Collider, Precision-Electroweak, and Leptonic-Flavor Fit Windows on a Declared CHC Gauge--Chiral Family

A phenomenological model is predictive only to the extent that its observables cannot be independently adjusted to their targets. Sharing a tuple of parameter symbols across channels is necessary but insufficient: the tuple may contain enough directions to interpolate every independent target. The appropriate tests are therefore the rank of the observable map, the structure of its null space, and performance on observables not used for calibration [citation].

The map studied below contains a weak scale $(v_*)$TeX(v_*), two gauge couplings, a neutral-current remainder, two width remainders, and a free matrix $U_\ell\in U(3)$TeXU_\ell\in U(3). We ask whether the stated observable basket identifies these quantities or leaves a saturated calibration family. This question precedes any claim of agreement: a fit obtained by algebraically solving one free parameter for each target has no independent predictive force for those targets.

The interval ranges below are retained to define the earlier calibration exercise, but their nonempty intersection is not treated as a statistical fit. A valid quantitative fit would require a joint likelihood, experimental and theoretical covariance, nuisance profiling, and a reported number of predictive degrees of freedom. The present paper establishes the mathematical obstruction and specifies the additional information required to remove it.

definition: Calibration family. The phenomenological calibration family is

\Thetafit:=\{(\vev,g,g',\deltaNC,\deltaZ,\deltaL,\Uell):\,\vev>0,\ g>0,\ g'>0,

|\deltaNC|\le \NCallow,\ |\deltaZ|\le \Gallow,\ |\deltaL|\le \Gallow,\ \Uell\in U(3)\}.

where:

- $\vev$TeX\vev is the single weak-boson loading scale carried by the admitted electroweak completion family; - $(g,g')$TeX(g,g') is the single gauge-coupling pair on the declared gauge sheet; - $\deltaNC$TeX\deltaNC is the bounded neutral-current remainder that converts the tree-level weak-mixing expression into the benchmark-facing neutral-current observables on the same family; - $(\deltaZ,\deltaL)$TeX(\deltaZ,\deltaL) is the shared width-block remainder pair for the total and leptonic $Z$TeXZ-width channels on the same family; and - $\Uell$TeX\Uell is the leptonic mismatch matrix on the admitted flavor subset.

The family is read with the subspace topology inherited from $\mathbb R^6\times U(3)$TeX\mathbb R^6\times U(3) and therefore has real dimension fifteen. Requiring a common tuple prevents literal channel-by-channel duplication, but it does not establish identifiability.

remark: Restricted flavor subset. The observable set uses only three functions of $U_\ell$TeXU_\ell. A full quark-sector flavor fit, CKM hierarchy, hadronic nuisance treatment, and global oscillation likelihood are outside the analysis.

definition: Observable set. The declared observable basket is

\Bfit=\Bew\cup\Bfl,

with precision-electroweak block

\Bew=\{m_W,\ m_Z,\ \seff,\ \Aell,\ \Gamma_Z,\ \Gammaell\}

and flavor block

\Bfl=\{\ttwelve,\ \sthree,\ \stwo\}.

The flavor-suppression ratio $\Rsup$TeX\Rsup is not an additional basket coordinate; it enters only through the explicit flavor-suppression gate defined in reference. The declared benchmark intervals adopted below are

\Iobs_{m_W}=[80.3506,\,80.3824]\ \mathrm{GeV}, 

\Iobs_{m_Z}=[91.1854,\,91.1896]\ \mathrm{GeV}, 

\Iobs_{\seff}=[0.23137,\,0.23169], 

\Iobs_{\Aell}=[0.1450,\,0.1520], 

\Iobs_{\Gamma_Z}=[2.4929,\,2.4975]\ \mathrm{GeV}, 

\Iobs_{\Gammaell}=[83.898,\,84.070]\ \mathrm{MeV}, 

\Iobs_{\ttwelve}=[0.49,\,0.66], 

\Iobs_{\sthree}=[0.02112,\,0.02264], 

\Iobs_{\stwo}=[0.529,\,0.593].

The asymmetry and width intervals are benchmark windows on the same neutral-current family, not a new world-average reanalysis. The solar-angle interval is taken directly from the declared KamLAND benchmark window rather than from a global oscillation fit. These windows are used only as declared basket anchors for the admitted-fit predicate; no full world-average reanalysis is attempted [citation].

