Gravity Analysis Spectral Convergence Spectrum Gap Persistence

Spectral gap persistence

In the mathematics of vibrating systems, a spectral gap is a separation between two frequencies, or eigenvalues, of a system. The classical question is whether this separation survives when you replace a continuous system, like a drumhead, with a discrete approximation, like a lattice of points. The theorem named spectrum_gap_persistence answers a simple version of that question: if two eigenvalue branches converge to distinct limits, then for all sufficiently fine approximations the branches remain ordered, with the lower one staying below the higher one.

The proof is short and rests on a standard fact about limits. If one sequence of numbers approaches a limit Λ and another approaches a larger limit Μ, then eventually every term of the first sequence is less than every term of the second. The declaration spectrum_gap_persistence (a proved theorem in the framework's machine-checked library of formal theorems) states this exactly: given two sequences lam and mu with limits Λ and Μ and Λ < Μ, it follows that eventually lam N < mu N for all large N. The proof applies the library's eventually_lt lemma, which packages this limit-ordering fact.

This result matters because it is the tool that a larger campaign uses to keep curved eigenvalue branches apart. The same library file proves a quantitative version of the convergence: for the flat discrete sine operator, the eigenvalue error is bounded by an explicit constant divided by N², where N is the lattice resolution. Feeding that rate into a squeeze lemma yields the qualitative limit, and then spectrum_gap_persistence guarantees that distinct limits do not collide at any finite resolution. The theorem itself, however, does not prove that any particular physical system has a spectral gap, nor does it give the size of the gap. It only preserves a gap that is already known to exist in the limit.

What the declaration does not claim is as important as what it proves. It does not establish that any specific operator, curved or flat, has distinct eigenvalue limits; that is the job of other theorems about particular systems. It does not quantify how large N must be before the separation holds, only that such an N exists. And it does not address the min-max principle for Hermitian matrices, which the campaign notes as future work. The theorem is a bridge: it carries an ordering from the limit back to the finite approximations, and nothing more.

/-- THEOREM (gap persistence). If two eigenvalue branches converge to
distinct limits `Λ < Μ`, then eventually `lam N < mu N`: spectral gaps
survive discretization for large `N`. The tool Phase 4 uses to separate
curved eigenvalue branches. -/
theorem spectrum_gap_persistence (lam mu : ℕ → ℝ) (Λ Μ : ℝ)
    (hlam : Filter.Tendsto lam Filter.atTop (nhds Λ))
    (hmu : Filter.Tendsto mu Filter.atTop (nhds Μ))
    (hlt : Λ < Μ) :
    ∀ᶠ N : ℕ in Filter.atTop, lam N < mu N :=
  hlam.eventually_lt hmu hlt

/-- THEOREM (gap persistence). If two eigenvalue branches converge to
distinct limits `Λ < Μ`, then eventually `lam N < mu N`: spectral gaps
survive discretization for large `N`. The tool Phase 4 uses to separate
curved eigenvalue branches. -/
theorem spectrum_gap_persistence (lam mu : ℕ → ℝ) (Λ Μ : ℝ)
    (hlam : Filter.Tendsto lam Filter.atTop (nhds Λ))
    (hmu : Filter.Tendsto mu Filter.atTop (nhds Μ))
    (hlt : Λ < Μ) :
    ∀ᶠ N : ℕ in Filter.atTop, lam N < mu N :=
  hlam.eventually_lt hmu hlt

What this page does not claim

The theorem does not prove that any particular physical system has a spectral gap. The theorem does not give the size of the gap or the resolution at which separation begins. The theorem does not address the min-max characterization of eigenvalues.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/Analysis/SpectralConvergence.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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