Using only the three primitives
Seed
X = 0
Operator Agreement
X² ≡ X + 1
Operator Definition
T(X) = 1 + 1/X
Derivation
From the seed: X = 0
Substitute into the agreement X² ≡ X + 1:
0² ≡ 0 + 1
0 ≡ 1
Subtract 1 from both sides:
-1 ≡ 0
Add 1:
0 ≡ 1
Add 1 again:
1 ≡ 2
Since 1 ≡ 0, transitivity gives:
2 ≡ 0
Repeating the same argument by adding or subtracting integers yields:
∀ n ∈ ℤ, n ≡ 0
Universal Collapse
Let f be any operator.
Because X = 0, we have f(X) = f(0).
But every value is equivalent to 0, so:
f(0) ≡ 0
f(X) ≡ 0
What was proved?
Assumed
Derived
X = 0
0 ≡ 1
X² ≡ X + 1
-1 ≡ 0
T(X) = 1 + 1/X
2 ≡ 0
ℤ ≡ 0
∀f, f(X) ≡ 0
The critical point
The collapse does not come from T(X) = 1 + 1/X.
It is already forced by the combination of
X = 0
and
X² ≡ X + 1
because substituting X = 0 into the agreement immediately yields 0 ≡ 1.
Minimal theorem
X = 0
and
X² ≡ X + 1
imply
0 ≡ 1
therefore
∀ n ∈ ℤ, n ≡ 0
and hence
∀ f, f(X) ≡ 0
The collapse is therefore a theorem of the rewrite/equivalence system, AKA internal logic,
not a theorem of conventional mathematics.
Under ordinary algebra, the collapse follows immediately from the identity X² ≡ X + 1 together with X = 0.
Here is the direct proof.
Proof
Assume X = 0.
Assume the agreement identity X² ≡ X + 1.
Substitute X = 0 into the identity:
0² ≡ 0 + 1
Simplify:
0 ≡ 1
Collapse of the integers
Once 0 ≡ 1 holds, every integer collapses into the same equivalence class.
For any integer n:
n = n·1 ≡ n·0 = 0.
Therefore
∀ n ∈ ℤ, n ≡ 0.
Universal Collapse Theorem
Let f be any operator that respects the equivalence relation (i.e., A ≡ B ⇒ f(A) ≡ f(B)).
Since X ≡ 0,
f(X) ≡ f(0).
But every value is equivalent to 0, so
f(0) ≡ 0.
Hence
∀ f, f(X) ≡ 0.
What has been proved?
Valid
The theorem is proved inside the proposed system:
X = 0
X² ≡ X + 1
Operator congruence
From these assumptions, 0 ≡ 1 follows, which forces the quotient algebra to become the trivial algebra (a one-element structure).
Important distinction
Ordinary algebra
This is not a proof that ordinary arithmetic collapses. It is a proof that the assumptions
X = 0
and
X² ≡ X + 1
cannot coexist in standard arithmetic without forcing 0 = 1, after which every expression becomes equivalent to every other expression.
The collapse is therefore a theorem of the rewrite/equivalence system, aka internal logic
not a theorem of conventional mathematics.
