Γ-BRT: Bounded Reflection Theorem — From Two Vacuum Constants to Universal Boundedness

Γ-BRT (Bounded Reflection Theorem) is a zero-free-parameter theoretical framework that derives all boundary phenomena from two vacuum constants: the electric permittivity ε₀ and the magnetic permeability μ₀. The central result is the Boundedness Theorem: the reflection coefficient Γ = (Z₂−Z₁) / (Z₂+Z₁) maps the entire non-negative impedance line 0, ∞) into the bounded interval [−1, +1. No physical observable can diverge; the universe is intrinsically bounded. Key Results Paper 0 — Bounded Reflection Theorem (Foundation) Derives c, Z₀, and α = Z₀/ (2RK) ≈ 1/137 from ε₀ and μ₀ alone Proves the Boundedness Theorem: |Γ| ≤ 1 for all impedance boundaries Establishes the Quarter-Energy Theorem: total reflection concentrates energy to 4u₀, providing a scale-independent phase-transition threshold Derives the cosmic energy budget Ω_Λ: ΩDM: Ωb = 68. 4: 26. 7: 4. 9, matching Planck 2018 to 0 at each boundary crossing defines the arrow of time Three meta-axioms (Γ ≠ 0, Γ → 0, Γ = 0 unreachable) generate all operational constraints including the Bode–Fano bandwidth bound Paper 1 — Physical Verification Cross-scale verification spanning acoustics, optics, electromagnetism, and gravitation Material spectra as exact cascaded-Γ solutions (r = 1. 000) Fine-structure correction α = Z₀/ (2RK) reduces residual MSE by 70% Bond-energy predictions from impedance ratios Semiconductor refractive indices from Γ-cascade Paper 2 — Bio-Impedance Engineering Neural architecture as impedance-matching networks Γ-GPT: language model replacing softmax attention with impedance reflection, matching standard GPT to within 1. 1% Bridge topology identifies the thalamus as a multi-arm Wheatstone bridge Memory as standing-wave resonance governed by cavity Q-factor Combinatorial Γ-channel capacity reaches 10⁶⁰⁰ states for 300 channels Framework vs Application Models The zero-parameter claim applies to the framework layer (Γ formula, boundedness, Quarter-Energy Theorem, cosmological ratios). Application models (bond energy, refractive index, cortical parcellation) introduce calibration constants (typically 1–2 per model), which is explicitly acknowledged in the paper.