A Proof in Economic Theory
The Execution Theorem
On the structural incompleteness of orthodox economic models and the necessary form of outcome production
Y = d(P) · S
I. Orthodox Model & Its Error
II. Lemma 1 — Causal Necessity of S
Y = F(d(P), S) — form to be determined
III. Lemma 2 — S is Signed
Standard economics writes Y = d(P) — decisions over a possibility space — implicitly fixing execution at unity. The proof below shows this is a formal structural error, not a simplification.
II. Lemma 1 — Causal Necessity of S
Lemma 1
A decision is not an outcome. d(P) requires execution S to travel to Y. If S = 0 then Y = 0 regardless of d(P). S is a necessary condition and must appear explicitly:
Setting S ≡ 1 eliminates the most consequential variable. □
III. Lemma 2 — S is Signed
Lemma 2
Y is signed. d(P) is non-negative. Therefore S must carry the sign.
Empirical fact
Outcomes are real-valued — profit or loss.
Y ∈ ℝ
Definition
Possibility spaces and decisions are non-negative.
d(P) ≥ 0
The sign of Y must be carried by S — the only candidate.
S ∈ ℝ : S>0 correct, S=0 none, S<0 destructiveEntailed by definition and observation — not assumed. □
IV. Theorem — Unique Multiplicative Form
Three conditions establish the unique form:
Condition 1 — Zero Necessity (Lemma 1)
F(x,0) = 0 ∀x F(0,s) = 0 ∀s — Neither factor substitutes for the total absence of the other.
Condition 2 — Sign Preservation (Lemma 2)
sgn(F(x,s)) = sgn(s) for x > 0 → Y = g(d(P)) · S for some function g.
Condition 3 — Cauchy Additivity + Continuity
g(a+b) = g(a)+g(b), g continuous → g(x) = kx. Unit normalisation k=1 → g(x) = x, therefore Y = d(P) · S.
Candidate forms eliminated against Conditions 1 and 2:
| Form | Cond. 1 | Cond. 2 | Status |
|---|---|---|---|
| Y = d(P) + S | Fails | — | ✗ |
| Y = d(P) · S² | Holds | Fails | ✗ |
| Y = d(P)S | Fails | Fails | ✗ |
| Y = min(d(P), S) | Holds | Fails | ✗ |
| Y = d(P) · S | Holds | Holds | ✓ |
Established
Y = d(P) · S
| 3 conditions + Cauchy | No arbitrary axioms — the form is forced |
| Orthodox Y = d(P) | Degenerate S ≡ 1 — structural error, not simplification |
| Policy implication | d(P) and S must be optimised jointly |