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

A coherent interaction family that reproduces one resonance mass but fails a width, an asymmetry, or a small mixing angle has not yet met experiment. The public benchmark is therefore not a single number but a fixed basket: weak-boson masses, neutral-current precision observables, leptonic width information, and flavor data must all survive on the same family without hidden retuning. A mathematically disciplined family can still be phenomenologically dead if it survives one channel only by resetting what another channel already fixed [citation]. Throughout, ``fit window'' means only a nonempty set-theoretic intersection of declared benchmark intervals on one fixed admitted family together with one explicit flavor-suppression gate; no global likelihood construction or unrestricted phenomenological survey is attempted.

A matter-coupled gauge sheet, an anomaly-admissible branch, and a restricted electroweak completion family are fixed on the admitted domain. What remains absent is a fit window on a declared benchmark basket. The missing question is therefore narrow and exact: can one fixed admitted family survive a minimal collider, precision-electroweak, and leptonic-flavor benchmark basket without hidden retuning across channels? The question is not whether can be globally tuned to every dataset, and it is not whether the admitted family is the Standard Model. It is whether one declared family supports one nonempty benchmark-window fit on one fixed basket together with one fixed same-family gate.

The front phenomena are therefore twofold. First, the same family must reproduce the weak-boson and neutral-current precision scales on one fixed sheet, including one asymmetry and one width block, without observable-specific redefinition of the family. Second, the same family must respect flavor mixing and flavor suppression without an independent reset of the flavor parameters channel by channel. The declared task is a fixed-basket fit-window problem rather than an unrestricted parameter scan. All benchmark intervals used below are declared windows on one fixed admitted family; no new world-average reanalysis, global likelihood construction, or unrestricted phenomenological survey is attempted.

definition: Admitted fit family. The admitted phenomenological fit 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.

No channel is allowed to carry its own private replacement for $\vev$TeX\vev, $(g,g')$TeX(g,g'), $\deltaNC$TeX\deltaNC, $(\deltaZ,\deltaL)$TeX(\deltaZ,\deltaL), or $\Uell$TeX\Uell. Any observable-specific counterterm that cannot be written as a single transformation of $(\vev,g,g',\deltaNC,\deltaZ,\deltaL,\Uell)$TeX(\vev,g,g',\deltaNC,\deltaZ,\deltaL,\Uell) defines a new family rather than a refinement of the same one. We read $\Thetafit$TeX\Thetafit with the subspace topology inherited from $\mathbb R^6\times U(3)$TeX\mathbb R^6\times U(3).

remark: Restricted flavor subset. The declared basket uses the realized leptonic-flavor subset of the admitted electroweak completion family. A full quark-sector flavor fit, complete CKM hierarchy, full hadronic nuisance treatment, and global flavor closure are deferred. The flavor block adopted here is therefore intentionally minimal and decisive rather than exhaustive.

definition: Declared observable basket. 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: Admitted-fit 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 fit window 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 interval form is the declared benchmark-window certification format for exact non-emptiness / emptiness claims on one fixed admitted family, with the flavor-suppression gate treated separately from the declared basket.

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 gate. The flavor-suppression ratio is

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

and is evaluated only on the admitted leptonic subset where $\stwo(\theta)>0$TeX\stwo(\theta)>0. On the admitted leptonic subset, the flavor-suppression gate 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: Flavor-Suppression Gate Proposition. The condition reference is a single family predicate on $\Uell$TeX\Uell. In particular, the admitted solar-angle window, the reactor-angle suppression, and the atmospheric entry are either supported jointly by the same mismatch matrix or not supported at all on the declared family. They cannot be recovered by assigning one matrix to $\ttwelve$TeX\ttwelve, another to $\sthree$TeX\sthree, and another to $\stwo$TeX\stwo.

proof. The three observables are functions of the same unitary matrix $\Uell$TeX\Uell. Therefore $\ttwelve$TeX\ttwelve, $\sthree$TeX\sthree, $\stwo$TeX\stwo, and $\Rsup$TeX\Rsup are fixed once $\Uell$TeX\Uell is fixed. Any separate reset of $\Uell$TeX\Uell for different observables leaves the admitted family $\Thetafit$TeX\Thetafit.

proposition: No-Retuning Across Channels Proposition. 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: No hidden per-channel counterterms. 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: Precision-Fit Compatibility Theorem on a Declared Observable Basket. 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).

