Foundation Coherence Exponent Uniqueness Both Equal 5 At 3

Two routes to five

The declaration both_equal_5_at_3 records a small piece of arithmetic about two counting rules. The first rule, called the Fibonacci deficit, takes a dimension D and returns 2^D minus D. The second, called the integration measure, takes D and returns D plus 2. For D equal to 3, the first rule gives 8 minus 3, which is 5, and the second gives 3 plus 2, which is also 5. The theorem states exactly this: both rules produce 5 at D equals 3.

The interesting part is that this agreement is unique. For D equal to 1, the rules give 1 and 3, which do not match. For D equal to 2, they give 2 and 4, which do not match. For D equal to 4, they give 12 and 6, which do not match. Only at D equals 3 do the two independent routes land on the same number. The library's theorem exponent_unique_at_D3 states this uniqueness for the dimensions 1 through 4: the two rules agree if and only if D equals 3.

In the Recognition Science framework, this agreement is not a coincidence. The framework's library of machine-checked formal theorems uses the number 5 as the coherence exponent, the power that appears in later results about the reduced Planck constant and the Einstein coupling. The declaration both_equal_5_at_3 is one step in that chain: it certifies that the two routes to the exponent agree, and that the agreement happens only in three dimensions.

The theorem does not claim that three dimensions are physically real or that the universe must have three spatial dimensions. It only establishes a fact about the two arithmetic rules. The physical bridge from this arithmetic agreement to actual space is a separate matter, and the framework leaves that step open. The declaration is a precise, narrow result: two formulas, one dimension, one number.

/-- Both equal 5 at D = 3. -/
theorem both_equal_5_at_3 : k_fib 3 = 5 ∧ k_int 3 = 5 := by decide

/-- Both equal 5 at D = 3. -/
theorem both_equal_5_at_3 : k_fib 3 = 5 ∧ k_int 3 = 5 := by decide

/-- 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

theorem coherenceExponent_eq_5 : coherenceExponent = 5 := rfl

What this page does not claim

The theorem does not assert that physical space has three dimensions. The theorem does not explain why the two rules were chosen or what they measure. The theorem does not derive the value of the reduced Planck constant or the Einstein coupling.

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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