MATH.7:5 - Archetypal Grounding

MATH.7:5.1 - Shifted numbers with a shifted addition law

Start with real numbers under addition. Let h(x)=x+1 and r(y)=y-1. Both inverse equations hold on all real numbers. The transported addition is:

u ⊕ v=h(r(u)+r(v))=u+v-1.

The source zero becomes h(0)=1, and source negation becomes neg_Y(u)=h(-r(u))=2-u. Thus u⊕1=u and u⊕(2-u)=1. Associativity follows directly because both groupings of three inputs give u+v+w-2; it also follows from the general transport argument.

To solve x+3=7, translate it as y⊕4=8. The receiving equation gives y=5, and decoding gives x=4. Using ordinary addition on the new labels would instead solve y+4=8 and return x=3, which fails the original equation.

The transported operation is useful as a representation of the source addition. If ordinary addition on Y is part of the receiving requirement, this h does not preserve that requirement; the construction has exposed a different operation.

The same map can transport real division, defined when the source denominator is nonzero. Its receiving formula is u⊘v=(u-1)/(v-1)+1, with domain v≠1. The label 1 decodes to the forbidden denominator zero; the label 0 decodes to the allowed denominator −1. For example, 3⊘0=-1 decodes to −2, the result of the source calculation 2/(-1). Using the familiar restriction v≠0 would exclude an allowed input and admit a forbidden one.

MATH.7:5.2 - Sets represented by bit vectors

Let X be the subsets of {a,b,c}. Map a subset to its membership vector in {0,1}^3; for example, {a,c} maps to (1,0,1). Decode by selecting the positions marked 1. These maps are inverse on all eight subsets and all eight vectors.

Take symmetric difference as the source operation: an element belongs to S△T when it belongs to exactly one of S and T. Its transported operation is coordinatewise XOR, whose output bit is 1 exactly when the two input bits differ. The empty set becomes (0,0,0).

For S={a,c} and T={b,c}, symmetric difference gives {a,b}. In coordinates:

(1,0,1) XOR (0,1,1) = (1,1,0).

Decoding returns {a,b}. This gives a way to perform the set operation in a binary representation and recover its result. The inverse map and operation formula explain why the method works for every subset pair; the eight-by-eight table is a finite check if needed.

If the next question asks for union instead, derive its operation: coordinatewise OR. Reusing XOR would discard an element present in both inputs. The same bijection supports both operations once each is defined.

MATH.7:5.3 - Addition survives a coordinate change while length needs a new formula

In X=R^2, let h(x,y)=(x+y,y). Its inverse is r(u,v)=(u-v,v). Transported vector addition coincides with ordinary coordinatewise addition because h is linear.

Now ask for squared Euclidean length, L_X(x,y)=x^2+y^2. The result is a real number whose interpretation stays unchanged, so its output map is the identity. The transported quantity is:

L_Y(u,v)=(u-v)^2+v^2.

The source vector (0,1) has squared length 1 and maps to (1,1). The transported formula still returns 1. The familiar formula u^2+v^2 would return 2.

A length bound L_X(x,y)≤1 consequently becomes (u-v)^2+v^2≤1. It does not become u^2+v^2≤1 through this coordinate map. The example separates an unchanged operation from a quantity whose receiving formula must be constructed. A later physical interpretation of these coordinates uses C.29 for its subject correspondence.