Generator-Resolved Anomaly-Cancellation Conditions on a Matter-Coupled Benchmark Sheet in the Framework

A gauge--chiral construction can fail in two logically distinct ways. The first failure is algebraic: a matter-coupled representation cell that looks locally plausible can still leak gauge or gravitational current once its signed generator moments are summed across the full chiral content [citation]. The second failure is dynamical: even an anomaly-closed sheet may be discarded if it requires wrong-sign kinetic sectors, cone-widening derivative terms, or a gauge reduction that does not yield a well-posed local evolution problem [citation]. For an admitted-branch gauge--chiral construction, these two failures must be tested on one and the same declared sheet.

The central idea developed below is that anomaly cancellation on a benchmark-facing gauge--chiral sheet is not an accidental list of unrelated coefficient equalities. On a declared matter-coupled one-generation cell, anomaly cancellation can be rewritten as exact closure of a generator-resolved phase-charge ledger. The same ledger that tracks the signed generator moments of the cell also determines whether replication of that cell is neutral or anomaly-producing. In that sense, generation is not used to repair an inconsistent construction; it is allowed only as a repetition of a cell that is already closed.

The front phenomena are therefore tightly linked: anomaly leakage on a one-generation gauge--chiral sheet and the persistence or failure of anomaly closure under repeated generation cells. The declared gauge group is

\Gbm=\SU(3)_c\times\SU(2)_{\chi}\times\Uone_X,

and the one-generation representation cell is fixed as

\Rcell=\{Q_{\LL},U_{\RR},D_{\RR},L_{\LL},E_{\RR},N_{\RR}\}.

The subscript $\chi$TeX\chi marks a benchmark-facing weak slot without identifying it with the completed electroweak sector, and $\Uone_X$TeX\Uone_X is a generator slot rather than an already completed hypercharge or electric-charge identity.

The new element is an admitted order-parameter doublet witness $\Hchi\sim(\mathbf1,\mathbf2)_{x_H}$TeX\Hchi\sim(\mathbf1,\mathbf2)_{x_H} that enters only through neutral bilinear interface contracts. This does not construct a BEH vacuum manifold, does not introduce a Yukawa completion, and does not identify the scalar sector with the Higgs field. It only states that, on one declared sheet, the benchmark-facing bilinear interfaces must be neutral under $\Uone_X$TeX\Uone_X. Once that neutrality is imposed, the perturbative anomaly ledger collapses sharply: the mixed color--$X$TeXX and mixed gravitational--$X$TeXX coefficients vanish identically, and the cubic $X^3$TeXX^3 coefficient becomes proportional to the single weak-balance factor $3x_Q+x_L$TeX3x_Q+x_L. The remaining global obstruction is the standard $\SU(2)$TeX\SU(2) Witten obstruction [citation].

On one declared matter-coupled benchmark sheet, the generator-resolved anomaly ledger is fixed, exact anomaly closure is proved on the same cell, and the same closed sheet is shown to satisfy the kinetic/coefficient positivity gate of the admitted principal class, to introduce no admitted principal cone-widening on the declared background cone, and to be conditionally locally well posed after the declared gauge reduction on an admitted background branch. The construction does not provide a full Standard Model, does not derive confinement, does not identify $\Uone_X$TeX\Uone_X with physical hypercharge except at the level of a benchmark-facing specialization up to normalization, and does not solve flavor mixing, ultraviolet completion, compactification, or vacuum selection.

