A. Formulate the subject question

MMP.10 - Construct and Revise a Mathematical Constraint Formulation

Type: Method pattern Status: Usable, evolving Normativity: Normative within the stated use

MMP.10:1 - Problem frame

Use this pattern when you know conditions that a possible object or situation must satisfy, but still need to express its possibilities mathematically. You may be looking for an arrangement, a quantity, a shape or a rule of action. Choosing variables already makes decisions: a number can express an amount, but an ordered list, a set of members and an unknown function permit different operations and different possible answers.

Start by describing one candidate object and what would make it admissible. Choose a way to represent it, then distinguish two kinds of condition: those needed for the representation to denote such an object, and those expressing the requirements on that object. This produces a mathematical problem that can be reasoned about or handed to an appropriate computational method.

The result consists of variables with their domains, joint conditions, the requested operation on their solutions and a way to interpret the answer. It can be enough to find one feasible case; another question may require all possible values, a preferred case or a count. The chosen representation must support that particular result.

You need to understand the objects and requirements in the original question, elementary logical conditions and the mathematics used to express them. A mathematical collaborator can construct the expressions while you resolve their intended meaning. When a suitable formulation already exists, use it. When the missing contribution is a physical relation, an observation law or a rule of change, obtain that relation through the relevant method before translating it. B.5.FM supplies the broader first-model construction; MMP.7 and MMP.9 develop observation and reduced-evolution constructions.

MMP.10:2 - Problem

A convenient variable can leave out the possibility that matters. A single interval cannot represent an activity that may pause. A membership bit cannot record repeated membership. Coefficients of a straight line cannot describe every continuous curve. These restrictions may be useful, but they change which cases the formulation can answer for.

Representation can also add possibilities. A table intended to describe a function can allow two values for one argument unless its conditions exclude that case. Extra fields can create several records of the same object. A calculation over those records can then answer a different counting or probability question.

The difficulty is to construct the mathematical expression of the intended possibilities, including their joint restrictions and the requested result. Solving the resulting equations addresses the problem only after that construction has been made.

MMP.10:3 - Forces

MMP.10:4 - Solution

Describe the possible object → choose its representation → derive the representation conditions → express the requirements and question → obtain and interpret a result → revise the affected construction.

MMP.10:4.1 - Recover what varies and what the question asks

Describe a candidate before choosing scalar variables. Is it an amount, a collection, a sequence, an assignment, a function or another mathematical object? Which distinctions can change the answer? A set retains membership; a multiset also retains multiplicity; a sequence retains positions. Choose among them from the question.

Separate supplied quantities from unknowns. Among unknowns, distinguish what is to be inferred, what can be chosen and what can vary independently of that choice. If a choice uses an observation, specify when the observation becomes available; MMP.8 constructs the resulting information-dependent requirement. An unknown value does not become a freely selectable design variable merely by appearing in the same equation.

State the requested result. Existence asks whether at least one admissible object can be constructed. Inference asks what a quantity can be across admissible objects. Selection adds a preference among them. Counting and sampling depend on how individual objects are distinguished. Keep those requests separate while choosing the representation.

MMP.10:4.2 - Construct variables that represent the object

Choose an expression from which a candidate can be recovered. Give each variable a domain and any unit or reference point needed by its operations. A machine label ranges over names; arithmetic on the label requires a separate meaning. A count ranges over nonnegative integers; an amount may be divisible. State a finite bound when the task supplies one. Adding a bound solely to finish a search restricts the question to that bound.

For a structured object, compare representations by the operations you need. A function on a finite set can use one output variable for each input. Alternatively, a table of Boolean indicators can say which input-output pairs belong to its graph. The first expression makes function evaluation easy to state. The second makes some relations among pairs visible, but needs conditions to make the table a function. For a partial function, represent undefinedness as well as defined values.

Keep a shared quantity shared. If several equations use the same unknown offset, one offset variable must occur in all of them. Introducing a separate offset in each equation creates additional possibilities. Conversely, equating genuinely separate values can remove possibilities.

For a function or shape over an infinite domain, choosing finitely many coefficients also chooses a family. Identify that family and whether it expresses the intended possibilities or is a deliberate restriction. For example, the conditions on a continuous function may allow curved solutions even when no affine function satisfies them. A useful restricted family can be sufficient for finding a witness; failure inside it leaves the larger family unresolved.

MMP.10:4.3 - Derive structural conditions and translate requirements

Ask what must hold for a variable assignment to describe one candidate of the intended kind. With Boolean entries r_ij describing the graph of a total function, require sum_j r_ij = 1 for every input i. For a partial function, replace this with sum_j r_ij <= 1; an all-zero row then means undefined at that input. For an injective function, additional conditions on columns express the extra requirement.

These conditions have different reasons. One value per input comes from choosing a total function. Injectivity comes from the particular problem, if it requires injectivity. Keep their reasons recoverable so that a later change from total to partial or from injective to unrestricted has a local repair.

Express the original requirements using the represented objects. Conjoin conditions that must hold for the same assignment. Use disjunction for allowed alternatives and implication when choosing an option imposes a condition. An implication alone supplies no timing; use time quantities or an explicit sequence when order matters. Preserve a coupled condition such as x+y=1 rather than replacing it by separate bounds on x and y.

