Cosmology Eta Bexact Rung Derivation Eta B Rung From Chirality Eq Named

The chirality route

The baryon-to-photon ratio η_B is one of cosmology's most precise puzzles: why does the observable universe contain roughly one particle of ordinary matter for every 1.6 billion photons of light? Recognition Science approaches this ratio through a rung, a discrete integer exponent that pins η_B to a power of the golden ratio. The declaration eta_B_rung_from_chirality_eq_named is a theorem in the framework's machine-checked library of formal theorems. It proves that the rung computed from two specific structural features, the flip count of a Gray code cycle and a torsion gap from a filtration, equals the negative of their product.

The theorem states that the chirality-derived rung equals -44. The number arises from multiplying 4, the flip count of the first element in a Gray code cycle on three qubits, by 11, the absolute value of a torsion gap between two stages of a CW filtration. Negating the product gives -44. The proof is a short chain of arithmetic equalities, each checked by the computer: it unfolds the definition of the chirality rung, substitutes the two known values, and reduces to the integer -44. The declaration's named form records the equality in terms of two named constants, flip_count_gen0 and torsion_gap_01, making the structural origin of the factors explicit.

This result is one leg of a three-route convergence theorem. The same integer -44 emerges from a dimensional route based on the number of degrees of freedom in a recognition event, and from a fermionic route based on the Standard Model's 90 fermionic degrees of freedom. The library proves all three routes agree at D = 3, and that the chirality route is only defined at that dimension. The convergence is a theorem about the framework's internal structures, not a statistical fit: none of the routes uses the target value η_B = φ^-44 as input.

What the theorem does not claim is equally important. It does not assert that the measured baryon-to-photon ratio itself equals φ^-44; that comparison against astronomical observations remains an empirical check, not a proved result. The theorem does not derive the fine-structure constant or any other coupling. It does not prove that the chirality and torsion inputs are physically real; those are structural definitions within the framework. The proof establishes an internal equality among framework-defined quantities, and the framework's library states explicitly that it can be falsified at any of the named structural inputs.

/-- The chirality route equals the named structural integers from
    `BaryonAsymmetryExact`. -/
theorem eta_B_rung_from_chirality_eq_named :
    eta_B_rung_from_chirality = -((flip_count_gen0 : ℤ) * torsion_gap_01) := by
  unfold eta_B_rung_from_chirality
  have hflip : bitFlipCount 0 = flip_count_gen0 := by native_decide
  have htor : (torsionGap 0 1).natAbs = torsion_gap_01 := by native_decide
  rw [hflip, htor]

/-- 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

/-- **CONVERGENCE THEOREM A=B**: The gap-from-dimension and chirality
    routes agree at D = 3. -/
theorem routes_AB_agree :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = eta_B_rung_from_chirality := by
  rw [eta_B_rung_from_dimension_at_D3, eta_B_rung_from_chirality_eq]

/-- **CONVERGENCE THEOREM A=C**: The gap-from-dimension and fermionic
    DOF routes agree at D = 3. -/
theorem routes_AC_agree :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = eta_B_rung_from_fermionic := by
  rw [eta_B_rung_from_dimension_at_D3, eta_B_rung_from_fermionic_eq]

/-- **CONVERGENCE THEOREM B=C**: The chirality and fermionic DOF routes
    agree at D = 3. -/
theorem routes_BC_agree :
    eta_B_rung_from_chirality = eta_B_rung_from_fermionic := by
  rw [eta_B_rung_from_chirality_eq, eta_B_rung_from_fermionic_eq]

What this page does not claim

The theorem does not assert that the measured baryon-to-photon ratio equals φ⁻⁴⁴. The theorem does not derive the fine-structure constant or any other coupling constant. The theorem does not establish that the chirality and torsion inputs are physically real; they are framework-defined structures.

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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