# UTFANSWF Items 03 + 06 — Global Heavy-Ratio / Projector Uniqueness Closeout
## 2026-08-19

**Scope:** private research closeout.  
**Production/public artifacts:** unchanged.

## Question

Does the full coupled UTFANSWF structure uniquely force

\[
r \equiv \frac{M_{H_C}}{M_X}
=\frac{M_\Sigma}{M_X}
=\frac{\sqrt2}5,
\]

or is \(\sqrt2/5\) only one self-consistent candidate point?

## Exact Z6 sector

The exact residual-Z6 selection rule remains

\[
\Delta B=3k.
\]

Isolated \(\Delta B=1\) and \(\Delta B=2\) remain forbidden. This result is independent of the numerical heavy ratio \(r\).

## Projector uniqueness

The projector identities allow

\[
D_5=a\mu P_3,\qquad D_\Sigma=b\mu P_{\rm comm},
\]

while retaining the same ranks, kernels and block structure. Therefore projector algebra alone does not force \(a=b=1\).

## Minimal-SUSY-SU(5) provenance correction

The heavy masses obey

\[
M_X^2=\frac5{12}g^2V^2,\qquad
M_{H_C}^2=\frac5{24}\lambda_2^2V^2,\qquad
M_\Sigma^2=\frac{15}{32}\lambda_1^2V^2.
\]

Hence

\[
\frac{M_{H_C}}{M_X}=\frac{\lambda_2}{\sqrt2 g},
\qquad
\frac{M_\Sigma}{M_X}=\frac{3\lambda_1}{2\sqrt2 g}.
\]

With \(g=1/\sqrt2\),

\[
\frac{M_{H_C}}{M_X}=\lambda_2,
\qquad
\frac{M_\Sigma}{M_X}=\frac32\lambda_1.
\]

Therefore choosing

\[
\lambda_2=\frac{\sqrt2}5,\qquad
\lambda_1=\frac{2\sqrt2}{15}
\]

is algebraically equivalent to choosing the desired heavy ratios unless those couplings are independently derived upstream.

No earlier saved derivation of these lambda values was found. The Aug-18 minimal-SU(5) calculation is therefore a **consistent embedding**, not an independent derivation of the ratio.

The one-loop wave-function/Konishi cancellation and matching-order audit remain valid conditional on the selected boundary; neither selects \(r\).

## Surviving one-parameter family

Even after imposing equal heavy masses,

\[
\boxed{\lambda_2=r,\qquad \lambda_1=\frac{2r}3,\qquad g=\frac1{\sqrt2}}
\]

gives

\[
M_{H_C}=M_\Sigma,\qquad
\frac{M_{H_C}}{M_X}=
\frac{M_\Sigma}{M_X}=r.
\]

A positive continuous modulus survives.

## Blind global sweep

For every tested \(r\), the sweep retains:

- \(g^2=1/2\);
- \(\alpha_U^{-1}=8\pi\);
- equal \(H_C\) and \(\Sigma\) masses;
- the same two-loop SM/MSSM gauge running;
- the same one-loop heavy-threshold weights;
- only \(\alpha_3(M_Z)=0.1184\) as the low-energy scale calibration.

No electromagnetic quantity is used to choose \(r\).

Representative results:

| r | MX [GeV] | MHC=MSigma [GeV] | alpha_em^-1(MZ) post-freeze | scaled D6 tau [yr] |
|---:|---:|---:|---:|---:|
| 0.100000000 | 6.669353e+16 | 6.669353e+15 | 130.648557 | 8.335e+38 |
| 0.200000000 | 2.596400e+16 | 5.192800e+15 | 128.516384 | 1.914e+37 |
| 0.250000000 | 1.916201e+16 | 4.790502e+15 | 127.829690 | 5.680e+36 |
| 0.282842712 | 1.619787e+16 | 4.581448e+15 | 127.449790 | 2.900e+36 |
| 0.350000000 | 1.211925e+16 | 4.241737e+15 | 126.793974 | 9.088e+35 |
| 0.500000000 | 7.456292e+15 | 3.728146e+15 | 125.695711 | 1.302e+35 |
| 0.700000000 | 4.714976e+15 | 3.300483e+15 | 124.659311 | 2.082e+34 |
| 0.800000000 | 3.930764e+15 | 3.144611e+15 | 124.247914 | 1.006e+34 |

The QCD equation re-closes continuously by shifting \(M_X\). QCD therefore does not select \(r\).

## Proton/heavy-scale bound

Using the existing dimension-6 diagnostic normalization,

\[
\tau_p^{D6}(r)
=
2.90\times10^{36}
\left[\frac{M_X(r)}{M_X(\sqrt2/5)}\right]^4
\,{\rm yr},
\]

the old \(10^{34}\) yr safety floor produces only

\[
\boxed{r\lesssim 0.800836913}.
\]

This is an interval, not a selector. The exact-Z6 rule itself is independent of \(r\).

## Flavor/Hodge sector

Item 04 inherited \(\epsilon=\sqrt2/5\) from the heavy boundary. It did not independently derive the number.

Replacing the inherited symbol with \(\epsilon=r\) leaves an ordered suppression hierarchy over a continuous \(0<r<1\) range. Using Item 04 to force \(r=\sqrt2/5\) would therefore be circular.

## Electroweak diagnostic — not a selector

Only after the blind sweep is frozen may the electromagnetic output be inspected.

The old held-out value

\[
\alpha_{em}^{-1}(M_Z)=127.94049864183329
\]

would be matched by

\[
r\approx 0.241159001,
\]

not by \(\sqrt2/5\).

This is **not accepted as a solution**. Selecting \(r\) from that holdout would be target fitting. It is recorded only as a diagnostic demonstration that the holdout does not independently select the candidate ratio.

## Global ledger

| Requirement | Status |
|---|---|
| Exact residual Z6 | PASS |
| Projector rank/kernel structure | PASS |
| Equal-heavy-mass SU(5) embedding | PASS over continuous family |
| QCD-only dimensional reclosure | PASS over continuous family |
| One-loop Z/Konishi ownership | PASS; does not select r |
| Matching-order consistency | PASS; does not select r |
| Legacy proton/heavy safety | PASS over finite interval |
| Flavor/Hodge consistency | compatible over interval; not independent selector |
| Unique target-free heavy ratio | **FAIL** |
| sqrt(2)/5 candidate | PASS_CANDIDATE |

## Final verdict

`PASS_EXACT_Z6_SELECTION_RULE__PASS_SQRT2_OVER5_SELF_CONSISTENT_EMBEDDING__FAIL_GLOBAL_UNIQUE_HEAVY_RATIO_SELECTOR__ONE_PARAMETER_R_MODULUS_SURVIVES`

### Item 03

**Closed as conditional/self-consistent candidate.**  
\(r=\sqrt2/5\) is viable and globally compatible, but current target-free structure does not uniquely force it.

### Item 06

**Exact Z6: CLOSED / PASS.**  
**Projector/heavy uniqueness: CLOSED / CLEAN FAIL.**

## Production consequence

None. No public prediction changed. No electromagnetic target was used to select \(r\). No failed branch was retuned.