We use the metric signature $(- + + +)$TeX(- + + +) and work on a spacetime domain $\UU$TeX\UU lying on an admitted background branch. The only imported background restriction is the small-gradient condition

\bgXi\ll 1,
\qquad
\frac{L}{\ell_{\mathcal H}}\ll1,

with $L$TeXL the probe scale and $\ell_{\mathcal H}$TeX\ell_{\mathcal H} the characteristic variation length of the admitted order-parameter background. No new gravitational or scalar field equation is introduced below.

definition: Declared benchmark sheet. A declared benchmark sheet is a tuple

(\UU,g_{\mu\nu},\HH,\Gbm,\Rcell,\Hchi,\Dadm)

consisting of an admitted background domain $\UU$TeX\UU, the fixed gauge group reference, the one-generation matter content reference, one order-parameter doublet witness $\Hchi\sim(\mathbf1,\mathbf2)_{x_H}$TeX\Hchi\sim(\mathbf1,\mathbf2)_{x_H}, and one declared operating window $\Dadm$TeX\Dadm on which all anomaly, positivity, causality, and well-posedness statements are evaluated. The cell is allowed to repeat only through generator-neutral family replication, meaning that every additional generation is a copy of $\Rcell$TeX\Rcell with the same generator data and no inter-generation generator asymmetry.

The representation content is fixed as center

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center

definition: Generator-resolved anomaly ledger. For the declared benchmark sheet, define the generator-resolved anomaly ledger

\Aled(\Rcell)
:=
\bigl(\mathcal A_{33X},\mathcal A_{22X},\mathcal A_{\mathrm{grav}\,X},\mathcal A_{XXX},\nu_W\bigr)

with entries

\mathcal A_{33X}:=2x_Q-x_U-x_D,

\mathcal A_{22X}:=3x_Q+x_L,

\mathcal A_{\mathrm{grav}\,X}
:=6x_Q-3x_U-3x_D+2x_L-x_E-x_N,

\mathcal A_{XXX}
:=6x_Q^3-3x_U^3-3x_D^3+2x_L^3-x_E^3-x_N^3,

\nu_W:=\#\{\text{left }\SU(2)_{\chi}\text{ doublets}\}\mod 2.

The coefficients reference--reference are the signed generator moments of the one-generation cell. The factor of $3$TeX3 in reference and the factor of $6$TeX6 in reference arise because the left quark doublet carries three color copies and two weak components. The datum $\nu_W$TeX\nu_W tracks the Witten obstruction for the compact weak slot.

definition: Neutral bilinear interface contract. The admitted bilinear interface contracts are called neutral when the $\Uone_X$TeX\Uone_X charges satisfy

x_U=x_Q+x_H,
 \qquad
 x_D=x_Q-x_H,
 \qquad
 x_E=x_L-x_H,
 \qquad
 x_N=x_L+x_H.

These are the neutrality conditions for the benchmark-facing bilinears attached to the witness $\Hchi$TeX\Hchi; they do not assert a completed Yukawa or electroweak identity.

theorem: Exact anomaly-closure theorem on the declared benchmark sheet. Assume the neutral interface conditions reference on the declared benchmark sheet. Then

\mathcal A_{33X}=0,

\mathcal A_{\mathrm{grav}\,X}=0,

\mathcal A_{XXX}=-6x_H^{2}\,\mathcal A_{22X}.

Consequently, perturbative anomaly closure on the declared sheet is equivalent to the single linear weak-balance condition

\mathcal A_{22X}=3x_Q+x_L=0.

Moreover, the global Witten obstruction is absent because one generation cell contains exactly four left $\SU(2)_{\chi}$TeX\SU(2)_{\chi} doublets.

proof. Using reference in reference gives

\mathcal A_{33X}=2x_Q-(x_Q+x_H)-(x_Q-x_H)=0.

Likewise, substituting reference into reference yields

\mathcal A_{\mathrm{grav}\,X}
=6x_Q-3(x_Q+x_H)-3(x_Q-x_H)+2x_L-(x_L-x_H)-(x_L+x_H)
\notag

=0.

For the cubic coefficient one finds

\mathcal A_{XXX}
=6x_Q^3-3(x_Q+x_H)^3-3(x_Q-x_H)^3+2x_L^3-(x_L-x_H)^3-(x_L+x_H)^3
\notag

=-6x_H^2(3x_Q+x_L)
\notag

=-6x_H^2\,\mathcal A_{22X}.