In particular, the declared direct basket together with the explicit flavor-suppression gate admits a nonempty interval-certified fit window on one fixed admitted family.

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: Explicit benchmark-point construction on the declared direct basket and gate. 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}).

theorem: Fixed-Basket No-Fit Theorem. 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.

In that case the declared basket together with the declared flavor-suppression gate is structurally inaccessible on the admitted family at the adopted thresholds.

proof. This is an immediate rewriting of reference.

proposition: Exact basket-killing element. 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 the exact basket-killing element on the declared family.

proof. The first condition says that every declared direct-basket constraint except $a_\star$TeXa_\star can be satisfied simultaneously on one admitted family point while the declared flavor-suppression gate still holds. The second says that adding $a_\star$TeXa_\star destroys the intersection. Therefore $a_\star$TeXa_\star alone kills the fixed-basket window.

proposition: Exact gate-killing condition. 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 declared flavor-suppression gate is the exact gate-killing condition on the admitted family.

proof. The first condition says that the direct benchmark basket alone remains jointly satisfiable on one admitted family point. The second says that adjoining the declared flavor-suppression gate destroys that intersection. Therefore the gate, rather than any direct basket element, is the exact killing condition on the declared family.

Public validation companion..

A separate bounded public support companion, CHC-PFW-VP1: Public Covariance, Contour, and ROOT Support Checks for Precision-Observable Windows, records reproducibility checks for the declared basket. Its companion source summary contains a scalar/contour lane and a covariance/contour support comparison: the latter uses a LEP/SLD public covariance subblock, checks KamLAND public $\Delta\chi^2$TeX\Delta\chi^2-map contour membership for the declared $\ttwelve$TeX\ttwelve window, uses T2K public contour information for the declared $\stwo$TeX\stwo window, and retains scalar-source reproduction for the remaining declared windows. These are public companion lanes only; they are not a global electroweak or flavor likelihood and do not enlarge the parent-paper claim.

Explicit non-claims..

The declared basket is not enlarged beyond reference: no full global collider/electroweak/flavor fit, complete LHC phenomenology, full QCD nuisance treatment, cosmological fit, or final proof is assigned here. Quark-sector flavor objects are not assumed unless built on the admitted completion family. The fit-window result concerns only the declared basket together with the declared flavor-suppression gate on one fixed family.

A theory that looks coherent on paper but survives only one observable at a time has not yet reached the declared benchmark basket. The fit-window result therefore fixes a narrower target: one admitted gauge--chiral / electroweak-facing family, one declared basket, one declared flavor-suppression gate, one interval-certified fit predicate, and no channel-by-channel retuning. On that restricted problem, the fit question becomes mathematical: is the admitted-fit set nonempty or empty?

The answer remains family-typed and benchmark-conditioned. If one reference family point exists with the required weak-boson loading scale, neutral-current remainder, shared width-block remainder pair, and leptonic mismatch matrix, then continuity yields a nonempty fit window on the same family. If no such point exists, the fixed-basket no-fit theorem states that the window is empty on the same declared domain, while the exact basket-killing and gate-killing propositions identify the obstruction when a unique direct-basket killer or gate killer exists. In either case the paper converts phenomenology into an exact fixed-family benchmark-window problem. The present result is a benchmark-window gate on one fixed admitted family. It is not a full collider likelihood, a global electroweak/flavor fit, a Standard-Model-identity claim, or a UV-completion verdict.

Funding and competing interests..

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