MATH.23 - Develop a Conjecture by Changing a Construction

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

MATH.23:1 - Problem frame

Use this pattern when a mathematical construction works in some cases, fails after a change, or reveals a relation that could support a new theorem. You need to formulate the next claim and a way to investigate it.

A conjecture is a mathematical assertion proposed for investigation. Developing it includes choosing its objects, assumptions, quantifiers and conclusion. The result can become a theorem, a refuted claim, or a more useful question. Its value lies in what the answer would let someone construct, infer, explain or change.

First useful move: change one mathematically consequential part of the construction and compare what happens. Replace a parameter by a variable, combine two outputs, remove a used assumption, or ask the same operation to preserve another property. Try the changed construction and identify the equation, condition or obstruction that controls its result.

The reader needs the mathematical operations used by the starting construction and elementary proof reasoning. The examples explain their additional operations; the last example develops a limit argument. The method can be performed by a person, an AI agent or cooperating contributors with the needed preparation.

Use an established result when it already answers the question. MATH.19 helps prove a sufficiently clear claim. B.5.QD develops questions more generally. Here the work develops a mathematical assertion through construction, variation, proof analysis and counterexamples.

MATH.23:2 - Problem

A list of successful cases leaves many possible general claims. A formula can match every computed value and fail beyond them. A proposed proof can depend on a property absent from the conjecture. Conversely, a failed conjecture can reveal a useful operation or a stronger question.

The difficulty is to extract a relation that can be investigated generally. Changing a number produces another instance; changing the construction law, allowed maps or quantifier can create a new problem. The latter change needs a mathematical reason and a feasible first attack.

MATH.23:3 - Forces

MATH.23:4 - Solution

Local mantra: recover the construction and its use; vary a consequential ingredient; express the suggested law; seek its reason and a separating case; revise the assertion or construction; choose and begin the next useful problem.

MATH.23:4.1 - Recover what makes the starting construction work

Write the inputs and the operations that produce the result. Identify a relation the result satisfies and where that relation is used. Separate what has been proved from what has only occurred in inspected cases.

Look inside the operation. Repeated updates may expose a recurrence. Combining two constructions may expose closure or an order dependence. A successful quotient may reveal which distinctions can be discarded. MATH.17 helps make the operations themselves available for this investigation.

When a program supplies examples, recover the class it actually generates. For instance, a generator that produces only trees cannot investigate a claim about arbitrary graphs unless the missing graph classes are handled elsewhere. The representation and evaluator are part of the available mathematical experiment.

MATH.23:4.2 - Construct a variation with a reason

Choose an ingredient whose change could affect the wanted relation. Useful mathematical moves include:

These are possible operations, not required stages. Choose the one selected by the construction. MATH.8/.11/.13 supplies symmetry or invariant reasoning; MATH.18 supplies interpretations; MATH.22 develops changed assumptions. Use their fuller methods when that mathematical contribution is needed.

Keep the original use visible. Requiring equal-sized groups repairs a mean-of-means formula, but leaves a request about arbitrary groups unanswered. The latter needs another construction, as in :5.2.

MATH.23:4.3 - State the proposed law and its answer form

Give the objects, domains, allowed operations, assumptions and proposed conclusion. In particular, make the quantifiers distinguishable. These statements ask different questions:

• For each input, some construction has the property.
• One construction works for every input.
• Every admitted construction has the property.

A finite pattern of results can suggest a formula. Derive it from the generating operation, a recurrence, an invariant or another mathematical relation when possible. If that derivation is missing, retain the formula as a conjecture and identify the step that would establish it.

Say what would resolve the present question. An explicit construction, a characterization of all admissible cases, a bound, or a counterexample can open different next uses. A conjectured optimal construction needs both an attainable value and an argument excluding better ones when optimality is the required conclusion.

Check whether the proposed law is already a known result under another description. MATH.18 can compare representations. An available theorem can close this question and expose a more useful next one.

MATH.23:4.4 - Alternate a proof attempt with revealing cases

Use MATH.19 to find intermediate claims connecting the construction to the proposed conclusion. Identify which mathematical property each step consumes. This can expose a condition that the conjecture has not yet stated.

At an uncertain step, construct a case that challenges that condition. Prefer one that distinguishes plausible accounts. To test whether a summary supports composition, find inputs with the same summary and place them in the same surrounding construction. If their required outputs differ, the summary has lost information that composition needs.

