**4. Conclusions**

Real numbers are the "ideal" framework for dealing with quantities associated with physical phenomena. However, real numbers are not attainable and "disappear" when the observer obtains the value of a quantity. The alternative is digital numbers but a measurement value becomes a point on a digital scale depending on the phenomenon itself, the accuracy, correctness, and errors of the devices used in the measurement, and the numbers of digits used to represent it on the digital scale.

A mark represents, in a consistent procedure, the point information provided by a digital scale. The system of marks has an internal structure that reflects not only the losses of information inherent in the readings on a digital scale and the evolution of the computations from them, but also the indiscernibility of the observed phenomena.

The computations performed using marks also reflect the gradual loss of information, due to numerical errors and truncations, and give relevant warnings for decision making, either on the acceptability of the results or on the usefulness of seeking more precision to achieve the necessary validity.

In conclusion, marks are an appropriate framework for any iterative process, within current research conditions and certain assumptions, where uncertainty is significant and can be generalized when needed. For example, if the process is a long simulation with many steps or an iterative approach with slow convergence, it is necessary to control the accumulation of experimental, scaling, and computational errors so as not to let it exceed the tolerance set by the observer.

The benchmark presented is a good example of the use of marks for an iterative process. Marks prove to be a correct and satisfactory tool for modeling physical systems with uncertainties in their variables and parameters. Marks provide a double contribution to any computational process: (1) the granularity as a good timely test for the validity of any result, and (2) they provide meaning to any valid result by means of the semantic theorem. The final semantics of a simulation using marks is just the one needed for problems like fault detection, control, or parameter identification (via optimization) of a mathematical model against a set of experimental data with uncertainties. This opens a wide field of applications for marks.

**Author Contributions:** Conceptualization, M.A.S., L.J., R.C., I.C and J.V.; methodology, M.A.S., L.J. and R.C.; software, M.A.S. and I.C.; validation, M.A.S. and I.C.; formal analysis, M.A.S., L.J. and R.C.; investigation, M.A.S., R.C., L.J. and J.V.; resources, M.A.S., R.C., L.J. and J.V.; data curation, M.A.S. and I.C.; writing—original draft preparation, M.A.S., R.C. and L.J.; writing—review and editing,M.A.S., L.J., R.C., I.C. and J.V.; visualization, M.A.S., L.J., R.C., I.C. and J.V.; supervision, M.A.S., L.J., R.C., I.C. and J.V.; project administration, R.C., I.C. and J.V.; funding acquisition, J.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by the Spanish Ministry of Science and Innovation through gran<sup>t</sup> PID2019-107722RB-C22/AEI/10.13039/ 501100011033 and the Government of Catalonia under 2017SGR1551.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available at reference [19].

**Conflicts of Interest:** The authors declare no conflict of interest.
