Restricted Electroweak Doublets, Yukawa Maps, and Flavor Hierarchies in the CHC Framework

One interaction channel remains long-ranged while its weak charged relatives are short-ranged and longitudinally loaded. At the same time, matter families repeat the same broad charge pattern without repeating the same masses or the same mixing angles. These are not bookkeeping curiosities. They are the most public electroweak facts: a protected neutral channel, loaded weak carriers, and structured family mixing [citation].

An admitted gauge sheet, a generator-resolved gauge--chiral admissibility closure, and an electroweak-facing non-identity statement fix the benchmark-facing interface. These results prevent false identification of the order-parameter object with the Standard-Model Higgs sector and isolate what a restricted interface contract can constrain on a declared family. They do not yet build a complex electroweak doublet, a vacuum manifold, matrix-valued Yukawa maps, CKM/PMNS mismatch matrices, or an explicit flavor hierarchy on an admitted family. The missing slot is therefore no longer comparison. It is completion on one fixed family [citation].

The constructive route is taken here. No identity between the backbone scalar/order-parameter object $\HH$TeX\HH and the Standard-Model Higgs field is asserted. What is constructed is an admitted completion object built from the admitted gauge--chiral / order-parameter data so that the weak-boson mass-loading grammar and the family-mismatch grammar appear on one declared branch while exact object-level non-identity remains intact.

A single radial loading amplitude can decide whether a channel is loaded or not, but it cannot by itself determine how two weak carriers mix, why one neutral direction stays protected, or why the same broad charge pattern can come with different masses in different families. Those questions require an exact weak-boson mass matrix with a protected neutral kernel, one charged weak pair, one loaded neutral channel, and map-valued family response data on the same admitted family. The completion problem is therefore not merely the existence of a doublet-like field; it is the simultaneous fixing of mass loading, neutral protection, unitary mismatch, and structural hierarchy on one branch.

definition: Benchmark electroweak completion grammar. The benchmark electroweak completion grammar is the tuple

\mathfrak B_{\mathrm{EW}}=(\Dfield_{\mathrm{SM}},\mathcal M_{\mathrm{vac}}^{\mathrm{SM}},\{\Yuk_f^{\mathrm{SM}}\}_{f\in\{u,d,e,\nu\}},\CKM^{\mathrm{SM}},\PMNS^{\mathrm{SM}}),

with the following benchmark properties:

- a complex two-component scalar doublet, - a vacuum orbit selected by a nonzero symmetry-breaking scale $\vstar$TeX\vstar, - weak-boson mass loading equation m_W^2=(g^2^2)/(4), m_Z^2=((g^2+g'^2)^2)/(4), m_gamma^2=0, equation - matrix-valued Yukawa maps with equation M_f=()/(\sqrt2) \Yuk_f^SM, equation - quark and lepton mismatch matrices equation ^SM=U_u,L^ U_d,L, ^SM=U_e,L^ U_nu,L, equation where the unitary matrices diagonalize the corresponding left sectors.

remark: Benchmark role. reference fixes what counts here as a doublet, a vacuum orbit, a weak-boson mass-loading grammar, a matrix-valued Yukawa map, and a family-mismatch matrix. It is the benchmark target, not yet the completion itself [citation].

definition: Admitted electroweak completion family. An admitted electroweak completion family is a tuple

\FEW=(\GEW,\Cchi,\Wload,\Brep)

consisting of:

- an admitted local gauge-facing sheet $\GEW$TeX\GEW carrying one declared $\SU(2)\times \Uone(1)$TeX\SU(2)\times \Uone(1)-facing benchmark action on one declared cell; - an admitted left/right response family $\Cchi$TeX\Cchi containing one normalized interface section $\sig\in\mathbb C^2$TeX\sig\in\mathbb C^2 with $\sig^\dagger\sig=1$TeX\sig^\dagger\sig=1; - an admitted loading window equation =(,rho,), equation where $\rho(\HH)\ge 0$TeX\rho(\HH)\ge 0 is a scalar loading map and $\vstar$TeX\vstar is one declared nonzero value in its image; - a representation-and-response family $\Brep$TeX\Brep carrying one declared bounded matrix-valued Yukawa family $\{Y_f\}_{f\in\{u,d,e,\nu\}}$TeX\{Y_f\}_{f\in\{u,d,e,\nu\}} on one fixed family index set, with explicit singular-value form fixed later in reference.

