No Proton Decay, Yes 𝑛 − 𝑛 ˉ : A Leaf-First 𝑍 48 Selection Theorem

We record a “leaf-first” construction that repeatedly emerged in our R^8 / Crystalline Axiverse work: a discrete bookkeeping clock in which a **non-split extension**\[0 \to \mathbb{Z}_{24}\to \mathbb{Z}_{48}\to \mathbb{Z}_2\to 0\]introduces a single parity bit that is **lost** under naive reduction mod 24. The lost bit is captured by a **projection-parity grade** \(g\in\mathbb{Z}_2\), defined for \(q\in\mathbb{Z}_{48}\) by\[q^{(24)} \equiv q \bmod 24,\qquad g(q)=\Big\lfloor \frac{q}{24}\Big\rfloor \bmod 2,\]so the visible projection is not just \(q\bmod 24\) but the pair \((q^{(24)},g)\). We interpret \(g=1\) as a **half-turn tag** (charge 24 in \(\mathbb{Z}_{48}\)), i.e. the residue distinguishing “0” from “24” that a coarse \(\mathbb{Z}_{24}\) view cannot detect. The framework is formulated on an eight-dimensional compact “leaf” background\[T^8=\mathbb{R}^8/\mathbb{Z}^8,\]with eight independent periodic identifications, \(\pi_1(T^8)\cong \mathbb{Z}^8\), and thus eight independent 1-cycles and holonomies. We use this topology as motivation for discrete holonomy-style bookkeeping (spin structure choices, Wilson lines, center data), not as a full dynamical derivation. A first pillar is an explicit “why quarks” consistency theorem: for one Standard Model generation with a Higgs doublet, demanding (i) gauge-invariant Yukawa couplings and (ii) cancellation of gauge and gravitational anomalies yields (up to scaling/sign conventions) the **unique minimal hypercharge assignment**. In integer units \(y=6Y\), a canonical solution is\[y_H=3,\quad y_Q=1,\quad y_{u^c}=-4,\quad y_{d^c}=2,\quad y_L=-3,\quad y_{e^c}=6,\]reproducing the standard electric charges via \(Q_{\rm em}=T_3+Y\). In this precise sense, “quark-like” representations are not optional decorations: they are required for an anomaly-free chiral electroweak sector with Yukawa masses. A second pillar is a family of discrete generators that combine \(\mathrm{SU}(3)\) **center triality** \(t\in\mathbb{Z}_3\), fermion number \(F\), hypercharge \(y\), and optional baryon/lepton terms. A \(\mathbb{Z}_{24}\) warm-up,\[q_{24}\equiv 8t+12F \pmod{24},\]reproduces confinement-style singlets: color-neutral composites land automatically in \(q_{24}\in\{0,12\}\), separating bosonic and fermionic singlets in the same spirit as \((-1)^F\). Extending with hypercharge mixing, \(q\equiv 8t+12F+k\,y\pmod{24}\), preserves Yukawas for all \(k\), while a “hardening” step (adding \(a(3B)+bL\)) can forbid dangerous operators without disrupting renormalizable Standard Model structure. The third pillar is the \(\mathbb{Z}_{48}\) upgrade, where the grade bit becomes essential. We study\[q_{48}\equiv 16t+24F+k\,y+a(3B)+bL \pmod{48},\]and emphasize that reducing to mod 24 can falsely classify half-turn operators as allowed. A particularly transparent clause is \((k,a,b)=(1,0,24)\), which yields a **baryon-violation hierarchy theorem**: for gauge- and Lorentz-invariant operators, triality neutrality and hypercharge neutrality remove the \(t\) and \(y\) contributions, leaving \(q_{48}(\mathcal{O})\equiv 24\,\Delta L\pmod{48}\). A parity lemma then implies \(\Delta B\) odd \(\Rightarrow\Delta L\) odd, hence all \(\Delta B=\pm1\) operators are **half-turn / grade-odd** (\(q_{48}=24\)) and are perturbatively forbidden at every operator dimension. The leading allowed baryon violation is \(\Delta B=\pm2\) at dimension nine (neutron–antineutron type), e.g. \(u^{c\,2}d^{c\,4}\). Phenomenologically, this inverts the usual GUT expectation: the selection rule says “look for \(\Delta B=2\), do not expect perturbative \(\Delta B=1\).” We translate neutron–antineutron bounds into an effective \(\Delta B=2\) scale using\[\delta m\sim \frac{C}{\Lambda_{\Delta B=2}^5}\langle \bar n|Q|n\rangle,\qquad \tau_{n\bar n}\sim \hbar/\delta m,\]so \(\Lambda_{\Delta B=2}\gtrsim\big(|\langle\bar n|Q|n\rangle|\,\tau_{n\bar n}/\hbar\big)^{1/5}\). Using lattice-QCD matrix elements \(|\langle\bar n|Q|n\rangle|\sim 10^{-3}\,\mathrm{GeV}^6\) and current bounds \(\tau_{n\bar n}\sim10^{8-9}\,\mathrm{s}\), a conservative envelope implies \(\Lambda_{\Delta B=2}\sim10^2\)–\(10^3\,\mathrm{TeV}\) for \(C\sim\mathcal{O}(1)\). To reconcile a low leaf scale \(M_c\sim\mathrm{TeV}\) with a large effective \(\Lambda_{\Delta B=2}\), we introduce an “R^8-like bank toll” parametrization\[C_{\rm eff}=\kappa\,g^p\Big(\frac{1}{16\pi^2}\Big)^L e^{-S_{\rm bank}},\qquad\Lambda_{\Delta B=2}=M_c/C_{\rm eff}^{1/5}.\]For \(M_c\sim3\,\mathrm{TeV}\), saturating current bounds requires \(C_{\rm eff}\lesssim10^{-13}\), equivalently \(S_{\rm bank}\gtrsim29\) for \(\kappa=g=1\) and no loop factor. This frames suppression as a single “bank toll” rather than a long perturbative chain, consistent with the leaf-first philosophy. Finally, we address strict discrete-gauge consistency. A lenient even-\(N\) sieve (cancellation modulo \(N/2\)) yields many viable \(\mathbb{Z}_{48}\) candidates built from the SM leaf, while strict mod-48 cancellation has no solutions for the SM leaf alone, signaling a UV completion requirement. A minimal fix is a heavy vectorlike \(\mathrm{SU}(2)\) doublet pair \((X,X^c)\) with half-turn total charge \(q_X+q_{X^c}\equiv24\pmod{48}\), which cancels the obstruction without altering the \(\Delta B\) grade theorem on the SM operator algebra. **Predictions / falsifiers.** (i) Observation of proton decay (\(\Delta B=1\)) falsifies the clause or grade conservation. (ii) Neutron–antineutron oscillation is the leading allowed baryon-violation channel and constrains the effective bank toll. (iii) Demanding a strict discrete gauge symmetry predicts additional UV states required by anomaly cancellation.