The Boundary Information Invariant of Quadratic Systems
B.D.
Abstract
Description Three mathematical ingredients — Gaussian smoothing (√2), binary distinction (ln 2), and quadratic curvature — produce a dimensionless sub-unity ceiling K_AUD = √2 × ln(2) ≈ 0.980. This document presents published data and exact algebraic relations showing the same three ingredients converge across independent domains: DESI DR2 cosmological distance ratios, atomic electron shell structure, chaos theory (Feigenbaum), information theory (Romeo 2025), and crystal field spectroscopy. Key results: • K_AUD = √2 × ln(2) is the unique sub-unity ceiling for K(n) = √n × ln(n) at integer n ≥ 2 • The √2-to-Shannon gap identity: 1/(2 ln 2) − 1/√2 = G/(2 ln 2) [exact algebraic identity] • The 35–37 threshold zone: Landauer crossing at n·G = ln 2 (n ≈ 35.11), geometric damping at n·G = 1/√2 (n ≈ 35.82), Shannon bound at n·G = 1/(2 ln 2) (n ≈ 36.54) • Crossing-zone self-reference: Δn₂₃ / Δn₁₂ = 1/K_AUD [exact] • Gap scaling: ρ = 400/11 − 1/2500 − 1/939939, agreement to 4×10⁻¹⁴ • DESI BAO distance ratios constructed from Table IV land within 0.03% of √2 • Testable prediction: DESI DR3 should preserve the √2 ratio within 0.1% This is a MAP, not a theory. It documents where the same invariant appears across published science. All data is from published sources. All algebra is independently verifiable. Companion to four previously published papers: • Paper 1: Coherence Ceiling (DOI: 10.17605/OSF.IO/5VZ2R) • Paper 2: Geometric Constants v2 (DOI: 10.17605/OSF.IO/SJBE9) • Paper 3: Complete Framework v3 (DOI: 10.17605/OSF.IO/QH5S2) • Paper 4: Gap Scaling 400/11 (DOI: 10.17605/OSF.IO/C4GK5) Interactive tools and plain-text versions: https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/about.html