-theoretic consistency requirements on ${\mathcal{B}}_{ϵ}$ independently of that gravitational compensator structure.

3. Compactness under Normalized Finite-Domain Group Averaging

3.1. The Functional Measure on a Bounded Domain

In Faddeev–Popov quantization, the path integral over gauge fields is written as [2,14]

$$Z=\int \mathcal{D}A\delta \left(f\left[A\right]\right)det{M}_{\mathrm{FP}}{e}^{iS[A,\psi ]},$$

(12)

where $f\left[A\right]=0$ is a gauge-fixing condition and $M_{\mathrm{FP}}$ is the Faddeev–Popov operator. Formally, and locally in field space before the Gribov ambiguity is addressed, the standard construction inserts the identity $1={\Delta }_F\left[A\right]{\int }_{\mathcal{G}}\mathcal{D}g\delta \left(f\left[A^g\right]\right)$ into the ungauge-fixed integral. The Faddeev–Popov determinant ${\Delta }_F\left[A\right]=det{M}_{\mathrm{FP}}$ is the Jacobian of the gauge-fixing map, the change-of-variables factor associated with the slice $f\left[A\right]=0$; it is not the object that cancels the group volume. The group volume $\mathrm{Vol}\left(\mathcal{G}\right)={\int }_{\mathcal{G}}\mathcal{D}g$ factors out of the integral and is divided away as a formal redundancy, or cancels between the numerator and denominator of a normalized expectation value. Under the normalized local group-averaging prescription adopted below, the regulated group average is required to be a finite invariant probability measure.

On a bounded domain ${\mathsf{\Omega }}_{\mathrm{loc}}$, the relevant group of gauge transformations is the based gauge group ${\mathcal{G}}_{{\mathsf{\Omega }}_{\mathrm{loc}},0}$ of Equation (8), with the boundary data specified by the localization prescription. The following statement is a condition of the regulated group-averaging prescription used here, not a universal theorem of BRST quantization.

Proposition 1

(Compactness under normalized local group averaging). Let a finite regulator of the bounded-domain gauge theory replace the local gauge group by a product of interior group factors ${\mathcal{G}}_N\supset G^N$, with one factor of G acting in each interior cell. If the gauge-orbit averaging in each factor is defined by a normalized invariant Haar probability measure, then G must be compact.

Proof 

(Proof). A locally compact group admits a nonzero left-invariant Haar measure, unique up to a positive scale. That measure has finite total mass, and can therefore be normalized to a probability measure, if and only if the group is compact [8]. At finite regulator level, the based boundary condition ${g|}_{\partial {\mathsf{\Omega }}_{\mathrm{loc}}}=e$ fixes the boundary gauge transformations but leaves the interior factor $G^N$ free. The assumed regulator assigns a normalized invariant probability measure to each copy of G. That prescription is available only when G is compact, and it fails for noncompact G for every $N\ge 1$, since no copy then admits a finite normalized invariant measure. Compactness of G is therefore necessary within the normalized local group-averaging prescription. □

The conclusion is tied to the prescription. It is not a no-go theorem for every quantization of a noncompact gauge symmetry: BRST treatments of Yang–Mills systems with noncompact gauge groups exist [9], with separate and unresolved questions of positivity, unitarity, and admissible matter representations. What the finite-domain construction supplies is a concrete physical setting for the compact-group lattice prescription adopted here: the gauge average over each local group factor is a normalized Haar integral, which exists precisely when G is compact. An analogous group-averaging requirement is implicit in many standard regulated definitions of the gauge-fixed measure built on a compact-group average; the windowed construction makes that local group-averaging step explicit on the bounded domain where the physical theory is posed.

3.2. Implications for the Gauge Algebra

Proposition 1 restricts G to compact groups under the stated prescription. Throughout, G is assumed connected; finite disconnected extensions are not classified here. Every connected compact Lie group is isomorphic to

$$G\cong {\displaystyle \frac{G_{\mathrm{sc}}\times \mathrm{U}{\left(1\right)}^m}{\mathrm{\Gamma }}},$$

(13)

where $G_{\mathrm{sc}}$ is a product of compact simply connected simple Lie groups, $\mathrm{U}{\left(1\right)}^m$ is a torus, and $\mathrm{\Gamma }$ is a finite central subgroup [15]. The simply connected simple factors are

$$\mathrm{SU}\left(N\right)(N\ge 2),\mathrm{Spin}\left(N\right)(N\ge 5),\mathrm{Sp}\left(N\right)(N\ge 1),G_2,F_4,E_6,E_7,E_8,$$

(14)

where $\mathrm{Sp}\left(N\right)$ denotes the compact symplectic group with $2N$-dimensional fundamental, so that $\mathrm{Sp}\left(1\right)\cong \mathrm{SU}\left(2\right)$ and $\mathrm{Sp}\left(2\right)\cong \mathrm{Spin}\left(5\right)$. This convention is used throughout. Noncompact groups such as $\mathrm{SL}(2,\mathbb{C})$, $\mathrm{SO}(1,3)$, $\mathrm{SO}(2,1)$, or noncompact forms of the exceptional algebras are excluded under the prescription of Proposition 1.

Equation (13) is the organizing classification for the rest of the paper. It ties the local Lie algebra, which controls compactness and the local anomaly, to the finite central quotient $\mathrm{\Gamma }$, which is part of the global-form data treated in Section 5.

4. Ordinary Anomaly Obstructions and Window-Dependent Boundary Terms

4.1. Gauge Anomalies and the Windowed Identity

In a theory with chiral fermions, the gauge current is not automatically conserved at the quantum level: triangle diagrams contribute an anomalous divergence [16,17]. For a gauge group G with generators $T_a$ in a chiral representation R, the leading anomaly coefficient is the totally symmetric cubic index

$$d_{abc}^{\left(R\right)}\equiv {\displaystyle \frac{1}{2}}{\mathrm{Tr}}_R\left[T_a\{T_b,T_c\}\right],$$

(15)

where ${\mathrm{Tr}}_R$ denotes the trace in the representation R. The trace returns a symmetric invariant tensor with three lower adjoint indices directly, so no invariant metric is needed to define $d_{abc}^{\left(R\right)}$. This tensor is the group-theoretic anomaly coefficient; it is not itself the cubic Casimir operator, which is the associated central element. Summed over the chiral content with a sign for chirality, the coefficient must vanish for the effective action to be gauge invariant [3,4]:

$$d_{abc}^{\left(\mathrm{tot}\right)}\equiv \sum _{\mathrm{Weyl}\mathrm{fermions}}{\chi }_f{d}_{abc}^{\left(R_f\right)}=0,$$

(16)

where ${\chi }_f=+1$ for left-handed and ${\chi }_f=-1$ for right-handed Weyl fermions.

The descent description of this anomaly uses differential-form language. To match the Hermitian-generator convention of Equation (1), define the anti-Hermitian matrix connection and its curvature

$$\mathcal{A}=-ig{A}^a{T}_a,\mathcal{F}=d\mathcal{A}+\mathcal{A}\wedge \mathcal{A},$$

(17)

so that $\mathcal{F}=-ig{F}_{\mu \nu }^a{T}_a\frac{1}{2}d{x}^{\mu }\wedge d{x}^{\nu }$ in components. The forms $\mathcal{A},\mathcal{F}$ are used throughout the descent discussion, in place of the component potential $A_{\mu }^a$, wherever the matrix-valued form convention is intended.

4.2. The Ordinary Anomaly Survives in the Interior

The windowed Ward identity (9) gives this requirement a geometric reading without changing its content. In the boundary layer ${\mathcal{B}}_{ϵ}$, the local Ward variation contains a boundary-layer contact contribution of the schematic form:

$${\left.{\delta }_{\alpha }{\mathrm{\Gamma }}_{\diamond }\left[A\right]\right|}_{{\mathcal{B}}_{ϵ}}\supset {\int }_{{\mathcal{B}}_{ϵ}}{d}^{4}x\sqrt{-g}({\nabla }_{\mu }\diamond )\langle J_a^{\mu }\rangle {\alpha }^a,$$

(18)

where ${\alpha }^a\left(x\right)$ is the gauge parameter and $\langle J_a^{\mu }\rangle$ is the expectation value of the current in the gauge-field background. This is one term in the local Ward variation, not the complete gauge variation of the gauge-fixed effective action and not, by itself, the consistent anomaly. For Equation (9) to hold as an operator equation in the quantized theory, the gauge variation of the effective action must be controlled by the consistent anomaly.