Inspect the status of a failure. A counterexample satisfying the conjecture’s premises refutes the conjecture. A failure of an auxiliary lemma may leave a different proof possible. A failed program may instead concern its representation or implementation. Recover the mathematical obstruction before revising the claim.

Computational search can propose constructions and counterexamples. Check a returned candidate against the mathematical conditions that its use requires. For a finite explicit object, this may be a small complete calculation. Sampled tests of an infinite family leave its universal claim to be justified.

MATH.23:4.5 - Revise the claim without hiding the lost use

Use the failure to choose the next mathematical change. Three common returns are:

1. Repair a hypothesis. State the condition that makes the failed lemma work, prove its sufficiency and check whether the intended cases satisfy it.
2. Repair the construction. Retain the original problem and change the objects, representation or operation so that the required property can be obtained.
3. Change the conclusion. Obtain a conditional answer, bound, obstruction or weaker property that still serves a useful question.

A sufficient condition can be worth developing before necessity is known. Label that difference: a proof that uniform convergence preserves continuity does not show that every continuous limit was obtained uniformly.

A counterexample can itself become a constructive ingredient. Ask what class it belongs to, which transformations preserve its obstruction, and which related conjectures it can test. The new question can concern the method of generating cases as well as the cases themselves.

Retain the result that survived. Changing the theorem should not erase a valid construction or an earlier unresolved requirement.

MATH.23:4.6 - Choose a next problem and begin its mathematical work

Compare the few continuations whose answers would change further work. A useful result can supply a missing lemma, a reusable combining operation, an explanation of failure, or a method that makes another class of questions approachable. Immediate engineering application is one possible use; developing a new mathematical operation can also enable later inquiry whose destination is not yet known.

Choose a first attempt that the available contributors can perform. For a broad characterization, this might be one direction of the theorem. For an extremal problem, it might be one explicit improved construction or a bound. For a limit question, it might be the estimate needed to justify one passage to the limit.

Return the conjecture, the mathematical basis that suggested it, the most revealing success or obstruction, and the next operation. An expression and a short argument can carry this result.

Use B.5.QD and C.40.CD when several questions and ways must develop together. E.10.INT helps distinguish the interest of further possible work from surprise, fame or a search score. C.36.RP supplies retention of the reconstructible method when continuing capability is at stake. Obtain further observations or checking only when their possible outcomes can change the next move enough to justify the effort.

MATH.23:5 - Archetypal Grounding

MATH.23:5.1 - From repeated instances to a family of operations

Repeatedly apply f(x)=a*x+b over real numbers. A few calculations give:

f²(x)=a²*x+(a+1)*b,

f³(x)=a³*x+(a²+a+1)*b.

The useful next question is how to represent any number of repetitions without expanding every substitution. The conjectured formula is:

f^n(x)=a^n*x+b*S_n, where S_0=0 and S_n=1+a+...+a^(n-1) for n>0.

Recover the generating rule. Applying f once more changes the coefficient of x from a^n to a^(n+1), and the constant from b*S_n to b*(a*S_n+1). The identity S_(n+1)=a*S_n+1 therefore supplies the induction step. At n=0, f^0 is the identity and S_0=0. The formula is proved for every natural n.

Now vary the problem: combine two potentially different maps f(x)=a*x+b and g(x)=c*x+d. Their composite is:

f(g(x))=(a*c)*x+(a*d+b).

This constructs an operation on coefficient pairs: (a,b) composed with (c,d) gives (a*c,a*d+b). The class is closed under composition. MATH.17 can study this operation; a computational method can use it to combine repetitions.

Order now becomes a useful question. Reversing the maps gives the same linear coefficient but constant c*b+d. Thus they commute exactly when d*(a-1)=b*(c-1). This condition says when reordered operations retain the result.

The progression has produced a general iteration formula, a closed combining operation and a condition for rearrangement. A further question about vector-valued affine maps needs its matrix composition law; scalar commutation cannot be silently carried into that new setting.

MATH.23:5.2 - A failed combination produces a better summary and another conjecture

A computation stores the arithmetic mean of each nonempty group of real numbers. The proposed combining operation takes the mean of those means. For groups [0] and [2,4], it returns (0+3)/2=1.5, while the combined group has mean 2.

Inspect the missing mathematical relation. Each group mean gives a total only when its group size is known. Equal group sizes make the proposed operation work, but arbitrary groups require another construction.

