MATH.20:4 - Solution

Local mantra: name the unknown and the needed comparison; construct a simpler comparator; prove its direction and scope; carry the comparison through the required operations; tighten or expose the gap; return the bounded result to use.

MATH.20:4.1 - Choose the quantity or object and its order

State what is unknown: a number, a function, a set, an error, an optimal value, or another mathematical object. State the comparison relation. Numerical order, pointwise order between functions and set inclusion support different conclusions.

For a numerical bound L≤q≤U, say which q is being bounded and under which assumptions. For an enclosure L⊆S⊆U, L consists of established members of S, while U contains every possible member under the stated description. Cardinality alone does not establish either inclusion.

For an optimum, distinguish the optimal value from an object attaining it. A bounded nonempty set of real values has an infimum and supremum, but an attaining object needs a further argument. For example, the interval (0,1) has infimum 0 and contains no minimum.

Select the needed direction and strength. To show q≤t, an upper bound at most t suffices. To refute that claim, a lower bound greater than t suffices. If the bounds straddle t, they leave this question open.

MATH.20:4.2 - Construct a comparator from available structure

Choose a construction whose relation to the unknown can be proved with available information.

Other constructions can use symmetry, a conserved quantity, a norm inequality or a comparison theorem. Select the property that supplies the missing direction. A familiar shape or similar-looking value is a candidate for investigation until that relation is established.

A comparator can be a different kind of object from the result. In :5.1 a node potential produces a numerical lower bound on every path length. The derivation makes those resulting values comparable; it does not compare a potential directly with a path.

MATH.20:4.3 - Establish the bound over its claimed domain

Prove the inequality or inclusion for every admitted case to which the result will apply. A feasible witness establishes one attainable value. A claim about all candidates needs a relation covering all of them, as the edge inequalities in :5.1 cover every path.

Track the assumptions used. For a relaxed feasible set, establish the original set’s inclusion in it and keep the objective unchanged on original candidates. For a norm estimate, specify the norm and the operator property that supports it. For a probabilistic inequality, retain its distributional conditions and probability claim.

Use a known comparison theorem when it settles this question. If it does not, MATH.19 can construct the missing intermediate inequality and MATH.6 can test a suspected overclaim with a separating case.

Where the comparison cannot be established, return the condition that is missing or the counterexample. A trial value may remain useful for exploration without being used as the unproved bound.

MATH.20:4.4 - Propagate the comparison through the needed operations

For an order-preserving map F, L≤q≤U implies F(L)≤F(q)≤F(U). For an order-reversing map, the endpoints exchange roles. Establish the relevant behavior on the admitted domain.

For example, multiplication by a nonnegative scalar preserves numerical order, while multiplication by a negative one reverses it. Squaring is increasing on nonnegative inputs; an interval crossing zero needs its minimum and maximum square computed on that interval. From -2≤x≤1, the bound is 0≤x²≤4.

For sets, image and preimage under a fixed function preserve inclusion; complement reverses it in a fixed ambient set. Thus an enclosure can be carried into a subsequent transformation without requiring the original set to be enumerated.

Keep shared variables and constraints when combining bounds. If x∈[0,1], separately bounding x and -x gives -1≤x+(-x)≤1, but retaining the dependency gives equality to zero. The loose result is valid; its loss comes from forgetting that the terms use the same x.

A bound computed with approximate arithmetic must preserve the claimed direction after that computation. A computational continuation can use error bounds or enclosing arithmetic for this purpose; the mathematical argument states the comparison it must maintain.

MATH.20:4.5 - Tighten the comparison or identify what prevents it

Inspect where the proof introduces slack. It may enlarge the feasible set, forget a dependency, replace a quantity by a worst-case value, or use one norm for effects that could be bounded separately.

Improve the part that can change the receiving result. Construct a better feasible candidate, strengthen the inequality, partition the domain, retain a lost relation, or add an available premise. A bound on a requested component can be enough when a bound on the whole object is costly or impossible.

Check attainability and equality conditions. When a feasible value meets a proved bound in the opposite direction, the optimum is established. When equality conditions conflict, the current bound can remain valid while failing to identify an attainable result. A sequence approaching a bound can establish an infimum or supremum without an attaining object.

If the gap reflects missing information, expose it. Two admissible constructions with different answers can show why the current assumptions do not determine a narrower result. That identifies a useful next assumption, observation or mathematical question.

MATH.20:4.6 - Use the bounded result and stop at the needed strength

Return the comparator, its relation to the requested result and the premises needed by the next use. A numerical interval, a set inclusion or a short inequality can carry this result.

For a minimization with feasible value U and lower bound L, the candidate is within U-L of the optimal value when these quantities are finite. The receiver can use that gap to decide whether further optimization matters. If only exclusion, existence or a threshold result was needed, stop once that result follows.

General choices about cost, value and further work use FPF’s existing characterization, Pareto and improvement methods. Componentwise mathematical bounds can inform them, but a candidate attaining one coordinate need not attain another. Preserve which constructions actually realize the compared results.

Reopen the affected bound when its domain, objective, available information or following operation changes. Preserve bounds whose proofs still apply.