How the constituents form a whole

One entry is a physical question. A physical explanation constrains which model relations can describe a phenomenon and what an observation can distinguish. Mathematical construction supplies consequences and proof conditions. A computational construction must obtain the required result within its resource and interaction conditions. Modelling relates that result back to the question; notation lets the participants perform, inspect and alter the constructions without silently changing their meaning.

These relations support revision as well as forward calculation. A counterexample can change a mathematical construction; an observational mismatch can reopen a physical premise; non-identifiability can change the requested inference; a computational limit can require a different representation or algorithm. The affected expressions and dependent conclusions then need to change with the premise.

The whole makes a construction or consequence jointly interpretable, obtainable and revisable while retaining the different grounds on which its parts rest. Mathematical validity, physical adequacy, computational feasibility and the intelligibility of an expression are different conditions. None substitutes for the others, and each applies only where that kind of claim is made: a pure proof needs no invented physical validation. The full account (Foundational.Reference:2.2) explains how these conditions constrain and revise one another.

For example, predicting an oscillating system requires a physical account of the interactions, a model with stated idealizations, a mathematical consequence, a way to compute the needed approximation and an expression whose variables and units remain interpretable. A measurement that defeats the idealization returns to the physical and modelling premises; running the same algorithm more accurately cannot repair that defect. Conversely, a numerical error need not invalidate the physical law. The linked DPFs teach the distinct constructions and returns.

This is a field architecture, not a requirement to use all five DPFs for every task. A pure mathematical construction can be complete without a physical example. Algorithm construction can begin with a required witness or cost bound. Notation design can begin with an ambiguous score, diagram or gesture and finish with a usable reading procedure; using notation in music does not bring musical performance into this Suite. A question can also begin with a failed model correspondence or an unexplained physical observation. The constituent repertoires remain independently usable, and applications outside the field remain applications.

For example, replacing one supplied value by two sensor reads changes the meaning of an expression. The identity q + q = 2*q remains true, but read() + read() need not equal 2*read(). Notation and computational interpretation expose which operation is intended; a measurement account supplies what a read means. This entry starts with expression use, not with a physical calculation awaiting its suppliers.

The field continues while the five kinds of constructive contribution and their relations remain available: deriving and warranting consequences, establishing their subject correspondence, forming physical accounts, obtaining results through effective operations, and constructing interpretable expressions. New theories, models, algorithms and notation can replace older ones when the affected grounds and couplings remain recoverable. Replacing a constituent requires an adequate contribution in these relations; merely removing its questions changes the field. A broader purpose or a new subject-specific field requires a fresh architecture judgement; a common word such as “model” or “thinking” is insufficient. The Reference (Foundational.Reference:2.3) develops the continuity rule and possible connections to larger fields.