Retain a sum s and count n. Combine (s,n) and (t,m) as (s+t,n+m), then return (s+t)/(n+m) when n+m>0. Associativity and commutativity follow from those of addition in both coordinates; (0,0) is an identity. Thus grouping and order of the combination do not change the final mean in real arithmetic. A finite-precision implementation needs its own error account.

The construction opens a broader mathematical question: Which summaries permit the summary of a combined input to be computed from the two summaries alone?

Let q map a collection of finite lists, closed under concatenation, to proposed summaries; let ++ concatenate lists. Means and medians here use nonempty lists, while sum and count also admit the empty list. A necessary condition is:

if q(u)=q(u’) and q(v)=q(v’), then q(u++v)=q(u’++v’).

It is also sufficient for defining a combining operation on attained summaries: choose lists representing the two summaries and apply q to their concatenation. The condition makes the result independent of those choices. This proves mathematical existence of the operation. Computing it from a concrete representation of the summaries still needs an effective rule. MATH.2 develops the compatible identification; MATH.12 and CMP handle obtaining the operation.

For means alone, take u=[0], u’=[0,0], and v=v’=[6]. The input summaries match, but the concatenated means are 3 and 2. The necessary condition fails. Sum and count repair it by an explicit operation.

Now ask whether median and count suffice. Take u=[0,1,100] and u’=[-100,1,2]. Each has count 3 and median 1. Append the same list [50,50]. The resulting five-element medians are respectively 50 and 2. This new conjecture is refuted by the same method.

The next useful problem concerns what more to retain. Sorted full lists are sufficient and can be merged, while a use requiring less storage can ask for a restricted input class or an approximate median with a stated error. The mathematical obstruction now informs a computational and modeling choice.

MATH.23:5.3 - A failed limit argument reveals a stronger condition

Consider the claim that a pointwise limit of continuous real functions is continuous. On [0,1], let f_n(x)=x^n for positive natural n. Each function is continuous. At any fixed x<1, x^n tends to zero; at x=1 it equals one. The limit f is zero below one and equals one at one, so it is discontinuous.

Locate the failed proof step. Pointwise convergence lets the approximation index depend on x. Continuity near a point needs control over nearby x together. Choosing a good approximation at the one point alone does not control its neighborhood.

This suggests a sufficient condition: uniform convergence. For every positive error allowance, one index makes all later approximations close to f at every point of the domain.

Work the proposed repair. Fix a point x0 and an error allowance e>0. Choose N so that the approximation error between f_N and f is less than e/3 everywhere. Continuity of f_N gives a neighborhood of x0 in which the difference between f_N(x) and f_N(x0) is less than e/3. The triangle inequality bounds the difference between f(x) and f(x0) by these three errors, hence by e. This proves continuity of f at x0.

The repaired theorem supplies a condition for transferring continuity. Here proof analysis discovered which quantifier change made the desired inference possible. A subsequent limit construction can use the proved condition to retain continuity.

Uniformity is sufficient, not necessary. On the domain [0,1), the same x^n sequence has the continuous limit zero, but convergence is not uniform: for every n there is an x<1 with x^n=1/2. The next question can therefore seek a weaker condition suited to the receiving use, rather than treating uniform convergence as the definition of every acceptable limit.

MATH.23:6 - Bias-Annotation

Small examples are chosen to expose a relation and permit a full calculation. In a larger inquiry, simple generators can miss structures with different behavior. Change the generating rule or inspect a contrasting class when that difference could defeat the proposed claim.

A solved or famous problem can attract attention while its reusable mathematical contribution remains unclear. Recover the construction and the questions it opens. The choice of a next problem depends on the work and contributors, not on a universal ranking of interesting statements.

MATH.23:7 - Conformance Checklist

For the conjecture being developed:

• The starting construction, established result and receiving use are recoverable.
• A mathematical change explains why the new question arises.
• Objects, domains, hypotheses, quantifiers and answer form distinguish the proposed claim.
• A proof attempt exposes its needed intermediate relation, and the chosen case tests a consequential condition.
• A counterexample is classified at the claim, lemma or implementation it actually defeats.
• A repaired hypothesis, construction or conclusion preserves the distinction between the new question and any earlier unsatisfied need.
• Generality follows from an argument over the stated class; tested cases retain their actual scope.
• The selected continuation has a useful attainable first operation and retains the earlier construction needed to perform it.

