This cement not is considered secondary deposition, but not also primary. The presence of calcite and aragonite in the cement of the beachrocks induces a preliminary question: who precipitated first? As this identification was not possible, because there is calcite and aragonite in the cement, and due to the great instability of aragonite, the answer is not possible. Geochemical tests with EDAX were performed without absolute success.
I advocate the theory that beachrocks cement derives from the dissolution of biogenic carbonate as well as the precipitation of seawater when they are in contact with fresh water. Analyzing the results of stable isotopes through modified diagrams of Milliman (1974), James & Choquette (1980b), Keith & Weber (1964), confirmed this hypothesis.
You have two issues to address. The first is the origin of the dissolved inorganic carbon, and the second is whether you have to consider active biological isotopic fractionation.
The degradation of organic matter plants C3 induces a CO2 with isotopic values at 13C of the order of -21. Its values can be seen on the analysis of the DIC.
Therefore the first is the origin of the dissolved inorganic carbon, and the second is biologically active isotopic fractionation.
As you are probably aware, there are two ways of forming calcite naturally: biogenically and chemically. Biogenic carbonate typically has a delta value of 0‰, but chemical carbonate will have a delta value matching that of the carbonate ions from which it is precipitated. In your case, you have a hot hydrothermal fluid causing the sea water at the ocean floor to precipitate CaCO3. The source of carbonate ions in that reaction will be from carbon dioxide produced by the respiration of microorganism which breakdown the biological carbon created at the surface of the ocean by photosynthesis. That carbon has a typical delta value of -25‰, and is carried to the sea floor by the biological pump.
Thus, I suspect that your carbonate is being produced chemically, not biogenically.
Yes, I know these heterotrophic organisms which can tolerate high temperatures. Now, I prefer microbially-induced carbonate, the organism use organic carbon as a carbon source and precipitate carbonate. During this process, the metabolic may cause the oxygen and carbon isotope fractionation.