Spectral Wall Arithmetic A: Hodge Theory of the Hofstadter Wall—A Rigorous Algebraic Model for Floquet Gap Closure

In the Harper–Hofstadter operator family, a rigorously treatable periodicallydriven quantum model, this paper establishes a complete algebraic theory of thewall–chamber structure. For any reduced rational flux p/q (q ≥ 2), three structuralconclusions are proved. First, the Chambers structural theorem: the fiber characteristic polynomial is completely determined by a single monic polynomial Q ofdegree q, the coefficients of Q lie in the real cyclotomic field Q(2cos(2πp/q)), anddet(λI − H(θ,φ)) = Q(λ)+4−2cos(qθ)−2cosφ.Second, the wall-position theorem: the wall set of the θ = 0 slice is exactly{Q =0}∪{Q=−4}∪{Q′ =0, Q∈(−4,0), E is a local extremum point of Q},hence an algebraic point set; the Q-degree of endpoint wall points does not exceedq · [K : Q], while the upper bound for interior critical points is sharper, namely(q − 1) · [K : Q]. Third, the reconstruction identity: the sum of the number ofconnected components of spectral bands and the total number of touch points isidentically equal to q, where touch points are counted separately as interior touchesand boundary tangent degenerate touches. The fiber-covering lemma provides arigorous basis for the fact that the θ = 0 slice loses no wall or gap information.The numerical machine checks of all structural assertions are layered by evidencetype: structural symmetries and gap labels cover 139 mutually inequivalent fluxpairs; value-form identities, wall-point classification, coefficient-field certification,and the reconstruction identity were verified at the symbolic level on 16 mutuallyinequivalent flux pairs with q ≤ 10; exact symbolic verification for q > 10 is absent,and the ledger explicitly marks symbolic absent. At the single-particle level, thismodel is one of the few instances possessing a rigorous algebraic structure and anexplicit wall-set characterization; spectral coverings and spectral curves provide analgebraic foundation for periodic arithmetic and variational Hodge structures, whilethe Abel–Jacobi inversion of the rotation number and the irrational-flux limit arelisted as open problems.