Foundational.Reference:2.2 - Relations that form the field
The discipline parts are the knowledge and reusable practices described by the five DPFs. A list of publications, shared vocabulary or frequent joint use would not establish their composition. The relations here connect what a construction is permitted to mean and do with how a result is obtained and how a failure changes the available knowledge or practice.
| Connected contributions | Knowledge constrains the operation | Results or failures return to the construction |
|---|---|---|
| Mathematical Thinking and Mathematical Modeling | The mathematical objects, operations and preservation laws limit which formulations and reductions retain the required consequence. The subject correspondence determines which mathematical distinctions matter. | A counterexample or failed answer recovery can expose an invalid reduction; non-identifiability can show that the formulation cannot determine the requested subject answer. MATH.2/.18 and MMP.10/.12/.14 develop these different repairs. |
| Physical Thinking and Mathematical Modeling | Interactions, preparation, conservation or response premises constrain an admissible physical model; the observation procedure constrains what its variables and data can mean. | A failed prediction, unresolved scale interaction or changed physical constraint can require a different physical account, observation or mathematical formulation. PHY’s prediction-and-observation construction and MMP.14/.16/.20 retain those returns. |
| Computational Thinking and mathematical construction or modeling | The required mathematical answer and allowed operations constrain correctness; access, representation, interaction and resource limits constrain what can be obtained. | An effective construction can make an existence result usable. A lower bound or unattainable resource demand can instead require another representation, weaker answer or changed formulation. CMP.1/.8/.10/.11 and the answer-under-limits case expose these differences. |
| Notational Engineering and the other four branches | Binding and interpretation determine which object, observation or operation an expression denotes. Mathematical laws, physical measurement meanings and computational semantics constrain permissible transformations. | A changed interpretation can invalidate a rewrite while leaving the mathematical identity intact. Difficult reading or editing can require a different expression, while the intended operation constrains what that redesign must preserve. NOT.2–.7 and CMP.12 supply the relevant constructions. |
These relations persist beyond a single Work occurrence. A proof establishes a transformation condition that later formulations and algorithms can reuse. A physical experiment can change the premises admitted by later models. A notation repair can change how later readers construct or inspect an argument. The field therefore includes the continued development of the contributions and their connections, not just an occasional team in which five specialists happen to participate.
A notational entry exposes a substantive difference. In the formula, graph and table case (NOT — NT-CHANGE-TOGETHER - Carry a changed request through several representations), q + q + b and 2*q + b agree when q denotes one supplied real value. If each occurrence becomes a fresh sensor read, successive readings 4 and 5 with b=1 give read() + read() + b = 10, whereas 2*read() + b = 9 using the first reading. The arithmetic law has not failed. NOT.3/.4 must distinguish the new interpretation and transformation conditions; CMP.12 can give the expression an effective interpreter, and the measurement account supplies what one read obtains. Binding one sample once or retaining two observations answers different questions. The returned distinction can change the notation, algorithm or measurement request.
A mathematical entry need not proceed to physics. The two-element selection case asks for a deterministic selection that respects renaming and proves the obstruction. That proof is already a mathematical result. Introducing an order or changing the guarantee to equal probabilities changes the construction; it does not repair the original theorem by a better implementation. The later dispatch example shows how the new construction can then alter a way of working without confusing the mathematical and organizational conditions.
An algorithmic entry can change what mathematics must retain. In the answer-under-limits case (CMP — CP-ANSWER-UNDER-LIMITS - Obtain the answer the work needs within available resources), asking for every optimum rather than one optimum changes the permitted pruning and the contents of a shared table. The optimum value may stay the same. The needed witness information and computational representation do not. This is a connection among answer meaning, representation and effective construction; no new physical theory is required to make it part of the field.
The physical prediction and operating-flow cases in Section 3 show further entries. Their calculations illustrate particular connections and retain their own conditions. None of the cases sets a narrower scope for all five branches.