Foundation Gap Derivation Coprimality Odd

The odd-dimension coprimality

In number theory, two integers are coprime when they share no common divisor greater than 1. For example, 8 and 9 are coprime; 8 and 12 are not, because both are divisible by 4. The declaration coprimality_odd states a precise fact of this kind: for every odd dimension d, the number 2^d and the product d²(d+2) are coprime. In symbols, gcd(2^d, d²(d+2)) = 1 for every odd d. The companion declaration coprimality_even_fails shows the opposite for even dimensions: for every even d ≥ 2, that same gcd is greater than 1.

The fact is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions. The proof runs by rewriting the product d²(d+2) for odd d into the form 2 times an integer plus 1, which forces it to be odd, and therefore it cannot share a factor of 2 with 2^d. For even d, the product is visibly even, so the common factor 2 appears. This is a theorem about natural numbers, not a physical measurement.

In Recognition Science, the framework uses this number-theory result as one of several arguments that the spatial dimension D must be odd. The framework defines a recognition event, a discrete record of a comparison, and counts its degrees of freedom. At dimension D, the framework sets the configuration dimension, the number of independent coordinates in a recognition event, to D + 2: D spatial, one temporal, one balance. The framework then forms the product D²(D+2), called the gap, and the coprimality result shows that this gap shares no factor with 2^D when D is odd. Since the framework also derives, from Alexander duality, that D must be 3, the gap at D = 3 becomes 3² × 5 = 45. The declaration coprimality_odd itself does not force D = 3; it only rules out even dimensions.

What the declaration does not claim is as important as what it proves. It does not claim that D must be odd; it states a conditional fact about any odd D. It does not claim that the gap is 45; that is a separate theorem, gap_at_D3, which holds only at D = 3. And it does not claim that the physical universe has three spatial dimensions; that conclusion requires the full framework chain, including the Alexander duality step, which is not part of this declaration. The coprimality fact is a clean piece of arithmetic, and the framework's use of it is an interpretation, not a consequence of the arithmetic alone.

For the reader, the practical takeaway is that the odd-dimension coprimality is a proved, checkable statement about numbers, and the framework's leap from that statement to a physical dimension is a separate, larger claim. The arithmetic holds on its own; the physical conclusion does not follow from this declaration alone.

/-- 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

/-- 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)

theorem gap_at_D3 : consciousnessGap D = 45 := by decide

What this page does not claim

The declaration does not prove that the spatial dimension must be odd. The declaration does not prove that the gap equals 45. The declaration does not establish that physical space has three dimensions.

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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