MATH.23:8 - Common Anti-Patterns and How to Avoid Them

Extrapolating the displayed values without the generating law. The iteration formula in :5.1 is justified by its recurrence and induction. Agreement at a few n values alone would leave other continuations possible.

Excluding the counterexample and abandoning the required class. Equal-sized groups rescue the mean-of-means formula in :5.2, but not a requirement to combine arbitrary groups. Sum and count answer that requirement.

Treating a convenient condition as necessary. Uniform convergence repairs the continuity argument in :5.3. The final countercase shows why necessity remains a different claim.

Losing the operation behind a successful answer. Keeping only the mean hides the combining structure. Retaining sum and count makes later aggregation possible; retaining the compatibility argument supports the next summary question.

MATH.23:9 - Consequences

A result becomes material for developing another mathematical problem. A failure can yield a condition, a construction or a new object of study. A success can reveal a family of operations and a way to transfer the result.

The method can stop with a well-formed conjecture and first attempt. Completing its proof, obtaining a computational procedure, learning the method and applying it to a physical subject have their own requirements. Keeping the connection to those uses makes partial progress useful without claiming their completion.

MATH.23:10 - Architectural Rationale

B.5.QD locates a consequential next question. The mathematical contribution here is to construct its proposed law: expose composition and closure, change a quantifier, characterize a compatible identification, or isolate a premise through proof analysis. The cases show how these operations produce statements that can be proved or refuted.

MATH.17/.18, MATH.19/.22 and the other mathematical patterns supply the operative constructions. This pattern assembles them around the development of a conjecture. A case can close with a new question, an answer, or a further obstruction; a universal fixed order of research stages would obscure these returns.

A correct answer and a reusable method have related but different uses. Either can be obtained by people or AI agents. Retaining the derivation, allowed transformations and failed conditions permits another prepared contributor to adapt the work and formulate its next problem.

MATH.23:11 - SoTA-Echoing

The working question is how to develop a mathematical conjecture from constructions and their failures, with a useful route into further work.

Adapt proof analysis. Lakatos’s Proofs and Refutations, Appendix 1 connects counterexamples to the hidden lemmas and concepts in an attempted proof. The adopted move changes :4.4-.5: inspect the failed inference and consider a revised condition or construction. It improves on simply excluding the offending example when that exclusion abandons the intended class. The method can require more work than proving a fixed, adequate claim; use the latter route when its premises and conclusion already serve the inquiry. Lakatos supplies a method for developing the conjecture; its use can still leave the conjecture unresolved.

Adapt construction search and generalization. Georgiev, Gómez-Serrano, Tao and Wagner’s Mathematical exploration and discovery at scale, §§1.3-1.6 and 4 distinguishes search for constructions from finding a program or formula that generalizes, and describes transitions to proof. It also reports evaluator defects and the limits of the tested search methods. This informs :4.1/:4.3-.4: inspect the generator and evaluator, recover a general construction, then establish the claim its use needs. Compared with direct object search, program search can retain reusable structure but adds generation and evaluation cost. A cheap direct search or existing theorem can be preferable. The paper supports these bounded choices, not a universal preference for AI search.

Retain methods as a usable result. The Math and AI declaration of 11 September 2026 argues that counting solved problems can obscure understanding and transmission of methods. Adopt that question in :4.6 and the relation to C.36.RP: preserve what another prepared contributor needs to change and use the construction. The declaration is a position on the purpose of mathematical work, not evidence that transmission must remain exclusively human. The construction-to-proof account above supplies a concrete alternative involving AI contributors.

Reopen the route when a new counterexample changes the admitted class, another representation makes the conjecture simpler, or the receiver needs a result the present construction cannot provide. Improved search tools change the means and cost of investigation; they leave the proposed mathematical assertion and its possible use to be stated.

MATH.23:12 - Relations

• B.5.QD and C.40.CD develop questions and obtaining ways across practices; this pattern supplies the mathematical conjecture-forming contribution.
• MATH.17/.18 supply operations on constructions and interpretations between accounts.
• MATH.8/.9/.11/.13 supply symmetry, selection and invariant methods that reveal general relations or obstructions.
• MATH.19/.6/.22 supply proof construction, countermodels and changed axioms.
• MATH.2 supplies the compatible identification used by composable summaries.
• C.29 and MMP interpret the mathematical result for a subject; CMP supplies effective search, construction and execution when needed.
• E.10.INT, C.36.RP and HCD/DOCA support interest, continuing availability of methods and acquisition of a missing capability.

MATH.23:End