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 Zₘax = 2k for assemblies of k pentachores T1; (3) the alpha-conjugate sequence A = 4k, Z = N = 2k saturating Zₘax (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 = 2kₙoble 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) = sumᵥ nᵥ counts the simultaneous reconstruction attempts at shared vertices, dual to mass mₖ/mₑ = u^nₖ² (u = exp (alpha*/4), P2) where n² 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 Zₘax. Mass and temperature share the same coupling alpha* = 1/ (4 ln 2) and the same positive exponential structure. The K5 thermal scale kB T* tau₀ / 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 (P12companionᵥ6. 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.
Jean-Baptiste BLATIERE (Sun,) studied this question.