Foundation Gap Derivation Constants E Coh Eq Config Dim

The coherence energy identity

The declaration Constants_E_coh_eq_configDim is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that the framework's constant coherence energy, a quantity written E_coh, equals the golden ratio φ raised to the negative of the configuration dimension of a recognition event. In plain terms, it connects a physical constant to a counting problem: how many independent degrees of freedom a single recognition event has.

A recognition event, in this framework, is a discrete record of a comparison between two states. The framework models such an event as having D + 2 independent degrees of freedom: D spatial dimensions, one temporal step, and one balance condition that keeps the ledger neutral. The declaration proves that at D = 3, the configuration dimension is 5, and therefore E_coh equals φ⁻⁵. This matches the framework's separately defined value for E_coh, which is the reduced Planck constant ħ in its native units.

The proof is part of a larger derivation called the Gap-45 derivation. The number 45 arises as the product of the parity count, D² = 9, and the configuration dimension, D + 2 = 5. The framework proves that this gap is coprime with 2^D when D is odd, and not coprime when D is even, which provides an argument that the spatial dimension D must be odd. Combined with other results, this leads to the claim that D = 3.

The theorem itself is a formal identity: it proves that two definitions, Constants.E_coh and φ^{−configDim(D)}, are equal. It does not prove that the physical universe has three spatial dimensions, nor that the coherence energy is measurable in any experiment. The framework's claims about the physical world rest on the interpretation of these formal objects, not on the theorem alone.

/-- The RS-native coherence energy equals `φ` to the minus configuration
dimension: `Constants.E_coh = φ^(-(D+2))`. -/
theorem Constants_E_coh_eq_configDim :
    Constants.E_coh = phi ^ (-(configDim D : ℤ)) := by
  have hcfg : (-(configDim D : ℤ)) = (-5 : ℤ) := by
    have := configDim_at_D3; omega
  rw [hcfg, ← Real.rpow_intCast phi (-5 : ℤ)]
  unfold Constants.E_coh Constants.cLagLock
  norm_num

theorem configDim_at_D3 : configDim D = 5 := 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)

What this page does not claim

The theorem does not prove that physical space has three dimensions; that requires the full forcing chain and the recognition-to-linking bridge. The theorem does not establish that the coherence energy is a measurable physical quantity. The declaration does not derive the value of the fine-structure constant or any other coupling constant.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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