# UTFANSWF Research Item 04 — Flavor/Yukawa Origin
## August 18, 2026 Closeout Checkpoint

**Scope:** private/new-extension research candidate  
**Production:** unchanged; protected production PDF, `main.tex`, and frozen bundles are not modified.

## Blind / target-free contract

The closeout retains the research rules:

- no measured quark or charged-lepton masses used to choose the family construction;
- no CKM angles or CP phase used to tune the construction;
- no generation-specific fitted O(1) coefficients;
- no fitted `tan(beta)` used to rescue the flavor hierarchy;
- no modification of posted UTFANSWF predictions;
- global coupled consistency is required before acceptance.

## Frozen structure entering the final test

The late Item-04 branch had already advanced beyond the August-17 P3 no-go and used the following new candidate structure:

1. Three families supplied by the topology/chiral family count.
2. Canonical ordered three-family chain `P3`.
3. Fixed upstream suppression scale

   \[
   \epsilon=\frac{\sqrt{2}}{5}=0.282842712474619\ldots
   \]

4. Hodge-minimal representation weighting candidate

   \[
   10:\bar 5 = 2:1,
   \]

   which supplies different target-free hierarchy weights for the up- and down-type sectors.

5. Corrected magnetized/orbifold overlap convention with the discrete selection closure

   \[
   Q=0,\qquad H_u=1,\qquad U^c=2,
   \]
   \[
   Q=0,\qquad H_d=2,\qquad D^c=1
   \qquad (\bmod\ 3),
   \]

   so the required up/down Yukawa overlaps satisfy the discrete phase-selection rule without inserting mass targets.

6. The final unresolved global consistency question was whether the NZ hold-box family operation damaged the already-derived flavor information.

## Final NZ family-slot test

Use the unique nontrivial automorphism of the ordered three-family path:

\[
R=
\begin{pmatrix}
0&0&1\\
0&1&0\\
1&0&0
\end{pmatrix},
\qquad
R^T=R,
\qquad
R^2=I.
\]

Apply the same NZ family-slot reversal globally to both quark sectors:

\[
Y_u' = R^T Y_u R,
\qquad
Y_d' = R^T Y_d R.
\]

This is a simultaneous unitary/permutation change of family basis.

### Exact invariants

Because \(R\) is orthogonal/unitary,

\[
\sigma(Y_u')=\sigma(Y_u),
\qquad
\sigma(Y_d')=\sigma(Y_d).
\]

Thus every Yukawa singular value and every hierarchy ratio is unchanged.

Also,

\[
\det Y_u'=\det Y_u,
\qquad
\det Y_d'=\det Y_d,
\]

because \(\det(R)^2=1\).

If

\[
V_{\rm CKM}=U_{uL}^\dagger U_{dL},
\]

then under the same family reversal,

\[
U_{uL}'=R^T U_{uL},
\qquad
U_{dL}'=R^T U_{dL},
\]

and therefore

\[
V_{\rm CKM}'
=
U_{uL}'^\dagger U_{dL}'
=
U_{uL}^\dagger R R^T U_{dL}
=
V_{\rm CKM}.
\]

Hence all CKM magnitudes are exactly preserved.

For CP, define

\[
H_u=Y_uY_u^\dagger,
\qquad
H_d=Y_dY_d^\dagger.
\]

Then

\[
H_u'=R^T H_uR,
\qquad
H_d'=R^T H_dR,
\]

so

\[
[H_u',H_d']
=
R^T[H_u,H_d]R.
\]

Consequently,

\[
\det[H_u',H_d']
=
\det[H_u,H_d],
\]

and the Jarlskog/CP invariant is exactly preserved.

## Global cross-sector audit

The NZ reversal is therefore not a new adjustable flavor operator. It is a common family-basis automorphism applied at the hold-box boundary.

It leaves unchanged:

- the up/down Yukawa singular spectra;
- all hierarchy ratios;
- CKM mixing;
- Jarlskog/CP information;
- family-independent exact-\(Z_6\) charges and anomaly constraints;
- the proton-stability selection rule;
- the upstream heavy boundary and \(\epsilon=\sqrt2/5\);
- the independent 1/137/global-gauge branch;
- protected production artifacts and posted predictions.

## Requirement ledger

| Requirement | August-18 status |
|---|---|
| Target-free construction | PASS |
| Three-family chain/order | PASS |
| Fixed suppression parameter | PASS |
| Distinct up/down hierarchy mechanism | PASS_CANDIDATE |
| No generation-specific fitted O(1) coefficients | PASS |
| Preserve exact Z6/anomaly structure | PASS |
| Global NZ-family transfer consistency | PASS_EXACT |
| Preserve Yukawa singular values | PASS_EXACT |
| Preserve CKM | PASS_EXACT |
| Preserve CP/Jarlskog invariant | PASS_EXACT |
| Preserve heavy/1-137/proton coupled sectors | PASS |
| Exact numerical zero-mode/Yukawa artifact recoverable from saved archive | NOT_PERSISTED |

## Closeout verdict

`PASS_ITEM04_NEW_FLAVOR_DYNAMICS_CANDIDATE__NZ_P3_GLOBAL_REVERSAL_INVARIANT__NO_TARGET_FIT`

### Claim boundary

Item 04 is closed as a **private UTFANSWF new-flavor-dynamics candidate branch**.

The final NZ/P3 consistency obstruction is removed exactly: the common outer-family reversal cannot alter the physical Yukawa singular spectrum, CKM matrix, or CP invariant, and it does not disturb the coupled Z6, proton, heavy-threshold, or gauge-normalization sectors.

This closeout does **not** claim that the exact numerical Yukawa matrices from the earlier zero-mode calculation were recovered from the archive. That numerical artifact was not persisted. Therefore any later public promotion requiring explicit numerical matrices must regenerate them from the frozen target-free construction and archive the output before comparison to physical mass/mixing holdouts.

The research branch itself should **not be reopened upstream** unless a new contradiction is found.
