The Discrete Hadron Mass Spectrum: Topological Generation of the Standard Model from Causal Sets

We derive the complete mass spectrum of stable hadrons from the Kirchhoff spanning tree count $\tau = \det(L^{(r)})$ of decorated $K_{3,3}$ subgraphs embedded in Poisson-sprinkled causal sets, with zero continuous free parameters. The bridge multiplicity vector $(n_1, n_2, n_5)$ of each hadron is constrained by three graph-theoretic principles: (i)~the minimum covering theorem (pion selection), (ii)~the $(\beta, \sigma)$ classification (parameter reduction), and (iii)~the degree homogenisation theorem (balanced allocation). For the non-strange sector, the proton bridge vector $(1,3,1)$ is proven to be the \emph{unique} solution to the Diophantine system $n_2 \geq 3$ (hybrid self-repair) $\wedge$ $\sigma \geq 2$ (complete covering) $\wedge$ $\beta = 9$. For the strange sector, the Phase Conservation Theorem uniquely determines all four Gen2 bridge vectors from $(I, S)$ alone. The dual-observable mass equation $M = \tau + \Sigma_{\mathrm{EW}}$ separates the strong force ($\det$, multiplicative, flavour-blind) from the electroweak self-energy ($\tr$, additive, flavour-dependent), yielding the neutron--proton splitting at $\Delta M / M_p = 5.71 \times 10^{-4}$ (physical: $1.378 \times 10^{-3}$, factor 2.4). The flavour hierarchy is generated by the $(1, n_2, 1)$ family: $\Lambda_c^+$ at $n_2 = 4$ ($+0.6\%$), $\Lambda_b^0$ at $n_2 = 5$ ($-4.2\%$). The confinement ceiling $P_{\mathrm{ac}}(n_2) = (4/5)^{\binom{n_2}{2}}$ restricts stable generations to $n_2 \in \{3, 4, 5\}$: exactly three. Quantum numbers are derived from the directed $\Kthree$ topology: spin $J = (|\sigma - 2| + B)/(1 + B)$, parity $P = (-1)^{1-B}$ (6/6 exact), the 8-gluon octet from the traceless edge-phase algebra (28 brackets, 56 Jacobi triples verified), and chirality $f_L = 0.4644$ from the arrow of time ($N = 10^7$, 20 realisations). The discrete S-matrix emerges from the engine's own $K_{2,3}$ rewrite rules when two hadron worldlines overlap in the Hasse DAG: 8,295 cross-hadron events detected over 500 ticks at $N = 20{,}000$ with 97\% baryon stability, requiring no new interaction laws. Every mass ratio in this paper is an exact rational number derived from integer arithmetic on graph Laplacians