The Geometric Quantization Framework: The Forbidden Zone as the Geometric Regulator of Energy and Time

This paper presents the Geometric Quantization Framework, demonstrating that the Forbidden Zone (FZ) of the Arithmetic Black Hole Model (ABHM) functions as a geometric regulator producing quantized energy and time from integer classification. From the Classification Law and ABHM shell geometry r (Ω) = 1/Ω, we derive: A fundamental energy scale EAXON = 1/L² = 4. 0 at the Monad limit (ε = 0) A dimensional coupling constant Λ = ER/4. 0 anchored to the Balmer threshold A temporal quantum Δt = ħ/ER ≈ 4. 838 × 10⁻¹⁷ seconds (Rydberg time unit) The ABHM shell structure generates 1/Ω² energy scaling which, when calibrated to the Rydberg energy at a single point, reproduces the hydrogen emission spectrum as a structural identity — verified at 0. 0000% error for all 15 tested transitions across the Lyman, Balmer, and Paschen series (Theorem 8. 1). Additional results include: The Dirac fine structure correction in ABHM form: f (Ω, ω) = −α²/Ω² × (Ω/ω − 3/4), verified for 20 quantum states with CV = 0. 000% The Droste Bridge Theorem: D (l, 1/3) = ±Lˡ for all l ≥ 0, proven via the Chebyshev product identity Sub-shell structure: each ABHM shell Ω partitions into exactly Ω sub-groups by ω (n), matching the quantum mechanical angular momentum degeneracy (verified to n = 1. 13 × 10⁹, shell 9) The four-way identity L (1+L) = 3/4, unique to L = 1/2, connecting the FZ width to the j (j+1) eigenvalue of SU (2) at j = 1/2 A conjectured closed form for the fine structure constant: α = L²/ (4πe + 1/10 + α²), matching CODATA 2022 to seven significant figures Helium screening σHe = (2/3) ³ = 8/27 from FZ boundary primes, yielding E = −78. 98 eV (experiment: −79. 005 eV, 0. 03% error) Multi-electron screening convergence for any finite Z, with each term bounded by (2/3) ᵏ FZ Width Monotonicity Lemma: L (ε) is strictly decreasing for ε ≥ 0 All claims carry explicit epistemic labels: 18 EXACT, 5 STRUCTURAL, 2 NUMERICAL, 1 NEGATIVE. The negative result (arithmetic-geometric pathway unification failure) is reported in full. Total: 18 exact results, 5 structural observations, 2 numerical conjectures, 1 negative result, 3 open problems. Supplementary material includes a fully self-contained HTML physics simulator (ABHM Geometric Quantization Simulator) with interactive 3D WebGL shell visualization, replayable verification suite, and all numerical experiments from the paper. This is the 14th and final paper in this AXONLang Labs research program series. It depends most directly on Papers 1 (Classification Law), 3 (ABHM), 5 (Hourglass Framework), 8 (Energy = Geometry × Speed), and 13 (The Forbidden Zone internal paper). INTEGER + GEOMETRY = TIME Related Identifiers (prior papers in the series): 10. 5281/zenodo. 18756471 — The Classification Law 10. 5281/zenodo. 18854321 — The Sieve Firewall 10. 5281/zenodo. 19078891 — The Arithmetic Black Hole Model 10. 5281/zenodo. 19119719 — ABHM Release Package 10. 5281/zenodo. 19120026 — The Hourglass Framework 10. 5281/zenodo. 19152117 — Three Consequences of the Hourglass 10. 5281/zenodo. 19154069 — We Are Not in a Simulation 10. 5281/zenodo. 19156617 — Energy = Geometry × Speed 10. 5281/zenodo. 19157299 — Toward a Metric on the Hourglass 10. 5281/zenodo. 19226378 — The Shape of the Universe 10. 5281/zenodo. 19226471 — One Hundred Years (Walter Russell) 10. 5281/zenodo. 19257572 — I Don't Know Anything. But God Does.