The family is admitted only if the exact non-identity of the backbone scalar/order-parameter object with the Standard-Model Higgs field remains explicit, and only if the same family respects the declared gauge--chiral admissibility filters imported from the earlier closure layer.

remark: Imported exact non-identity. The previously established non-identity of the backbone scalar/order-parameter object with the Standard-Model Higgs/Yukawa sector is retained throughout. The construction below is strictly narrower: one admitted family is tested for an electroweak-facing completion object built from that backbone object together with additional interface data, without collapsing object-level non-identity into practical identity.

definition: Complex doublet completion object. Given an admitted family $\FEW$TeX\FEW, define the complex doublet completion object by

\Dfield(x):=\frac{\rho(\HH(x))}{\sqrt2}\,\sig(x),
\qquad \sig(x)^\dagger\sig(x)=1.

The object $\Dfield$TeX\Dfield is not identical to the backbone scalar/order-parameter object $\HH$TeX\HH. It is an interface completion object built from the radial loading map $\rho(\HH)$TeX\rho(\HH) and the normalized orientation section $\sig$TeX\sig on the admitted family.

On the declared gauge-facing sheet we use the local action

\sig\mapsto e^{i\alpha_Y/2}U_2\sig,
\qquad U_2\in \SU(2),

with covariant derivative

\Dmu\Dfield=
\left(\partial_\mu-
\frac{i g}{2}\tau^a W_\mu^a-
\frac{i g'}{2}B_\mu\right)\Dfield.

The completion potential is taken in the admitted family to be

V_{\mathrm{EW}}(\Dfield)=\lambda\left(\Dfield^\dagger\Dfield-\frac{\vstar^2}{2}\right)^2,
\qquad \lambda>0.

This is a declared interface potential. It is not a global identification of the CHC backbone scalar with the Standard-Model Higgs field.

remark: Weak-boson mass loading versus scalar identity. The radial map $\rho(\HH)$TeX\rho(\HH) controls weak-boson mass loading on the admitted family, but it does not by itself create a doublet. The doublet appears only after pairing the radial loading map with the normalized interface section $\sig$TeX\sig. This is the central CHC-native move: weak-boson mass loading is a completion problem on an admitted family, not a direct scalar-vev identity.

proposition: Vacuum-Manifold and Goldstone-Pattern Proposition. Let $\FEW$TeX\FEW be an admitted family and let $\Dfield$TeX\Dfield and $V_{\mathrm{EW}}$TeXV_{\mathrm{EW}} be given by reference. Then:

- the raw vacuum set is equation = C^2:\ ^=(^2)/(2) S^3; equation - for the representative vacuum $\Dfield_0=(0,\vstar/\sqrt2)^T$TeX\Dfield_0=(0,\vstar/\sqrt2)^T, the residual neutral generator equation :=T_3+(Y)/(2) equation stabilizes $\Dfield_0$TeX\Dfield_0; - the gauge-orbit directions associated with the broken generators produce the usual three Goldstone directions, while the radial direction remains physical; - the gauge-boson mass term extracted from $|\Dmu\Dfield_0|^2$TeX|\Dmu\Dfield_0|^2 is equation (^2)/(8)[g^2((W_mu^1)^2+(W_mu^2)^2)+(gW_mu^3-g'B_mu)^2], equation so that equation m_W^2=(g^2^2)/(4), m_Z^2=((g^2+g'^2)^2)/(4), m_gamma^2=0; equation - in the real basis $(W_\mu^1,W_\mu^2,W_\mu^3,B_\mu)$TeX(W_\mu^1,W_\mu^2,W_\mu^3,B_\mu), the vector mass matrix is equation M_V^2=(^2)/(4) pmatrix g^2 & 0 & 0 & 0\\ 0 & g^2 & 0 & 0\\ 0 & 0 & g^2 & -gg'\\ 0 & 0 & -gg' & g'^2 pmatrix, equation whose neutral block has rank one with a one-dimensional kernel spanned by the protected neutral direction, while the charged/neutral channel decomposition contains exactly one charged weak pair and one loaded neutral channel.

