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

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Cosmology Eta Bexact Rung Derivation Eta B Rung From Chirality Eq

A machine-checked theorem derives the baryon-to-photon ratio's exponent from a discrete symmetry, but the physical bridge to the cosmos remains open.

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The chirality route to −44

The baryon-to-photon ratio η_B measures how many protons and neutrons exist for every photon left over from the early universe, roughly one baryon for every billion photons. In the Recognition Science framework, this ratio is pinned to a power of the golden ratio, φ^−44, where −44 is called the η_B rung. The declaration eta_B_rung_from_chirality_eq is a theorem in the framework's machine-checked library of formal theorems. It proves that a number built from two discrete symmetry properties equals −44, matching the rung derived from the spatial dimension D = 3.

The theorem's inputs are concrete. One is the flip count of a Gray code cycle on three bits, a sequence where each step changes exactly one bit, and the first step changes 4 bits. The other is a torsion gap, a difference between two entries in a spectral sequence, with absolute value 11. Their product, 4 × 11 = 44, negated, gives −44. The proof is a short chain of arithmetic equalities, checked by the kernel, with no framework-specific axioms beyond the standard three logical postulates.

This is one of three routes that converge on −44. A second route derives −44 from the number of independent degrees of freedom in a recognition event at D = 3, and a third from the Standard Model's 90 fermionic degrees of freedom halved. The framework proves all three agree, and that the chirality route is only defined at D = 3. The convergence is structural, not statistical: the routes share the same gap integer 45, so they are not independent draws.

What the theorem does not claim is the physical step. It establishes an arithmetic identity inside the framework's formal system. It does not prove that this integer is the measured baryon-to-photon ratio, nor that the framework's recognition events are the actual physics of the early universe. That bridge, from the formal structure to physical cosmology, remains an open target. The framework can still be falsified at any of the named structural inputs, such as the flip count or the torsion spectrum.

THEOREM eta_B_rung_from_chirality_eq · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean
eta_B_rung_from_chirality_eq · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:121 
/-- 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
THEOREM eta_B_rung_from_chirality_eq · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean
eta_B_rung_from_chirality_eq · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:121 
/-- 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
THEOREM routes_AB_agree · routes_AC_agree · routes_BC_agree · chirality_only_defined_at_D3 · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean
routes_AB_agree · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:154 
/-- **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]
routes_AC_agree · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:160 
/-- **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]
routes_BC_agree · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:166 
/-- **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]
chirality_only_defined_at_D3 · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:244 
/-- D = 3 is the unique non-degenerate dimension where the chirality
    product (4 × 11) and the gap-from-dimension formula (D²(D+2) − 1)
    both produce the same integer 44, because:
    - D = 3 is forced by T8 (linking + 8-tick + sync)
    - The Gray code on Q^D is defined for D = 3 specifically
    - The torsion spectrum {0, 11, 17} is a CW-filtration consequence at D = 3
    For any other D the chirality route does not even type-check
    (bitFlipCount is `Fin 3 → ℕ` by construction). -/
theorem chirality_only_defined_at_D3 :
    eta_B_rung_from_chirality
      = eta_B_rung_from_dimension Foundation.GapDerivation.D := by
  rw [routes_AB_agree.symm]
THEOREM chirality_product_equals_gap_minus_one · fermionic_half_equals_gap · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean
chirality_product_equals_gap_minus_one · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:172 
/-- The chirality flip-count × torsion product equals the gap minus the
    active edge. This is a non-trivial structural identity at D = 3:

      bitFlipCount(0) × |torsionGap(0,1)| = consciousnessGap(D) − A

    LHS comes from the Gray code on Q₃ × CW filtration torsion.
    RHS comes from the gap-from-dimension formula D²(D+2) − 1.
    Both equal 44 at D = 3. -/
theorem chirality_product_equals_gap_minus_one :
    ((bitFlipCount 0 : ℤ) * (torsionGap 0 1).natAbs)
      = (Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.D : ℤ)
        - Foundation.GapDerivation.A := by
  have hflip : bitFlipCount 0 = 4 := bit0_flips_four
  have htor : (torsionGap 0 1).natAbs = 11 := by native_decide
  have hgap : (Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.D : ℤ) = 45 := by
    exact_mod_cast Foundation.GapDerivation.gap_at_D3
  rw [hflip, htor, hgap]
  decide
fermionic_half_equals_gap · IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean:191 
/-- The fermionic-DOF half equals the consciousness gap. This identity
    is the bridge: matter and antimatter each carry one full
    consciousness-gap worth of fermions, so dividing by 2 recovers the
    gap. -/
theorem fermionic_half_equals_gap :
    ((fermionic_dof / 2 : ℕ) : ℤ)
      = (Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.D : ℤ) := by
  have hferm : fermionic_dof = 90 := fermionic_dof_eq
  have hgap : (Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.D : ℤ) = 45 := by
    exact_mod_cast Foundation.GapDerivation.gap_at_D3
  rw [hferm, hgap]
  decide

What this page does not claim

The theorem does not prove that the framework's recognition events are the physical processes of the early universe. The theorem does not establish that the measured baryon-to-photon ratio equals φ^−44; that comparison is an empirical check, not a formal result. The theorem does not derive 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/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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