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

definition: Interval-compatibility predicate. Let $\Mmap: \Thetafit\to \mathbb R^9$TeX\Mmap: \Thetafit\to \mathbb R^9 be the direct observable map defined in reference. The suppression ratio $\Rsup$TeX\Rsup is evaluated only on the admitted leptonic subset where $\stwo(\theta)>0$TeX\stwo(\theta)>0. The admitted-fit predicate is

\mathcal A_{\rm fit}(\theta;\sigsup)=1
\iff
\Mmap_a(\theta)\in \Iobs_a\ \text{for every}\ a\in\Bfit
\ \text{and}\
\Rsup(\theta)\le \sigsup.

The interval-compatible set is

\Wfit(\sigsup):=\{\theta\in \Thetafit:\ \mathcal A_{\rm fit}(\theta;\sigsup)=1\}.

For any $\theta\in\Thetafit$TeX\theta\in\Thetafit, the basket-violation set is

\mathcal K(\theta):=\{a\in\Bfit: \Mmap_a(\theta)\notin\Iobs_a\}.

This predicate records interval membership only. It is not a likelihood and contains no covariance, uncertainty model, or complexity penalty.

On the declared admitted family, the electroweak-facing map is

m_W(\theta)=\frac{g\vev}{2}, 

m_Z(\theta)=\frac{\sqrt{g^2+g'^2}\,\vev}{2}, 

\sw(\theta)=\frac{g'^2}{g^2+g'^2}, 

\seff(\theta)=\sw(\theta)+\deltaNC, 

\Aell(\theta)=\frac{2\bigl(1-4\seff(\theta)\bigr)}{1+\bigl(1-4\seff(\theta)\bigr)^2}, 

\Gamma_Z(\theta)=\Gamma_Z^{(0)}(\vev,g,g')+\deltaZ, 

\Gammaell(\theta)=\Gamma_{\ell}^{(0)}(\vev,g,g',\seff(\theta))+\deltaL.

Here $\Gamma_Z^{(0)}$TeX\Gamma_Z^{(0)} and $\Gamma_{\ell}^{(0)}$TeX\Gamma_{\ell}^{(0)} are the declared total and leptonic width functionals on the same family. They are not a full hadronic simulation and do not by themselves constitute a global QCD nuisance treatment. The role of $\deltaNC$TeX\deltaNC and $(\deltaZ,\deltaL)$TeX(\deltaZ,\deltaL) is likewise narrow: they are shared family-level benchmark remainders, not per-observable tuning knobs.

remark: Weak-boson loading, asymmetry, and shared width block. The same loading scale $\vev$TeX\vev enters both $m_W$TeXm_W and $m_Z$TeXm_Z. The same coupling pair $(g,g')$TeX(g,g') enters $\sw$TeX\sw, $\seff$TeX\seff, $\Aell$TeX\Aell, and the width functionals. The asymmetry channel is therefore not an independently retuned sector, and the width block is controlled by one shared remainder pair on the same family. Any attempt to fit masses, asymmetry, or widths on different families exits $\Thetafit$TeX\Thetafit by definition.

The present direct flavor block uses the leptonic mismatch subset of the admitted family. Let the unitary matrix $\Uell$TeX\Uell be parametrized by three mixing angles and phases. The reduced observables are

\ttwelve(\theta):=\frac{|(\Uell)_{e2}|^2}{|(\Uell)_{e1}|^2},

\sthree(\theta):=|(\Uell)_{e3}|^2,

\stwo(\theta):=\frac{|(\Uell)_{\mu 3}|^2}{1-\sthree(\theta)}.

definition: Flavor-suppression condition. The flavor-suppression ratio is

\Rsup(\theta):=\frac{\sthree(\theta)}{\stwo(\theta)}.

and is evaluated only where $\stwo(\theta)>0$TeX\stwo(\theta)>0. The condition is satisfied when

\ttwelve(\theta)\in \Iobs_{\ttwelve},
\qquad
\sthree(\theta)\in \Iobs_{\sthree},
\qquad
\stwo(\theta)\in \Iobs_{\stwo},
\qquad
\Rsup(\theta)\le \sigsup.

proposition: Common-matrix dependence. The condition reference is a predicate on one matrix $U_\ell$TeXU_\ell. This common dependence does not reduce the three mixing-angle degrees of freedom: every interior triple in the physical domain can be realized by a unitary matrix.