Thus $\mathcal A_{XXX}=0$TeX\mathcal A_{XXX}=0 if and only if $\mathcal A_{22X}=0$TeX\mathcal A_{22X}=0 on the declared sheet. Finally, the left-handed $\SU(2)_{\chi}$TeX\SU(2)_{\chi} doublets are the three color copies of $Q_{\LL}$TeXQ_{\LL} together with one copy of $L_{\LL}$TeXL_{\LL}, so the total number is $3+1=4$TeX3+1=4, which is even. Hence $\nu_W=0$TeX\nu_W=0 and the Witten obstruction is absent.

corollary: Affine anomaly-closed family. If reference holds, the charge sheet is of the form

(x_Q,x_U,x_D,x_L,x_E,x_N)
=\bigl(a,\,a+b,\,a-b,\,-3a,\,-3a-b,\,-3a+b\bigr)

for real parameters $(a,b)=(x_Q,x_H)$TeX(a,b)=(x_Q,x_H).

proof. Set $a:=x_Q$TeXa:=x_Q and $b:=x_H$TeXb:=x_H. Then reference gives $x_L=-3a$TeXx_L=-3a, and reference yields the remaining entries.

corollary: Neutral-singlet specialization and benchmark charge pattern. If, in addition, the singlet is neutral,

x_N=0,

then $b=3a$TeXb=3a and the affine family reference reduces to

(x_Q,x_U,x_D,x_L,x_E,x_N)
=\bigl(a,\,4a,\,-2a,\,-3a,\,-6a,\,0\bigr).

After the normalization $a=\frac16$TeXa=\frac16, this is exactly the standard one-generation benchmark charge pattern with a neutral right-handed singlet.

proof. From reference one has $x_N=-3a+b$TeXx_N=-3a+b. Setting $x_N=0$TeXx_N=0 gives $b=3a$TeXb=3a, and substitution yields reference. The normalization $a=\frac16$TeXa=\frac16 is then immediate.

remark. The specialization in reference does not identify $\Uone_X$TeX\Uone_X with physical hypercharge. The benchmark pattern emerges from anomaly closure, admitted bilinear neutrality, and a neutral-singlet specialization on the declared sheet. This is a benchmark-facing emergence statement, not an electroweak identity theorem.

corollary: Closed-cell replication. If one declared generation cell satisfies reference, reference, and $\nu_W=0$TeX\nu_W=0, then every finite generator-neutral replication of that same cell remains anomaly closed. The perturbative anomaly coefficients scale linearly with the number of copies, while the global Witten datum remains zero because the total number of left $\SU(2)_{\chi}$TeX\SU(2)_{\chi} doublets is $4n$TeX4n for $n$TeXn copies.

proof. Each perturbative anomaly coefficient is additive under disjoint union of identical generations, so every local coefficient is multiplied by $n$TeXn. If the one-generation cell is closed, the replicated coefficients remain zero. For the global Witten datum, one generation cell contains four left $\SU(2)_{\chi}$TeX\SU(2)_{\chi} doublets; hence $\nu_W^{(n)}=4n\mod2=0$TeX\nu_W^{(n)}=4n\mod2=0.

proposition: Anomaly closure does not select the benchmark matter content. The anomaly equations on the declared sheet do not determine the charge normalization, the ratio $x_H/x_Q$TeXx_H/x_Q, the number of replicated generation cells, or the choice of the declared field content. The standard benchmark pattern follows only after the additional neutral-singlet condition and a normalization convention are imposed.

proof. By reference, the neutral-bilinear and anomaly conditions admit the two-parameter family $(a,b)=(x_Q,x_H)$TeX(a,b)=(x_Q,x_H). Removing an overall nonzero normalization still leaves the ratio $b/a$TeXb/a undetermined. The condition $x_N=0$TeXx_N=0 is an additional equation, not an anomaly equation, and selects $b/a=3$TeXb/a=3; the value $a=1/6$TeXa=1/6 is then a normalization choice. By reference, every positive integer number of identical cells is also anomaly closed. Finally, all equations were computed after the representation list in the benchmark sheet had been stipulated. Thus closure tests that chosen sheet but does not derive its field content or generation multiplicity.