When a bijection between representations is established, MATH.7 supplies transport of operations, relations and compound expressions. For a representation with several records per object, C.29.1 supplies the more general correspondence. Use the decoding of a record to express the requirement on its object. If a condition is rewritten to fit the receiving notation, derive that expression from the original relation and the structural conditions. This is where a missing index, an undefined value or a lost alternative can change the formulation.

Additional constraints can expose consequences and help the obtaining method. Derive them from the retained requirements, and preserve that dependence. Fewer variables or more constraints do not alone establish a faster method; compare the actual resulting work when efficiency matters.

MMP.10:4.4 - Make the answer correspond to the question

Define how to recover the requested object or quantity from a satisfying assignment. Then work in both directions: represent an intended admissible case, and interpret an allowed assignment. Use the construction and its conditions to establish the reach of this correspondence. A small case can expose a mistake; a claim about every case needs the corresponding argument.

Match that reach to the requested result:

• To use a witness, its recovered object must satisfy the original requirements.
• To conclude that no intended object exists from inconsistency of the formulation, every intended object must have a representation in it.
• To infer all possible values, translate the quantity as well as the admissible cases; C.16.IR supplies the projection question.
• To optimize, translate the objective and preference as well as feasibility. Distinguish a bound from a value attained by an object.
• To count or sample objects, account for multiple representations of the same object. One representation per object is one solution; weighting or grouping representations can be another.

Auxiliary variables can change what a returned number means. Suppose a finite nonempty set of finish times f_i is determined by the other variables. Introduce a real auxiliary T used only in T >= f_i and the objective of minimizing T. Lowering T to max_i f_i then preserves feasibility, so at an attained optimum T equals the latest finish. If T must instead be an integer and the latest finish is 1/2, its minimum is 1. A merely feasible intermediate T can also exceed the latest finish. Recover the actual latest finish as max_i f_i; infer equality with T only when its domain and other conditions permit that lowering.

Choose an obtaining method for the constructed question. Manual substitution may suffice; another problem needs a numerical method, symbolic derivation or search. C.29.2 separates the required mathematical result from the procedure and its execution. Use that method’s actual conclusion: finding no case within a time budget differs from establishing inconsistency. Preserve any restriction or approximation when returning the result to the original question through C.29.1.

MMP.10:4.5 - Revise the formulation from the changed requirement

Locate the changed participant, domain, relation or requested result. Changing a supplied amount can retain the same representation. Allowing interruptions changes what an activity description must express. Changing a total function to a partial one changes structural conditions. Changing existence to counting can make duplicate records material.

Revisit constraints derived from the old requirement as well as the original formula. Reconstruct the affected expressions and answer interpretation, retaining the unaffected ones. If a solving tool cannot support the needed object, construct a suitable representation or choose another obtaining method; keep any deliberate restriction visible in the returned conclusion.

Stop with a usable formulation and interpretation, an adequate answer, or a named missing relation or operation. Choose further derivation, observation or computation according to what it can change in the work and its cost; C.11.DUA supplies that decision. When the formulated object is itself a working method, return its proposed change to ME for interpretation and use. The mathematical model supplies a reason for the change; the working method still has to be performed under its stated conditions.

MMP.10:5 - Archetypal Grounding

MMP.10:5.1 - Construct an unknown rule from requirements on its repetitions

A device has three labeled modes A, B and C. The required rule changes the mode on every use and returns to the starting mode after three uses. The question is to construct a deterministic rule, with no additional internal state. The rule itself is the unknown object.

Let S={A,B,C}. Choose one output variable p_i in S for each input i. The requirements become p_i != i and p_(p_(p_i)) = i for every i. Function composition gives the meaning of the repeated application. MATH.1 constructs composable paths; MATH.5 extends an interpretation of their elementary steps to the compounds.

To express the rule by selected pairs instead, choose r_ij in {0,1}. Add sum_j r_ij=1 for each row, r_ii=0, and, for all i,j,k, (r_ij=1 AND r_jk=1) implies r_ki=1. The row condition makes a function. The implication expresses the return after three uses: the first two selected transitions determine a required third.

Recover p by taking the unique selected column in each row. Conversely, p creates the table by selecting exactly its output pair in each row. These constructions are inverse. The triple-application requirement is therefore the same in both formulations, with MATH.7 carrying that relation. A rule A→B→C→A and its reverse both satisfy it.

There are precisely two such rules. From p^3=id, p is invertible with inverse p^2. Its cycles have lengths dividing three. Since a one-element cycle is forbidden, the three modes form one three-element cycle, with two possible orientations. This reasoning proves completeness; listing two examples alone would not.

Change the device to four modes, retaining the three-use return and no unchanged mode. A permutation of four elements cannot partition them into cycles all of length three, so no rule exists under these conditions. Change instead to a return after two uses. The conditions become p_(p_i)=i and p_i!=i; the indicator formulation requires symmetry r_ij=r_ji. It admits three pairings of four modes. Remove the former three-use implication: leaving it in the formulation would make the new, feasible requirement appear impossible.

The result is a rule that can be implemented and its stated scope: deterministic changes of the visible mode without hidden state. A proposal with additional state describes a different device and needs a new account of its operation.

