Cosmology Eta Bexact Rung Derivation Matches Existing Eta B Rung

The convergence theorem

The baryon-to-photon ratio η_B measures how many protons and neutrons exist compared with photons in the observable universe, roughly one baryon for every 1.6 billion photons. In the Recognition Science framework, this ratio is tied to a discrete integer rung on a phi-power ladder. The declaration matches_existing_eta_B_rung is a machine-checked theorem stating that the rung derived from the dimension of space equals the previously established η_B rung. Concretely, the theorem proves that the expression 1 − D²(D+2), evaluated at D = 3, equals −44.

This is not a single lucky coincidence. The framework's library contains three separate derivations that all converge on the same integer. The dimensional route computes the rung from the number of independent degrees of freedom in a recognition event. The chirality route multiplies a bit-flip count of 4 by a torsion gap of 11, giving 44, then negates it. The fermionic route takes the Standard Model's 90 fermionic degrees of freedom, divides by 2 for matter and antimatter, and subtracts from 1. Each route lands on −44, and the machine-checked theorems routes_AB_agree, routes_AC_agree, and routes_BC_agree formally record that all three agree.

The convergence theorem is not a statistical claim. The routes are not independent draws from a hat; the fermionic route re-encodes the same gap integer 45 that appears in the dimensional route. The agreement is a structural identity, not a probabilistic coincidence. The framework can still be falsified at any of the named structural inputs, for example if the bit-flip count or the torsion gap were different. The theorem also includes counterfactual checks: at D = 1 the rung would be −2, at D = 2 it would be −15, and at D = 5 it would be −174, showing that the −44 result is specific to three dimensions.

What the theorem does not claim is equally important. It does not claim that η_B itself has been measured or derived from first principles; it only establishes that the rung integer matches an existing definition. It does not claim that the framework's structural inputs, such as the consciousness gap or the fermionic degree count, are themselves derived from the core cost function. Those inputs are definitional choices within the framework. The theorem certifies internal consistency, not empirical verification. The comparison against the observed baryon-to-photon ratio remains an empirical check, not a proved consequence.

/-- The derived rung matches the existing definition `BaryonAsymmetryExact.eta_B_rung`. -/
theorem matches_existing_eta_B_rung :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = eta_B_rung := by
  rw [eta_B_rung_from_dimension_at_D3]
  rfl

/-- At D = 3, the gap-from-dimension route yields −44. -/
theorem eta_B_rung_from_dimension_at_D3 :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = -44 := by
  unfold eta_B_rung_from_dimension
  have hgap : (Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.D : ℤ) = 45 := by
    exact_mod_cast Foundation.GapDerivation.gap_at_D3
  rw [hgap]
  decide

/-- The chirality route yields −44. -/
theorem eta_B_rung_from_chirality_eq : eta_B_rung_from_chirality = -44 := by
  unfold eta_B_rung_from_chirality
  have hflip : bitFlipCount 0 = 4 := bit0_flips_four
  have htor : (torsionGap 0 1).natAbs = 11 := by native_decide
  rw [hflip, htor]
  decide

/-- The fermionic DOF route yields −44. -/
theorem eta_B_rung_from_fermionic_eq : eta_B_rung_from_fermionic = -44 := by
  unfold eta_B_rung_from_fermionic
  have hferm : fermionic_dof = 90 := fermionic_dof_eq
  rw [hferm]
  decide

/-- **Counterfactual at D = 1**: gap-from-dimension would give η_B rung
    = 1 − 1²×3 = 1 − 3 = −2 (very different from observed). -/
theorem D1_counterfactual_rung :
    eta_B_rung_from_dimension 1 = -2 := by
  unfold eta_B_rung_from_dimension Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.A
  decide

/-- **Counterfactual at D = 2**: gap-from-dimension would give η_B rung
    = 1 − 4×4 = 1 − 16 = −15. -/
theorem D2_counterfactual_rung :
    eta_B_rung_from_dimension 2 = -15 := by
  unfold eta_B_rung_from_dimension Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.A
  decide

/-- **Counterfactual at D = 5**: gap-from-dimension would give η_B rung
    = 1 − 25×7 = 1 − 175 = −174. -/
theorem D5_counterfactual_rung :
    eta_B_rung_from_dimension 5 = -174 := by
  unfold eta_B_rung_from_dimension Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.A
  decide

What this page does not claim

The theorem does not claim that η_B has been measured or derived from first principles. The theorem does not claim that the framework's structural inputs are derived from the core cost function. The theorem does not claim empirical verification of the baryon-to-photon ratio.

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

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