B.5.TC - Compare Theoretical Accounts for a Working Question

Type: Method-description pattern Status: Stable Normativity: Normative unless marked informative

B.5.TC:1 - Problem frame

Use this when two theoretical accounts appear to give different answers, explanations or instructions for the work you are doing, and you need to decide how to use them. They may employ different objects, preserve different information or obtain a result by different operations. Comparing their vocabulary or final numbers has left the practical difference unclear.

The work here is to compare particular uses of those accounts. A theory supplies objects, relations and ways of reasoning; an application adds a question, an interpretation and the premises needed to obtain an answer. Compare those applications while keeping the underlying accounts understandable on their own terms.

First useful move. Ask one concrete question of each account. Supply the same relevant situation, recover how each answer is obtained, and locate the first difference that matters to the use. A route description that says where travel is possible and one that gives minimum travel cost can both be useful. The question determines which information and operations you need.

The result explains what each account contributes, the conditions of agreement or difference, and which use or inquiry should follow. You may select one sufficient account, combine complementary contributions, or retain an unresolved difference together with what remains usable.

Use a familiar comparison directly when its question and conditions already fit. If only one application is unclear, B.5.TU helps construct it. If the disputed object is a numerical implementation of an understood model, begin with C.29.2. This pattern becomes useful when it is still necessary to compare the accounts or locate the source of their apparent disagreement.

B.5.TC:2 - Problem

Two descriptions can use the same word for different quantities, or different words for a shared construction. One can answer a weaker question. They can agree on observations while disagreeing about an intervention. An approximate computation can also obscure agreement between the theories from which the computations were derived.

A useful comparison must recover enough of each application to distinguish these situations. It must also leave room for an account that opens a question the original comparison did not ask. The practical problem is to determine what the difference permits, prevents or changes for the work, without requiring a verdict on the theories in every possible use.

B.5.TC:3 - Forces

B.5.TC:4 - Solution

Fix the working question → construct each account’s application → align the case and the meaning of its answer → locate the consequential difference → use what the comparison establishes, or pursue the difference that matters.

The sequence is revisable. Reuse an established application or correspondence. Return to the question when the accounts reveal that it combines different needs. Stop when the available result suffices for the intended use.

B.5.TC:4.1 - State the answer the work needs

Say what you want to explain, predict, construct or decide, and what would count as an answer. Specify the conditions that can alter that answer: for example, the initial state and time of a prediction, the allowed routes and quantity to minimize, or which intervention is contemplated.

Keep the account and its proposed use separately visible. Comparing a theory’s possible reach with a rival’s already implemented calculation is a comparison of unequal contributions. Either construct the relevant application of both or state which contribution is still unavailable. An unavailable application can be a reason to use another account now while retaining the first for development.

If the accounts answer different questions, identify their shared question, if any. For the remaining questions, compare their usefulness as complementary contributions. A newly expressible question can itself be a valuable result: explain what inquiry it opens and why that inquiry matters. B.5:4.4 connects this result to problem development.

B.5.TC:4.2 - Recover an application of each account

For each account, recover the objects and relations used, what is supplied or assumed, the operation that obtains the answer, and how that answer is interpreted. Follow the reasoning far enough to reproduce the step on which the comparison turns. B.5.TU, B.5.RC and B.5.RA supply application, construction and argument recovery when needed.

Preserve the source’s distinctions during this work. If two accounts use state differently, explain which information each state retains and what continuation it determines. If a symbol has no counterpart, say what the account does with it before proposing a correspondence. Translate the required objects and operations; a word-to-word substitution alone cannot settle their relation.

An account may need information the present work lacks. Separate a supplied premise, a proposed premise and a derived result in the explanation. Continue conditionally when that is useful: “Under this independence assumption the minimum is 3; if continuations depend on the incoming route, we must retain that dependency.”

A collaborator or tool can supply a calculation or proof. Recover the interpretation and conditions needed to compare its contribution. Learn or request the missing operation when you cannot yet follow the decisive step; a broad study of both theories is unnecessary when a smaller contribution settles the use.

B.5.TC:4.3 - Work the same case and align the answers

Choose a small case that exercises the disputed operation or distinction. Give each account the corresponding inputs, derive its answer, and express the answers in the terms of the working question. Explain the correspondence when the inputs or outputs have different forms.