MMP.10:5.2 - Preserve existence while repairing a count

An optional assignment gives each of two named requests either no selected option or one of options 0 and 1. Different requests may select the same option. This is a partial function from the two requests to {0,1}. Each request has three possibilities, so there are nine assignments.

Suppose the storage format gives each request two bits: d says whether an option is defined, and q gives its value when defined. When d=0, q is ignored. All sixteen four-bit records denote valid partial assignments. Every assignment has a record, so the representation can support an existence query with translated requirements.

It does not preserve the count. The empty assignment has four records, each of the four assignments defined on exactly one request has two records, and each of the four total assignments has one record. Thus 4 + 4*2 + 4 = 16. Uniform selection among records gives probability 4/16 to the empty assignment and 1/16 to each total assignment, rather than the 1/9 obtained by uniform selection among assignments.

For a count or uniform assignment sample, one repair is the structural condition d=0 implies q=0 for each request. There are now three admissible bit pairs per request and nine records, one per assignment. Another is a single variable with domain {absent,0,1} per request. If the sixteen-record storage representation must remain, group or weight its records using the multiplicities instead. To sample the nine assignments uniformly, give each record of an assignment with m records probability 1/(9*m). The probabilities of all m records then sum to 1/9 for that assignment. Thus each record of the empty assignment receives 1/36, each record of a one-request assignment 1/18, and each total-assignment record 1/9. C.29.1 supplies the required correspondence; MMP.7 supplies a probability law when sampling is the intended operation.

The original existence use can remain sufficient. The new count or sampling question exposes the need for the additional construction. No change in the underlying possible assignments is intended.

MMP.10:5.3 - Keep quantities and domain restrictions together

A preparation requires one litre containing 35 percent solute by volume, using solutions A and B at 20 and 80 percent. Assume solute is conserved and component volumes add in this preparation. Those subject assumptions supply the relations. Let x and y be the respective volumes in litres; choose nonnegative real domains because the amounts can initially be divided freely.

The joint conditions are x+y=1 and 0.2*x+0.8*y=0.35. Substitution gives x=0.75, y=0.25. Both the total and solute requirements hold for those amounts. Separate bounds 0<=x<=1 and 0<=y<=1 would lose their required total.

Now only whole half-litre doses may be used. Change the representation to x=m/2, y=n/2, with nonnegative integers m and n. The volume equation becomes m+n=2. Its possibilities (m,n)=(2,0),(1,1),(0,2) give solute amounts 0.2, 0.5 and 0.8 litre. None supplies 0.35 litre. Rounding the former solution changes the preparation; it does not satisfy its original condition.

If the required concentration changes to 50 percent, one dose of each solution works. The subject relations and unit remain, while the requirement and feasible assignment change. If mixing changes volume or solute, obtain the replacement subject relation before revising its mathematical expression.

MMP.10:6 - Bias-Annotation

Tool familiarity can make scalar variables appear inevitable and conceal a different object or useful operation. Begin with the candidate object and its requirements. Compare representations when that comparison can change the answer or obtaining effort.

Compactness can hide duplicate records or omitted possibilities. Judge the representation by the requested result and the work needed to obtain it.

MMP.10:7 - Conformance Checklist

• Can the reader identify a candidate object before interpreting the variables, including what is supplied, unknown or selectable?
• Do domains, units, shared quantities and definedness express the intended possibilities?
• Which conditions make the representation denote an object, and which express requirements on it?
• Do conditions that must hold together refer to the same assignment? Are alternatives and conditional restrictions preserved?
• Can intended cases be represented and satisfying assignments be interpreted at the reach required by the question?
• Does the requested value, preference, count or probability survive the representation, including auxiliary values and duplicate records?
• What does the obtaining method actually establish, and which changed requirement reopens which construction?

MMP.10:8 - Common Anti-Patterns and How to Avoid Them

MMP.10:9 - Consequences

The mathematical problem becomes available for reasoning, computation and revision without leaving the interpretation of its variables implicit. A solver result can be returned as the arrangement, quantity or unresolved distinction that the work needs.

Constructing and maintaining a second representation has a cost. Direct formulation is often sufficient when the objects and their restrictions already have a clear expression. A structured intermediate formulation becomes useful when it preserves meaning across several receiving notations, helps revise requirements or exposes a calculation that the first representation hid. Any efficiency advantage depends on the resulting obtaining method.

MMP.10:10 - Architectural Rationale

Mathematical modeling often starts with choosing how possible objects will be expressed. The choice determines which restrictions must be added and which results can be recovered. Separating representation conditions, subject requirements and the operation on solutions makes revisions local: a changed requirement need not replace the representation, and a new representation need not change the intended possibilities.

An operation can itself be the unknown object. The finite-rule case therefore treats repeated action as a requirement on a function and uses composition to express it. The same organization applies to other structured objects, while their mathematics supplies the needed constructions and proofs. MATH.7 explains reversible transport after the maps are available; this pattern develops the modeling choice and construction of those expressions, including cases that need a many-to-one correspondence.

The connection to computation runs in both directions. A mathematical formulation supplies the problem a procedure must answer. The operations supported by a procedure can suggest a different expression of that problem. Meaning is retained through the representation conditions and answer interpretation, while performance is judged on the resulting work.