Thus one protected neutral direction, one charged weak pair, and one loaded neutral channel arise on the admitted family.

proof. The minima of reference satisfy $\Dfield^\dagger\Dfield=\vstar^2/2$TeX\Dfield^\dagger\Dfield=\vstar^2/2, giving reference. Choosing $\Dfield_0=(0,\vstar/\sqrt2)^T$TeX\Dfield_0=(0,\vstar/\sqrt2)^T, one checks directly that the generator $\Qem=T_3+Y/2$TeX\Qem=T_3+Y/2 annihilates $\Dfield_0$TeX\Dfield_0, so a neutral stabilizer remains. Expanding $|\Dmu\Dfield|^2$TeX|\Dmu\Dfield|^2 around $\Dfield_0$TeX\Dfield_0 yields reference. Passing to

W_\mu^{\pm}=\frac{W_\mu^1\mp iW_\mu^2}{\sqrt2},
\qquad
 Z_\mu 
 A_\mu 
=

\cos\theta_W  -\sin\theta_W

\sin\theta_W  \cos\theta_W

 W_\mu^3 
 B_\mu ,
\qquad
\tan\theta_W=\frac{g'}{g},

then reference follows. The neutral combination $A_\mu$TeXA_\mu remains unloaded, while $W^\pm$TeXW^\pm and $Z$TeXZ are loaded. The three broken orbit directions are the Goldstone directions and the radial mode is the remaining scalar mode.

remark: Completion-object vacuum-orbit status. The vacuum orbit fixed in reference is the orbit of the declared completion object $\Dfield$TeX\Dfield under the admitted interface potential on one admitted family. It is not promoted here to a universal vacuum-selection law.

corollary: Neutral-channel protection. On the admitted family, the long-range neutral channel is protected by the residual stabilizer $\Qem$TeX\Qem, whereas the charged weak channels and the loaded neutral channel are mass-loaded by the same completion scale $\vstar$TeX\vstar.

corollary: Weak-boson mass-matrix rank and neutral protection. Let $M_V^2$TeXM_V^2 be the vector mass matrix reference. Then the neutral $(W_\mu^3,B_\mu)$TeX(W_\mu^3,B_\mu) block has rank one and a one-dimensional kernel generated by the protected neutral direction. Equivalently, after passing to the charged/neutral channel basis $(W_\mu^+,W_\mu^-,Z_\mu,A_\mu)$TeX(W_\mu^+,W_\mu^-,Z_\mu,A_\mu), the loaded channel space has rank two, represented by one charged weak pair and one loaded neutral channel, while the photon direction remains exactly massless on the admitted family.

definition: Matrix-valued Yukawa family. For each fermion sector $f\in\{u,d,e,\nu\}$TeXf\in\{u,d,e,\nu\}, let

Y_f:=L_f\Lamf R_f^\dagger,
\qquad
\Lamf=\diag(y_{f,1},y_{f,2},y_{f,3}),
\qquad
0\le y_{f,1}\le y_{f,2}\le y_{f,3},

with $L_f,R_f$TeXL_f,R_f unitary. On the admitted family the Yukawa sector is the interaction

\mathcal L_Y
= -\bar Q_L Y_d \Dfield \, d_R
   -\bar Q_L Y_u \, \tilde{\Dfield} \, u_R 

\quad -\bar L_L Y_e \Dfield \, e_R
   -\bar L_L Y_{\nu} \, \tilde{\Dfield} \, \nu_R + \mathrm{h.c.},

where $\tilde{\Dfield}=i\tau_2 \Dfield^*$TeX\tilde{\Dfield}=i\tau_2 \Dfield^*. The neutrino sector is treated only in this minimal Dirac-type benchmark grammar; no Majorana or seesaw completion is claimed.

At the vacuum scale $\vstar$TeX\vstar, the mass matrices are

M_f=\frac{\vstar}{\sqrt2}Y_f,

so the singular values of $M_f$TeXM_f are $\vstar y_{f,i}/\sqrt2$TeX\vstar y_{f,i}/\sqrt2. The ordered singular spectrum $(y_{f,1},y_{f,2},y_{f,3})$TeX(y_{f,1},y_{f,2},y_{f,3}) is therefore the declared family hierarchy on the admitted branch. This hierarchy is restricted: it is carried by one admitted matrix family, not by a claim of universal flavor closure.

remark: Flavor differentiation as map-valued sheet response. The CHC-native gain is not that flavor is explained by an arbitrary texture dropped in by hand. The gain is that family differentiation is represented by a declared matrix-valued response map on one admitted family. The common loading scale $\vstar$TeX\vstar is supplied by the admitted loading window, but the matrix-valued Yukawa family carries independent admitted family response data and is not determined by $\rho(\HH)$TeX\rho(\HH) alone.