For a mathematical application, compare the retained distinctions and operations. C.29.1 supplies the preservation argument when a result is to be transferred. For a physical application, keep the phenomenon, operating conditions and meaning of the predicted quantity aligned. For a computation, separate the mathematical formulation from approximation, numerical tolerance and execution.

Begin with an understood case so a transcription or implementation error can be found. Then use a changed case that exercises the claimed difference: a shared resource, a constraint, an intervention or another relevant variation. Derive a general relation when the intended conclusion ranges beyond the cases. A finite example establishes its own result; a counterexample can refute a general assertion whose premises it satisfies.

Agreement of reported numbers is informative only after their meanings and conditions are aligned. Conversely, different intermediate objects can lead to the same answer through an explained correspondence. Record only the derivation or explanation the receiving use needs; the comparison does not require a standard table or separate report.

B.5.TC:4.4 - Locate what causes the difference

Follow each answer back to the first step where the applications cease to agree. Use the following distinctions where they resolve that case.

Several differences can interact. Aligning one premise can remove an apparent conflict; it can also expose a remaining one. Keep the remaining question named. If the answer depends on a missing correspondence or inaccessible operation, return that missing contribution rather than ranking the theories from their summaries.

Additional evidence is useful when it can change the needed conclusion. First use an available derivation, counterexample, observation or established result. If the difference remains consequential, compare the possible gain from a discriminating inquiry with its cost and feasibility under C.11.DUA. A difference that cannot change the present use can remain unresolved. When it does change the use and cannot be settled, retain the conditional answers or obtain a sufficient weaker result.

B.5.TC:4.5 - Choose the use and preserve the reason

Return the comparison in the form the work can use. A short explanation can state the question, the decisive construction, what agrees or differs and the resulting continuation.

Common outcomes are:

• One sufficient account. Use the account that answers the question under acceptable conditions and effort. State the limitation that would make another account necessary.
• Agreement for a stated range. Reuse either account for that result, choosing by effort, available tools or the further construction it makes easier. Preserve the relation that supports the agreement.
• Complementary uses. Give each account its question and contribution. When combining their results, establish the compatibility needed at the connection.
• A consequential unresolved difference. Keep the conditional answers and the premise or observation that would discriminate them. Continue with an unaffected result, a sufficient bound or the worthwhile inquiry selected for that difference.

Accuracy, reach, computational effort, learning effort and the ability to change the construction can matter differently across uses. Compare the values that can change this choice. If their trade-off is unresolved, C.11 supplies the decision method; a single “best theory” score would hide it.

Reopen when the question, premises, interpretation, available operation or relevant evidence changes. B.5.RR helps revise the affected reasoning. Reuse the comparison where its dependencies remain intact.

B.5.TC:4.6 - Separate recognition from assurance

Recognize the need from an unresolved disagreement or a choice between accounts. One worked comparison can already reveal an omitted premise or a useful complementary answer.

Qualify the result for its intended use. A mathematical equivalence needs a derivation over its claimed domain. A physical prediction needs the corresponding model and observation basis. An implemented approximation needs an error account adequate for the question. Use the relevant subject Methods and B.3 for these claims. Broader reliance is a reason to establish the broader claim, while an ordinary local use can stop with its sufficient local result.

B.5.TC:5 - Archetypal Grounding

B.5.TC:5.1 - Routes, reachability and minimum cost

A mathematical account keeps routes as distinct objects. Routes p and q go from X to Y and cost 1 and 4. A route r goes from Y to Z and costs 2. Routes with matching endpoints can be concatenated; the cost of a concatenation is the sum of its costs. The two routes from X through Y to Z therefore cost 3 and 6.

A second account keeps only whether travel between endpoints is possible. Both X-to-Y routes become the answer “yes”. Joining that answer to Y-to-Z reachability gives “yes” for X-to-Z. This account answers whether travel is possible. It cannot give the cost of the route actually taken: p followed by r and q followed by r have the same reachability summary and different costs.