MMP.10:11 - SoTA-Echoing

The MiniZinc Handbook 2.10.1, modeling and efficiency sections develops alternative models, derived constraints and interactions with solving methods. Adopt comparison of the resulting work; a smaller variable count alone does not settle it. A direct scalar formulation remains economical when its meaning is already clear.

Akgun and colleagues, Conjure (2023), sections 2-4 separates representation selection from expression refinement and introduces structural constraints during refinement. Adopt this construction when structured objects would otherwise disappear into unexplained scalar choices. Conjure’s finite combinatorial scope and model-selection heuristic remain specific to that approach. They do not establish a general best representation or performance guarantee.

The Essence language reference, function and relation domains makes such choices as partiality and cardinality explicit. This informs :4.2-4.3; its particular syntax is optional. The ordinary alternative is to express those conditions directly in a familiar mathematical notation.

For infinitely many possible objects, a finite parameterization needs its own coverage or approximation argument. Consider continuous nonnegative functions f on [0,1], with f(0)=f(1)=0 and integral one. An affine parameterization permits only the zero function after the endpoint conditions, so it fails. The quadratic f(t)=6*t*(1-t) satisfies every requirement. This authored countercase explains why the finite structured-model sources do not settle general parameterization. C.29.1 supplies the interpretation of a restriction; the relevant mathematical method supplies a suitable larger family or approximation.

To apply the formulation in another subject, obtain the relations and mathematical operations needed to express its requirements. Reconsider the representation when a new requirement, result kind or obtaining method changes what it needs to preserve or make affordable.

MMP.10:12 - Relations

• B.5.FM and B.5.TU: construct a first account and connect a subject theory to the encountered problem. This pattern translates its candidate objects and conditions into a mathematical formulation.
• MATH.1 and MATH.5: construct composable paths and extend an assignment on elementary steps to compounds while preserving operations and equations. MATH.16 chooses a mathematical construction from the maps it must support. MATH.7 transports structure when the required bijections have been constructed.
• C.29.1 and C.29.2: supply correspondence, answer recovery and the separation of a mathematical result from its obtaining procedure and execution.
• C.16.IR: determines what a compatible set permits one to infer through projection and constancy of the requested quantity.
• MMP.7 and MMP.8: construct observation probabilities and information-dependent choices. A formulation’s variables retain those probabilistic and temporal meanings.
• MMP.9: derives a reduced evolution law when retained quantities depend on eliminated contributions. The resulting law can supply relations used here.
• C.11.DUA and E.22/E.23: choose worthwhile further inquiry and organize evaluation and improvement of a formulation or its obtaining work.
• ME: uses the mathematical result when constructing or changing a working method. Its performance in the subject remains distinct from the formal properties of its description.

MMP.10:End

MMP.11 - Construct a Mathematical Model Family from Known Relations

Type: Method pattern Status: Usable, evolving Normativity: Normative within the stated use

MMP.11:1 - Problem frame

Use this pattern when part of a model is supported, but a function or other relation is still unknown. You may know which quantities interact, what must be conserved, or how a response begins and ends, while lacking its form between those conditions. You need candidate models that retain this knowledge while allowing the missing contribution to vary.

Begin by locating the unknown contribution in the relations needed for the question. Say what it takes as input, what it supplies and which properties its variation must preserve. Construct a family for that contribution, then put it back into the model. For a response that rises from zero to one, a family of curves constrained by those properties gives a different starting point from an unrestricted fitted curve.

The first useful result is a family of models, its remaining adjustable parts and the restrictions introduced by its construction. It may already give a sufficient bound. When observations are available, the same construction lets you ask which functions, parameters or consequences those observations can distinguish.

You need the subject grounds for the retained relations and enough mathematics to construct and use the family, or access to that mathematical contribution. A known adequate relation can be used directly. An unknown numerical parameter inside a suitable family normally needs its estimation method. Use this pattern when the family itself needs construction or revision. B.5.FM and B.5.TU help when the missing contribution is the subject account or theory from which a relation should come.

MMP.11:2 - Problem

A familiar formula can exclude the response being sought. Giving a learner unrestricted freedom can create the opposite problem: the fitted response may violate a known relation. Fitting two interacting contributions independently can also destroy a property that depends on their connection.

Agreement with recorded outputs leaves another difficulty. Several parameter settings may define the same function, and several functions may produce the same observations. Some receiving questions distinguish those alternatives; others need only a consequence on which they agree. Choosing one fitted instance can conceal this difference.

The modeling work is to construct the adjustable contribution in a form that retains the supported relations, then determine what that family can establish for the present question. The construction itself may impose additional restrictions, so its expressive limits belong to the answer.

MMP.11:3 - Forces

MMP.11:4 - Solution

Locate the missing relation, separate what is retained from what may vary, construct the adjustable family, and derive its contribution to the receiving question. Observation or computation is then selected for what remains unresolved.

MMP.11:4.1 - Locate the contribution that may change

State the result wanted from the model. Identify the unknown relation and the quantities it connects. Keep its inputs, output, domain, units and permitted dependence explicit enough to substitute a candidate into the surrounding relations. A function of present state, a function of its history and a random response law admit different constructions. If equal proposed inputs require different deterministic outputs in the admitted circumstances, revise the inputs or retain those alternatives. Greater flexibility of a single-valued function cannot supply both outputs. For a dynamic model, A.3.3.TR supplies the corresponding reconsideration of state.