Gauge invariance of the anomalous effective action is governed by the Wess–Zumino consistency condition [18]. Define the anomaly functional $\mathcal{A}(\alpha ;A)\equiv {\delta }_{\alpha }\mathrm{\Gamma }\left[A\right]$. The integrability of the anomalous Ward identity is the statement that $\mathcal{A}$ is consistent under successive gauge variations,

$${\delta }_{{\alpha }_1}\mathcal{A}({\alpha }_2;A)-{\delta }_{{\alpha }_2}\mathcal{A}({\alpha }_1;A)=\mathcal{A}([{\alpha }_1,{\alpha }_2];A),$$

(19)

or, in BRST notation with the ghost $c^a$ replacing the parameter ${\alpha }^a$,

$$s\mathcal{A}=0,\mathcal{A}\sim \mathcal{A}+s\Xi +d{\rm Y},$$

(20)

so that the anomaly is a ghost-number-one element of the local BRST cohomology $H^1\left(s\right|d)$ [19,20].

Lemma 1

(Interior anomaly cancellation). For the renormalized windowed Slavnov–Taylor identity to hold for ghost support contained in the window interior, the representation content must satisfy

$$d_{abc}^{\left(\mathrm{tot}\right)}\equiv \sum _f{\chi }_f{d}_{abc}^{\left(R_f\right)}=0.$$

(21)

Proof 

(Proof). Let $U⋐{\mathsf{\Omega }}_{\mathrm{loc}}$ be an open set contained in the window interior, so that $\diamond =1$ and $d\diamond =0$ on U. Choose the BRST ghost $c^a$ with compact support in U. On that support, the windowed action and its local BRST complex reduce exactly to those of the ordinary chiral gauge theory, so a nontrivial element of the local BRST cohomology $H^1\left(s\right|d)$ survives unchanged in the windowed theory [20]. The local Slavnov–Taylor identity can hold for arbitrary interior ghost support only if the ordinary perturbative anomaly coefficient vanishes, which is Equation (21). Terms containing $d\diamond$ are supported only in ${\mathcal{B}}_{ϵ}$ and cannot cancel an anomaly tested by a ghost supported entirely in U. □

Remark 1.

The window leaves the ordinary perturbative anomaly obstruction unchanged in the interior. The boundary layer can carry additional representative-level contact terms, but it cannot relax the ordinary anomaly-cancellation condition. The anomaly is therefore not created or concentrated at ${\mathcal{B}}_{ϵ}$. For the observed constant-coefficient fermion content, $d_{abc}^{\left(\mathrm{tot}\right)}$ is a representation-theoretic number, independent of spacetime point, and the interior test above is the natural place to read it off, since the anomaly density is present wherever $\diamond =1$.

The condition $d_{abc}^{\left(\mathrm{tot}\right)}=0$ controls perturbative local anomalies. A complete BRST quantization also requires the absence of global gauge anomalies [21], since a global anomaly makes the fermion determinant multivalued on gauge-orbit space even when the local anomaly polynomial vanishes. This is an obstruction to single-valuedness on the full gauge-orbit space, not a local density concentrated at ${\mathcal{B}}_{ϵ}$, and it must be checked separately from the local descent. For the observed SM matter content, the number of left-handed $\mathrm{SU}{\left(2\right)}_L$ fundamental doublets per generation, one lepton doublet plus three color copies of the quark doublet, totals four; summed over three generations this is twelve, an even number, so the Witten $\mathrm{SU}\left(2\right)$ global anomaly is absent.

4.3. Boundary Transgression of Anomaly Representatives

The boundary layer carries window-dependent terms in the anomaly representative. Their support and sharp-window limit can be stated cleanly. Their complete BRST-consistent form requires a calculation not performed here, as the following makes explicit.

Let the six-form anomaly polynomial of the chiral content be, up to an overall convention-dependent normalization,

$$I_6\left(R\right)={\displaystyle \frac{1}{6}}{d}_{abc}^{\left(\mathrm{tot}\right)}{\mathcal{F}}^a\wedge {\mathcal{F}}^b\wedge {\mathcal{F}}^c,$$

(22)

where the components ${\mathcal{F}}^a$ are those of Equation (17). This polynomial contains only the pure non-Abelian cubic gauge term. Mixed gauge-gravitational contributions, which involve curvature factors $\mathrm{tr}(R\wedge R)$, are not displayed and are treated as separate conditions. The canonical Stora–Zumino descent of $I_6$ in four dimensions is

$$I_6=d{Q}_5^{\left(0\right)},s{Q}_5^{\left(0\right)}+d{Q}_4^{\left(1\right)}=0,s{Q}_4^{\left(1\right)}+d{Q}_3^{\left(2\right)}=0,$$

(23)

where the superscript is the ghost number and the subscript is the form degree [19,22,23]. The consistent anomaly of the unwindowed theory is $\mathcal{A}(c,A)=2\pi i{\int }_{M_4}{Q}_4^{\left(1\right)}$, a ghost-number-one cocycle. The three-form $Q_3^{\left(2\right)}$ immediately below it carries ghost number two, not one.

Proposition 2

(Support of window-dependent anomaly terms). Let $\diamond \in C^{\infty }\left(M_4\right)$ be gauge-singlet and BRST-inert, and let $Q_5^{\left(0\right)},Q_4^{\left(1\right)},Q_3^{\left(2\right)}$ be the canonical descent forms of Equation (23). Consider the window-weighted four-form $\diamond Q_4^{\left(1\right)}$. Then:

(i)

wherever $\diamond =1$, the ordinary consistent anomaly $Q_4^{\left(1\right)}$ is recovered exactly;

(ii)

the window-weighted four-form is not, by itself, BRST closed when $d\diamond \ne 0$. Applying the BRST operator and using $s\diamond =0$ and Equation (23) gives, up to an outer-boundary term,

$$s{\int }_{M_4}\diamond Q_4^{\left(1\right)}={\int }_{M_4}d\diamond \wedge Q_3^{\left(2\right)},$$

(24)

with $Q_3^{\left(2\right)}$ of ghost number two and $d\diamond \wedge Q_3^{\left(2\right)}$ supported on ${\mathcal{B}}_{ϵ}$;

(iii)

in a regular sharp-window limit $\diamond \to \Theta \left(f\right)$, so that $d\diamond \to \delta \left(f\right)df$, any $d\diamond$-weighted boundary term becomes a codimension-one surface term over $\Sigma =\{f=0\}$.

A BRST-consistent completion, if it exists within the chosen boundary field content, would take the form of additional ghost-number-one boundary functionals ${\mathcal{B}}_{\diamond }^{\left(1\right)}$ satisfying

$$s{\mathcal{B}}_{\diamond }^{\left(1\right)}=-{\int }_{M_4}d\diamond \wedge Q_3^{\left(2\right)}$$

(25)

up to outer-boundary terms. Whether such functionals exist within the chosen boundary field content, and, if they do, their explicit form and coefficients, must be determined by the boundary-layer BV/BRST calculation deferred to subsequent work.