The anomaly ledger alone does not define an admissible branch. The same sheet must also survive the kinetic/coefficient positivity, causality, and local well-posedness tests on the same declared domain [citation].

The gauge fields of the declared sheet are $G^A_{\mu}$TeXG^A_{\mu} for $\SU(3)_c$TeX\SU(3)_c, $W^a_{\mu}$TeXW^a_{\mu} for $\SU(2)_{\chi}$TeX\SU(2)_{\chi}, and $B_{\mu}$TeXB_{\mu} for $\Uone_X$TeX\Uone_X, with field strengths

\mathcal G^A_{\mu\nu}
=\partial_{\mu}G^A_{\nu}-\partial_{\nu}G^A_{\mu}+g_3 f^{ABC}G^B_{\mu}G^C_{\nu},

W^a_{\mu\nu}
=\partial_{\mu}W^a_{\nu}-\partial_{\nu}W^a_{\mu}+g_2\epsilon^{abc}W^b_{\mu}W^c_{\nu},

B_{\mu\nu}
=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}.

The covariant derivative on any field $\Psi$TeX\Psi of charges $(R_c,R_{\chi},x_{\Psi})$TeX(R_c,R_{\chi},x_{\Psi}) is

D_{\mu}\Psi
=
\nabla_{\mu}\Psi
-i g_3 G^A_{\mu}T^A_{c}\Psi
-i g_2 W^a_{\mu}T^a_{\chi}\Psi
-i g_X x_{\Psi} B_{\mu}\Psi.

The declared branch action is

S_{\Dadm}
=
\int_{\UU}\dd^4x\,\sqrt{-g}\Bigg[
-\frac{\kthree}{4}\mathcal G^A_{\mu\nu}\mathcal G^{A\mu\nu}
-\frac{\ktwo}{4}W^a_{\mu\nu}W^{a\mu\nu}
-\frac{\kX}{4}B_{\mu\nu}B^{\mu\nu}

+\sum_{\Psi\in\Rcell} i Z_{\Psi}\,\bar\Psi\gamma^{\mu}D_{\mu}P_{s_{\Psi}}\Psi
+ \Zh\,(D_{\mu}\Hchi)^{\dagger}D^{\mu}\Hchi
- U_{\chi}(\Hchi,\HH)
+ \Lag_{\mathrm{adm}}^{\mathrm{int}}
\Bigg],

with positive coefficients

\kthree>0,
\qquad
\ktwo>0,
\qquad
\kX>0,
\qquad
Z_{\Psi}>0,
\qquad
\Zh>0,

and with $\Lag_{\mathrm{adm}}^{\mathrm{int}}$TeX\Lag_{\mathrm{adm}}^{\mathrm{int}} restricted to zero-derivative or first-order interface terms. The admitted bilinear subset is

\Lag_{\mathrm{adm}}^{\mathrm{int}}\supset
-y_U\,\bar Q_{\LL}\widetilde\Hchi U_{\RR}
-y_D\,\bar Q_{\LL}\Hchi D_{\RR}
-y_E\,\bar L_{\LL}\Hchi E_{\RR}
-y_N\,\bar L_{\LL}\widetilde\Hchi N_{\RR}
+\mathrm{h.c.},

where $\widetilde\Hchi$TeX\widetilde\Hchi denotes the conjugate $\SU(2)_{\chi}$TeX\SU(2)_{\chi} doublet with charge $-x_H$TeX-x_H.

definition: Same-sheet admissibility gate. The same-sheet admissibility predicate on $\Dadm$TeX\Dadm is