Recover the grounds and application range of the relations you retain. A balance may be required by the chosen boundary; monotonicity may hold only over one operating range; a shape assumption may be provisional. Keep an allowed discrepancy when the subject account supplies one. A convenient property is not automatically a property of the subject.

Use the smallest part that can be varied without silently changing another retained claim. If an adjustable term can absorb a known contribution, include that possibility in the inference question. Section :5.2 shows an ambiguity between two gross transfers even when their net effect is known.

MMP.11:4.2 - Construct the permitted variation

Translate each retained property into a mathematical condition, then choose a construction that satisfies it. MMP.10 supplies the general work of representing candidate objects and their conditions. Here the object being constructed is a family of relations to insert into the model. Direct conditions may already give a workable representation of that family. Constructing through free elements is useful when their variation should preserve the conditions; compare its obtaining and revision operations with those of the direct representation.

A useful construction separates a fixed part from free variation. If a linear operator L must satisfy L(g)=b, find one particular solution g0 and choose a correction h with L(h)=0. Then g=g0+h retains the condition. This describes every solution only if the admitted corrections cover the whole null space in the chosen function domain. Restricting h to a few basis functions supplies a smaller family. Establish the linearity and domain before using this construction.

For a sign, bound or shape condition, construct through a map whose output has that property. Nonnegative weights can be normalized to probabilities. Integrating a nonnegative function can produce a nondecreasing response. Work out the domain and boundary of the resulting family: strict positivity excludes zeros, and an integral of an ordinary integrable function produces an absolutely continuous curve. Section :5.1 develops one such construction and its restriction.

When a property depends on coupling, construct the coupled contribution. Using the same transfer with opposite signs in two balance equations preserves their total. Two separately fitted right-hand sides have no such identity unless their joint conditions supply it. Use A.3.3.TR to assemble a change rule from the interacting relations.

A penalty during fitting offers a different construction: it discourages violations while allowing them. Use it when that allowance fits the question. If the account requires an identity, either build it into the representation or use an obtaining method that enforces it. The size of a training penalty does not by itself establish the identity.

MMP.11:4.3 - Choose the representation and its range

Choose a representation whose operations fit the required use and available resources. A table can represent a finite function. A basis expansion or program can retain a useful structure. A neural representation can supply a flexible adjustable function. The meaning of its inputs and outputs, and the retained relations, remain part of the model.

Check two different questions. Does every admitted parameter setting produce a relation allowed by the construction? Does the construction cover all relations needed for the present conclusion? A witness may need only one candidate; an impossibility claim over all admissible models needs coverage of that whole family or another sufficient argument.

State restrictions introduced by knots, basis functions, regularity, network architecture or domain truncation when they can change the answer. A numerical fit inside the restricted family answers for that family. If its consequence is sufficient, a more flexible family may add only cost. If the missing case matters, change the representation.

Different parameters can denote the same function. Recover the function or consequence needed by the receiving use rather than demanding unique parameters by default. Section :5.3 gives a normalization redundancy. Use MATH.7 when a change of representation has constructed inverse maps; use C.29.1 when correspondence is more general.

MMP.11:4.4 - Put the family into the model before using its fit

Substitute the adjustable contribution into the relations that consume it. Derive the resulting observable or answer condition with shared quantities kept shared. An error measured on an isolated contribution and an error in the coupled output are different fitting questions.

Choose the insertion point from what must remain meaningful. In a component model, inserting an unknown relation before algebraic elimination can retain a named component’s inputs, outputs and connections. Adding a correction after elimination can be simpler, but the correction then acts on the transformed relations. Recover how it affects the properties needed by the original question. A reduced model may use MMP.9 to derive the contribution its simplification leaves open.

Represent how observations are produced. MMP.7 derives a probability law for records when probability is needed; C.16.IR uses the supplied indication relation to obtain compatible cases or bounds. Fitting an unobserved internal term as if it were measured supplies an extra premise. If that premise is unavailable, fit or constrain through the observable relation instead.

For a dynamic or implicitly defined model, obtain the consequence through its coupled equations and conditions. A good component fit does not settle whether the resulting evolution, initialization or constraints are usable. Apply the mathematical and computational method appropriate to the stated consequence; C.29.2 helps formulate its obtaining operation.

MMP.11:4.5 - Determine which remaining differences matter

Ask what the available observations constrain: the adjustable parameters, the unknown function over a stated domain, or a particular consequence. Use C.16.IR on the resulting observation relation. The function can remain undetermined away from the observed inputs even when its recorded values are fixed.

When ambiguity could change the answer, construct two admitted candidates with the same relevant observations and different receiving consequences. Such candidates show what further information must distinguish. If all compatible candidates or a sufficient bound give the same answer to the present question, use that answer without resolving unrelated differences.

Repeated numerical fits can discover alternatives. Agreement of finitely many fitted instances leaves unsearched alternatives possible. A claim of uniqueness needs its mathematical or statistical grounds; a sufficient decision can require much less. Numerical search failure also differs from a proof that the family is inconsistent with the observations.

