Al-Hindara Project — v7.0.0 Release

This release marks a major milestone in the Al-Hindara project, transitioning from exploratory frameworks to a formalized audit draft (v7.7.2). It introduces rigorous domain analysis for the spectral problem and presents stable numerical evidence for a spectral gap, clearly separating analytically proven results from open questions.

📄 What's Included in this Release?

1. The Consolidated Manuscript (v7.7.2 Audit Draft)

The LaTeX source (all-frames2-tex.tex) for the updated manuscript, integrating all four frameworks.

• Analytical Corrections: Establishes the formal domain inclusion (g \in \mathcal{D}(L_F)) and proves that the kernel is strictly one-dimensional ((\dim \ker L_F = 1)).
• Numerical Variational Audit: Presents a constrained B-spline Rayleigh-Ritz computation. The numerical evidence strongly supports a positive spectral gap (discrete eigenvalue (\lambda_{1,\mathrm{RR}} \approx 1.7685)), proving stability across quadrature refinements and endpoint truncations.
• Strict Evidence Classification: The §Z Audit section rigorously categorizes every claim as Proven, Numerically Verified, or Open.
• Open Problems: Explicitly formulates the remaining open questions regarding the continuous self-adjoint realization and spectral convergence.

2. Numerical Verification Code (Jupyter Notebook)

• Includes all-frames2.ipynb, containing the full Python implementation for the B-spline Rayleigh-Ritz method, piecewise Gauss-Legendre quadrature, and eigenvalue computation.

📌 Current Status & Remaining Work

To maintain scientific integrity, the following items are explicitly documented as Open in the manuscript:

• The continuous spectral gap: While numerically stable at (\lambda_{1,\mathrm{RR}} \approx 1.7685), proving this analytically for the continuous operator remains open.
• Compact resolvent: The compactness of the embedding (D(q_F) \hookrightarrow L^2(0, 2\pi)) requires further rigorous treatment.
• The self-adjoint domain of (L) on ((0, 2\pi)).

"This document serves to preserve a starting point, not to serve as a final verdict..."