Cosmology Theta Crit From Dimension Eta B Derived Matches Old

The derived baryon scale

The baryon asymmetry of the universe is the observed imbalance between matter and antimatter. In the Recognition Science framework, this asymmetry is represented by a scale, denoted η_B, which sits on a ladder of powers of the golden ratio φ. The declaration eta_B_derived_matches_old establishes that a newly derived way of computing this scale agrees exactly with the framework's earlier, separately defined value.

Previously, the exponent for η_B was set by hand. The new derivation starts from the framework's result that space has three dimensions. From that single fact, the framework computes a quantity called the consciousness gap, equal to D²(D+2) = 3²(3+2) = 45. The baryon scale is then placed at rung 1 − 45 = −44, meaning η_B = φ^(−44). The declaration proves that this derived value equals the old definition, removing the need for the exponent to be an unexplained input.

The same derivation also produces a companion quantity, the consciousness saturation threshold Θ_crit = φ^45. The two sit on opposite rungs, and their product is exactly φ. This is not a trivial identity; both exponents −44 and 45 are consequences of the dimension calculation, not choices. The declaration eta_B_derived_matches_old is one link in this chain, confirming that the new route reproduces the established scale.

What the declaration does not do is connect η_B to any measured value of the baryon asymmetry. The framework's scale is a mathematical object derived from its own axioms; whether it matches astronomical observations is a separate empirical question the declaration does not address. The result also depends entirely on the framework's prior claim that space is three-dimensional, a theorem within the framework but not a statement about physical space outside it.

/-- Bridge: derived η_B matches the existing definition. -/
theorem eta_B_derived_matches_old :
    eta_B_derived = BaryonAsymmetryExact.eta_B_phi_scale := by
  unfold eta_B_derived BaryonAsymmetryExact.eta_B_phi_scale
  rw [eta_B_rung_derived_eq]

theorem eta_B_rung_derived_eq : eta_B_rung_derived = -44 := by
  unfold eta_B_rung_derived A
  norm_num [gap_at_D3]

/-- The matter-consciousness duality, derived from D = 3 alone.
    η_B × Θ_crit = φ^(A − gap) × φ^(gap) = φ^A = φ.

    This is NOT the trivial identity φ^(-44) × φ^45 = φ.
    The exponents −44 and 45 are both DERIVED from
    consciousnessGap D = D²(D+2) = 45 at D = 3. -/
theorem duality_from_dimension :
    eta_B_derived * theta_crit_derived = phi := by
  unfold eta_B_derived theta_crit_derived eta_B_rung_derived A
  have hg : (↑(consciousnessGap D) : ℤ) = 45 := by exact_mod_cast gap_at_D3
  rw [hg, ← zpow_add₀ (ne_of_gt phi_pos)]
  have : (1 : ℤ) - 45 + 45 = 1 := by norm_num
  rw [this, zpow_one]

What this page does not claim

No connection is made to any measured value of the baryon asymmetry. The result does not assert that physical space is three-dimensional outside the framework. The declaration does not prove the consciousness gap itself; it uses it as an input.

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

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