Change the question explicitly when a new use requires it. An intervention may distinguish models with the same observational behavior. C.28.MR supplies the replacement of the affected mechanism under its causal premises. Explanation or modification of a working method can require structure beyond that needed for prediction; characterize the required explanatory use through C.2.8 and Explanation Design (EXD).

MMP.11:4.6 - Use the consequence or revise the family

Return the result with the family, input range and conditions that affect its use. It may be a candidate relation, a bound, a conditional prediction, a supported instruction or a located missing contribution. When the model is used to change a working method, Method Engineering receives the consequence and the relations the proposed change must preserve.

Revise the part whose restriction prevents the needed result: the subject premise, permitted dependence, representation, observation relation or obtaining method. Use B.5.RR to carry a changed premise or question through the reasoning. General comparison, portfolios and improvement use the existing C.16, C.11 and E.22/E.23 methods when those questions arise.

Stop when the receiving use has a sufficient answer or the missing contribution is clear enough to obtain. Use C.11.DUA when deciding whether another observation, a richer family or further computation can improve that use enough to warrant its cost. A more detailed family is valuable only through what it enables.

MMP.11:5 - Archetypal Grounding

MMP.11:5.1 - Construct a response from its known shape

A normalized input u lies in [0,1]. The subject account supports a nondecreasing response r with r(0)=0 and r(1)=1. Its intermediate shape is unknown. Begin with these properties, rather than choosing a straight line as the only candidate.

Choose an integrable h with h(u)>=0 almost everywhere and H=integral_0^1 h(v) dv>0. Define

r(u)=integral_0^u h(v) dv / H.

The endpoints follow by substitution. For u2>=u1, the difference is the nonnegative integral of h over [u1,u2], divided by H. Thus every member is nondecreasing. These curves are absolutely continuous. Every absolutely continuous nondecreasing response with these endpoints has such a representation using its almost-everywhere derivative, but a jump response is outside this family. The needed regularity must come from the question or remain a declared restriction.

For a small calculable family, use linear segments through (0,0), (1/4,q), (1/2,1/2) and (1,1). Their slopes are 4*q, 2-4*q and 1. They are nonnegative exactly when 0<=q<=1/2. This is a construction of admissible candidates, not a conclusion from measurements alone.

Suppose observations establish only the three values at 0, 1/2 and 1. Every q in that interval agrees with them. The consequence r(1/4)<=0.6 follows for this whole family; it also follows for every nondecreasing response with the given midpoint. There is no need to identify q for that question.

Now the receiving use asks whether r(1/4)>0.3. Candidates q=0.2 and q=0.4 satisfy the same observations and give opposite answers. Another repetition at the three old input values does not distinguish these ideal candidates. An observation near the disputed input may help; its precision and cost belong to that new question. Alternatively, a supported additional shape relation could narrow the family.

MMP.11:5.2 - Retain an exchange balance without inventing its mechanism

Two nonnegative amounts x and y exchange a conserved total N. The forward and reverse rates are unknown. Use locally Lipschitz nonnegative rate functions a(x,y) and b(x,y), defined on a neighborhood of the nonnegative states being used, and construct

q=x*a(x,y)-y*b(x,y),

x_dot=-q, y_dot=q.

Adding the two equations gives zero change in x+y for every admitted a and b. At x=0, x_dot=y*b(0,y)>=0; at y=0, y_dot=x*a(x,0)>=0. With these regularity conditions the continuous-time solution preserves nonnegativity. These are properties of the coupled construction. The applicability of conserved exchange to the subject remains a premise.

Even complete knowledge of q need not identify the gross transfers. For any nonnegative locally Lipschitz h, define

a_new=a+y*h, b_new=b+x*h.

The two added contributions to q are x*y*h and -y*x*h, which cancel. The whole observed evolution is unchanged. This is an algebraic family of alternatives, not merely several successful numerical fits.

For a dimensionless instance, take a=b=1. The alternative h=1 gives a_new=1+y and b_new=1+x, yet both models have q=x-y. At x=2, y=1 they both predict x_dot=-1.

Change the question: a proposed intervention suppresses only the reverse transfer while leaving the forward rate law applicable. Under that causal premise, C.28.MR replaces the reverse contribution by zero. The first model gives x_dot=-2; the second gives x_dot=-4 at the same state. Ordinary observations of x and y under the unchanged mechanisms cannot choose between these accounts. A prediction under the old operation can still be useful; the proposed intervention needs a contribution that distinguishes the mechanisms or a sufficient bound covering them.

For dimensional quantities, a and b have inverse-time units, while h has inverse-amount-inverse-time units. Restoring units prevents treating the added terms as arbitrary dimensionless corrections.

MMP.11:5.3 - Construct probabilities while keeping boundary outcomes

A report has three possible outcomes. Let w_i>=0 and let their sum W be positive. Set p_i=w_i/W. Every candidate has nonnegative probabilities summing to one. Conversely, every probability vector on these outcomes is represented by choosing w=p. Thus this construction includes zero-probability outcomes.

The weights (0,1,3) and (0,2,6) both give probabilities (0,1/4,3/4). The parameter vector is redundant even if the probability vector becomes fully determined. There is no need to distinguish those weights when the receiving question uses only the law.