Proof 

(Proof). Part (i) is immediate, since $\diamond =1$ and $d\diamond =0$ reduce $\diamond Q_4^{\left(1\right)}$ to $Q_4^{\left(1\right)}$. For part (ii), apply s and use that ⋄ is BRST-inert: $s(\diamond Q_4^{\left(1\right)})=\diamond s{Q}_4^{\left(1\right)}=-\diamond d{Q}_3^{\left(2\right)}$ by Equation (23). Integrating over $M_4$ and using ${\int }_{M_4}\diamond d{Q}_3^{\left(2\right)}={\int }_{M_4}d(\diamond Q_3^{\left(2\right)})-{\int }_{M_4}d\diamond \wedge Q_3^{\left(2\right)}$, the total derivative gives an outer-boundary term, leaving Equation (24). The form $Q_3^{\left(2\right)}$ has ghost number two, and the product $d\diamond \wedge Q_3^{\left(2\right)}$ is supported where $d\diamond$ is nonzero, namely on ${\mathcal{B}}_{ϵ}$. For part (iii), distributionally $d\diamond \to \delta \left(f\right)df$ as $\diamond \to \Theta \left(f\right)$, so any $d\diamond$-weighted integral collapses to an integral over the level set $\Sigma =\{f=0\}$. □

Equation (24) provides the decisive point. A window with $d\diamond \ne 0$ multiplied into the consistent anomaly does not produce a BRST cocycle, because the descent below the consistent anomaly carries ghost number two. The boundary layer therefore supports a window-dependent transgression term at the level of representatives, and in the sharp-window limit that term is surface-supported, in the manner of an inflow-like boundary contribution. It is not, without the deferred completion, the canonical anomaly-inflow surface term of the Callan–Harvey mechanism [24], which arises from a physical higher-dimensional Chern–Simons sector that is absent here. The density $c^a{d}_{abc}^{\left(\mathrm{tot}\right)}{\mathcal{F}}^b\wedge {\mathcal{F}}^c$ is moreover the covariant-form schematic; the consistent anomaly representative differs from it by the Bardeen–Zumino local polynomial and contains additional connection-dependent terms [19,25]. Because ⋄ is BRST-inert, none of these representative-level boundary terms can trivialize a nonzero element of $H^1\left(s\right|d)$. Vanishing of the ordinary perturbative, mixed, and global anomaly coefficients removes the corresponding standard anomaly obstructions. Proposition 2 does not by itself establish the complete all-order boundary-layer Slavnov–Taylor completion, whose local boundary functionals are deferred to subsequent work.

4.4. Which Representations Are Anomaly-Free

For a compact gauge group G in four dimensions, the cubic index of Equation (15) is a group-theoretic invariant [26,27]. The relevant criterion is whether the Lie algebra admits an independent symmetric cubic invariant in adjoint indices. A representation of complex type, equivalently a non-self-conjugate representation, is necessary for a nonzero perturbative cubic gauge anomaly, but it is not sufficient. The simple compact algebras fall into a short list:

• $\mathfrak{su}\left(N\right)$ for $N\ge 3$ (the $A_{N-1}$ series, including the accidental isomorphism $\mathfrak{so}\left(6\right)\cong \mathfrak{su}\left(4\right)$): an independent cubic adjoint invariant exists; $d_{abc}^{\left(R\right)}$ can be nonzero for representations of complex type, and in particular it is nonzero for the fundamental $\mathbf{N}$, while it vanishes identically for real and pseudoreal representations. These are the only simple algebras with a nonzero pure cubic gauge anomaly [25,27].
• $\mathfrak{su}\left(2\right)$: no independent cubic invariant, so the perturbative cubic anomaly vanishes for every representation. Its representations are real or pseudoreal. Pseudoreal representations can still participate in the global $\mathrm{SU}\left(2\right)$ anomaly [21], so vanishing of the perturbative cubic coefficient does not by itself imply global anomaly freedom.
• $\mathfrak{so}\left(N\right)$ for $N\ge 5$ and $N\ne 6$, and the compact symplectic algebras $\mathfrak{sp}\left(N\right)$: no independent cubic adjoint invariant, so $d_{abc}^{\left(R\right)}=0$ for every representation [26].
• The exceptional algebras ${\mathfrak{g}}_2,{\mathfrak{f}}_4,{\mathfrak{e}}_6,{\mathfrak{e}}_7,{\mathfrak{e}}_8$: no independent cubic adjoint invariant, so $d_{abc}^{\left(R\right)}=0$ for every representation, even where some representations are complex. The algebra ${\mathfrak{e}}_6$ is the instructive case: its $\mathbf{27}$ is complex, yet the pure $E_6$ gauge anomaly vanishes, since the familiar cubic invariant of $E_6$ lives in the matter indices of ${\mathbf{27}}^{\otimes 3}$, not in the adjoint indices that carry the gauge anomaly [25,27].

The Abelian factor $\mathfrak{u}\left(1\right)$ is not simple and is treated separately: its cubic anomaly coefficient is $d_{\mathrm{U}\left(1\right)}^{\left(R\right)}\propto {\sum }_f{Q}_f^3$, where the $Q_f$ are the charges, and mixed anomalies involving $\mathrm{U}\left(1\right)$ factors are separate conditions.

For the SM matter content, three generations of quarks and leptons with the SM hypercharge assignments, the cubic condition ${\sum }_f{Y}_f^3=0$, the mixed gravitational condition ${\sum }_f{Y}_f=0$, and the mixed $\mathrm{SU}{\left(3\right)}^2\mathrm{U}\left(1\right)$ and $\mathrm{SU}{\left(2\right)}^2\mathrm{U}\left(1\right)$ conditions all hold [10,11]. The point developed in Section 6 is that the SM gauge group is the minimal solution to these conditions for the observed matter content, under a named minimality criterion.

5. Gribov Geometry and Global Gauge Data on Bounded Domains

5.1. Euclidean Gauge-Fixed Setup

The Gribov analysis is a Euclidean, gauge-fixed, operator-domain statement. It is performed after Wick rotation on a bounded Euclidean domain, in Landau-type gauge ${\partial }_{\mu }{A}^{\mu }=0$, with boundary conditions that make the ghost operator self-adjoint and elliptic. The precise operator domain is part of the gauge-fixing prescription. Positivity, the lowest eigenvalue, and convexity of the Gribov region are Euclidean spectral notions, and they are stated only after this Wick rotation, not for the Lorentzian oscillatory integral of Equation (12).

Two operators must be kept distinct, because the corpus gates the ghost sector with the window. The orbit-map Faddeev–Popov operator

$$M\left[A\right]={\left.{\displaystyle \frac{\delta f\left[A^g\right]}{\delta \alpha }}\right|}_{\alpha =0}=-{\partial }_{\mu }{D}^{\mu }\left[A\right]$$

(26)

determines infinitesimal gauge copies and acts in the ordinary $L^2$ inner product on the domain. The window-weighted ghost Hessian comes from the windowed ghost term $S_{\mathrm{gh},\diamond }=\int d^{4}x\diamond {\partial }_{\mu }{\overline{c}}^a{D}^{\mu }{c}^a$; integrating by parts gives the operator

$$M_{\diamond }=-{\partial }_{\mu }(\diamond D^{\mu }\left[A\right])$$

(27)

in the ordinary $L^2$ pairing. In the ⋄-weighted pairing, the same ghost term is represented, where $\diamond >0$, by

$${\tilde{M}}_{\diamond }={\diamond }^{-1}{M}_{\diamond }=-{\diamond }^{-1}{\partial }_{\mu }(\diamond D^{\mu }\left[A\right])=M\left[A\right]-({\partial }_{\mu }ln\diamond )D^{\mu }\left[A\right].$$

(28)

Neither $M_{\diamond }$ nor ${\tilde{M}}_{\diamond }$ is identical to the orbit-map operator $M\left[A\right]$ on ${\mathcal{B}}_{ϵ}$; the three operators coincide where $\diamond =1$. The local Gribov statements below refer to the orbit-map operator $M\left[A\right]$ in the ordinary inner product, whose kernel characterizes infinitesimal gauge copies and whose positivity defines the first Gribov region.

5.2. Local Gribov Geometry

In non-Abelian gauge theories, the Faddeev–Popov procedure is complicated by gauge-equivalent configurations that satisfy the same gauge condition, the Gribov copies [28,29]. For a compact non-Abelian group on a sphere, no global continuous gauge-fixing section exists, so the ambiguity cannot be removed by any single continuous choice [30]. The Gribov region ${\mathsf{\Omega }}_G$ is the set of gauge-fixed configurations on which the orbit-map Faddeev–Popov operator $M\left[A\right]$ is positive [31]; it is a region of gauge-fixed configuration space, not of orbit space.