proposition: Declared hierarchy law on the admitted family. For each fermion sector $f\in\{u,d,e,\nu\}$TeXf\in\{u,d,e,\nu\}, the ordered singular spectrum

\mathbf h_f:=(y_{f,1},y_{f,2},y_{f,3}),
\qquad
0\le y_{f,1}\le y_{f,2}\le y_{f,3},

defines the declared family hierarchy law of the admitted branch. The spectrum $\mathbf h_f$TeX\mathbf h_f is invariant under all biunitary rephasings and common family relabelings that preserve the diagonal ordering. Hence any strict hierarchy condition

y_{f,1}<y_{f,2}<y_{f,3}
 \quad\text{or more generally}\quad
 \frac{y_{f,i+1}}{y_{f,i}}\ge \kappa_{f,i}>1

for the nonzero singular values is a structural property of the admitted map family and not a removable basis artifact.

proof. The ordered singular values are determined uniquely by the positive spectrum of $M_f^\dagger M_f$TeXM_f^\dagger M_f, hence by $Y_f^\dagger Y_f$TeXY_f^\dagger Y_f. Biunitary rephasings leave that spectrum unchanged, and a simultaneous family relabeling can only permute the singular values before the imposed ordering is restored. Therefore the ordered spectrum $\mathbf h_f$TeX\mathbf h_f and any inequality or gap condition written directly in terms of its entries are invariant data of the admitted family.

proposition: Matrix Yukawa and Mixing-Mismatch Proposition. Let $Y_f$TeXY_f be the matrix-valued Yukawa family of reference. Then:

- each mass matrix $M_f$TeXM_f admits a biunitary diagonalization equation U_f,L^ M_f U_f,R=(m_f,1,m_f,2,m_f,3), equation with $m_{f,i}=\vstar y_{f,i}/\sqrt2\ge 0$TeXm_{f,i}=\vstar y_{f,i}/\sqrt2\ge 0; - the quark and lepton mismatch maps equation :=U_u,L^ U_d,L, :=U_e,L^ U_nu,L equation are unitary on the admitted family; - if the left diagonalizers coincide, the corresponding mismatch map is the identity, so nontrivial mixing occurs exactly when the diagonalization sectors do not align.

Thus CKM/PMNS-like objects enter as mismatch maps between diagonalization sectors, not as independent external decorations.

proof. Equation reference is a singular-value decomposition, so reference implies the biunitary diagonalization reference with nonnegative singular values. The products in reference are unitary because they are products of unitary matrices. If $U_{u,L}=U_{d,L}$TeXU_{u,L}=U_{d,L} then $\CKM=\mathbf 1$TeX\CKM=\mathbf 1; similarly for $\PMNS$TeX\PMNS. Therefore nontrivial family mixing is exactly the mismatch of left diagonalization sectors on the admitted family.

corollary: Unitary mismatch theorem on the admitted family. On the admitted family, the mismatch maps $\CKM$TeX\CKM and $\PMNS$TeX\PMNS are unitary and remain unchanged under any simultaneous family relabeling applied equally to both compared left sectors. Consequently, once a mismatch map is nontrivial, its nontriviality cannot be removed by a common relabeling of families that preserves the declared diagonal spectra.

proof. If a common relabeling is represented by a unitary matrix $S$TeXS, then the compared left diagonalizers transform as $U_{u,L}\mapsto S U_{u,L}$TeXU_{u,L}\mapsto S U_{u,L} and $U_{d,L}\mapsto S U_{d,L}$TeXU_{d,L}\mapsto S U_{d,L}, or analogously in the lepton sector. Hence

(SU_{u,L})^\dagger (SU_{d,L})=U_{u,L}^\dagger U_{d,L}=\CKM,

and similarly for $\PMNS$TeX\PMNS. Therefore a common relabeling cannot turn a nontrivial mismatch map into the identity.

theorem: Restricted Electroweak Completion Theorem. Let $\FEW$TeX\FEW be an admitted electroweak completion family in the sense of reference. Assume:

- one normalized interface section $\sig\in\mathbb C^2$TeX\sig\in\mathbb C^2 exists on the declared family; - the loading map $\rho(\HH)$TeX\rho(\HH) attains one nonzero value $\vstar$TeX\vstar on the admitted order-parameter window; - the potential reference is used with $\lambda>0$TeX\lambda>0; - the matrix-valued Yukawa family $\{Y_f\}$TeX\{Y_f\} is bounded on the same declared family.

