The Geometric Quantization Theorem

The Geometric Quantization Theorem: The Forbidden Zone as the Geometric Regulator of Energy and Time This paper demonstrates that the Forbidden Zone (FZ) of the Arithmetic Black Hole Model (ABHM) produces quantized energy and time from the geometry of integer classification, without importing quantum mechanics. The Forbidden Zone is a provably empty band in integer classification space between the DARK region (primes and semiprimes) and the ALIVE region (composites with three or more prime factors). No integer can occupy this band. This paper treats the FZ as a resonant cavity and derives measurable physical quantities from its geometry alone. Results: 1. A fundamental energy unit is derived from the FZ cavity: EAXON = k/L², yielding a minimal energy gap of exactly 4. 0 at the Monad (the theoretical limit where the deformation parameter epsilon equals zero). 2. A dimensional coupling constant Λ = ER/4. 0 anchors this abstract unit to the Rydberg energy. The value of Λ equals the absolute energy of the hydrogen n=2 state (3. 4014 eV), the threshold of the Balmer series. 3. Applying the energy-time uncertainty principle yields a temporal quantum ΔtAXON = ℏ/ER ≈ 4. 838 × 10⁻¹⁷ seconds, the Rydberg time unit. 4. The ABHM shell radii r (Ω) = 1/Ω generate energy levels E (Ω) = EAXON/Ω², producing transition energies that recover the Rydberg formula exactly: ER (1/m² − 1/n²). 5. Empirical validation against NIST hydrogen spectral data across the Lyman, Balmer, and Paschen series yields 0. 0000% error for all 15 tested transitions. 6. The Hourglass equation (INTEGER + GEOMETRY = TIME) is realized as a computable identity linking integer classification geometry to measurable atomic physics. All claims are labeled EXACT, INTERPRETIVE, or CONJECTURAL per AXONLang epistemic standards. A complete reproducibility script (verifyₜheorem. py) is included as supplementary material. It requires only Python 3. 6+ with no external packages and independently verifies every numerical claim in the paper in under one second. Supplementary files included: - verifyₜheorem. py (reproducibility script) - nistₕydrogen. csv (raw NIST spectral data with provenance) - constants. txt (all physical constants used, with CODATA sources and precision notes) - README. md (instructions for reproduction) In Plain Language: Every whole number (1, 2, 3, 4,. . . ) can be described by how many prime building blocks it has. The number 12 = 2 × 2 × 3 has three prime factors. The number 7 has one (itself). When you sort all numbers this way, a gap appears between the simple numbers (few prime factors) and the complex numbers (many prime factors). No number can sit inside this gap. It is structurally forbidden. This paper shows that the shape of that gap produces real, measurable physics. The gap acts like a tiny resonant chamber. When you calculate the energy of its simplest vibration mode and connect it to hydrogen (the simplest atom), the math produces the exact same formula that physicists have used for over a century to predict the colors of light that hydrogen emits. Every prediction matches laboratory measurements to the precision of the data. The formula was not borrowed from physics. It fell out of the arithmetic. The conclusion: the structure of whole numbers, organized by their prime factors, contains the energy and time scales of atomic physics within its geometry. Prior Work: This paper builds on the following series, all published under CC BY 4. 0: Paper 1: The Classification LawDOI: 10. 5281/zenodo. 18756471 Paper 2: The Sieve FirewallDOI: 10. 5281/zenodo. 18854321 Paper 3: The Arithmetic Black Hole Model (ABHM) DOI: 10. 5281/zenodo. 19078891 Paper 4: ABHM Release Package (Calculator + Presentation) DOI: 10. 5281/zenodo. 19119719 Paper 5: The Hourglass Framework: INTEGER + GEOMETRY = TIMEDOI: 10. 5281/zenodo. 19120026 Paper 6: Three Consequences of the Hourglass FrameworkDOI: 10. 5281/zenodo. 19152117 Paper 7: We Are Not in a SimulationDOI: 10. 5281/zenodo. 19154069 Paper 8: Energy = Geometry × SpeedDOI: 10. 5281/zenodo. 19156617 Paper 9: Toward a Metric on the Hourglass Operator SpaceDOI: 10. 5281/zenodo. 19157299 Paper 10: The Shape of the UniverseDOI: 10. 5281/zenodo. 19226378 Paper 11: One Hundred Years: Walter Russell's Universal One (1926) and the Arithmetic Foundation of the Same Structure (2026) DOI: 10. 5281/zenodo. 19226471 Paper 12: I Don't Know Anything. But God Does. DOI: 10. 5281/zenodo. 19257572 AXONLang Labs LLCDaniel N. Santiago, CEO & FounderSoli Deo Gloria