# main.tex — Row 28 Proton-Stability Change Sheet (READ-ONLY AUDIT)

**Protection rule:** this file is guidance only. `main.tex` was not modified by the integration run.

## Replacement logic

- New protected candidate branch: exact gauged `Z6` gives `DeltaB = 3 k`; ordinary `DeltaB=1` proton decay and `DeltaB=2` baryon violation are forbidden.
- Conventional finite-lifetime SU(5) `X,Y` / color-triplet calculations should not be silently deleted. Preserve them as **legacy/conventional comparison branch** where historically useful.
- Any line currently labeling a finite proton lifetime as a UTFANSWF `PASS` must be revised if the protected branch is adopted.
- The protected branch has a separate two-loop unification compatibility diagnostic (~0.03523% spread with pinned v22 low-energy couplings), not an independent blind prediction.
- Keep the independent primitive `alpha_U^-1=8*pi` blind miss (~1.14564%) visible as an open UV-boundary issue.
- Do not claim a trinucleon lifetime: the `DeltaB=3` Wilson coefficient/scale is not fixed.

## Executive / forecast summaries

- **Line 157:** `\newcommand{\MHC}{M_{H_C}}`
- **Line 360:** `\item Nuclear / proton decay — Unified layer`
- **Line 715:** `Unification \& proton decay (compact forecast)`
- **Line 716:** `& \(M_X\!\sim\!10^{16}\,\mathrm{GeV};\ \tau_p^{(D6)}\!\sim\!10^{35}\,\mathrm{yr};\ \tau_p^{(D5)}\!\sim\!10^{34\text{--}36}\,\mathrm{yr}\)`

## Early unified-layer claims and status tables

- **Line 1338:** `\paragraph{Proton decay constraints.}`
- **Line 1339:** `A key prediction of GUT models is baryon number violation. Dimension-5 operators mediated by the heavy Higgs triplet take the form`
- **Line 1361:** `\includegraphics[width=0.7\textwidth]{figs/proton_lifetime_vs_MHC.pdf}`
- **Line 1362:** `\caption{Median proton lifetime $\tau_p$ vs heavy Higgs triplet mass $M_{HC}$ (scaled).}`
- **Line 1365:** `Experimental bounds on proton lifetime therefore impose strong lower limits on $M_{HC}$ and constrain viable models.`
- **Line 1391:** `\item proton decay rates,`
- **Line 1668:** `\item Core gates: FRG fixed point / positivity / PPN / GW / proton lifetime remain satisfied (A and B1).`
- **Line 2018:** `Proton lifetime`
- **Line 2019:** `& $\tau_p \gtrsim 10^{34}\,\mathrm{yr}$ with $\tau_p \propto M_{HC}^2$`
- **Line 2054:** `Inflation \((n_s,r)\): Pass (\(r\) falls back to A/B1 if strict TCC enforced); Seesaw: Pass; Axion DM: Pass for mid-band \(f_a\); BFB/Unitarity cone: Pass; Proton decay: median \(\tau_p\propto M_{HC}^2 r^{-2}\) requires threshold sweep (open if needed).`

## Other proton-decay references

- **Line 2344:** `\item proton decay scaling tied to unification thresholds,`
- **Line 2364:** `\item refine proton decay limits to test unification thresholds,`
- **Line 2673:** `Baryon number & Proton lifetime & \(\tau_p\gtrsim 10^{34}\,{\rm yr}\), trending \(\propto M_{HC}^2 r^{-2}\) \\`
- **Line 2706:** `Because all candidates satisfy baseline consistency conditions, failures occur only at the level of data mismatch (CMB/BAO/SN/GW, axion searches, proton decay, etc.). This sharply separates:`
- **Line 2959:** `SU(5)/SO(10) thresholds & Solve Eqs.\ in main text with $\tau_p{>}10^{34}$ yr & Open (needs sweep) \\`
- **Line 2960:** `Proton decay & $\tau_p \propto M_{HC}^2/r^2$ within limits & Open (needs sweep) \\`