proof. The first statement follows from reference--reference. Conversely, for any $x\ge0$TeXx\ge0, $0\le z<1$TeX0\le z<1, and $0\le y\le1$TeX0\le y\le1, choose the standard three-angle unitary parameterization with $\tan^2\theta_{12}=x$TeX\tan^2\theta_{12}=x, $\sin^2\theta_{13}=z$TeX\sin^2\theta_{13}=z, and $\sin^2\theta_{23}=y$TeX\sin^2\theta_{23}=y. Equations reference--reference then return $(x,z,y)$TeX(x,z,y), independently of the remaining phases.

proposition: Common-parameter evaluation. Let $\theta\in\Thetafit$TeX\theta\in\Thetafit. Then the electroweak block $(m_W,m_Z,\seff,\Aell,\Gamma_Z,\Gammaell)$TeX(m_W,m_Z,\seff,\Aell,\Gamma_Z,\Gammaell), the direct leptonic-flavor block $(\ttwelve,\sthree,\stwo)$TeX(\ttwelve,\sthree,\stwo), and the explicit gate quantity $\Rsup$TeX\Rsup are evaluated on the same tuple $(\vev,g,g',\deltaNC,\deltaZ,\deltaL,\Uell)$TeX(\vev,g,g',\deltaNC,\deltaZ,\deltaL,\Uell). Any procedure that changes one of these entries for only one observable sector does not refine the same admitted family; it defines a different family point outside the declared no-retuning problem.

proof. This is immediate from reference. The observable map $\Mmap$TeX\Mmap is defined on one fixed domain $\Thetafit$TeX\Thetafit, not on an observable-indexed product of family copies.

corollary: Observable-specific counterterms change the model. Let $\{c_a\}_{a\in\Bfit}$TeX\{c_a\}_{a\in\Bfit} be observable-specific counterterms. If there is no single transformation

(\vev,g,g',\deltaNC,\deltaZ,\deltaL,\Uell)\longmapsto (\tilde v,\tilde g,\tilde g',\tilde\deltaNC,\tilde\deltaZ,\tilde\deltaL,\tilde U_{\ell})

whose induced observable map absorbs all $c_a$TeXc_a simultaneously, then the corrected basket does not belong to the same admitted family.

proof. A single admitted family is, by definition, one point of $\Thetafit$TeX\Thetafit. Observable-specific counterterms that cannot be represented by one transformed family point therefore create an observable-indexed family copy and violate reference.

theorem: Rank bound and non-identifiability. At every differentiability point of $\Mmap$TeX\Mmap, the Jacobian satisfies

\operatorname{rank}D\Mmap\le 8,
\qquad
\dim\ker D\Mmap\ge 7.

Consequently, the fifteen-dimensional parameter vector in reference is nowhere locally identifiable from the nine-observable set in reference.

proof. The electroweak block contains six outputs, but $\Aell$TeX\Aell is the composition of the scalar function

a(s)=\frac{2(1-4s)}{1+(1-4s)^2}

with $\seff$TeX\seff. Hence $d\Aell=a'(\seff)d\seff$TeXd\Aell=a'(\seff)d\seff and the six electroweak rows have rank at most five. The flavor block contains three real functions of $U(3)$TeXU(3) and therefore has rank at most three. The blocks depend on disjoint parameter coordinates, so the full rank is at most $5+3=8$TeX5+3=8. Since $\dim(\mathbb R^6\times U(3))=6+9=15$TeX\dim(\mathbb R^6\times U(3))=6+9=15, rank--nullity gives nullity at least seven. Local identifiability of a smooth fifteen-parameter model would require column rank fifteen, which is impossible.

theorem: Constructive target interpolation. Let target values $(y_W,y_Z,y_s,y_Z^\Gamma,y_\ell^\Gamma,x,z,y)$TeX(y_W,y_Z,y_s,y_Z^\Gamma,y_\ell^\Gamma,x,z,y) satisfy

0<y_W<y_Z,\qquad x\ge0,\qquad 0\le z<1,\qquad 0<y\le1.

