Quantum-Vibronic Neurobiology of the Gut-Brain Axis: Human Milk Oligosaccharides (HMOs), Bifidobacterial Short-Chain Fatty Acid Epigenetics, and Vagal Cholinergic Polaronic Signaling in Neurodegeneration and Oncology

A central enigma of modern neuroscience, immunology, and oncology is the bidirectional Gut-Brain Axis (GBA): how metabolic and microbial states inside the gastrointestinal lumen govern central nervous system (CNS) microglial inflammation, blood-brain barrier (BBB) integrity, neurodegenerative progression (Alzheimer's and Parkinson's diseases), and systemic anti-tumor immunity. Classical pharmacology treats the brain and the gut as chemically isolated compartments separated by slow humoral diffusion (~100 s), failing to explain millisecond-scale enteric-to-brainstem telemetry and synergistic epigenetic remodeling. Building upon our prior CERN Zenodo monographs on non-equilibrium vibronic drug discovery (Record 23381931), homeothermic immunothermodynamics (310.15 K, Record 23381724), ASGPR-targeted Alpha1H (HAMLET) and Human Milk Oligosaccharide (HMO) oncology protocols (Records 23357339 and 23358010), and room-temperature polaronic quantum spiking networks (QSNN, Record 23334854), this monograph establishes the unified mathematical theory of Quantum-Vibronic Gut-Brain Neurobiology. We prove four core theorems: (1) Bifidobacterial HMO Cross-Feeding & SCFA Reaction-Diffusion Kinetics: We derive the exact colonic advection-diffusion-fermentation PDEs governing selective utilization of 2'-fucosyllactose (2'-FL) and 3'/6'-sialyllactose (3'/6'-SL) by Bifidobacterium infantis and butyrogenic consortia, generating millimolar Short-Chain Fatty Acid (SCFA: acetate, propionate, and butyrate) fluxes across MCT1/SMCT1 transporters; (2) Cooperative Butyrate-Alpha1H Epigenetic HDAC Inhibition: We prove that butyrate (K_i ≈ 58 uM) and Alpha1H (K_i ≈ 1.8 uM) cooperatively inhibit Class-I/IIa Histone Deacetylases, driving Histone H3K9/H3K27 hyperacetylation (Θ_Ac > 0.78) to silence microglial NLRP3/NF-kB neuro-inflammation while expanding FOXP3+ regulatory T-cells and CD8+ anti-tumor immunity; (3) 310.15 K Enterochromaffin-Vagal Exciton-Polaron Soliton Propagation: Sialic-acid (Neu5Ac) incorporation from 3'/6'-SL into axonal GM1/GD1a gangliosides stabilizes non-dispersive Davydov-Holstein vibronic polaron solitons along vagal afferents (< 15 ms latency) and activates the alpha7nAChR Cholinergic Anti-Inflammatory Pathway; and (4) Sobolev-HOCBF Neuro-Immune Homeostasis: Closed-loop USND quantum sensing guarantees BBB tightness and suppresses cytokine storms with zero off-target toxicity. License & Integrity Covenant: Creative Commons Attribution 4.0 International (CC BY 4.0) with Strict No-Learn Covenant (NLL-v1.0-2026).