## Heavy-threshold and D=5/D=6 derivations

- **Line 3234:** `\subsection*{4. Proton decay: scaling relations}`
- **Line 3235:** `\paragraph{Dimension-6 (gauge boson exchange).} For $p\to e^+\pi^0$ mediated by $(X,Y)$ gauge bosons,`
- **Line 3237:** `\tau_p^{(D6)} \;\simeq\; \frac{4\pi}{\alpha_U^2}\,\frac{M_X^4}{m_p^5}\,\frac{1}{\mathcal{A}_R^2\,|\alpha_H|^2}\,\frac{1}{\mathcal{F}_{\rm chiral}},`
- **Line 3239:** `where $\alpha_H\equiv\langle\pi^0|(ud)_Ru_L|p\rangle$ is the hadronic matrix element, $\mathcal{A}_R$ the short-distance renormalization factor from $M_X$ to $\sim$GeV, and $\mathcal{F}_{\rm chiral}$ encodes chiral Lagrangian dressing. The key scaling is $\tau_p^{(D6)}\propto M_X^4/\alpha_U^2$.`
- **Line 3241:** `\paragraph{Dimension-5 (color-triplet Higgs exchange).} In SUSY-like slices with color-triplet mass $M_{HC}$, the leading $p\to K^+\bar\nu$ amplitude scales as`
- **Line 3247:** `\Gamma_{p\to K^+\bar\nu}^{(D5)}\;\propto\; \frac{r^2}{M_{HC}^2} \quad \Longrightarrow\quad \boxed{\ \tau_p^{(D5)}\;\propto\; \frac{M_{HC}^2}{r^2}\ }.`
- **Line 3258:** `\item \textbf{Proton decay:} evaluate $\tau_p^{(D6)}$ via Eq.~\eqref{eq:tauD6} and $\tau_p^{(D5)}$ via Eq.~\eqref{eq:tauD5} with your hadronic/RG inputs; reject points below experimental bounds.`
- **Line 3278:** `\noindent\textbf{Summary.} The combinations in Eqs.~\eqref{eq:comboHC}--\eqref{eq:comboX} isolate the heavy thresholds $(M_{HC},M_X,M_\Sigma)$ directly from the running couplings. Mapping these to $\tau_p$ via the scaling laws \eqref{eq:tauD6} and \eqref{eq:tauD5} provides a clean, solver-ready path to populate Fig.~2 and enforce the proton-decay gate in UTFANSWF.`
- **Line 3710:** `% UTFANSWF v22 – SU(5) Threshold Sweep and Proton Decay`
- **Line 3732:** `\tau_p^{(D6)} \propto \frac{M_X^4}{\alpha_U^2}.`
- **Line 3736:** `dimension--6 proton lifetime across rows 1--8.`
- **Line 3746:** `\tau_p^{(D5)} \propto \frac{M_{HC}^{2}}{r^{2}},`
- **Line 4103:** `UTFANSWF embeds the Standard Model into unified gauge structures (minimal SU(5), SO(10), heterotic extensions), and the real check here comes from heavy thresholds and proton decay.`
- **Line 4131:** `\subsection{Proton Decay Scaling Laws}`
- **Line 4135:** `\paragraph{Dimension-6 gauge-boson exchange.}`
- **Line 4138:** `\tau_p^{(D6)} \;\;\propto\;\; \frac{M_X^4}{\alpha_U^2 m_p^5}.`
- **Line 4143:** `\paragraph{Dimension-5 triplet-Higgs exchange.}`
- **Line 4146:** `\tau_p^{(D5)} \;\;\propto\;\; \frac{M_{HC}^2}{r^2},`
- **Line 4162:** `\tau_p^{(D6)} &\sim 10^{35}\,\text{yr}, \\`
- **Line 4163:** `\tau_p^{(D5)} &\sim 10^{34-36}\,\text{yr}.`
- **Line 4166:** `So we’re above current Super-K limits ($\tau_p > 10^{34}\,\text{yr}$), but still sitting right in the range that Hyper-K and DUNE can probe.`
- **Line 4175:** `\item Proton lifetimes are consistent with existing bounds and remain testable.`
- **Line 4188:** `Proton decay (D=6) & $\tau_p^{(D6)} > 10^{34}$ yr & $\sim 10^{35}$ yr & Pass \\`
- **Line 4189:** `Proton decay (D=5) & $\tau_p^{(D5)} > 10^{34}$ yr & $\sim 10^{34-36}$ yr & Pass \\`

## Gate/status/falsification summaries

- **Line 4480:** `\item Proton lifetime above current experimental limits.`
- **Line 4520:** `Proton Decay & $\tau_p \gtrsim 10^{34} \, {\rm yr}$ (median scaling) & Conditional (threshold sweep) \\`
- **Line 4533:** `\item \textbf{Proton decay:} requires completing the heavy-threshold sweep to lock in $\tau_p$ scaling.`
- **Line 5939:** `\item \textbf{Proton decay exclusion:} No $(M_{HC},M_X,M_\Sigma)$ solution consistent with $\tau_p > 10^{34}$ yrs.`
- **Line 5977:** `Proton decay exclusion & No viable threshold solution & Open \\`