For every $v_*>0$TeXv_*>0, define

g=\frac{2y_W}{v_*},
g'=\frac{2\sqrt{y_Z^2-y_W^2}}{v_*},

\delta_{\rm NC}=y_s-\left(1-\frac{y_W^2}{y_Z^2}\right),
\delta_Z=y_Z^\Gamma-\Gamma_Z^{(0)}(v_*,g,g'),

\delta_L=y_\ell^\Gamma-\Gamma_\ell^{(0)}(v_*,g,g',y_s).

Choose $U_\ell$TeXU_\ell with $\tan^2\theta_{12}=x$TeX\tan^2\theta_{12}=x, $\sin^2\theta_{13}=z$TeX\sin^2\theta_{13}=z, and $\sin^2\theta_{23}=y$TeX\sin^2\theta_{23}=y. Whenever the three remainders obey the bounds in reference, the map reproduces exactly

(m_W,m_Z,\seff,\Gamma_Z,\Gamma_\ell,\ttwelve,\sthree,\stwo)
=(y_W,y_Z,y_s,y_Z^\Gamma,y_\ell^\Gamma,x,z,y).

The asymmetry is then fixed only through $\Aell=a(y_s)$TeX\Aell=a(y_s).

proof. Substitution of reference into reference--reference gives $(y_W,y_Z)$TeX(y_W,y_Z). It also gives $\sw=1-y_W^2/y_Z^2$TeX\sw=1-y_W^2/y_Z^2, so reference yields $\seff=y_s$TeX\seff=y_s. The two remainder definitions give the target widths identically. The standard unitary three-angle parameterization realizes $(x,z,y)$TeX(x,z,y) by reference. Finally, reference contains no remaining independent parameter after $y_s$TeXy_s is fixed.

corollary: Calibration saturation. Agreement of the eight quantities in reference cannot test the model when the same quantities determine the parameters by reference--reference. The only relation in the stated basket that is not separately interpolated is $\Aell=a(\seff)$TeX\Aell=a(\seff); its evidential value requires a joint covariance analysis and data not used to set $\seff$TeX\seff.

proof. The first statement is the constructive inverse map of reference. The last statement follows from the unique row dependence established in reference.

proposition: Topological interior lemma. Assume that:

- the admitted family $\Thetafit$TeX\Thetafit is nonempty and the map $\Mmap$TeX\Mmap is continuous on it; - there exists a reference point $\theta_\bullet\in\Thetafit$TeX\theta_\bullet\in\Thetafit such that every direct observable lies in the interior of its declared interval, equation \Mmap_a(\theta_) int(\Iobs_a) for all a, equation and $\Rsup(\theta_\bullet)<\sigsup$TeX\Rsup(\theta_\bullet)<\sigsup; - the same $\theta_\bullet$TeX\theta_\bullet obeys the no-retuning condition of reference.

Then there exists an open neighborhood $\mathcal U\subset\Thetafit$TeX\mathcal U\subset\Thetafit of $\theta_\bullet$TeX\theta_\bullet such that

\mathcal U\subseteq \Wfit(\sigsup).

This conclusion establishes only local interval compatibility; it supplies no likelihood value or predictive degree of freedom.

proof. Each component of $\Mmap$TeX\Mmap is continuous on $\Thetafit$TeX\Thetafit, hence the preimage of every open interval $\mathrm{int}(\Iobs_a)$TeX\mathrm{int}(\Iobs_a) is open. The strict inequality $\Rsup<\sigsup$TeX\Rsup<\sigsup also defines an open preimage on the admitted leptonic subset where $\stwo>0$TeX\stwo>0. The finite intersection of these open preimages contains $\theta_\bullet$TeX\theta_\bullet, and any sufficiently small neighborhood around $\theta_\bullet$TeX\theta_\bullet inside that intersection is contained in $\Wfit(\sigsup)$TeX\Wfit(\sigsup).

corollary: Numerical instance of target interpolation. Let $v_{\bullet}=246.22\,\mathrm{GeV}$TeXv_{\bullet}=246.22\,\mathrm{GeV}, and define

g_\bullet:=\frac{2\times 80.3665\ \mathrm{GeV}}{v_{\bullet}},
\qquad
g'_\bullet:=\sqrt{\frac{4\times (91.1875\ \mathrm{GeV})^2}{v_{\bullet}^2}-g_\bullet^2}.

