# UTFANSWF v23 — SANSWF Symmetry / Stability / Unitarity Preflight

## Purpose

This module makes the SANSWF layer executable after ANSWF and before the unified UTFANSWF embedding. Harness execution label **0.3 / SANSWF** is an ordering label only; historical UTFANSWF row numbering is preserved.

## Upstream dependency

SANSWF requires:

`PASS_ANSWF_EFT_POSITIVITY_TO_SANSWF_HANDOFF`.

## SU(5) gauge-consistency check

The gate constructs all 24 Hermitian fundamental generators of `SU(5)` with

`Tr(T^a T^b) = 1/2 delta^{ab}`

and computes the structure constants from the commutator. For the canonical adjoint breaking direction proportional to

`diag(2,2,2,-3,-3)`,

the gate reconstructs the adjoint mass operator in the index-safe form

`R_ad = f^{acd} v_c`,  `M^2 = g^2 R R^T`.

The structural test sets `g=1`; this is a normalization probe, not a GUT-scale mass prediction. `M^2` must be positive semidefinite with 12 zero modes (`SU(3)xSU(2)xU(1)`) and 12 positive broken-generator modes.

This replaces the compact manuscript expression with a reproducible index-safe implementation.

## Exact BFB geometry for the minimal adjoint potential

For

`V = lambda_H (H^dag H)^2 + lambda_A Tr(Sigma^4) + lambda_p (H^dag H) Tr(Sigma^2)`,

define

`r = Tr(Sigma^4)/(Tr(Sigma^2))^2`

for nonzero traceless Hermitian `5x5` adjoint directions. At extrema with fixed trace and norm, the Lagrange equation is cubic, so stationary eigenvalue patterns have at most three distinct values.

Two-value multiplicities `(k,5-k)` give the endpoint candidates:

- `k=1 or 4`: `r=13/20`;
- `k=2 or 3`: `r=7/30`.

Three-distinct-root stationary multiplicities give `r=1/2` or `r=1/4`, both internal to that interval. Hence the executable minimal-potential interval is

`7/30 <= r <= 13/20`.

Writing `x=H^dag H>=0`, `y=Tr(Sigma^2)>=0`, `t=x/y`, the quartic form becomes

`V/y^2 = lambda_H t^2 + lambda_p t + lambda_A r`.

The exact minimal-potential BFB condition therefore uses `r_min=7/30`, not an implicit `r=1`. For the current benchmark `(0.13,0.20,0.10)`, the portal is positive and the minimum occurs at `t=0`, leaving a strictly positive adjoint margin.

## Full tree-level contact matrix

The gate does not use the legacy summary values `|a0|~0.12` or `lambda_eig~1.2`, because those values are mutually inconsistent and the earlier appendix left the group factors `kappa_A,kappa_p` unevaluated.

Instead it derives the fourth derivatives of the stated potential in canonical real components:

- 4 Higgs real components;
- 24 SU(5) adjoint real components;
- 28 real scalar components total;
- 406 unordered symmetric two-particle channels.

The two-particle basis uses a `sqrt(2)` normalization factor for identical-component channels, and the partial-wave convention is explicitly

`a0 = M_const/(16*pi)`.

At `(lambda_H,lambda_A,lambda_p)=(0.13,0.20,0.10)`, the projected singlet matrix is

`[[0.78, 0.4898979486], [0.4898979486, 1.88]]`,

with eigenvalues approximately `0.5934540069` and `2.0665459931`. The full 406-channel diagonalization independently confirms that the global maximum absolute eigenvalue is the same:

`max |lambda_eig| = 2.0665459931`,

so

`max |a0| = 0.04111262624 < 1/2`.

This closes the missing tree-level group contractions under the declared normalization.

## Layer boundary

SANSWF closes gauge consistency, minimal-potential BFB stability, and benchmark tree-level scalar partial-wave unitarity. It does not claim to close:

- unified anomaly cancellation;
- heavy-threshold gauge-coupling unification;
- seesaw parameter consistency;
- proton-sector predictions;
- loop-level scalar unitarity;
- BFB conditions after adding new independent quartic invariants.

Those belong to the downstream UTFANSWF embedding or a future extended scalar-sector gate.

## PASS rule

SANSWF is **Critical / Blocking**. PASS requires the ANSWF handoff, SU(5) algebra/mass checks, corrected BFB check, and all 406 tree-level scalar contact channels to satisfy the declared unitarity bound.
