The atom and its temperature from the pentachoron

We show that the combinatorial structure of the complete graph on five vertices (K5, the pentachoron) encodes both the atom and the origin of temperature. A binary colouring of vertices yields, with zero free parameters: (1) the alpha particle as the unique nucleus within one tetrahedron [T1]; (2) the nuclear mass number A = 4k - s and maximum charge Z_max = 2k for assemblies of k pentachores [T1]; (3) the alpha-conjugate sequence A = 4k, Z = N = 2k saturating Z_max (4He, 8Be, 12C, 16O, 20Ne, 24Mg, 28Si, 32S, 36Ar, 40Ca for k = 1-10) [T1]; the doubly-magic subset (4He, 16O, 40Ca) follows by intersection with the shell-model magic numbers {2, 8, 20} accepted as external input [T2]; (4) all stable isotopes from H to Ne reproduced from colourings of k = 1-5 pentachores [T1]; (5) electron shell capacities 2(l+1)^2 from the uniqueness of d = 4 [T1]; (6) valences 1,0,1,2,3,4,3,2,1,0 for H-Ne, exact 10/10, from the distance to the nearest noble gas Z = 2k_noble [T2]; (7) chemical bonding as shared tetrahedral faces between pentachoric assemblies [T2]; (8) temperature as total reconstruction conflict: T(S) proportional to exp(+alpha* C(S)), where the integer C(S) = sum_v n_v counts the simultaneous reconstruction attempts at shared vertices, dual to mass m_k/m_e = un_k^2 (u = exp(alpha*/4), P2) where n^2 counts spectral channels [T1, mass formula from P2]. The instability of 8Be is derived as a theorem: two nuclear tetrahedra sharing zero triangular faces fragment into 2alpha despite saturating Z_max. Mass and temperature share the same coupling alpha* = 1/(4 ln 2) and the same positive exponential structure. The K5 thermal scale k_B T* tau_0 / hbar = rho* = 4 ln 2 follows from P15 and anchors T(S) on the fundamental tick without free parameters. Every quantitative claim is verified by a companion script (P12_companion_v6.py, 56 tests, 0 failures); the alpha-conjugate sequence is verified by exhaustive enumeration up to k = 7 (28Si). No continuous free parameter enters the construction.