Set

\delta_{\mathrm{NC},\bullet}:=0.23153-\frac{g_\bullet'^2}{g_\bullet^2+g_\bullet'^2},
\qquad
\delta_{Z,\bullet}:=2.4952\ \mathrm{GeV}-\Gamma_Z^{(0)}(v_{\bullet},g_\bullet,g'_\bullet),

and

\delta_{L,\bullet}:=83.984\ \mathrm{MeV}-\Gamma_{\ell}^{(0)}(v_{\bullet},g_\bullet,g'_\bullet,0.23153).

Choose $U_{\ell,\bullet}\in U(3)$TeXU_{\ell,\bullet}\in U(3) so that $\ttwelve(U_{\ell,\bullet})\in \Iobs_{\ttwelve}$TeX\ttwelve(U_{\ell,\bullet})\in \Iobs_{\ttwelve}, $\sthree(U_{\ell,\bullet})\in \Iobs_{\sthree}$TeX\sthree(U_{\ell,\bullet})\in \Iobs_{\sthree}, and $\stwo(U_{\ell,\bullet})\in \Iobs_{\stwo}$TeX\stwo(U_{\ell,\bullet})\in \Iobs_{\stwo}. If, in addition,

\Aell(\theta_\bullet)\in\Iobs_{\Aell}
\qquad\text{and}\qquad
\Rsup(U_{\ell,\bullet})<\sigsup,

then the hypotheses of reference are satisfied for

\theta_\bullet=(v_\bullet,g_\bullet,g'_\bullet,\delta_{\mathrm{NC},\bullet},\delta_{Z,\bullet},\delta_{L,\bullet},U_{\ell,\bullet}).

proof. The definitions of $g_\bullet$TeXg_\bullet and $g'_\bullet$TeXg'_\bullet are the inverse solution of the two mass equations. The three $\delta$TeX\delta definitions are residuals chosen to reproduce their respective central targets. Existence of $U_{\ell,\bullet}$TeXU_{\ell,\bullet} follows from reference. The two remaining displayed inequalities are precisely the dependent-asymmetry and suppression conditions. Thus this construction is an instance of reference; it is not an independent prediction of the target values.

Let $y$TeXy denote a vector of measurements with positive-definite covariance $C$TeXC, and let $m(\theta,\eta)$TeXm(\theta,\eta) contain parameters of interest $\theta$TeX\theta and nuisance parameters $\eta$TeX\eta. Define whitened Jacobians

J_\theta=C^{-1/2}\frac{\partial m}{\partial\theta},
\qquad
J_\eta=C^{-1/2}\frac{\partial m}{\partial\eta},

and let $P_\eta$TeXP_\eta be the orthogonal projector onto $\operatorname{col}(J_\eta)$TeX\operatorname{col}(J_\eta).

theorem: Profiled identifiability criterion. The parameters of interest are locally identifiable after nuisance profiling only if

\operatorname{rank}\bigl[(I-P_\eta)J_\theta\bigr]=\dim\theta.

Equivalently, the profiled Fisher matrix

F_{\rm prof}=J_\theta^{\mathsf T}(I-P_\eta)J_\theta

must be positive definite.

proof. Profiling removes the component of an infinitesimal signal variation lying in the nuisance tangent space. The remaining variation is $(I-P_\eta)J_\theta\,d\theta$TeX(I-P_\eta)J_\theta\,d\theta. A nonzero vector in its kernel changes $\theta$TeX\theta without changing the profiled prediction to first order, so local identifiability fails. Full column rank is therefore necessary and makes the displayed Gram matrix positive definite; conversely, positive definiteness implies a trivial kernel and full column rank.

For the present calibration family, reference violates the corresponding full-parameter rank condition before experimental covariance is considered. A predictive successor must reduce the parameterization by deriving at least some of $(\delta_{\rm NC},\delta_Z,\delta_L,U_\ell)$TeX(\delta_{\rm NC},\delta_Z,\delta_L,U_\ell) from independently constrained dynamics, fit a joint covariance likelihood, and reserve observables not used in that derivation for validation.

proposition: Set-theoretic emptiness identity. The fit window $\Wfit(\sigsup)$TeX\Wfit(\sigsup) is empty if and only if

\bigcap_{a\in\Bfit}\Mmap_a^{-1}(\Iobs_a)\ \cap\ \Rsup^{-1}(({-}\infty,\sigsup])=\varnothing.

