MATH.5:5 - Archetypal Grounding
MATH.5:5.1 - Evaluate expressions, then test an equation
Build expressions from x,y,0,1,+,*. Assign x=2 and y=3, and use ordinary integer addition and multiplication. Evaluation returns E((x*y)+1)=7. The same rules evaluate every finite expression and preserve its two operations.
Now let the source identify expressions using the usual commutative-semiring laws: associativity, the two identities, commutativity of both operations, distributivity, and multiplication by zero. Integer arithmetic satisfies those laws, so evaluation also defines a map from the identified expressions.
Change the assignment to matrices:
A=[[0,1],[0,0]], B=[[0,0],[1,0]].
Use matrix addition and multiplication, the zero matrix and identity matrix. This still evaluates freely formed expressions. But:
A*B=[[1,0],[0,0]]
B*A=[[0,0],[0,1]].
The source’s equation x*y=y*x therefore fails under this assignment. A substitution into the commutative quotient would give two answers for one source element.
If ordered multiplication is needed, use expressions whose equations retain its order. Matrix addition, multiplication, zero and identity support the remaining semiring laws. If commutative multiplication is essential to the original question, retain a target and assignment that satisfy it instead. The failed equation identifies the mathematical change required.
MATH.5:5.2 - Calculate the combined effect of a word of operations
Let a word contain commands I and D. Their mathematical effects on an integer are:
I(x)=x+1, D(x)=2*x.
Represent an affine transformation x -> s*x+t by the pair (s,t). Thus I receives (1,1) and D receives (2,0).
For (s,t) followed by (u,v), define:
(s,t) star (u,v)=(u*s,u*t+v).
Substitution derives this rule: u*(s*x+t)+v=(u*s)*x+(u*t+v). The identity is (1,0). For a third pair (w,z), either grouping gives (w*u*s,w*u*t+w*v+z), so the rule is associative.
Extend the two generator assignments to words. Then:
I;Dreceives(2,2), meaningx -> 2*x+2;D;Ireceives(2,1), meaningx -> 2*x+1;I;D;Ireceives(2,3), meaningx -> 2*x+3.
The third word sends 10 to 23. The pair describes its effect for every integer, allowing it to be composed with another affine operation without expanding the whole word again.
A different map can send each generator to cost 1 and concatenate by addition. It gives both I;D and D;I the value 2. This is a valid cost homomorphism but loses their different effects. Requiring the source equation I;D=D;I would preserve that cost map while preventing the stated effect map. The receiving question decides which distinction must remain.
If a command’s availability depends on intermediate state, the all-words construction is no longer the intended source. Use state-sensitive paths under MATH.1 and preserve their interfaces when constructing the interpretation.
MATH.5:5.3 - Interpret paths with different kinds of values
Take source objects A and B, with generators u:A -> B and v:B -> A. Interpret A as the integers and B as pairs of integers. Assign u_target(n)=(n,0) and v_target(n,m)=n. The empty path at A becomes the identity on integers; the empty path at B becomes the identity on integer pairs.
Extension gives E(u;v)(n)=n, whereas E(v;u)(n,m)=(n,0). The first composite therefore equals E(id_A). The second differs from E(id_B): it sends (7,5) to (7,0). The interpretation can descend through the source equation u;v=id_A, together with the equations generated from it by permitted composition. It keeps the information that the return path loses the second component.
Now require v;u=id_B as well. The same assignment fails that new equation. Keep the free-path interpretation or the first quotient, change the assigned maps, or change the required identification. Both composites are valid paths; the failed assertion concerns their values, not whether they can be formed. This distinction lets a representation carry construction followed by recovery without claiming recovery in both directions.