NOVELTY - Biological abiotic waste treatment system provides conjugate degradation and reduction of carbanaceous biological oxygen demand/chemical oxygen demand (BOD/COD), toxic recalcitrant organics, and nitrogen and phosphorus and minimizes excess biosolids.    USE - The methods provide simple, efficient, reliable, stable and economical systems for removal of organics, nutrients, and other target constituents, while generating little or no waste biological solids. Conjugate degradation and reduction of carbonaceous BOD and COD, toxic and recalcitrant organics and nitrogen and phosphorus and minimization of excess biosolids is enabled.    DETAILED DESCRIPTION - An oxidation-reduction treatment of waste water for producing treated effluent in a system with biomass and involving terminal reducing agents and primary oxidizers comprises:    (a) charging recuperable oxidation-reduction mediator specie(s) in the system;    (b) oxidizing terminal reducing agent(s) with reduction of recuperable oxidation-reduction specie(s);    (c) reducing primary oxidizing agent(s) with oxidation of the recuperable oxidation-reduction mediator specie(s); and    (d) retaining recuperable oxidation-reduction mediator specie(s) in the system in the form which is insoluble in the effluent, at least a portion of the waste water together with the recuperable oxidation-reduction specie(s) being repeatedly treated in the steps (b) and (c).    INDEPENDENT CLAIMS are also included for:    (1) a method of biological waste water treatment in a system with biomass for removal of nitrogen oxyions and producing treated effluent, comprising:    (a) charging recuperable oxidation-reduction mediator specie(s) in the system;    (b) oxidizing terminal reducing agent(s) with reduction of the mediator specie(s);    (c) reducing the nitrogen oxyions with oxidation of the mediator specie(s) whereby nitrogen oxyions form primary oxidizers; and    (d) retaining recuperable oxidation-reduction mediator specie(s) in the system in the form insoluble in the effluent, at least a portion of the waste water together with the recuperable oxidation-reduction specie(s) being repeatedly treated in the steps (b) and (c);    (2) a method of biological waste water treatment in a system with biomass, whereby the biomass generation is minimized and treated effluent produced, comprising:    (a) charging recuperable oxidation-reduction mediator specie(s) in the system;    (b) a biomass oxidation step, whereby the biomass forms a terminal reducing agent;    (c) reducing primary oxidizer(s); and    (d) retaining recuperable oxidation-reduction mediator specie(s) in the system in the form insoluble in the effluent, whereby the biomass is repeatedly treated in steps (b) and (c); the step of reducing the primary oxidizer is a step of oxidizing the mediator specie(s) and the biomass oxidation step is a reduction of the mediator specie(s);    (3) a method of biological waste water treatment comprising hydrocarbons in a system with biomass, whereby the hydrocarbons are oxidized and treated effluent produced, the method comprising:    (a) charging recuperable oxidation-reduction mediator specie(s) in the system;    (b) hydrocarbon oxidation step, whereby the hydrocarbon forms a terminal reducing agent;    (c) reducing primary oxidizer(s); and    (d) retaining recuperable oxidation-reduction mediator specie(s) in the system in the form insoluble in the effluent, whereby the biomass is repeatedly treated in steps (b) and (c); the step of reducing the primary oxidizer is a step of oxidizing the mediator specie(s) and the hydrocarbon oxidation step is a reduction of the mediator specie(s); and    (4) a method of biological wastewater treatment in a system with biomass for removal of toxic and recalcitrant organics and for producing treated effluent, the method comprising:    (a) charging recuperable oxidation-reduction mediator specie(s) in the system;    (b) oxidizing terminal reducing agent(s) while reducing the mediator specie(s);    (c) reducing the toxic and recalcitrant organics while oxidizing the mediator specie(s) to form primary oxidizers; and    (d) retaining recuperable oxidation-reduction mediator specie(s) in the system in the form insoluble in the effluent, whereby the biomass and mediator specie(s) are repeatedly treated in steps (b) and (c).