This identity is a restatement of the definition and has no model-selection content by itself.

proof. This is an immediate rewriting of reference.

proposition: Minimal interval obstruction. Suppose there exists an observable $a_\star\in\Bfit$TeXa_\star\in\Bfit such that

\bigcap_{a\in\Bfit\setminus\{a_\star\}}\Mmap_a^{-1}(\Iobs_a)\ \cap\ \Rsup^{-1}(({-}\infty,\sigsup])\neq\varnothing,

but

\bigcap_{a\in\Bfit}\Mmap_a^{-1}(\Iobs_a)\ \cap\ \Rsup^{-1}(({-}\infty,\sigsup])=\varnothing.

Then $a_\star$TeXa_\star is a minimal single-observable obstruction to interval compatibility.

proof. The first condition gives compatibility without $a_\star$TeXa_\star and the second gives incompatibility after adjoining it. This is exactly the stated minimal-obstruction property.

proposition: Suppression-condition obstruction. Suppose the direct basket constraints are jointly satisfiable on the declared family,

\bigcap_{a\in\Bfit}\Mmap_a^{-1}(\Iobs_a)\neq\varnothing,

but every such point fails the declared flavor-suppression gate,

\bigcap_{a\in\Bfit}\Mmap_a^{-1}(\Iobs_a)\ \cap\ \Rsup^{-1}(({-}\infty,\sigsup])=\varnothing.

Then the suppression inequality is the additional obstruction to interval compatibility.

proof. The first condition gives a point satisfying the direct intervals, while the second excludes every such point after the suppression inequality is imposed.

Companion calculation..

CHC--PFW--VP1 reproduces selected public scalar intervals and covariance or contour subblocks from LEP/SLD, KamLAND, and T2K. It is a source-reproduction calculation, not a joint electroweak--flavor likelihood. Its results are interpreted only as checks of the numerical inputs used above.

Scope..

No full collider, electroweak, or flavor likelihood is evaluated here. Quark-sector observables, QCD uncertainties, and cosmological constraints are absent. The interval calculations cannot be used as evidence for a broader CHC construction.

Let the frozen calibration family define $F:\mathbb R^q\to\mathbb R^m$TeXF:\mathbb R^q\to\mathbb R^m. At a regular point with $\operatorname{rank}DF=r<m$TeX\operatorname{rank}DF=r<m, its local predictions occupy an $r$TeXr-dimensional compatibility manifold and obey $m-r$TeXm-r independent equations. Their tangent form is $w^{\mathsf T}\delta y=0$TeXw^{\mathsf T}\delta y=0 for every $w\in\ker(DF)^{\mathsf T}$TeXw\in\ker(DF)^{\mathsf T}. This makes rank deficiency productive only when the null directions are evaluated on observables not used to construct the fit.

The converse is equally decisive. If nuisance enlargement gives full row rank $m$TeXm, the response map is locally onto and no equality constraint survives. Such a family may interpolate the basket but cannot predict it. The audit must therefore freeze parameters and observable definitions, propagate covariance into the left-null residuals, and distinguish underidentification of parameters from overidentification of observables. Only the latter supplies a falsifiable cross-sector relation.

The closure test for the precision electroweak and flavor windows 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 electroweak pole observables, lepton masses, oscillation quantities, and flavor ratios. Let $a$TeXa range over the independent constitutive inputs comprising precision remainders, mixing parameters, covariance, and theoretical error model.

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 the nuisance family is frozen before fitting and the held-out basket has positive profiled predictive codimension.

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 a saturated calibration family can match the intervals while making no equality prediction. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The stated observable family is a saturated calibration model. Its Jacobian has rank at most eight on a fifteen-dimensional parameter space, and eight independent target coordinates can be reproduced by explicit inversion. Consequently, a nonempty interval-compatible set is expected and does not measure predictive success. The relation between $A_\ell$TeXA_\ell and $s_{\rm eff}^2$TeXs_{\rm eff}^2 is the sole dependent relation in the nine-entry set, and it is inherited from the adopted electroweak formula rather than derived from CHC-specific dynamics.

A predictive version of the model must replace freely adjustable remainders and mixing parameters by dynamical relations, demonstrate full profiled rank for the retained parameters, propagate the joint experimental and theoretical covariance, and evaluate at least one held-out observable. Until those conditions are met, the construction establishes parameterization compatibility only and supplies no precision-electroweak or leptonic-flavor evidence for CHC.

Funding and competing interests..

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