Spectral asymptotics near annular Type-I singularities in the parabolic Bernoulli problem

This work studies quantitative asymptotics near Type-I singularities for the parabolic Bernoulli free-boundary problem when the unique tangent flow is the radial annular self-shrinker.Building on the Type-I compactness, annular tangent-flow uniqueness, and spectral sign classification established by Liu, Muñoz, and Zhao, the paper develops a quantitative nonlinear asymptotic theory for actual flows converging to the annular shrinker. The analysis combines a time-dependent partial-hodograph transformation, interpolation from the Type-I Hessian bound, hyperbolic invariant-manifold theory, spectral decomposition, and validated Sturm–Liouville computation.The first step upgrades qualitative convergence to the annular tangent flow into convergence in the natural Sobolev–Slobodetskii dynamical phase space. The time-dependent partial hodograph transforms the moving free-boundary problem into a uniformly parabolic quasilinear problem on a fixed annular domain with a uniformly oblique nonlinear Bernoulli boundary condition. Hyperbolicity then places the orbit on the local stable manifold and yields exponential convergence.The nonlinear remainder is quadratic in the natural maximal-regularity spaces. This allows the construction of sequential spectral asymptotic amplitudes. If A_j is the first nonzero stable amplitude, then the solution has a leading asymptotic term of the formη(s) = exp(−μ_j s) A_j + o(exp(−μ_j s)).The vanishing of the slowest amplitude defines a positive-codimension submanifold of the local stable manifold. Consequently, for a generic convergent orbit, the slowest stable eigenspace determines the first nonlinear geometric correction.The annular spectral classification reduces the slowest stable candidate to three sectors:(ℓ, k) = (0, 3), (2, 2), and (4, 1).A validated computer-assisted Sturm–Liouville analysis determines their ordering in finite dimensions, while analytic min–max estimates close the large-dimensional regime. For 2 ≤ n ≤ 154, the finite computation uses interval shooting, Krawczyk enclosure, certified Sturm nodal counts, and two independent directed-rounding implementations. For n ≥ 155, analytic estimates determine the ordering without numerical continuation. C_A_I_FINAL_REVISED_V2The resulting slowest stable sector is(ℓ*, k*) = (4, 1) for 2 ≤ n ≤ 8,and(ℓ*, k*) = (2, 2) for n ≥ 9.Thus the transition occurs exactly between dimensions eight and nine. C_A_I_FINAL_REVISED_V2The main geometric consequence is that, for an open dense subset of nearby initial states on the local stable manifold, the first correction to both components of the annular free boundary has spherical-harmonic degree four in dimensions 2 through 8 and degree two in dimensions 9 and higher. In this sense, dimension nine marks a genuine transition in the leading geometry of annular Type-I singularities. C_A_I_FINAL_REVISED_V2The accompanying reproducibility package contains the computational and validation material associated with the finite-dimensional spectral ordering, including replay scripts, stored results, independent verification components, audit records, and checksums.