# Custodial Alignment and Exact \(\rho\)-Parameter Sensitivity

**Date:** 17 October 2026  
**Branches:** B05, B06 and B11  
**Claim status:** exact tree-level gauge-mass relation for the aligned-orientation, unequal-norm vacuum; observational fitting and loop corrections remain open

## 1. Unequal trace and traceless vacuum values

Use the orientation

\[
u=ke_1,qquad v=ke_2,qquad a=we_3,
\]

but do not assume \(w=k\). The complex charged triplet has vacuum scale \(k\), while the neutral real triplet supplied by the Jordan trace axis has scale \(w\).

## 2. Exact gauge masses

The two charged weak generators have equal mass

\[
m_W^2=g^2(w^2+k^2).
\]

The neutral mass block depends only on the charged complex-triplet scale:

\[
M_0^2
=2k^2
\begin{pmatrix}
g^2&-gg_E\\
-gg_E&g_E^2
\end{pmatrix},
\]

so

\[
m_Z^2=2k^2(g^2+g_E^2),
\qquad
\cos^2\theta_K=\frac{g^2}{g^2+g_E^2}.
\]

Therefore

\[
\boxed{
\rho(w,k)
=\frac{m_W^2}{m_Z^2\cos^2\theta_K}
=\frac{w^2+k^2}{2k^2}.
}
\]

The dependence on gauge couplings cancels.

## 3. Misalignment expansion

Let

\[
\frac{w}{k}=1+\varepsilon.
\]

Then the result is exactly

\[
\boxed{
\rho-1=\varepsilon+\frac12\varepsilon^2.
}
\]

Near the custodial point, the deviation is linear rather than quadratic. The equal-vacuum condition must therefore be protected with the same precision demanded of \(\rho\).

For the executable checkpoint \(w=2k\),

\[
\rho=\frac52.
\]

This is paired with a locally stable radial stationary point of the general Jordan-native potential, proving that symmetry-allowed dynamics can sit far from custodial alignment.

## 4. Observable meaning

The mass ratio is the first clean discriminator generated by the full-carrier branch:

- exact custodial alignment gives \(\rho=1\);
- the original two-column lock corresponds effectively to \(w=0\) and gives \(\rho=1/2\);
- independent trace/traceless vacuum values move continuously between and beyond these cases.

This turns “custodial symmetry must be controlled” into a quantitative requirement on a specific vacuum ratio.

## 5. Branch decision

**Stop-everything positive:** the Jordan/Furey dynamics predicts \(w/k=1\), or a calculable tiny departure consistent with the complete electroweak fit.

**Negative:** generic determinant-sensitive terms yield an order-one \(w/k-1\), or loop corrections require repeated retuning.

**Current result:** the full-carrier repair is exact at one aligned point but linearly sensitive to the trace/traceless vacuum mismatch allowed by native symmetry.

**Unsafe shortcut:** quote \(\rho=1\) from the field content without solving the vacuum-alignment equations.

## 6. Certificate

The [rho-misalignment certificate](../../../verification/verify/verify-full-lock-rho-misalignment.mjs) builds the exact gauge masses at unequal \(w,k\), verifies the coupling-independent formula and checks its \(\varepsilon\)-expansion.

## Companion documents

- [General Jordan-native full-lock potential](89_jordan_native_full_lock_potential.md)
- [Full-Jordan-carrier custodial completion](87_full_jordan_carrier_custodial_completion.md)
- [Complex-triplet custodial obstruction](85_locking_complex_triplet_and_custodial_obstruction.md)
- [Standard Model as stress–energy](01_standard_model_as_stress_energy_full_analysis.md)
- [Custodial scalar-loop closure](93_full_lock_custodial_loop_closure.md)
- [Full-lock custodial Hessian](94_full_lock_custodial_hessian_and_charge_spectrum.md)