Foundation Gap Derivation Config Dim At D3

Configuration dimension at three spatial dimensions

The declaration configDim_at_D3 is a machine-checked theorem stating that the configuration dimension of a recognition event equals 5 when the spatial dimension D equals 3. In plain terms: a recognition event, the discrete record of an event in the framework's ledger, has D + 2 independent degrees of freedom. With D = 3, that gives 3 + 2 = 5. The theorem is proved by direct computation in the framework's machine-checked library of formal theorems, and it closes a boundary item by showing that the coherence energy exponent equals D + 2.

The number 5 is not an arbitrary choice. The framework defines the configuration dimension as configDim(d) = d + 2, where the two extra degrees of freedom come from one temporal tick (the advance of the ledger's clock) and one balance (the neutrality condition J(x) = J(x⁻¹) that keeps the ledger symmetric). At D = 3, the theorem configDim_at_D3 yields 5. From this, the framework derives a gap: the parity count is D² = 9, and multiplying by the configuration dimension gives D²(D+2) = 9 × 5 = 45. The theorem gap_at_D3 states this directly: consciousnessGap D = 45.

The framework also proves coprimality results around this gap. The theorem coprimality_odd states that for all odd D, gcd(2^D, D²(D+2)) = 1, while coprimality_even_fails states that for all even D ≥ 2, the gcd exceeds 1. This provides a fourth argument that D must be odd. Combined with Alexander duality, which selects D = 3, the gap of 45 follows from D = 3 alone. The framework uses this to connect to the coherence energy: E_coh = φ^(−5) at D = 3, matching the constant hbar = φ^(−5).

What the declaration does not claim is worth stating plainly. configDim_at_D3 does not prove that D = 3; it assumes D = 3 as a definition (def D : ℕ := 3) and computes the consequence. The theorem does not establish the physical recognition-to-linking bridge that would justify why spatial dimension is 3; that bridge remains open. It does not derive the fine-structure constant or any coupling constant. The coprimality results are about natural numbers, not about physical measurements. The gap of 45 is a structural number within the framework, not a measured quantity.

The significance of the declaration is that it makes the gap computation explicit and checkable. A reader can verify by hand that 3² × 5 = 45, and the machine-checked proof confirms it. The framework's claim is not that 45 is a universal constant of nature, but that within its model, the gap follows from the definition of configuration dimension and the choice of D = 3. The theorem is a building block in a larger chain, and its value is in the clarity of its statement and the transparency of its proof.

theorem configDim_at_D3 : configDim D = 5 := by decide

theorem gap_at_D3 : consciousnessGap D = 45 := by decide

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

/-- The RS-native action quantum has exponent equal to the configuration
dimension: `Constants.hbar = φ^(-(D+2))`. The exponent `5` is the forced
`D + 2`, not a free parameter. -/
theorem hbar_exponent_eq_configDim :
    Constants.hbar = phi ^ (-(configDim D : ℤ)) := by
  have hcfg : (-(configDim D : ℤ)) = (-5 : ℤ) := by
    have := configDim_at_D3; omega
  rw [hcfg, Constants.hbar_eq_phi_inv_fifth, ← Real.rpow_intCast phi (-5 : ℤ)]
  norm_num

What this page does not claim

The declaration does not prove that D = 3; it assumes D = 3 as a definition. The theorem does not establish the physical recognition-to-linking bridge justifying why spatial dimension is 3. The gap of 45 is a structural number within the framework, not a measured physical quantity.

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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