Foundation Gap Derivation

The 45 gap

The number 45 appears in Recognition Science as the product of two smaller counts. The first count is the square of the spatial dimension, 3 squared equals 9. The second is the configuration dimension of a recognition event, which the framework defines as the spatial dimension plus 2, giving 5 at dimension 3. Their product, 9 times 5, is the gap, and the framework proves this identity holds at dimension 3.

The configuration dimension itself has a plain meaning. A recognition event, a discrete record of a comparison, carries one degree of freedom for each spatial dimension, one for the advance of a tick, and one for the balance condition that the cost of recognition is symmetric. At three spatial dimensions that sums to five independent degrees of freedom. The framework's machine-checked library of formal theorems proves that the coherence energy, the energy associated with a locked recognition state, equals the golden ratio raised to the negative of this configuration dimension, which at dimension 3 is phi to the minus 5.

The gap also satisfies a parity condition. The framework proves that 2 to the power of the dimension is coprime to the gap when the dimension is odd, and shares a common factor when the dimension is even. This gives a fourth argument that the spatial dimension must be odd, joining other structural results in the framework. At dimension 3, the gap is 45, and the coprimality holds: 8 and 45 share no common factor.

In Recognition Science, this derivation closes a specific boundary item in the framework's development. The result links the spatial dimension, the configuration dimension, and the coherence energy into one arithmetic statement. The gap also appears in a balance identity: phi raised to (1 minus 45) times phi raised to 45 equals phi, a relation the framework calls the matter-consciousness link. The framework's status is 0 sorry and 0 axiom, meaning every theorem in it is checked by the machine.

theorem configDim_at_D3 : configDim D = 5 := by decide

theorem gap_at_D3 : consciousnessGap D = 45 := by decide

theorem E_coh_gap_eq : E_coh_gap = phi ^ (-5 : ℤ) := by
  unfold E_coh_gap configDim D; 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

/-- 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)

What this page does not claim

The gap derivation does not prove that spatial dimension must be 3; it assumes dimension 3 and derives the gap. The coherence energy formula is a definition within the framework, not a measured physical quantity. The parity condition alone does not force odd dimension; it is one of several arguments.

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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