analytic torsion under bounded geometry and spectral gap in odd dimensions

we prove a quantitative structural bound for ray–singer analytic torsion on closed oriented odd-dimensional riemannian manifolds under bounded geometry of finite order and a uniform positive hodge spectral gap. we expand the original argument by: (i) providing a self-contained proof of the short-time heat-kernel expansion under bounded geometry; (ii) establishing explicit bounds on the heat-kernel coefficients via gilkey’s universal polynomials; (iii) proving the long-time semigroup estimate from first principles via the spectral theorem; (iv) computing the tail function φn precisely for n = 3, 5, 7 via explicit integration by parts, deriving the correct closed forms and the complete asymptotic expansion; (v) working through the determinant-line formalism in full algebraic detail; (vi) providing four explicit classes of examples: hyperbolic rational homology spheres, berger spheres, nilmanifolds, and arithmetic hyperbolic 3-manifolds; (vii) proving a quantitative version of the structural dichotomy with explicit growth-rate estimates; (viii) providing the quantitative bridge between analytic torsion and integral torsion homology.