Cosmology Theta Crit From Dimension Eta B Rung Derived Matches Old

The derived baryon rung

The baryon asymmetry is the observed fact that the universe contains far more matter than antimatter. In the Recognition Science framework, this asymmetry is represented by a number called eta_B, which sits on a specific rung of a logarithmic ladder based on the golden ratio. The declaration eta_B_rung_derived_matches_old is a machine-checked theorem establishing that a newly derived value for this rung is identical to the value that was previously defined directly.

Classically, the framework's earlier work simply set the baryon rung at negative 44. The new theorem derives this same number from the spatial dimension. The derivation works through a chain: with three spatial dimensions, a configuration dimension is 5, a parity count is 9, and their product gives a consciousness gap of 45. The baryon rung is then defined as 1 minus this gap, which equals negative 44. The theorem eta_B_rung_derived_matches_old formally proves that this derived rung equals the previously defined one, confirming the new derivation reproduces the old value exactly.

This matters because it converts a seemingly arbitrary number into a consequence of the framework's account of space. The number 44 is no longer a free input; it follows from the framework's derivation that three spatial dimensions force the baryon rung to be negative 44. The theorem also supports a broader duality statement: the derived baryon scale multiplied by a derived consciousness threshold equals the golden ratio itself, a relationship that holds because the two exponents, negative 44 and 45, sum to 1.

What the theorem does not claim is equally important. It does not prove that the framework's account of space is physically correct, nor does it establish that the baryon asymmetry in the real universe is actually described by this number. The derivation is internal to the framework's axioms and definitions. The theorem also does not claim that the number negative 44 matches any measured value of the baryon asymmetry; it only shows consistency between two ways of defining the same quantity within the framework. The physical interpretation of this rung, and its connection to actual cosmological measurements, remains a separate question.

/-- Bridge: the derived rung matches the existing definition. -/
theorem eta_B_rung_derived_matches_old :
    eta_B_rung_derived = BaryonAsymmetryExact.eta_B_rung := by
  rw [eta_B_rung_derived_eq]
  rfl

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

The theorem does not prove that the framework's derivation of three spatial dimensions is physically correct. The theorem does not establish that the baryon asymmetry in the real universe is described by this derived number. The theorem does not connect the derived rung to any measured value of the baryon asymmetry.

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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