\Ajoint(\Rcell,\Dadm)
:=
\mathsf A\wedge\mathsf P\wedge\mathsf C\wedge\mathsf W,

where

- $\mathsf A$TeX\mathsf A is the exact ledger closure of reference, - $\mathsf P$TeX\mathsf P is the kinetic/coefficient positivity gate encoded in the coefficients of reference, - $\mathsf C$TeX\mathsf C is the absence of higher-derivative cone-widening operators in the principal part, - $\mathsf W$TeX\mathsf W is the existence of a standard Lorenz-type gauge reduction whose principal symbol is strongly hyperbolic on the admitted background window.

proposition: Same-sheet admissibility on the declared principal class. Assume the declared benchmark sheet, the action reference, the positivity conditions reference, and the exclusion of all higher-derivative or cone-widening operators from the principal part on $\Dadm$TeX\Dadm.

Then, after a standard Lorenz-type gauge reduction and on constraint-compatible initial data, the leading principal symbol is the direct sum of the normally hyperbolic Yang--Mills principal symbols for $\SU(3)_c$TeX\SU(3)_c, $\SU(2)_{\chi}$TeX\SU(2)_{\chi}, and $\Uone_X$TeX\Uone_X, together with first-order Dirac symbols for the chiral matter fields and the normally hyperbolic doublet symbol for $\Hchi$TeX\Hchi. Consequently, the gauge and scalar characteristic sets are metric-null at principal order, while the Dirac blocks have the standard first-order hyperbolic characteristic structure whose squared principal symbol is metric-null. The reduced system is conditionally locally well posed on the declared window in the Sobolev class and gauge reduction used for the benchmark sheet.

proof. Under the exclusion of higher-derivative or cone-widening operators, the gauge-boson principal part comes only from the quadratic field-strength terms in reference; after Lorenz-type gauge fixing these are normally hyperbolic wave operators on the fixed background metric. The chiral sector contributes only first-order Dirac symbols weighted by positive coefficients $Z_{\Psi}$TeXZ_{\Psi}, and the witness field $\Hchi$TeX\Hchi contributes a standard covariant wave operator weighted by $\Zh>0$TeX\Zh>0. Hence the principal symbol is block diagonal at leading order, the characteristic covectors satisfy the background cone condition, and the reduced system lies within the standard local hyperbolic Cauchy theory [citation].

theorem: Admissible benchmark-sheet branch theorem on the declared principal class. On the declared benchmark sheet, assume the neutral interface conditions reference, the weak-balance condition reference, the positivity conditions reference, and the exclusion of higher-derivative cone-widening operators from the principal part. Then the same matter-coupled gauge--chiral sheet is anomaly closed, free of the global Witten obstruction, satisfies the kinetic/coefficient positivity gate in reference, introduces no admitted principal cone-widening on the declared background cone, and is conditionally locally well posed after the declared gauge reduction on constraint-compatible initial data. The same conclusion holds for every finite generator-neutral replication of the closed one-generation cell under the same declared principal-class assumptions.

proof. Combine reference.

The admitted witness $\Hchi$TeX\Hchi is introduced only through neutral interface contracts. It is not identified with a completed electroweak Higgs doublet, it does not by itself define a BEH vacuum manifold, and it does not supply a full Yukawa or flavor map. The same-sheet theorem is therefore logically prior to any restricted mass-window analysis: the anomaly ledger depends only on representation content and generator data, while mass parameters can modify the admitted low-energy window only if they do not alter that content or those data.

Accordingly, the following are explicit non-claims.

- No confinement law or QCD phenomenology is established. - No electroweak gauge-boson identity is established. - No BEH vacuum manifold, Goldstone sector, or full Yukawa completion is built. - No flavor-mixing law or generation-hierarchy fit is derived. - No collider or precision-electroweak phenomenology is attempted. - No ultraviolet completion, compactification closure, or vacuum-selection theorem is provided.