A third account keeps minimum costs. For finite nonempty sets of allowed first and second segments, assume every first segment can be followed by every second segment and costs add. Then the least combined cost is the sum of the two least costs. Every combined cost is at least that sum, and combining the two minimizing segments attains it. Here the answer is 1 + 2 = 3. The calculation can therefore answer the minimum-cost question without retaining every route. Choosing an actual route also requires the minimizing segments to be recoverable.

Now change the allowed combinations. Let the second segments be r with cost 2 and s with cost 10. Only p followed by s, and q followed by r, are allowed. The true minimum is min(1 + 10, 4 + 2) = 6. Independently minimizing the two stages gives 3, which no allowed route attains. The failure is the omitted compatibility relation. Retain the incoming-route distinction or calculate over the allowed pairs.

The comparison yields three useful accounts with different retained information. Its changed case also identifies a condition for the minimum-cost composition rule. No physical interpretation is needed for this mathematical result. Using cost as travel time or expenditure adds the corresponding interpretation and additivity conditions.

B.5.TC:5.2 - The same oscillator, different formulation and computation

Consider an ideal one-dimensional mass on a linear spring, with mass m > 0, spring constant k > 0, displacement q and velocity v. Friction and external forcing are absent. The question is its motion from q(0) = a and v(0) = 0.

The force account gives m q’’ = -k q. The variational account uses L(q,v) = m v²/2 - k q²/2. Its Euler-Lagrange equation is d/dt(∂L/∂v) - ∂L/∂q = 0; substitution gives m q’’ + k q = 0. Both therefore give q(t) = a cos(√(k/m) t) under these premises. For this motion question, the two constructions agree. The variational formulation can become preferable when another coordinate choice simplifies constraints; the force formulation may be the quicker account for this simple case.

Suppose a computation appears to disagree. Take m = k = a = 1 and use the forward Euler updates q_next = q + h v and v_next = v - h q. From q = 1, v = 0 and h = 0.1, it returns q_next = 1, v_next = -0.1. The energy (q² + v²)/2 rises from 0.5 to 0.505. In fact these updates multiply energy by 1 + h² at each step, whereas the differential equation conserves it.

The discrepancy is produced by the approximation used in the computation. Compare an adequate step size or another numerical method under C.29.2 for the requested horizon and error. If the real oscillator loses energy, examine friction and other physical interactions; that changes the model question. The equality of the two ideal derivations remains available within its premises.

B.5.TC:5.3 - Agreement in observation, disagreement under intervention

An indicator X and an output Y always follow a binary controller command U. Two causal hypotheses explain the observed pairs: A uses X := U and Y := X; B uses X := U and Y := U. In A, the indicator drives the output. In B, the controller drives both directly. Both yield X = Y = U in ordinary operation.

The question is now what happens when X is forced to 1 while U remains 0. Represent this intervention by replacing the assignment to X. A gives Y = 1; B gives Y = 0. The same observed pairs leave that difference unresolved.

The next contribution could be the circuit description, an already available intervention result or a worthwhile discriminating test. If the work only predicts the observed indicator from U in unchanged operation, this causal difference need not be settled for that use. If it proposes controlling Y through X, retain the unresolved consequence until the needed causal premise is supplied. C.28 governs that intervention claim.

B.5.TC:6 - Bias-Annotation

A shared question can favor the account that originally framed it. Retain a rival’s newly expressible question when it opens useful inquiry; compare that contribution on its own terms. Conversely, unfamiliar notation can make an ordinary operation appear novel. Recover the operation before crediting or rejecting the account.

The cases use small formal constructions so their decisive steps can be inspected. They demonstrate the comparison method, with domain premises stated in each case. A difficult real application can need specialist knowledge, uncertain premises and more than one iteration. Human, AI and collective contributions can participate in the comparison; claims about their learning or reliability require their own basis.

B.5.TC:7 - Conformance Checklist

• The comparison names a working question and explains what its answer changes.
• Each application has recoverable meanings, inputs, premises, decisive operations and an interpreted answer.
• The shared case corresponds across accounts; a missing correspondence remains visible.
• Agreement or disagreement is traced to the relevant question, premise, operation, inference, implementation or physical account.
• The conclusion has the scope established by its derivation, cases and applicable evidence.
• A sufficient cheaper account, complementary use or useful conditional result remains selectable.
• Further inquiry has a consequence for the intended use that can justify its burden.
• The result states the next use and the change that would reopen the comparison.