Proposition 3

(Local Gribov geometry on bounded domains). On a bounded Euclidean domain ${\mathsf{\Omega }}_{\mathrm{loc}}\subset {\mathbb{R}}^4$ with smooth boundary, in Landau-type gauge with self-adjoint elliptic ghost boundary conditions, and assuming that the standard Dell’Antonio–Zwanziger hypotheses extend to the specified bounded elliptic operator domain, the first Gribov region ${\mathsf{\Omega }}_G$ for a compact non-Abelian gauge group G is a bounded convex region of gauge-fixed configuration space [32,33], open under strict positivity of $M\left[A\right]$ and closed when the first Gribov horizon is included. This geometry depends on the Lie algebra and the local operator $M\left[A\right]$, not on the global form of G: within the same local bundle sector and with identical adjoint boundary conditions, a group G and a finite central quotient $G/\mathrm{\Gamma }$ (for instance $\mathrm{SU}\left(N\right)$ and $\mathrm{SU}\left(N\right)/{\mathbb{Z}}_N$) share the same local adjoint Faddeev–Popov operator and the same first Gribov region. Restriction to ${\mathsf{\Omega }}_G$ removes configurations beyond the first horizon but does not by itself select a unique representative.

Proof 

(Proof). The statement follows, under the stated assumption, from the analysis of Dell’Antonio and Zwanziger [32] applied to a bounded Euclidean domain. In Landau-type gauge the orbit-map operator $M\left[A\right]=-{\partial }_{\mu }{D}^{\mu }\left[A\right]$ is elliptic and, with the stated boundary conditions, self-adjoint; its first region of positivity is bounded by the first Gribov horizon, where the lowest eigenvalue vanishes, and is convex. The operator $M\left[A\right]$ depends only on the adjoint action of the Lie algebra, which is identical for a group and its finite central quotient, so the local geometry is insensitive to the global form. □

The fundamental modular region, the set of absolute minima of the gauge-fixing functional $\parallel A^g{\parallel }^2$ along each orbit, gives a representative almost everywhere, subject to boundary identifications and degeneracies on its boundary [29]. Unqualified uniqueness does not hold; the boundary of the modular region carries identified configurations.

Abelian gauge factors, including $\mathrm{U}\left(1\right)$, are exempt from this local obstruction. Because Abelian gauge transformations act linearly on the gauge field, the Abelian Faddeev–Popov operator is field-independent, $M^{\mathrm{U}\left(1\right)}=-{\partial }^2$, and develops no Gribov horizon [29]. The non-Abelian Gribov-copy obstruction therefore applies only to the non-Abelian simple factors. The exemption is from the field-dependent Gribov obstruction alone; it is not a claim that Abelian gauge fixing on a bounded domain is automatic. Constant zero modes, harmonic representatives on topologically nontrivial domains, large gauge transformations, and boundary edge modes all remain present for $\mathrm{U}\left(1\right)$ factors and must be fixed by admissible boundary conditions or retained as physical boundary data.

5.3. Global Sector Specification

The local operator $M\left[A\right]$ does not fix the global data of the gauge-fixed theory. For any gauge group on a bounded domain, the construction also requires a principal-bundle sector, the allowed boundary gauge transformations, a line-operator spectrum or equivalent global-observable specification, possible topological terms, and a Gribov/modular-region prescription. These data depend on the global form of G, controlled by ${\pi }_1\left(G\right)$ and the finite central quotient $\mathrm{\Gamma }$ of Equation (13), and on the topology of the domain and its boundary conditions [34,35].

Proposition 4

(Global data on bounded domains). On a bounded domain ${\mathsf{\Omega }}_{\mathrm{loc}}$ with the based boundary condition of Equation (8), the finite-domain gauge-fixed theory is not fully specified by the local Faddeev–Popov operator alone. The allowed bundles, the discrete center identifications, the genuine line-operator content, the continuous theta parameters, and, where permitted, discrete theta-like data all depend on the global form of G [34,35]. Moreover, even for a simply connected G, the based maps on a four-ball carry large gauge classes

$$[B^4,\partial B^4;G,e]\cong {\pi }_4\left(G\right),$$

(29)

so simple connectedness does not remove all large gauge transformations. The relevant homotopy classes and bundle sectors must be specified as part of the gauge-fixing data; they are invisible to a purely local slice.

Proof 

(Proof). The bundle, center, and line-operator data are global invariants of the gauge-fixed configuration space and are determined by the global form of G, not by the local gauge condition [34,35]. For the homotopy statement, based maps $B^4\to G$ sending $\partial B^4$ to the identity descend to based maps $S^4\to G$, whose homotopy classes are ${\pi }_4\left(G\right)$. For $G=\mathrm{SU}\left(2\right)$ this group is ${\mathbb{Z}}_2$, which is exactly the class detected by the Witten global anomaly. Hence simple-connectedness controls ${\pi }_1$ but not ${\pi }_4$, and the large gauge sectors remain part of the global specification. □

Equation (29) is the reason the Witten $\mathrm{SU}\left(2\right)$ anomaly remains relevant on a bounded simply connected domain: the obstruction it detects lives in ${\pi }_4$, which is not removed by based boundary conditions. The general lesson is that the simply connected cover is a convenient reference presentation, not the only admissible global form; finite central quotients are admissible once their associated global-sector data are supplied [34,35,36].

5.4. Finite-Size Spectral Estimate

Remark 2

(Finite-size estimate under elliptic boundary conditions). Under Dirichlet or comparable elliptic ghost boundary conditions, the lowest eigenvalue of the orbit-map operator $M\left[A\right]$ at zero gauge field on a domain of characteristic size L scales as ${\lambda }_{min}^{\left(0\right)}\sim L^{-2}$ [31]. Perturbatively, the gauge-field-dependent part of $M\left[A\right]$ shifts this eigenvalue by $O(g\parallel A\parallel /L)$, so the first Gribov horizon, where the lowest eigenvalue vanishes, lies at field amplitudes of order $\parallel A\parallel \sim {\left(gL\right)}^{-1}$. This inferred scale is norm-, geometry-, and gauge-dependent and is recorded only as a heuristic finite-size estimate, not as a sharp statement about the gauge-fixed measure.

6. Finite-Domain Gauge Admissibility and the Standard Model Application

6.1. Statement of Constraints

Combining Section 3, Section 4 and Section 5, the finite-domain gauge-fixed construction imposes three necessary structural conditions on the gauge data of a chiral gauge theory on a localization-bounded domain:

(C₁)

Under the normalized local group-averaging prescription of Proposition 1, G is compact.

(C₂)

The representation content is anomaly-free: the pure and mixed-gauge perturbative coefficients vanish ($d_{abc}^{\left(\mathrm{tot}\right)}=0$, with indices ranging over the full reductive algebra), the mixed gauge–gravitational coefficients vanish, and global gauge anomalies are absent, by the interior BRST test of Lemma 1 and the separate single-valuedness requirement on the fermion determinant.

(C₃)

The global form of the non-Abelian gauge group, the allowed bundle and line-operator sectors, the boundary gauge group, and the Gribov/modular-region prescription must be specified. The simply connected cover provides a convenient reference presentation, but it is not the only admissible global form; finite central quotients are admissible when their associated global-sector data are supplied. Abelian $\mathrm{U}\left(1\right)$ factors carry no field-dependent Gribov horizon and are exempt from the non-Abelian part of this condition after ordinary zero modes are fixed.

These conditions do not share a single boundary-layer locus. Compactness is a whole-domain regulator condition; the local anomaly is already testable in the interior; the global form and the Gribov structure concern the full gauge-fixed orbit space.

These constraints are not specific to the SM. They apply to any chiral gauge theory placed on a bounded domain with the windowed construction.

6.2. Finite-Domain Gauge-Admissibility Theorem

The three constraints can be promoted to a theorem that identifies them as necessary consistency conditions on the gauge data of any chiral gauge theory quantized across the localization boundary, under explicitly named assumptions.

Theorem 1

(Finite-domain gauge admissibility). Let ${\mathsf{\Omega }}_{\mathrm{loc}}\subset {\mathbb{R}}^4$ be a bounded smooth localization domain and let $\diamond \in C^{\infty }\left({\mathbb{R}}^4\right)$ be a fixed gauge-singlet, BRST-inert window with $0\le \diamond \le 1$, $\diamond =1$ on ${\mathsf{\Omega }}_{\mathrm{core}}$, and ${\nabla }_{\mu }\diamond \ne 0$ only in the boundary layer ${\mathcal{B}}_{ϵ}$, equipped with the matching conditions of the windowed-action construction. Let G be the internal gauge group of a four-dimensional chiral gauge theory with matter representation content R, and assume:

(A1)

the boundary gauge redundancy is the based gauge group of Equation (8), or an explicit edge-sector replacement is supplied;

(A2)

the regulated local group averaging uses a normalized invariant Haar measure on each local group factor, as in Proposition 1;

(A3)

the gauge-fixed Euclidean operator problem uses a specified Landau-type gauge with elliptic, self-adjoint ghost boundary conditions;

(A4)

the renormalized effective action and its local Slavnov–Taylor breaking functional are well defined, so that the breaking can be classified by local BRST cohomology.