The branch is rejected if any of the following occurs on the declared window:

- the bilinear contracts reference cannot be made neutral on the same sheet, - the weak-balance equation reference fails, - the number of left $\SU(2)_{\chi}$TeX\SU(2)_{\chi} doublets becomes odd, - positivity requires a nonpositive kinetic coefficient in reference, - causality or local well-posedness can be maintained only by higher-derivative or cone-widening operators outside the admitted principal class, - generator-neutral replication fails because inter-generation generator asymmetry is introduced.

The generator-resolved anomaly equations remain algebraic consistency conditions on the admitted gauge representations. Compact target topology of the root scalar does not alter their traces or supply missing chiral matter. If gauge coefficients depend on $\mathcal H$TeX\mathcal H, the resulting phase-loading functional must additionally have a formally self-adjoint Fr\'echet derivative to descend from a local scalar action. Anomaly cancellation and variational integrability are independent requirements.

Across several anomaly-free observables, a common phase-dependent gauge action yields a shared sensitivity matrix. Any left-null direction is a first-order compatibility equation that must hold after the gauge and phase parameters are fixed. Generator-specific nuisance couplings may remove these restrictions but then abandon the claimed common loading. Thus anomaly closure is necessary for the sheet, Helmholtz integrability is necessary for its phase lift, and lower-than-observable response rank is necessary for predictive cross-sector content.

The closure test for the generator-resolved anomaly cancellation is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of all gauge, mixed, and gravitational anomaly coefficients plus matter observables. Let $a$TeXa range over the independent constitutive inputs comprising representations, chiral charges, phase-dependent coefficients, and flavor response.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if exact anomaly cancellation is imposed before any variational phase loading or predictive-rank claim is evaluated.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because no phase-loading term can compensate a nonzero gauge anomaly in the quantum measure. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

A gauge--chiral extension becomes mathematically admissible only when one and the same matter-coupled benchmark sheet closes its anomaly ledger and survives the admissibility gates on one declared domain. On the declared sheet, the neutral bilinear interface conditions force the mixed color--$X$TeXX and mixed gravitational--$X$TeXX anomaly coefficients to vanish identically and reduce the cubic $X^3$TeXX^3 coefficient to the single weak-balance factor $3x_Q+x_L$TeX3x_Q+x_L. Exact perturbative anomaly closure is therefore equivalent to one linear ledger condition rather than a list of unrelated cancellations. The global Witten obstruction is absent because one generation cell contains four left $\SU(2)_{\chi}$TeX\SU(2)_{\chi} doublets, and any generator-neutral replication of the same closed cell remains anomaly closed.

The same sheet also survives the dynamical gates. With positive gauge and matter kinetic coefficients, standard first-order chiral kinetic terms, and no higher-derivative cone-widening operators in the principal part, the same matter-coupled gauge--chiral sheet satisfies the kinetic/coefficient positivity gate of the admitted principal class in reference, introduces no admitted principal cone-widening on the declared background cone, and is conditionally locally well posed after the declared Lorenz-type gauge reduction on constraint-compatible initial data. The neutral-singlet specialization then reproduces the standard one-generation benchmark charge pattern up to normalization without identifying $\Uone_X$TeX\Uone_X with physical hypercharge. Proposition reference shows that anomaly cancellation alone neither selects this specialization nor fixes the representation list, charge normalization, or generation count.

What is established is therefore an exact anomaly-closure and same-sheet admissibility result on one declared benchmark sheet, stronger than a minimal surviving-family existence note and weaker than a Standard-Model completion. The result fixes one generator-resolved anomaly ledger, one matter-coupled benchmark sheet, one closed-cell replication rule, and one same-sheet admissibility theorem. What it does not establish is equally explicit: no confinement law, no electroweak identity, no BEH/Yukawa/flavor completion, no phenomenological fit, no ultraviolet closure, and no Standard-Model completion.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.