MMP.Preface:3 - Solution - Connect the contributions the question needs

MMP.Preface:3.1 - Form the first account and recover its interpretation

If there is no mathematical question yet, use B.5.FM: identify the participants, propose the relations relevant to the difficulty, work a small consequence and return it to the question. B.5.TU helps when an available theory supplies those relations. Mathematical Thinking provides constructions of objects, operations, representations and arguments when the needed mathematics itself must be developed.

Recover what a request to use a model means in this situation. A structure satisfying stated axioms answers a different question from a representation used to predict an observed process. Both can be useful mathematical work. E.10 clarifies the intended use when the word model conceals it; keep the subject’s established vocabulary when it is already clear.

A drawing, formula, program or learned representation also needs an operation that obtains the requested answer. A person may reason from a diagram; software may solve equations; an experimental arrangement may exhibit a behavior. Make the required interpretation available when another participant must continue the work. A.6.3.RT.OE helps make an expression usable for that operation, and C.29.2 develops a computational formulation when computation is needed. The obtaining procedure can itself become a subject of mathematical investigation.

MMP.Preface:3.2 - Represent admissible objects and construct missing relations

MMP.10 starts from intended candidate objects and their requirements. It chooses a representation, derives the conditions that make the representation valid and expresses the required answer. These steps matter for sets, sequences, functions and quantitative objects alike. If several records describe one object, a count or probability over records can require correction before it answers the subject question.

MMP.11 starts with supported relations and an unknown contribution among them. It constructs adjustable families that retain the needed properties and inserts that contribution into the connected model. A known total, a monotone response and a normalized probability law require different mathematical constructions. Their common modeling question is how to permit the unknown variation while retaining what is supported.

Choose the form from the next operation. Direct constraints may already describe all useful candidates. A parameterization can make changes preserve the constraints automatically, but may introduce duplicate descriptions or omit parts of the allowed family. Retain that difference when interpreting a fitted value, an impossibility result or an observed agreement.

MMP.Preface:3.3 - Construct change and observation together when they interact

For an evolving situation, A.3.3.TR constructs a state-change rule from the contributing relations, including simultaneous constraints, alternatives and events. If situations assigned the same state need different continuations under the same modeled inputs, revise the state or retain the alternatives. The computation’s chosen solution order need not be the represented order of physical change.

MMP.7 derives a probability law for the record produced by an observing procedure. Compose the source and recording laws, retain shared unknowns, sum or integrate unrecorded alternatives, and account for selection. The resulting law can supply a statistical inference or prediction method. C.16.IR addresses what a supplied observation relation resolves, including bounds and consequential ambiguity.

MMP.12 constructs recovery from a forward relation. Find which changes to the unknown leave the records unchanged or change them too little for stable recovery. A restriction or penalty can make a useful reconstruction possible; derive both the error it suppresses and the target detail it may remove. A bound that already settles the question can end the work before regularization.

MMP.13 constructs the inferential claim. Choose whether the use needs a repeated-sampling guarantee, a posterior probability or a prediction for a new outcome; derive that result under the record law and the additional assumptions it requires. Propagate joint uncertainty to the quantity the receiver actually needs. A regularized optimum alone supplies neither a posterior distribution nor a coverage guarantee.

Observation can also change the situation. C.28 and C.28.MR help formulate that intervention and replace the affected mechanism. MMP.15 asks whether its requested consequence is determined by the available laws and causal assumptions. It derives an identifying expression, a sufficient bound or an unresolved difference between compatible causal accounts. MMP.13 can then estimate an identified quantity from finite records; an observational fit alone does not identify it.

A question about the same case under another action needs a further connection. MMP.19 constructs the underlying conditions shared by its alternatives, uses MMP.7/.13 to recover what the factual record says about them, then applies the changed mechanisms. Separately known intervention distributions can leave their same-case relation ambiguous. The result can be a conditional response or a bound; a choice based only on expected outcomes can remain settled without resolving that ambiguity.

MMP.Preface:3.4 - Turn uncertainty into a question about available action

MMP.8 separates circumstances from choices and specifies when information arrives. It then formulates whether the work needs a fixed decision or an instruction depending on an available report, and whether performance is required for each admitted circumstance or on average under a stated probability law.

This formulation can expose a change needed in the work itself: observe earlier, distinguish another circumstance, permit another action or revise the requirement. Method Engineering (ME) develops the observing and acting methods and their composition. Their timing and resource demands return as premises of the mathematical question.