Then the following are necessary:

(i)

G is compact, under assumption [ass:A2](A2);

(ii)

the ordinary local perturbative gauge-anomaly obstructions of the supplied chiral matter vanish. The pure and mixed-gauge cubic coefficients satisfy

$$d_{abc}^{\left(\mathrm{tot}\right)}\equiv \sum _f{\chi }_f{d}_{abc}^{\left(R_f\right)}=0,{\chi }_f=+1\left(\mathrm{L}\right),{\chi }_f=-1\left(\mathrm{R}\right),$$

(30)

with $d_{abc}^{\left(R_f\right)}$ as in Equation (15) and the indices $a,b,c$ ranging over the full reductive gauge algebra; the mixed gauge–gravitational coefficients vanish,

$$\sum _f{\chi }_f{\mathrm{Tr}}_{R_f}\left(T_a\right)=0,$$

(31)

and global gauge anomalies are absent;

(iii)

the global form, bundle sectors, line-operator content, boundary gauge group, and Gribov/modular-region prescription are specified, as in Propositions 3 and 4; Abelian $\mathrm{U}\left(1\right)$ factors are exempt from the non-Abelian part after ordinary zero modes are fixed.

These conditions are necessary, not sufficient: they do not by themselves construct a nonperturbative continuum quantum field theory.

Proof 

(Proof). Because ⋄ is a gauge singlet and BRST-inert, it introduces no new gauge charge, no new ghost number, and no new representation of G. The classical Noether identity of the windowed theory is ${\mathcal{D}}_{\mu }(\diamond J_a^{\mu })=0$, Equation (9), with the ordinary covariant Ward identity recovered in the interior, where ${\nabla }_{\mu }\diamond =0$.

For compactness, under assumption [ass:A2](A2), Proposition 1 applies: the regulated interior factor $G^N$ requires a normalized invariant Haar measure on each copy of G, which exists only for compact G. This gives [thm-fd:i](i).

For anomalies, under assumption [ass:A4](A4), the quantum Ward identity is replaced by a Slavnov–Taylor identity for the renormalized effective action ${\mathrm{\Gamma }}_{\diamond }$, with anomaly functional satisfying $s{\mathcal{A}}_{\diamond }=0$. By Lemma 1, choosing ghost support in the window interior reduces the local BRST complex exactly to that of the ordinary chiral theory, so the renormalized Slavnov–Taylor identity holds for arbitrary interior ghost support only if Equation (30) holds. Because the indices $a,b,c$ in that equation range over the full reductive gauge algebra, the same interior-support argument covers the mixed gauge anomalies involving the Abelian factors as components of the same tensor. Applying the analogous interior test to the gauge–gravitational triangle gives the mixed gauge–gravitational condition ${\sum }_f{\chi }_f{\mathrm{Tr}}_{R_f}\left(T_a\right)=0$ stated above. By Proposition 2, $d\diamond$-supported terms are confined to ${\mathcal{B}}_{ϵ}$ and cannot cancel an interior-supported anomaly, and they do not, by themselves, form a BRST cocycle. Global gauge anomalies must also be absent, since otherwise the fermion determinant is not single-valued on gauge-orbit space and the BRST quantization is globally inconsistent. This is a separate condition on the orbit space, not a local density. Together these give [thm-fd:ii](ii).

For global topology and Gribov structure, under assumption [ass:A3](A3) and by Propositions 3 and 4, the gauge-fixed orbit space requires a specified global form, bundle and line-operator sectors, boundary gauge group, and Gribov/modular-region prescription; the local Faddeev–Popov slice does not fix these. For Abelian factors the gauge transformation is linear, $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\lambda$, the Faddeev–Popov operator is field-independent, and the non-Abelian Gribov obstruction is absent after ordinary zero modes are fixed. This gives [thm-fd:iii](iii) and completes the proof. □

6.3. Matter Admissibility and Absence of Window-Based Selection

The finite-domain gauge admissibility theorem constrains the gauge group and the allowed anomaly class of the chiral matter, but it does not by itself determine the observed quark-lepton spectrum. The reason is that a gauge-singlet, BRST-inert scalar window supplies no representation-selecting functional.

Theorem 2

(Matter anomaly admissibility and absence of window-based selection). Let G be a compact internal gauge group satisfying the regulator, boundary, and global-sector assumptions of Theorem 1, and let the matter fields be ordinary finite-component local fields in a finite set of finite-dimensional continuous representations $\left\{R_f\right\}$ with chirality signs ${\chi }_f=\pm 1$. Consider

$$S_{\diamond ,\psi }=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\sum _f{\overline{\psi }}_{f}i{\gamma }^{\mu }{\mathcal{D}}_{\mu }^{\left(R_f\right)}{\psi }_f,$$

(32)

with ${\delta }_g\diamond =0$ and ${\delta }_{\mathrm{BRST}}\diamond =0$. If the renormalized windowed Slavnov–Taylor identity holds for ghost support in the window interior, and the fermion determinant is single-valued on the specified gauge-orbit space, then:

(i)

compactness of G allows each $R_f$ to be chosen unitary, by averaging an invariant inner product;

(ii)

all perturbative gauge anomalies vanish,

$$\sum _f{\chi }_f{d}_{abc}^{\left(R_f\right)}=0;$$

(33)

(iii)

all mixed gauge, mixed gauge-gravitational, and global anomalies are absent.

The scalar window supplies no additional functional that selects among otherwise admissible representation sets. It does not distinguish two representation sets beyond the ordinary dynamical and anomaly data already carried by their currents, couplings, and functional determinants.

Proof 

(Proof). Finite-dimensional local multiplets are an input of the ordinary finite-component field theory. Once G is compact, any finite-dimensional continuous representation can be made unitary by averaging an inner product over G with normalized Haar measure, which is the standard unitarization construction; the window plays no role in this step, giving [thm-matter:i](i). Because ⋄ is gauge-singlet and BRST-inert, it carries no internal gauge index, ghost number, or representation label. The total current is the sum of fermion bilinears,

$$J_a^{\mu }=\sum _f{\overline{\psi }}_f{\gamma }^{\mu }{T}_a^{\left(R_f\right)}{\psi }_f,$$

(34)

and, as in Equation (9), the windowed identity is covariant, ${\mathcal{D}}_{\mu }(\diamond J_a^{\mu })=0$. Representation data enter only through the ordinary generators $T_a^{\left(R_f\right)}$ and their quantum anomaly coefficients. Lemma 1 gives the pure perturbative local condition in Equation (33). The mixed local conditions follow from the corresponding interior anomaly polynomials. Global anomalies are excluded separately by Theorem 1, through the requirement that the fermion determinant be single-valued on gauge-orbit space. This proves [thm-matter:ii](ii) and [thm-matter:iii](iii).

These conditions do not determine a unique spectrum. The scalar profile $\diamond \left(x\right)$ depends on no $R_f$ and contains no operator that counts generations, forbids gauge singlets, or distinguishes one anomaly-free chiral set from another. Sterile singlets ${(1,1)}_0$ carry no SM gauge charge. Vectorlike pairs $R\oplus \overline{R}$ have canceling anomaly contributions. Complete SM generations,

$${(3,2)}_{1/6}\oplus {(\overline{3},1)}_{-2/3}\oplus {(\overline{3},1)}_{1/3}\oplus {(1,2)}_{-1/2}\oplus {(1,1)}_1,$$

(35)

are anomaly-free in the left-handed Weyl convention [10,11], and each contains four left-handed $\mathrm{SU}{\left(2\right)}_L$ doublets once color multiplicity is counted, so the Witten anomaly is absent for any integer number of complete generations [21]. The windowed axioms therefore permit one, two, three, or more complete generations unless an additional selection principle is supplied. □

Remark 3.