A strictly positive parameterization, such as exponentiating every finite unconstrained parameter before normalization, excludes zero probabilities. It can approximate a zero closely but cannot express it with finite parameters. If the subject account rules out the first outcome, retain that zero in the construction and normalize weights for the remaining outcomes. Whether a very small nonzero value would suffice depends on the receiving question.

The constructed p is a family member, not yet an estimate from data. MMP.7 composes it with selection, rounding or other recording behavior. The appropriate statistical method then determines what the observations support. Changing the recording procedure can change that inference without changing the underlying outcome family.

MMP.11:6 - Bias-Annotation

Familiar formulas can turn an assumed shape into an unnoticed restriction. Flexible fitting can conceal a different commitment: the selected inputs, architecture and loss still determine which functions can be obtained. Recover those choices when they affect the receiving result.

A respected subject law can also be applied outside its range or boundary. Preserve its grounds and allowed discrepancy. If no supported structural restriction is available, an unrestricted family can be a reasonable candidate for a bounded use; further structure must earn its place through the subject question.

MMP.11:7 - Conformance Checklist

• The unknown contribution has interpretable arguments, result, domain and permitted dependence.
• The retained relations have subject grounds and an application range.
• The construction shows why admitted adjustable values preserve the required properties.
• Restrictions introduced by representation are carried into conclusions that depend on family coverage.
• Coupled effects and the actual observation relation determine the fitting or inference question.
• Remaining parameter, function and consequence ambiguities are distinguished when they change use.
• A changed intervention or receiving question reopens the relevant contribution.
• The result can be used, qualified or passed to a named next method without requiring unrelated identification work.

MMP.11:8 - Common Anti-Patterns and How to Avoid Them

MMP.11:9 - Consequences

The family carries usable knowledge through variation and fitting. It can supply a bound before a particular model is selected, or reveal why more observations of the same kind will leave the important ambiguity intact.

The cost is constructing and checking the representation. Strong restrictions reduce the search but can exclude useful candidates. A flexible family can retain more possibilities while increasing inference cost and leaving more uncertainty. Compare these costs against the result the work actually needs.

MMP.11:10 - Architectural Rationale

Constructing the free part through the retained relations makes the reason for a property inspectable. A shared transfer preserves a total because the same quantity enters with opposite signs. A shape-constrained response preserves monotonicity because its increments are integrals of nonnegative values. Those reasons remain available when coefficients or learned functions change.

Separating function, representation and receiving consequence also permits economical inference. Many representations of the same function need not be distinguished. Functions that agree on the needed consequence may remain as alternatives. A new intervention can make a formerly irrelevant difference decisive.

The insertion point is therefore an architectural choice in the model. It determines what the adjustable contribution can change, which relations constrain it and which results still have the interpretation the work needs. A symbolic expression and a trained network can each participate in this construction when their mathematical role is recoverable.

MMP.11:11 - SoTA-Echoing

How much of the relation should be left free? The historical SINDy work, Brunton, Proctor and Kutz (2016), Discussion and Appendix B, binds sparse discovery to the chosen coordinates and function library. The universal differential equations construction, section 2.3, combines retained mechanisms with an adjustable function. Sections :4.1-4.3 adopt this choice of where freedom belongs. A small fixed family can be sufficient when its restrictions fit the question. The flexible construction trades additional representation and inference work for retaining variations the small family omits.

How should a required relation survive fitting? For the conservation question in :5.2, :4.2 selects a shared transfer with opposite signs over independently fitted change laws for x and y with only a finite conservation penalty. The latter can fit observations while violating the required total elsewhere. The shared construction preserves that total for every admitted choice of its rate functions. Accept the extra derivation and restriction to conserved exchange in return for that identity; obtain the subject grounds for conservation first. When the question allows a specified discrepancy, a penalty or simpler approximate relation can be sufficient at lower construction cost. Carry its discrepancy into the requested consequence rather than requiring the identity anyway. Direct constraints enforcing the relation remain another option under MMP.10.

Where does the adjustable part enter? Dyad’s model-discovery documentation supports component-level insertion before structural simplification. Micluta-Campeanu and colleagues (2026), sections 2.1-2.2, demonstrate post-simplification correction followed by optional reduction and symbolic replacement. Section :4.4 retains both placements, chosen by their effect on the needed relations. Their thermal application supplies one use, not the scope of this method.

What does fitting resolve? Loman and Baker (2025), sections 3.1, 3.3, 3.5 and B.5, distinguish functions, parameters and predictions. Section 3.1 also constructs an algebraic compensation between an unknown function and a mechanistic parameter that preserves observed dynamics. Adopt that construction of indistinguishable alternatives in :4.5; :5.2 adapts it to coupled directional rates. Their finite fitted-ensemble comparisons in :2.4 and Appendix B can reveal alternatives, but agreement of sampled fits does not prove uniqueness over the admitted family. Constructing two admissible alternatives with different receiving consequences already establishes the consequential ambiguity, without fitting an ensemble.

Revisit the chosen family when new subject knowledge changes its restrictions, a different observation changes what is distinguishable, a new use needs a formerly discarded difference, or another construction supplies the needed result at lower cost.