B.5.TC:8 - Common Anti-Patterns and How to Avoid Them

B.5.TC:9 - Consequences

The practitioner gains a reasoned division of use between accounts, or a located disagreement that can guide the next inquiry. That result also supports collaboration: contributors can work on the missing mathematical operation, physical premise or computational approximation instead of repeating the whole dispute.

Reconstruction costs effort. Reuse an applicable comparison and stop at the explanation needed for the work. A comparison remains dependent on its question and premises; a changed use can make previously irrelevant distinctions decisive.

B.5.TC:10 - Architectural Rationale

The unit of comparison is an application because a theory’s objects alone leave its useful inference undetermined. Reconstructing that inference makes differences in representation, premises and operations comparable without first imposing a common ontology on every source concept.

Working a shared case provides a place to connect the accounts. Following the dependency that causes a difference then supports a broader argument or a discriminating example. These operations complement one another: a derivation establishes scope, while an example can reveal that the chosen scope misses the work’s question.

Separating agreement, complementary use and unresolved difference keeps the result productive. An account may remain preferable for a cheap prediction and insufficient for an intervention. Another may be harder to use now yet expose a construction worth developing. This preserves the connection between obtaining knowledge, applying it and improving the available ways of inquiry.

B.5.TC:11 - SoTA-Echoing

For comparing what different mathematical accounts let a practitioner compute, adopt the operation-based comparison illustrated by Fong and Spivak’s Seven Sketches in Compositionality (2019), §2.5.2–§2.5.3. Reachability and cost use different operations for combining steps and choosing among alternatives. Against comparison by shared route terminology, this makes the lost answer and composition condition visible in :4.3 and :5.1. The finite compatibility example is a construction here. Reopen its application when the quantity or allowed composition changes. Author manuscript.

For physical formulations that may agree, adopt the constructive comparison in Sussman and Wisdom’s Structure and Interpretation of Classical Mechanics, second edition (2015), §1.6: derive the equations under the stated force and potential assumptions. This answers more than a numerical fit and often costs less than a new simulation. Against selecting a formulation solely by familiarity, preserve the advantage of coordinates adapted to constraints. Sections :4.3–:4.4 and :5.2 separate that choice from numerical approximation. Reopen when interactions, constraints or the requested result change. Publisher’s text.

For comparing causal uses, adopt intervention construction from Pearl’s Causal inference in statistics: An overview (2009), §3.2.1, and adapt it here to expose the particular consequence on which two accounts disagree. Agreement in the observed distribution, a serious comparator for prediction, can leave that intervention consequence undecided. This changes :4.4 and :5.3. C.28 supplies the broader causal method. Reopen when the causal premises or intended intervention change. Author’s paper.

For prediction, a serious alternative to choosing a single account is combining predictive distributions. Yao, Vehtari, Simpson and Gelman’s stacking method (2018) chooses weights through predictive performance when the candidate set need not contain the data-generating model. Adapt that question-relative choice into :4.5’s permission to retain combinations. Reject using such predictive weights as a verdict about causal mechanism or mathematical equivalence; those are different comparison questions. The statistical method remains with its domain, including its scoring and validation conditions. Reopen when prediction under another distribution or a different use is required. Authors’ paper.

This pattern combines these constructive contributions into a common comparison method. Its first result is the comparison obtained under the stated conditions.

B.5.TC:12 - Relations

• B.5 coordinates the reasoning cycle and return to a changed question. B.5.TU, B.5.RC and B.5.RA recover an application, construction or argument; B.5.RR revises the affected reasoning.
• C.29 selects mathematical lens use; C.29.1 constructs the correspondence needed to transfer a result. C.29.2 and C.29.3 separate computational formulation from physical execution.
• B.5.MPC connects mathematical, physical and computational contributions. C.28 governs causal and intervention claims.
• C.38 constructs comparable ways to obtain one result. This pattern supplies a theoretical comparison when such a way depends on an unsettled account. C.11 and C.11.DUA govern a consequential choice and the worth of further inquiry.
• F.0.2 uses a comparison of accounts when forming a semantic synthesis across sources. A.6.3.RT supports operative expression, and A.15.9 obtains a missing specialist contribution.

B.5.TC:End