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
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
The neutral mass block depends only on the charged complex-triplet scale:
so
Therefore
The dependence on gauge couplings cancels.
3. Misalignment expansion
Let
Then the result is exactly
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 \(w=2k\),
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 builds the exact gauge masses at unequal \(w,k\), verifies the coupling-independent formula and checks its \(\varepsilon\)-expansion.