## Numerical SU(5) threshold/lifetime benchmarks

- **Line 6063:** `We solve for the heavy thresholds and check whether proton decay stays safely above experimental bounds.`
- **Line 6072:** `= -\,\frac{1}{2\pi}\,\frac{12}{5}\,\ln\!\left(\frac{\mu}{\MHC}\right),`
- **Line 6082:** `\ln\!\left(\frac{\mu}{\MHC}\right) &= -\frac{5\pi}{6}\,C_1(\mu), \\`
- **Line 6108:** `\MHC &\simeq \SI{1.08e18}{\giga\electronvolt}.`
- **Line 6117:** `\paragraph{Dimension-6 (gauge bosons).}`
- **Line 6120:** `\tau_p^{(D6)} \propto \frac{\MX^4}{\alphU^2}`
- **Line 6126:** `\paragraph{Dimension-5 (color-triplet Higgs).}`
- **Line 6129:** `\tau_p^{(D5)} \propto \frac{\MHC^2}{r^2}`
- **Line 6132:** `With $\MHC \sim 10^{18}\,\GeV$ and typical SUSY dressing $r \lesssim 0.3$, this channel is also strongly suppressed.`
- **Line 6150:** `$\MX\simeq \SI{1.11e16}{GeV}$,\; $\MSig\simeq \SI{1.73e16}{GeV}$,\; $\MHC\simeq \SI{1.08e18}{GeV}$`
- **Line 6154:** `& $\tau_p^{(D6)}\!\propto\! \MX^4/\alphU^2$;\; $\MX\sim 10^{16}\GeV$;\; clears $>10^{34}\yr$ bounds`
- **Line 6158:** `& $\tau_p^{(D5)}\!\propto\! \MHC^2/r^2$;\; $\MHC\sim 10^{18}\GeV$;\; safe for $r\lesssim 0.3$`
- **Line 7047:** `\subsection{Proton Decay Lifetime}`
- **Line 7051:** `\tau_p \sim 10^{35\text{--}36}~{\rm yrs}.`
- **Line 7056:** `\noindent\textbf{Takeaway:} No proton decay expected before Hyper-K / DUNE sensitivity scales`
- **Line 7113:** `\item Proton decay: $\tau_p \sim 10^{35\text{--}36}$ yrs`

## Pedagogical proton-decay workflow

- **Line 8965:** `if the couplings don’t meet tightly enough, or if the proton decays too fast, the slice fails.`
- **Line 9033:** `If that’s not satisfied, we’re done—no need to even talk about proton decay.`
- **Line 9035:** `\subsection*{C.\;Proton decay: dimension-6 channel}`
- **Line 9051:** `\tau_p=\frac{1}{\Gamma}.`
- **Line 9056:** `\boxed{\;\tau_p \ge \tau_{\rm exp}\;}`
- **Line 9062:** `There’s also the dimension-5 channel. UTFANSWF either:`
- **Line 9070:** `\tau_p \text{ must still pass the same bound.}`
- **Line 9082:** `\item If yes, compute $\tau_p$.`
- **Line 9083:** `\item Check: does $\tau_p$ pass?`

## Later PASS/lifetime benchmark claims

- **Line 9891:** `\item \textbf{Proton lifetime:} safely above bounds at $\tau_p \approx 1.54e+36~\mathrm{yr}$.`
- **Line 9912:** `Proton lifetime (dim-6) & PASS & $\tau_p\approx 1.54e+36~\mathrm{yr} \ge 1.6\times10^{34}~\mathrm{yr}$. \\`

## Late appendices / falsification constraints

- **Line 15105:** `\subsection{Proton lifetime scaling (dimension-6)}`
- **Line 15107:** `At low energies, baryon violation appears through dimension-6 operators suppressed by $\Lambda^2$, giving`
- **Line 15109:** `\tau_p \sim \frac{\Lambda^4}{\alpha_H^2\,m_p^5}\times \mathcal{F}_{\rm RG}\big(\{\alpha_i(\mu)\}\big),`
- **Line 16460:** `\item \emph{Proton Decay}`
- **Line 16464:** `\tau_p(\theta)\ge 10^{34}\,\text{yr}.`
- **Line 16496:** `- Proof that SU(5)/SO(10) thresholds cannot satisfy the unification equations \eqref{eq:unif1}–\eqref{eq:unif2} while also satisfying $\tau_p$`

## Suggested manuscript-level structure (for Ron to implement)

1. Add a new proton-stability subsection before conventional GUT lifetime estimates.
2. State the ancestry: `Ngen=3 -> anomaly-free U(1)_X -> Higgsing by charge ±6 singlets -> exact Z6 -> DeltaB=3k`.
3. Give the residual charge/operator table and anomaly/cobordism checks.
4. Introduce the internal order-3 `3+2` split as the candidate unification implementation that projects out physical `X,Y` and `H_C` mediators.
5. Move the old D=5/D=6 lifetime material into a clearly labeled conventional/legacy comparison subsection.
6. Replace finite-lifetime UTFANSWF predictions with the protected-branch falsification rule: any verified `DeltaB=1` or `DeltaB=2` event falsifies the exact-Z6 branch; `DeltaB=3` remains allowed.
7. Reference executable Row 28 and keep its `PRIVATE_RESEARCH` status until the manuscript integration and independent audit are complete.