Theorem 2 is a statement against overclaiming, not a physical indistinguishability claim. Two anomaly-free spectra are not physically indistinguishable: their beta functions, correlators, spectra, and interactions can differ. A derivation of the generation number or the observed quark-lepton representations would require an additional input, such as a family index theorem, a unification representation, a spectral-triple structure, or a new boundary degree of freedom beyond the gauge-singlet window. No such selector is present in the fixed scalar-window axioms used here; this is the precise content of the statement, and it does not assert that every conceivable selector must modify interior dynamics.

6.4. Classification of Admissible Minimal Gauge Groups

Under constraints [con:C1]($C_1$) and [con:C3]($C_3$), the admissible connected gauge groups are the finite central quotients of Equation (13),

$$G\cong {\displaystyle \frac{(G_1\times G_2\times \dots \times G_k)\times \mathrm{U}{\left(1\right)}^m}{\mathrm{\Gamma }}},$$

(36)

where each $G_i$ is a compact simply connected simple Lie group from the list (14), so that $G_{\mathrm{sc}}=G_1\times \dots \times G_k$, and $\mathrm{\Gamma }$ is a finite central subgroup whose global-sector data are supplied as in Section 5.3. Constraint [con:C2]($C_2$) then selects which algebras admit anomaly-free fermion representations.

6.5. Standard Model Anomaly Conditions

Take the observed matter content in the left-handed Weyl convention, in which each right-handed field is represented by its left-handed charge conjugate: one generation is $Q_L\sim {(3,2)}_{1/6}$, $u^c\sim {(\overline{3},1)}_{-2/3}$, $d^c\sim {(\overline{3},1)}_{1/3}$, $L_L\sim {(1,2)}_{-1/2}$, and $e^c\sim {(1,1)}_1$. Anomaly sums use these left-handed charge-conjugate fields $u^c,d^c,e^c$; the Yukawa section of Section 6.8 returns to the physical right-handed fields $u_R,d_R,e_R$. Each Weyl field is counted with its color and weak multiplicity, so per generation there are six fields in $Q_L$, three in each of $u^c$ and $d^c$, two in $L_L$, and one in $e^c$. The perturbative triangle and mixed gauge-gravitational anomalies must cancel [10,11]:

$$\begin{array}{cc}\hfill {\left[\mathrm{U}{\left(1\right)}_Y\right]}^3:& 6Y_Q^3+3Y_{u^c}^3+3Y_{d^c}^3+2Y_L^3+Y_{e^c}^3=0,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill \mathrm{U}{\left(1\right)}_Y{]\left[\mathrm{gravity}\right]}^2:& 6Y_Q+3Y_{u^c}+3Y_{d^c}+2Y_L+Y_{e^c}=0,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill \mathrm{SU}{\left(3\right)}_c{]}^2\mathrm{U}{\left(1\right)}_Y:& 2Y_Q+Y_{u^c}+Y_{d^c}=0,\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \mathrm{SU}{\left(2\right)}_L{]}^2\mathrm{U}{\left(1\right)}_Y:& 3Y_Q+Y_L=0,\hfill \end{array}$$

(40)

where the common Dynkin index $T\left(\mathrm{fund}\right)=\frac{1}{2}$ has been factored out of Equations (39) and (40). The purely non-Abelian cubic anomalies do not appear as independent constraints. The ${\left[\mathrm{SU}{\left(2\right)}_L\right]}^3$ coefficient vanishes identically, since $\mathrm{SU}\left(2\right)$ has no independent cubic invariant. The ${\left[\mathrm{SU}{\left(3\right)}_c\right]}^3$ coefficient cancels within each generation, since the color-triplet quark doublet contributes $+2$ and the two color-anti-triplet singlets contribute $-1$ each. The mixed $\left[\mathrm{SU}{\left(3\right)}_c\right]{\left[\mathrm{gravity}\right]}^2$ and $\left[\mathrm{SU}{\left(2\right)}_L\right]{\left[\mathrm{gravity}\right]}^2$ anomalies vanish because $\mathrm{tr}{T}_a=0$ for a simple group. One condition remains, and it is topological rather than diagrammatic. Absence of the global $\mathrm{SU}{\left(2\right)}_L$ anomaly requires an even number of fundamental $\mathrm{SU}{\left(2\right)}_L$ doublets [21],

$$\mathrm{SU}{\left(2\right)}_L(\mathrm{Witten}\mathrm{global}):n_{\mathrm{doublets}}\equiv 0(mod2),$$

(41)

where the simple mod-two count applies to fundamental doublets; more general $\mathrm{SU}\left(2\right)$ representations require the appropriate mod-two index. Because ${\pi }_4\left(\mathrm{SU}\left(2\right)\right)={\mathbb{Z}}_2$, an odd doublet count makes the fermion determinant double-valued, independently of the perturbative coefficients. For the observed content each generation contributes four doublets, so three generations give twelve and the condition holds.

The four perturbative equations do not, without further input, fix a single branch. Solving the linear conditions (38)–(40) gives

$$Y_L=-3Y_Q,Y_{e^c}=6Y_Q,Y_{d^c}=-2Y_Q-Y_{u^c},$$

(42)

and the cubic condition (37) then factorizes as

$$-18Y_Q(Y_{u^c}-2Y_Q)(Y_{u^c}+4Y_Q)=0.$$

(43)

The three branches are: (1) $Y_{u^c}=-4Y_Q$, hence $Y_{d^c}=2Y_Q$, the SM all-left-handed branch; (2) $Y_{u^c}=2Y_Q$, hence $Y_{d^c}=-4Y_Q$, the up/down relabeling branch; and (3) $Y_Q=0$, with $Y_L=Y_{e^c}=0$ and $Y_{d^c}=-Y_{u^c}$, a degenerate anomaly-free branch on which the doublet and lepton hypercharges vanish while the two weak-singlet quark hypercharges are opposite. The SM branch is selected by imposing a nontrivial chiral hypercharge assignment, $Y_Q\ne 0$, together with the up/down singlet identification, or equivalently by imposing the observed electric-charge pattern and Yukawa compatibility. With the conventional normalization $Y_Q=1/6$, branch (1) gives the familiar hypercharges $(Y_Q,Y_{u^c},Y_{d^c},Y_L,Y_{e^c})=(1/6,-2/3,1/3,-1/2,1)$.

6.6. Application to Observed Quark-Lepton Matter

Corollary 1

(Minimal Standard Model solution for observed quark-lepton matter). Assume the observed three-generation matter content in the following form: the quark fields occur in irreducible complex three-state internal multiplets while the leptons are singlets under that interaction, the left-handed quark and lepton fields occur in irreducible two-state multiplets while the corresponding charged fields are singlets, a nonzero overall charge normalization is fixed, and the up-type and down-type weak-singlet quark multiplets are identified as observed. Impose minimality in the named sense: family universality, no additional chiral matter introduced solely to cancel anomalies, and no additional gauge factors beyond those acting nontrivially on the observed multiplets. Then the minimal gauge structure satisfying Theorem 1 is the gauge algebra

$${\mathfrak{g}}_{\mathrm{SM}}=\mathfrak{su}{\left(3\right)}_c\oplus \mathfrak{su}{\left(2\right)}_L\oplus \mathfrak{u}{\left(1\right)}_Y,$$

(44)

with direct-product covering presentation $\mathrm{SU}{\left(3\right)}_c\times \mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y$, of rank four, up to the allowed finite central quotients of Section 6.7.

Proof 

(Proof). The proof applies the anomaly classification of Minahan, Ramond, and Warner [10] and Geng and Marshak [11] within the admissibility conditions of Theorem 1. Condition [thm-fd:i](i) restricts the local factors to compact Lie algebras under the stated group-averaging prescription. The observed irreducible three-state and two-state multiplets then select the local algebras $\mathfrak{su}\left(3\right)$ and $\mathfrak{su}\left(2\right)$. Their global forms are treated separately in Section 6.7.