Then the same declared family supports all of the following simultaneously:

- a complex doublet completion object $\Dfield$TeX\Dfield on the admitted family; - a completion-object vacuum-orbit grammar and one protected neutral direction; - the benchmark weak-boson mass-loading grammar $m_\gamma=0$TeXm_\gamma=0, $m_W>0$TeXm_W>0, $m_Z>m_W$TeXm_Z>m_W; - one declared bounded matrix-valued Yukawa family, unitary CKM/PMNS-like mismatch matrices, and a declared hierarchy law on the same branch; - exact object-level non-identity between the backbone scalar/order-parameter object $\HH$TeX\HH and the completion object $\Dfield$TeX\Dfield.

Hence a restricted electroweak-facing completion exists on the declared family without identifying the backbone scalar with the Standard-Model / Yukawa sector.

proof. Items (i)--(iii) follow from reference. Item (iv) follows from reference. For item (v), reference shows that $\Dfield$TeX\Dfield is built from the pair $(\rho(\HH),\sig)$TeX(\rho(\HH),\sig), not from $\HH$TeX\HH alone. The backbone object $\HH$TeX\HH is a scalar order-parameter object, whereas $\Dfield$TeX\Dfield is a complex doublet completion object carrying the declared local action. Since the two objects differ in both field type and representation content, no object-level identity is licensed. Therefore the declared family supports a restricted electroweak completion while preserving exact non-identity with the backbone scalar/order-parameter object.

remark: Restricted status. reference is family-conditioned and benchmark-facing. It does not assert that every branch carries such a completion, that the resulting family is phenomenologically unique, or that the backbone scalar is globally identical to the Standard-Model Higgs field. The theorem fixes existence and status on one declared admitted family only.

The declared completion fails if any of the following occurs:

- no normalized interface section $\sig\in\mathbb C^2$TeX\sig\in\mathbb C^2 can be given on the declared family; - no nonzero loading value $\vstar$TeX\vstar exists on the admitted order-parameter window; - no completion-object vacuum-orbit grammar with residual neutral stabilizer can be defined; - the benchmark weak-boson mass-loading grammar requires identification of $\HH$TeX\HH itself with a Higgs doublet; - no bounded matrix-valued Yukawa family exists on the same admitted branch; - CKM/PMNS-like mismatch requires additional off-family data not carried by the admitted family; - the construction violates previously admitted gauge--chiral consistency restrictions; - flavor hierarchy can be stated only by unconstrained texture insertion rather than by one declared matrix family.

The result does not establish:

- global identity between the backbone scalar and the Standard-Model Higgs field; - universal Standard-Model completion outside the declared family; - baryogenesis, CP-violation closure, or anomaly closure beyond previously imported gauge--chiral admissibility; - confinement, hadronization, or QCD dynamical closure; - ultraviolet completion, compactification, or vacuum-selection closure; - a full phenomenological fit to electroweak or flavor data.

One long-range neutral channel, short-range loaded weak carriers, and family mixing do not by themselves force scalar identity. They instead force a more precise statement: if a gauge--chiral / order-parameter family is to face the electroweak benchmark honestly, it must carry a complex doublet completion object, a completion-object vacuum-orbit grammar with a protected neutral direction, one declared bounded matrix-valued Yukawa family, and mismatch matrices that are all defined on the same declared branch.

That constructive route is realized here in restricted form on one admitted family. The backbone scalar/order-parameter object remains non-identical to the Standard-Model Higgs field, but an admitted interface completion object built from the same branch can reproduce the benchmark weak-boson mass-loading grammar and a matrix-valued family-mixing grammar. Nothing in the present construction weakens the exact non-identity firewall fixed at the interface layer. The realized completion remains family-conditioned and completion-object-level, not backbone Higgs identity.

What remains open is equally explicit. Collider, precision-electroweak, and flavor-data fit windows are not part of the present theorem set; nor does the declared family close anomaly questions beyond the imported gauge--chiral admissibility layer or supply ultraviolet completion. What is fixed here is the missing object slot between exact non-identity and a realized electroweak-facing completion on one declared admitted family.

Funding and competing interests..

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