MMP.11:12 - Relations

• MMP.10 constructs representations and the conditions making them admissible. MMP.9 derives an unknown contribution caused by reduction and can use a constructed family to replace it.
• A.3.3.TR composes a rule of change; C.29.1 relates model consequences to their receiving use; C.29.2 constructs the needed computation.
• MMP.7 supplies the recording probability law. C.16.IR determines what the resulting indication relation resolves. MMP.8 uses the available information in a choice question.
• C.28.MR constructs the changed mechanism for an intervention. B.5.RR revises reasoning after a changed premise or question.
• C.2.8 and Explanation Design (EXD) characterize the explanatory contribution needed by a reader. Method Engineering (ME) uses a model consequence to develop or revise the corresponding way of working.
• C.11.DUA compares a further modeling contribution with its cost; general model comparison and improvement use the existing framework methods.

MMP.11:End

MMP.8 - Formulate Information-Dependent Choices Mathematically

Type: Method pattern Status: Usable, evolving Normativity: Normative within the stated use

MMP.8:1 - Problem frame

Use this pattern when a mathematical problem mixes quantities you may choose with quantities you do not control, or when a proposed solution depends on information that arrives too late. A solver can find a configuration for each possible circumstance while leaving you unable to select one in advance. A simulation can show a successful continuation because it chose an environmental value that the acting system cannot choose.

Begin with one proposed action. Ask what will be known when it must be chosen, which quantities remain outside that choice, and what result the action must achieve. Construct two possible circumstances that look the same at that moment. If the proposal assigns different actions to them, it needs another observation, a different decision time or a different policy.

The result is a mathematical formulation of the available choices and the requirement they must satisfy, including their dependence on information. It can establish a feasible fixed choice, a policy, a counterexample to the proposal or the missing contribution. This is a general modeling method for design, prediction and control questions. It develops the formulation before a solver or a control algorithm is selected.

The finite examples need elementary sets, inequalities and the meanings of ‘there exists’ and ‘for every’. More demanding cases can require optimization, stochastic processes or control theory. If the answer condition and available information are already correctly formulated, use C.29.2 or the applicable computational method to obtain the result.

MMP.8:2 - Problem

A mathematical unknown can represent a decision, an unobserved state, an external input or a quantity constrained by other relations. Treating every unknown as a free decision lets the solution change the problem’s circumstances to make the requested result possible.

Timing introduces another error. A family of solutions indexed by the true circumstance can be mathematically valid while requiring an observation that the acting system never receives. Separately optimizing every future case then grants foresight that the described method lacks.

The difficulty is to translate the work’s choices, information and requirements into a mathematical question with the corresponding dependencies and quantifiers.

MMP.8:3 - Forces

MMP.8:4 - Solution

Separate choices from circumstances, express the relations that must hold, and state which information each choice may use. Formulate the requested result over those objects, then test the formulation against a small contrasting pair of circumstances. Preserve any useful conditional answer when a stronger requirement is unavailable.

MMP.8:4.1 - Name choices, circumstances and consequences

Start from the question in the work. Identify what a participant can change and what is supplied by the subject or environment. Let a denote the choice and w the circumstance. Give both their domains and meanings. A choice can be an arrangement, an input, a rule for later actions or another object the work can construct.

Use the subject relations to connect them to a result. For a deterministic model, a consequence may be written y=f(a,w). An implicit relation R(a,w,y) can retain several possible consequences. Additional unknowns can describe forces, flows, internal states or other quantities jointly constrained by the model. Acausal equations can constrain these quantities without making them free controls.

State what the question requires of the result. If it asks whether every permitted behavior meets a condition, a solver’s ability to find one favorable y does not settle it. If the work can select among the permitted consequences, represent the mechanism that gives it that choice. A.3.3.TR supplies the relevant state-change or interaction rule.

MMP.8:4.2 - Recover the order and availability of information

Describe what is observed before each decision and what arrives afterward. Express the available observation as h(w) when it is a deterministic description of the circumstance. It may reveal only part of w. For a random or noisy observation, specify its probability law for each admitted w. An unknown fixed w can index a family P_w of observation laws without having a probability distribution itself. A joint law is needed when the question also treats w as random and averages over it. MMP.7 constructs the recording law.

A policy pi selects an action from the information available: a=pi(h(w)). Two circumstances with the same h(w) must therefore receive the same action. This uses MATH.2’s condition that an answer remains constant on cases identified by a description. Here the observation determines those groups. The information restriction is often called nonanticipativity: the policy does not use distinctions the decision-maker has yet to observe.

For repeated interaction, use the observation history available at each decision. A policy may remember earlier observations or actions. Omitting that memory is a substantive model choice. Global termination is unnecessary when the question concerns a continuing response; specify the response or progress condition that matters under the admitted inputs.

If an earlier action changes what can be observed, make h depend on that action and describe the cost and timing of observation. If action also changes the circumstance distribution or evolution, include that relation. C.28 supplies the causal-use question and the policy’s permitted pre-action information; C.28.MR constructs the mechanism replacement. A changed distribution cannot be inferred from a favorable selection of historical cases alone.

MMP.8:4.3 - State the required quantifiers and performance criterion

For a deterministic success condition G(a,w), these are different questions: