One Missing Axiom for the Complete Closure of the Heterotic Landscape: Resonance Classifier of a Vibrating String Creates the Typical Particle Classes

Whether the vibrational picture of elementary particles; now nearly half a century old; determines the spectrum we observe has remained without a settled answer. We close the question. At a fixed compactification of the heterotic string, the entire low-energy content follows from the geometry of the chosen Calabi–Yau manifold. Across compactifications, no internal rule singles out the one nature uses: the choice depends on input from outside the framework, of a kind we make explicit. Once that input is supplied, masses and mixings are computable; without it, no derivation is possible. Background. The picture of elementary particles as vibrational modes of a string is forty years old, and a steady catalogue of explicit heterotic constructions has shown that the observed gauge group and three-family chiral content can be reproduced. What no construction has settled is the converse question: does the resonance–particle correspondence, taken as a classification programme, determine the observed spectrum, or merely admit it? The literature has answered case by case; a framework-level resolution has been absent. Methods. We separate the framework into an unconditional mathematical layer (index theorem, Dolbeault cohomology, slope-polystability, Calabi–Yau metric existence, unobstructedness, BRST classification, anomaly cancellation) and a selection layer (choice of compactification datum). The achievable range of topological and cohomological selectors over the resulting landscape is computed by Kodaira–Spencer deformation theory and the special geometry of the complex-structure moduli space. Results. At fixed admissible datum the perturbative massless spectrum is fully determined by bundle cohomology and representation branching (positive direction). No topological or cohomological rule singles out the observed vacuum; the obstruction is a nontrivial class on a positive-dimensional moduli component (negative direction). A closure-completeness theorem unifies the two directions; universality, maximality, rigidity, stable-obstruction, categorical-impossibility, and quantitative-dimension theorems show the result holds for every framework whose predictive data is cohomological and is not improvable from within. A corrected audit lemma, with its converse, identifies the singleton condition any external selector must satisfy. Five residual closure theorems — selector completion, internal no-go, fixed-vacuum Yukawa computability, algebraic exotic lifting with post-stabilisation threshold, and ensemble finiteness — reduce the residual problems to one external axiom.