Foundation Coherence Exponent Uniqueness Disagreement At 2

The disagreement at D = 2

The declaration disagreement_at_2 is a small but load-bearing piece of a larger argument. It states that two different formulas for a quantity called the coherence exponent give different answers when the dimension D equals 2. In plain terms: if you compute the exponent two different ways in a two-dimensional setting, the results do not match.

The two formulas come from different starting points. The first, the Fibonacci deficit, is k_fib(D) = 2^D - D. At D = 2 it gives 2^2 - 2 = 2. The second, the integration measure, is k_int(D) = D + 2. At D = 2 it gives 2 + 2 = 4. Since 2 is not equal to 4, the two routes disagree. This is not a bug; it is the point. The disagreement at D = 2 is one of the three counterexamples that make the uniqueness claim meaningful.

The full theorem, exponent_unique_at_D3, states that among the dimensions 1, 2, 3, and 4, the two formulas agree if and only if D = 3. At D = 1 they give 1 and 3; at D = 2 they give 2 and 4; at D = 4 they give 12 and 6. Only at D = 3 do both give 5. The disagreement_at_2 declaration is the D = 2 case of that uniqueness theorem, proved by direct computation in the machine-checked library of formal theorems.

What disagreement_at_2 does not claim is just as important. It does not say that two dimensions are impossible or forbidden. It says only that two specific formulas disagree there. It does not say anything about what the coherence exponent means physically, nor does it claim that D = 3 is the only dimension that could ever work. The theorem is a statement about arithmetic, not about the world. The step from this arithmetic to a claim about physical space is a separate matter, and it is not made here.

/-- Disagreement at D = 2. -/
theorem disagreement_at_2 : k_fib 2 ≠ k_int 2 := by decide

/-- Fibonacci deficit: k_fib(D) = 2^D - D. -/
def k_fib (D : ℕ) : ℕ := 2^D - D

/-- Integration measure: k_int(D) = D + 2. -/
def k_int (D : ℕ) : ℕ := D + 2

/-- D = 3 is the unique dimension in {1,2,3,4} where both routes agree. -/
theorem exponent_unique_at_D3 :
    ∀ D ∈ ({1, 2, 3, 4} : Finset ℕ), k_fib D = k_int D ↔ D = 3 := by
  decide

What this page does not claim

The declaration does not say that two dimensions are physically impossible. The declaration does not define what the coherence exponent represents. The declaration does not claim that D = 3 is the only dimension where any two formulas could ever agree.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/CoherenceExponentUniqueness.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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