MATH.2:5 - Archetypal Grounding

MATH.2:5.1 - Absolute value fails, parity works for addition

Suppose integers are identified when they have the same absolute value. Then 1 and -1 are identified. Add the same integer 1 to both: the results are 2 and 0, which have different absolute values. This equivalence relation therefore cannot support addition inherited from integer representatives.

For a question about parity, choose a different relation: n~m when n-m is even. Reflexivity, symmetry and transitivity follow from the corresponding facts about differences divisible by 2.

If n~n' and m~m', then (n+m)-(n'+m')=(n-n')+(m-m') is even. Addition therefore preserves the relation. There are two classes:

The class of 7 plus the class of 4 is Odd. This answers the parity question using two classes. It does not determine whether the sum is 11 or another odd integer; a request for the sum itself requires the operands or more information.

Now request multiplication as well. The earlier addition argument does not settle the new operation. Here a separate calculation does:

n*m-n'*m'=(n-n')*m+n'*(m-m').

Both terms on the right are even when the corresponding inputs have the same parity. The quotient can therefore also support multiplication. The changed request led to a new compatibility argument while preserving the earlier addition result.

MATH.2:5.2 - A location class loses availability

Let V0 and V1 be states at location V, without and with permission. A partial transition r is defined at V1 and returns T; it is undefined at V0.

The proposed location-only relation identifies V0~V1. Output comparison alone would find no conflicting pair of returned values, because one value does not exist. The domain test detects the failure: V1∈D_r and V0∉D_r.

Refine the relation to retain permission. The two states are now in different classes, and the inherited transition is available only on the class containing V1. In MATH.1’s route case, this retains the permitted q;r of cost 6 and excludes the apparent p;r of cost 3.

A convention that declares a class enabled whenever any representative is enabled would answer a different question: a transition is possible from some member. To execute it from the actual state, that convention still needs a suitable member or an enabling step. The present construction preserves the availability of the given operation at the represented state.

MATH.2:5.3 - Identifying words changes the question they answer

Take the one-generator paths a^n from MATH.1, with concatenation a^m;a^n=a^(m+n). Impose a^2~a^0 and choose the smallest equivalence relation compatible with concatenation that contains this equation.

Compatibility propagates this equation under concatenation. Adding one a gives a^3~a, and repeated deletion of a pair reduces every even-length word to the empty path and every odd-length word to a. These two groups remain distinct: parity itself is a compatible relation satisfying the imposed equation, as :5.1 shows, so the smallest such relation cannot identify opposite parities. This yields two classes and a composition table identical to the addition table above.

The quotient can describe the parity of repeated toggling. It discards the number of toggles. If each use takes time, elapsed cost cannot be recovered from those two classes alone. Retain the length or accumulated cost for a question that consumes it.

MATH.2:5.4 - Normalize labelled values before identifying them

A temperature-checking method accepts values labelled Celsius or kelvin. It converts them to kelvin, then tests membership in the inclusive interval [273.15,303.15]. Grouping inputs by their numeral alone loses the answer: 20 Celsius becomes 293.15 kelvin and passes, while 20 kelvin fails.

Let n(v,C)=v+273.15 and n(v,K)=v. Identify inputs when their n-values agree. Every input has a normalization result, and identified inputs have equal results; both the normalization and the following interval test therefore descend to these classes. This uses the separate input and output sorts of :4.1.

A later question about the original unit cannot be answered from the class alone. Retain the label when that question matters. If the description of the working method places comparison before normalization, Method Engineering ME.12:4.4 helps locate and repair that contradiction. The mathematical compatibility test and the repair of the described work answer different parts of this example.