Foundation Gap Derivation Coprimality Even Fails

The parity condition

The declaration coprimality_even_fails is a proved theorem about a simple arithmetic expression. For any positive even number, the theorem shows that the number 2 raised to that even number shares a common factor with the expression (that number squared) times (that number plus two). In plainer terms, the two quantities are never coprime: they always have a common divisor greater than 1. The proof is direct and mechanical, showing that 2 divides both quantities.

This fact matters in the Recognition Science framework because of how it pairs with a companion theorem. For any odd number, the same two quantities are always coprime, meaning their greatest common divisor is exactly 1. Together, the two theorems create a sharp divide: the coprimality property holds for all odd numbers and fails for all even numbers. The framework uses this divide as one of several arguments that the spatial dimension D must be odd, not even.

In the framework's account, the expression D²(D+2) arises from counting the degrees of freedom of a recognition event. At the framework's chosen dimension D = 3, this expression evaluates to 9 × 5 = 45, a number called the gap. The theorem coprimality_even_fails shows that no even D can produce a gap coprime to 2^D, while the companion theorem coprimality_odd shows that every odd D can. This parity filter is one step in a chain that selects odd dimensions, with Alexander duality cited as the further step that selects D = 3 specifically.

The theorem is a formal statement about natural numbers and divisibility. It does not claim that D must be 3, nor that the framework's derivation of three spatial dimensions is complete. It establishes only the negative result for even dimensions, leaving the positive selection of D = 3 to other arguments in the framework's library.

/-- For even D = 2k (k ≥ 1), D²(D+2) is even, so gcd(2^D, D²(D+2)) > 1. -/
theorem coprimality_even_fails (k : ℕ) (hk : 0 < k) :
    ¬ Nat.Coprime (2 ^ (2 * k)) ((2 * k) ^ 2 * (2 * k + 2)) := by
  intro h
  have h1 : 2 ∣ 2 ^ (2 * k) := dvd_pow (dvd_refl 2) (by omega)
  have h2 : 2 ∣ (2 * k) ^ 2 * (2 * k + 2) := ⟨2 * k ^ 2 * (2 * k + 2), by ring⟩
  have h3 := Nat.dvd_gcd h1 h2
  rw [h] at h3
  exact absurd h3 (by norm_num)

/-- For odd D = 2k+1, D²(D+2) is odd (product of odd numbers),
    hence coprime with any power of 2. -/
theorem coprimality_odd (k : ℕ) :
    Nat.Coprime (2 ^ (2 * k + 1)) ((2 * k + 1) ^ 2 * (2 * k + 3)) := by
  suffices h : Nat.Coprime 2 ((2 * k + 1) ^ 2 * (2 * k + 3)) from h.pow_left _
  show Nat.gcd 2 ((2 * k + 1) ^ 2 * (2 * k + 3)) = 1
  have hodd : (2 * k + 1) ^ 2 * (2 * k + 3) =
      2 * (4 * k ^ 3 + 10 * k ^ 2 + 7 * k + 1) + 1 := by ring
  rw [hodd]
  set n := 4 * k ^ 3 + 10 * k ^ 2 + 7 * k + 1
  rw [Nat.gcd_rec]
  have : (2 * n + 1) % 2 = 1 := by omega
  rw [this]
  decide

theorem gap_at_D3 : consciousnessGap D = 45 := by decide

What this page does not claim

The theorem does not prove that D = 3 is the only possible dimension. The theorem does not establish that any even dimension is physically realized. The theorem does not derive the value of the coherence energy exponent.

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

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