(i)

The strong-interaction factor must accommodate an irreducible three-dimensional representation of complex type, the observed color triplet. Among compact simple Lie algebras, $\mathfrak{su}\left(3\right)$ is the unique algebra admitting a three-dimensional irreducible representation of complex type, so the complex-triplet requirement forces the color algebra to $\mathfrak{su}\left(3\right)$. Larger candidates are excluded by minimality: $\mathrm{SU}\left(4\right)\cong \mathrm{Spin}\left(6\right)$ has smallest complex-type representation the $\mathbf{4}$, so hosting the observed quarks there would require enlarged color multiplets or additional matter. The algebra $\mathfrak{su}\left(2\right)$ has no three-dimensional irreducible representation of complex type, and $\mathrm{U}\left(1\right)$ has no three-dimensional irreducible representation at all, since all its irreducibles are one-dimensional.

(ii)

The weak-interaction factor must contain an irreducible two-state representation accommodating the left-handed doublets with nontrivial chiral structure. Among compact connected non-Abelian simple groups admitting a nontrivial irreducible two-dimensional representation, the local Lie algebra is uniquely $\mathfrak{su}\left(2\right)$, whose fundamental is the pseudoreal doublet $\mathbf{2}$. Minimality therefore selects $\mathrm{SU}{\left(2\right)}_L$ as the reference covering factor.

(iii)

After $\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)$ is specified, the perturbative anomaly conditions of Equations (37)–(40) fix the hypercharge assignments up to the branch selection of Section 6.5, with the SM branch chosen by the nontrivial chiral assignment and up/down identification. A single $\mathrm{U}{\left(1\right)}_Y$ factor is both necessary, to assign distinct hypercharges to quarks and leptons, and sufficient.

The minimality condition then selects the gauge algebra of Equation (44) under the stated assumptions. The rank is $2+1+1=4$. □

Remark 4.

Corollary 1 does not derive the matter spectrum, generation number, or minimality criterion from the scalar window alone; those are taken as input, and steps [cor-sm:i](i)–(iii) reproduce the results of [10,11]. Given those inputs, the corollary derives the SM gauge algebra as the minimal finite-domain admissible solution. The windowed construction gives the admissibility conditions of Theorem 1 a finite-domain origin: they are necessary requirements for the regulated gauge-fixed quantization of the theory on a localization-bounded domain, not additional axioms. The construction supplies no new algebraic classification of the observed hypercharges; it gives a finite-domain consistency reading of why the anomaly-free classification is required.

On extra Abelian factors, the precise statement is as follows. A second $\mathrm{U}\left(1\right)$ proportional to Y is redundant, since it is a rescaling of the same charge. An independent, nontrivial, family-universal $\mathrm{U}\left(1\right)$ extension that is anomaly-free, such as gauged $B-L$, requires relaxing a named assumption: $B-L$ is anomaly-free only once right-handed neutrinos are added, introducing chiral matter beyond the observed spectrum, so it violates the no-additional-chiral-matter criterion. A family-nonuniversal extension such as $L_{\mu }-L_{\tau }$ violates family universality. Both are therefore excluded by the named minimality criterion, not by the windowed construction on structural grounds.

6.7. Standard Model Global Form

The local anomaly analysis fixes the gauge algebra and the direct-product covering presentation, but not the global form. The covering presentation $\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$ is the direct-product presentation associated with the SM gauge algebra; the term universal cover does not apply to a product retaining $\mathrm{U}\left(1\right)$, whose universal cover is $\mathbb{R}$, not $\mathrm{U}\left(1\right)$. The admissible global forms are

$$G_{\mathrm{SM},\mathrm{\Gamma }}={\displaystyle \frac{\mathrm{SU}\left(3\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)}{\mathrm{\Gamma }}},\mathrm{\Gamma }\subseteq {\mathbb{Z}}_6,$$

(45)

with $\mathrm{\Gamma }=1,{\mathbb{Z}}_2,{\mathbb{Z}}_3,{\mathbb{Z}}_6$ the standard possibilities for the observed spectrum [34,35]. The diagonal ${\mathbb{Z}}_6$ subgroup of the center ${\mathbb{Z}}_3\times {\mathbb{Z}}_2\times \mathrm{U}\left(1\right)$ acts trivially on every SM representation, which is why all four quotients are consistent with the observed matter. Which global form is realized is a choice of exactly the bundle and line-operator data of Section 5.3, and the present local anomaly analysis does not fix it.

Remark 5

(Rank under the named minimality criterion). The rank-four value is a restatement of the named minimality criterion applied to the selected minimal algebra, not an independent localization theorem. Once ${\mathfrak{g}}_{\mathrm{SM}}$ of Equation (44) is selected, $\mathrm{rank}=2+1+1=4$ follows arithmetically. Relaxing family universality or admitting anomaly-spectator matter permits anomaly-free gauge factors of higher total rank, so no prescription-independent ceiling on the rank is claimed.

6.8. Scalar-Sector Compatibility and the Minimal Higgs Doublet

The window also cannot carry electroweak charge or simulate symmetry breaking. Once the observed chiral matter and the minimal SM gauge algebra [37,38,39] are taken as input, the minimal scalar sector is fixed by ordinary renormalizable gauge invariance and by the requirement that electromagnetism remain unbroken. Throughout this subsection, representations are written in the order ${(\mathrm{SU}{\left(3\right)}_c,\mathrm{SU}{\left(2\right)}_L)}_Y$, family indices and Yukawa matrices are suppressed, and the Yukawa couplings are written with the physical right-handed fields $u_R,d_R,e_R$.

Proposition 5

(Minimal scalar-sector compatibility). Assume the gauge algebra and observed chiral matter content of Corollary 1. Require a scalar sector that is compatible with ${\delta }_{\mathrm{BRST}}\diamond =0$, generates renormalizable Yukawa masses for the observed charged fermions, and leaves exactly $\mathrm{U}{\left(1\right)}_{\mathrm{em}}$ unbroken. Then the minimal scalar representation is, up to conjugation,

$$H\sim {(1,2)}_{+1/2},$$

(46)

with

$$V\left(H\right)=-{\mu }^2{H}^{†}H+\lambda {\left(H^{†}H\right)}^2,{\mu }^2>0,\lambda >0,\langle H\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}0\\ v\end{array}\right),$$

(47)

so that

$$\mathrm{SU}{\left(2\right)}_L\times \mathrm{U}{\left(1\right)}_Y⟶\mathrm{U}{\left(1\right)}_{\mathrm{em}}.$$

(48)

The localization window does not replace or modify the Higgs mechanism [40,41,42]; it only multiplies the scalar action and localizes the corresponding Ward/Slavnov contact terms to ${\mathcal{B}}_{ϵ}$.

Proof 

(Proof). The windowed scalar-Yukawa action is

$$S_{\diamond ,H}=\int d^{4}x\sqrt{-g}\diamond \left(x\right)\left[{\left({\mathcal{D}}_{\mu }H\right)}^{†}{\mathcal{D}}^{\mu }H-V\left(H\right)-{\mathcal{L}}_Y\right],$$

(49)

with the Yukawa couplings

$${\mathcal{L}}_Y=y_d{\overline{Q}}_{L}H{d}_R+y_u{\overline{Q}}_L\tilde{H}{u}_R+y_e{\overline{L}}_{L}H{e}_R+\mathrm{h}.\mathrm{c}.,\tilde{H}=i{\sigma }^2{H}^{*},$$

(50)

where family indices and Yukawa matrices are suppressed. Because ${\delta }_{\mathrm{BRST}}\diamond =0$, BRST invariance of $S_{\diamond ,H}$ is equivalent to ordinary gauge invariance of the scalar-Yukawa density. The scalar current

$$J_{H,a}^{\mu }=i{H}^{†}{T}_a{\mathcal{D}}^{\mu }H-i{\left({\mathcal{D}}^{\mu }H\right)}^{†}{T}_{a}H$$

(51)

adds to the fermion current in the windowed Ward identity,

$${\mathcal{D}}_{\mu }\left[\diamond \left(J_{f,a}^{\mu }+J_{H,a}^{\mu }\right)\right]=0,$$

(52)

so the window supplies no scalar charge.