An information-dependent instruction needs both an obtainable report and a participant able to act on it at the stated time. Check this condition for the combined method. A correct reporting law and a correct decision calculation can still describe an instruction that the proposed work cannot perform.

MMP.16 constructs the records that alternatives would produce under feasible designs. Compare the distinction an observation could resolve with its contribution to the receiving question and its burden. An informative report can still leave the preferred action unchanged.

MMP.8.SD develops continuing choice: retain a state or belief sufficient for the proposed decisions, construct transitions and observations, and derive how a present choice and its continuation determine the accumulated consequence. This joins observation design to action when timing, information or changes caused by observing matter.

MMP.Preface:3.5 - Simplify, diagnose and revise for the required consequence

MMP.9 starts with a source evolution law and quantities to retain. It derives their change, identifies the contribution that depends on removed detail and constructs a replacement through elimination, memory, added state, approximation or a sufficient bound. Initial conditions, inputs and the requested horizon determine whether the replacement serves the question. MMP.11 can supply a family for a still-unknown replacement relation.

MMP.17 constructs a cheaper supplier for selected source-model responses. Choose the responses and input region from their intended use, build the replacement, and refine or return to the source where approximation changes the answer. Retaining a value can be insufficient when the receiver needs a derivative, tail event or explanation.

MMP.18 connects models through the quantities and conditions they exchange. Translate between their quantities and scales, supply any omitted influence needed by the connection, and retain shared information. Preserving a total amount, a constant field or a joint probability law requires different conditions on that connection. Matching software inputs and outputs does not establish those conditions.

MMP.14 helps when available observations reveal a consequential prediction failure, or a proposed use makes a comparison worth performing. Choose a discrepancy relevant to that use, derive predictions for comparable records, locate the mismatch and change the implicated relation or assumption. Recalculate the receiving consequence. Several repairs may explain one discrepancy; predictive improvement alone does not identify its cause. Existing records, an algebraic comparison or a restricted use can be sufficient.

When a premise or question changes, B.5.RR identifies the reasoning that depends on it and derives the revised consequence. If the argument must first be recovered, use B.5.RA. B.5.MPC.R develops the comparison when the difficulty concerns a connection among physical, mathematical and computational accounts. The repair may belong to subject assumptions, observation, mathematical formulation or computation.

These contributions admit several entry points and returns. A changed reporting procedure can require a new probability law while leaving the choice criterion intact. A changed criterion can require another instruction while leaving the reporting law intact. A changed intervention can require revising the model’s mechanisms. Preserve each still-useful result and reconsider the part whose conditions changed.

MMP.Preface:3.6 - Use the model-forming result across the Suite

The Foundational Thinking Suite Reference explains the shared architecture and worked combinations. Mathematical Thinking constructs the objects and arguments a formulation needs. Physical or other subject methods supply the supported relations. Computational methods obtain the consequence, and the subject interpretation determines what that consequence lets the work do. Notational methods make these operations recoverable by their participants.

A particularly useful boundary concerns an unknown relation. MMP.11 can construct a family respecting supported conditions without selecting one fitted member. MMP.7 supplies a probabilistic recording law when that is the question, while C.16.IR can expose what a supplied indication or bound resolves. The appropriate next step can therefore be inference, a discriminating observation, an already sufficient bound, or action under the remaining uncertainty.

The Readme illustrates two further connections: carrying intervention consequences through a diagnostic report into a continuing instruction, and using cheaper components while retaining a sufficient combined answer. The needed intermediate result chooses the next method. General portfolio comparison and improvement remain with the existing FPF methods.

MMP.Preface:3.7 - Constituent actions in ongoing work

While constructing a model of recorded observations, summing over unrecorded alternatives can be part of forming the observation law, within the modeling inquiry already under way. If the observing procedure changes from recording a below-threshold result to omitting the case entirely, that local operation must follow the changed selection rule. Mathematical calculation and knowledge of the subject may both be available while the intermediate description of observation is missing. MMP.7 helps construct the recorded-data law; obtaining the procedure’s actual conditions still requires the appropriate source or contributor.

FPF B.1.5.EW helps recover these constituent–whole connections; B.1.5.RS examines a proposed replacement. Use the parts of the vertical that can change the present result. A Method described here can require additional capability, available support and compatible resources at other grains.