The Spectrum of a Compact Internal Space I: Gauge Structure and a Scalar Mass Gap

# The Spectrum of a Compact Internal Space I: Gauge Structure and a Scalar Mass Gap[Archival DOI](https://doi.org/10.5281/zenodo.22067117)## A common construction for geometry, matter and responseThis paper develops a generating route from minimal compositional change to compact internal geometry, gauge structure, matter representations and a positive spectral gap. Its guiding question is how far these structures can be obtained as compatible stages of one construction. The continuation rules are stated through the response data they preserve, so that the classification at each stage has a definite mathematical meaning.The source begins with a least-dimensional faithful irreducible real orthogonal carrier of nontrivial connected compact change. It selects the rotation plane. Independent composition, exact same-shape scaling and unit covariance determine square-root scaling and a centered Gaussian law. Local information and total normalized correlation give its two characteristic lengths. Within the specified local separated boundary class, preservation of constants selects reflecting Neumann closure.Gaussian ladder structure, matrix units and coherent comparisons organize the compact process spectrum. The round parent is associated with SU(2), while complete center-invariant observations give the projective space RP3. The internal geometry therefore enters through the representation and observation structure of the generated source.## Gauge symmetry, matter and the scalar connectorThree classification results determine the gauge-matter continuation. In the stated class preserving adjoints, exchanges, source phase and reversal, the effective faithful gauge group is The full complex matter carrier has dimension sixteen, including a neutral right line. Its charged quotient has dimension fifteen. Reversal and the source phase constrain the elementary chiral connector to one complex weak doublet with quantum numbers. The same construction fixes the representation conventions needed to interpret the surviving gauge action faithfully.These results give the framework its structural force. The gauge group, matter carrier and connector are classified together under the continuation rules. Their dimensions and charges enter subsequent response calculations through this common construction. Preservation of adjoints fixes the unitary comparison class for the matter representations, and vacuum transport controls the relative phases and primitive charge normalization.## The generated Yang-Mills gapThe paper then establishes a positive gap for the generated equilibrium Yang-Mills response. A nonnegative collective connection intensity changes the Gaussian precision through adjoint-Casimir feedback. The resulting relative information functional is coercive and strictly convex. Its derivative determines a unique positive equilibrium once the internal length passes the critical value. In source units the critical length is, and the canonical infrared length is.The source information metric fixes the radial normalization. The complete coexact connection response retains the angular connection modes. In the notation of the resulting operator decomposition, the squared gap is Here the radial restoring coefficient and the positive shift are evaluated at the generated equilibrium. The spectral bottom is attained. The equilibrium response completion has a unique vacuum and a first positive excitation. Complete curvature observables retain both direct variance and the collective susceptibility, making the connection-level response accessible beyond a single scalar approximation.This is a constructive mass-gap theorem for the specified generated Yang-Mills operator. It supplies the operator, its equilibrium, its normalization and its positive spectral bottom in the same chain.## From internal structure to fieldsThe first Hermitian two-by-two corner provides a Lorentzian event carrier through its determinant. The positive cone gives the future orientation, and the induced action yields the Lorentz transformation structure. Translated Gaussian composition and the source phase organize affine events and a relational clock.Free-field reconstruction connects the internal response semigroup to an external one-particle mass operator. The scalar channel has a positive isolated pole, and multiparticle channels give the corresponding continua. An interacting gauge-matter construction is developed as a formal perturbative series with causal factorization and BRST compatibility at finite internal resolution.The paper thus establishes a substantial sequence of results: a generated compact geometry, a classified faithful gauge group, the sixteen-dimensional matter carrier, the weak scalar doublet, a rigorous positive equilibrium Yang-Mills gap, and compatible spacetime and field realizations. Formal proof libraries accompany the indexed foundational interfaces, while the computational framework evaluates the finite spectral responses. The distinctive contribution is the mathematical connection among these outputs through their common Gaussian source and response-preserving continuations.