The Yukawa terms in Equation (50) require H to be an $\mathrm{SU}{\left(2\right)}_L$ doublet, since $Q_L$ and $L_L$ are doublets and $u_R,d_R,e_R$ are singlets. Hypercharge neutrality of each term gives, with $Y\left(Q_L\right)=1/6$, $Y\left(u_R\right)=2/3$, $Y\left(d_R\right)=-1/3$, $Y\left(L_L\right)=-1/2$, and $Y\left(e_R\right)=-1$,

$$\begin{array}{cc}\hfill -Y\left(Q_L\right)+Y\left(H\right)+Y\left(d_R\right)& =0,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill -Y\left(Q_L\right)-Y\left(H\right)+Y\left(u_R\right)& =0,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill -Y\left(L_L\right)+Y\left(H\right)+Y\left(e_R\right)& =0,\hfill \end{array}$$

(55)

and each equation gives $Y\left(H\right)=1/2$ [1,2]. With ${\mu }^2>0$ and $\lambda >0$, the potential (47) is minimized at $H^{†}H={\mu }^2/2\lambda =v^2/2$. The lower component of the vacuum has $T_3=-1/2$ and $Y=+1/2$, so the generator $Q=T_3+Y$ annihilates $\langle H\rangle$ while the orthogonal electroweak generators are broken. The unbroken subgroup is therefore $\mathrm{U}{\left(1\right)}_{\mathrm{em}}$. □

Remark 6.

Proposition 5 is a minimality result, not a no-go theorem against extended scalar sectors. Additional doublets, singlets, triplets, or other scalar representations may be introduced if their quantum numbers and couplings are specified consistently. Localization does not alter the ordinary minimal Higgs logic.

7. Discussion

7.1. What Is Established Conditionally

The finite-domain gauge-fixed construction imposes three necessary conditions on the gauge data of a chiral gauge theory on a localization-bounded domain, collected in Theorem 1. Compactness follows under the normalized local group-averaging prescription of Proposition 1; it is conditional on that prescription and is not a universal theorem of BRST quantization. Anomaly freedom follows from the interior BRST test of Lemma 1 and the separate single-valuedness requirement on the fermion determinant. Global form and Gribov data must be specified, by Propositions 3 and 4.

7.2. What Remains Ordinary Gauge Theory

The window does not create new anomaly-cancellation or global-form laws. The perturbative anomaly obstruction is present wherever $\diamond =1$ and is read off in the interior; it is not concentrated at ${\mathcal{B}}_{ϵ}$. Proposition 2 shows that the window-weighted four-form is not BRST closed when $d\diamond \ne 0$, because the descent below the consistent anomaly carries ghost number two, so the boundary layer carries a representative-level transgression term whose full BRST completion is deferred. The window induces boundary-supported transgression terms in the window-weighted anomaly representative; those terms do not by themselves form a complete BRST cocycle, and the window supplies a finite-domain reason these ordinary conditions are required, not relaxed.

7.3. What the Standard Model Application Assumes

The SM application takes the observed three-generation chiral matter content as input and applies the anomaly classification of Minahan, Ramond, and Warner [10] and Geng and Marshak [11]. Given the observed irreducible multiplet dimensions, a nonzero charge normalization, the observed up- and down-type singlet identification, and the named minimality criterion, Corollary 1 recovers the gauge algebra $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ and its covering presentation, up to the allowed finite central quotients of Equation (45). Proposition 5 then selects the minimal Higgs doublet. Extended gauge groups, additional $\mathrm{U}\left(1\right)$ factors, right-handed neutrinos, vectorlike matter, and enlarged Higgs sectors are not excluded; each requires its own anomaly, global-form, and boundary-consistency checks.

7.4. What the Paper Does Not Establish

The paper does not derive the matter spectrum, generation number, or minimality criterion, or electroweak symmetry breaking, from localization alone; given those inputs, it derives the SM gauge algebra as the minimal finite-domain admissible solution. By Theorem 2, the gauge-singlet window is representation-blind: it requires anomaly-free matter without counting generations or forbidding sterile singlets and vectorlike pairs. The local anomaly analysis fixes the gauge algebra and covering presentation but not the global form. Compactness is conditional on the group-averaging prescription. The all-order boundary-layer BV/BRST completion is not proved in the corpus. Relation to grand unification is complementary: any proposed low-energy or UV gauge group must satisfy the same bounded-domain admissibility conditions, and the simply connected forms of $\mathrm{SU}\left(5\right)$, $\mathrm{Spin}\left(10\right)$, and $E_6$ satisfy the compactness condition and provide natural reference global forms [5,6,43], while their matter, bundle, boundary, line-operator, and large-gauge sectors must still be specified. The relation to anomaly inflow is limited: in the Callan–Harvey mechanism [24] a higher-dimensional Chern–Simons sector cancels a boundary anomaly, whereas the present four-dimensional theory has no such sector and the boundary layer carries only a representative-level transgression term.

7.5. Limitations and Calculations Deferred to Subsequent Work

This subsection collects the open limitations of the present construction and indicates the calculations that would close them. In summary: Compactness is conditional on the normalized group-averaging prescription; the based boundary gauge group is an assumption and excludes an edge-mode analysis; the complete boundary-layer BV/BRST completion is not proved; global sectors are specified as required data, not classified; and matter selection and minimality are not derived.

The first deferred calculation is an explicit gauge-sector extension of the bulk/boundary renormalization result already established in companion work [44], which gives the bulk/boundary counterterm split with ordinary interior beta functions and boundary-response coefficients and a scalar boundary example. In choosing a Euclidean bounded domain, a Landau-type gauge, a smooth window, and BRST-compatible boundary conditions, one can compute the orbit-map Faddeev–Popov spectrum, the window-weighted ghost Hessian of Equation (27), the one-loop Slavnov–Taylor breaking functional, and the boundary-response counterterm coefficients. That calculation would extend the established split to an explicit non-Abelian gauge model and determine whether the required ghost-number-one boundary completion functionals of Proposition 2 exist and, if so, compute them. A second direction is to apply the admissibility criterion to specified nonminimal models, such as $\mathrm{U}{\left(1\right)}_{B-L}$ with right-handed neutrinos, $\mathrm{U}{\left(1\right)}_{L_{\mu }-L_{\tau }}$, two-Higgs-doublet sectors, and simple-group embeddings such as $\mathrm{SU}\left(5\right)$, $\mathrm{Spin}\left(10\right)$, or $E_6$: state the matter content and global form, evaluate the perturbative and global anomalies, specify the quotient or bundle sectors, and determine whether the bounded-domain construction remains well defined. Both directions extend the admissibility framework without enlarging its present scope: the first completes the boundary-layer calculation deferred above, and the second tests the same admissibility criterion against matter content and gauge sectors beyond the SM.

8. Conclusions

The finite-domain construction does not replace the ordinary gauge-consistency conditions, and it does not derive the matter spectrum, generation number, or minimality criterion from the scalar window alone. Under the stated regulator and boundary assumptions, it supplies a common admissibility test for any chiral gauge theory physically instantiated on a bounded spacetime domain: normalized local group averaging requires a compact gauge group; the ordinary perturbative and global anomaly obstructions, read off in the window interior and on the full gauge-orbit space, must vanish; and the global form, bundle and line-operator sectors, boundary gauge group, and Gribov/modular-region prescription must be specified. These necessary conditions are collected in Theorem 1.

The window induces boundary-supported transgression terms in the window-weighted anomaly representative, not a new cancellation mechanism; those terms do not by themselves form a complete BRST cocycle, since the window is BRST-inert and the descent below the consistent anomaly carries ghost number two (Proposition 2).

Applied to the observed three-generation quark-lepton matter under a named minimality criterion, the admissibility conditions recover the SM gauge algebra $\mathfrak{su}\left(3\right)\oplus \mathfrak{su}\left(2\right)\oplus \mathfrak{u}\left(1\right)$ and its direct-product covering presentation, up to allowed finite central quotients, and select the minimal Higgs doublet. The matter spectrum itself is not derived: by Theorem 2, the gauge-singlet window requires anomaly-free matter without selecting the generation number or excluding sterile singlets and vectorlike pairs. The boundary layer is not the locus of compactness, the global anomaly, or the global form; those conditions concern the whole regulated domain and the full gauge-fixed orbit space. Said simply, the boundary layer and windowed approach do not force the consistency conditions themselves, only their expression in a windowed theory.

The scope of this work’s core result is therefore bounded: finite-domain localization does not derive the SM from first principles, but it gives the familiar compactness, anomaly, and global-form requirements a common finite-domain implementation. Under the stated matter and minimality inputs, those requirements recover the SM gauge algebra as the minimal admissible solution.