IndisputableMonolith.Gravity.Analysis.SpectralConvergence
Elementary spectral-convergence toolkit for discrete lattice modes in the gravity analysis stack. Supplies sine remainder bounds, discrete eigenvalue expansions and limits as mesh N o∞, uniform-bound passage to continuum eigenvalues, and spectrum-gap persistence. Downstream continuum-limit and underdetermination modules import it for Freudenthal energy and one-mode cylinder preflights. Arguments are standard real-analysis estimates over naturals.
claimHelper lemmas for discrete-to-continuum spectra: $C/N^2\to 0$ as $N\to\infty$; cube bounds $|\sin x-x|\lesssim |x|^3$ and companions; expansions and limits of discrete sine eigenvalues to continuum values; if eigenvalues are uniformly bounded then the continuum limit exists; strict spectral gaps persist under the $N\to\infty$ passage.
background
In the RS gravity analysis stack, continuum limits of frozen quadratic energies on lattices require control of discrete spectra. Axis modes of a flat lattice Laplacian (or 1D sine modes) have eigenvalues built from $\sin(\pi k/N)$ factors; as the mesh $1/N\to 0$ these must approach continuum Laplacian eigenvalues without collapsing gaps.
This module is the elementary real-analysis layer for that passage. It packages the $C/N^2\to 0$ helper, cube-order sine remainders, discrete-sine eigenvalue expansions and tendsto statements, a uniform-bound-to-limit lemma, and spectrum-gap persistence. Downstream continuum campaigns import the package rather than re-proving trig limits inline.
The setting is flat, componentwise lattice analysis: no curvature endomorphism is present here.
proof idea
Definition-and-lemma library, not one master theorem. Typical flow: prove cube inequalities comparing $\sin$ to its linear part; rewrite discrete sine eigenvalues in expanded form; apply Mathlib tendsto and squeeze lemmas for $N\to\infty$ (including $C/N^2\to 0$); lift a uniform bound on a family of eigenvalues to existence of the continuum limit; obtain gap persistence from continuity of the limiting spectrum plus a strict inequality on the gap.
why it matters in Recognition Science
Feeds three gravity-analysis parents. FreudenthalEnergyLimit (QG Phase 2b, panel Test G stage 2 / candidate C8) needs continuum spectral control for the action-level limit of the frozen quadratic energy on the Freudenthal family. OneModeCylinderPreflight (C10 toy preflight) uses the same limits for a one-mode Gaussian cylinder-measure setup over that energy. CurvedOperatorUnderdetermination (Gap 4 blocker) cites the narrow reach of flat certified spectra: continuum Lichnerowicz values introduced from the flat reduction $\Delta_L=-\Delta$ still lack Riemann coupling, so flat spectral convergence alone cannot fix curved-background operators. This module is the spectral half of that discrete-to-continuum bridge.
scope and limits
- Does not treat curved backgrounds or Riemann-coupled Lichnerowicz operators.
- Does not construct path-sum or full multi-mode cylinder measures.
- Does not by itself prove existence of continuum action or energy limits.
- Does not address anisotropic tensor structure beyond scalar/axis sine modes.
- Does not claim uniqueness of continuum couplings from flat spectra.