Cosmology Baryon Asymmetry Derivation Derivation Chain Complete

What the chain proves

The universe is made of matter, not antimatter. This asymmetry, called the baryon asymmetry, is one of the deepest facts in cosmology. The Recognition Science framework's ledger, a discrete record of recognition events, provides a chain of formal reasoning that derives the existence of this asymmetry from first principles. The declaration derivation_chain_complete is the master theorem that assembles the entire chain into a single machine-checked statement.

The chain begins with the framework's foundational cost function, which forces the golden ratio φ as a universal scaling constant. From this, the framework derives three spatial dimensions and a specific structure for the strong force gauge group Q₃. This leads to a Gray code with a particular chirality, meaning a handedness that distinguishes matter from antimatter. This chirality produces a nonzero Jarlskog invariant, a measure of CP violation, which is the difference in behavior between particles and antiparticles. Combined with the Sakharov conditions, the three requirements for generating a baryon asymmetry, the chain concludes that the baryon asymmetry is positive. In plain language, matter exists.

What the chain does not claim is just as important. The theorem proves only the sign of the asymmetry, not its magnitude. The structural formula eta_B_structural, which is proportional to the Jarlskog invariant divided by the number of relativistic degrees of freedom, gives a value about 500 times larger than the observed baryon-to-photon ratio of about 6.1 × 10⁻¹⁰. This scaffold is explicitly not a prediction. The missing physics is the detailed dynamics of the electroweak phase transition, including transport and washout effects, which remains an open problem.

The observed magnitude is instead matched by a separate hypothesis: the baryon asymmetry is approximately φ⁻⁴⁴, which is about 6.38 × 10⁻¹⁰, within about 4.5 percent of the measured value. A prefactor (1−φ⁻⁸)² brings it into the Planck band. This is an empirical rung match, not a theorem, and it carries a named falsifier: a tightened cosmic microwave background measurement outside the φ⁻⁴⁴·(1−φ⁻⁸)² band would refute it. The integer 44 appears in several guises, but these are re-expressions of the same number, not independent derivations.

The framework's library, a machine-checked collection of formal theorems, establishes the sign chain rigorously. The declaration derivation_chain_complete assembles the five key facts: three generations, chirality, positive CP violation, baryon number violation, and a positive baryon asymmetry. This is the derived content. The framework models the existence of matter, and it predicts a specific magnitude as a falsifiable hypothesis. The distinction between what is proved and what is conjectured is the central honesty of the framework.

/-- The chain to the SIGN of the asymmetry is complete: from the Sakharov
    conditions + J_CP > 0, a positive baryon asymmetry follows.

    RS-derived ingredients in the chain:
    - 3 generations → from D = 3 (face_pairs)
    - chirality → from Gray code [0,1,3,2,6,7,5,4]
    - flip asymmetry → [4,2,2] from the specific Gray code path
    - torsion → {0, 11, 17} from CW filtration
    - J_CP → from Berry phase × torsion overlap
    - Sakharov → from ledger + J_CP + EW transition

    NOT in this chain: the magnitude (the −44 rung is HYPOTHESIS; the
    structural J_CP/g_★ scaffold is ~500× too large — see header). -/
theorem derivation_chain_complete :
    face_pairs 3 = 3 ∧                          -- 3 generations
    IsChiral grayFlipCounts ∧                    -- chirality
    jarlskog_structural > 0 ∧                    -- CP violation
    deltaB_per_sphaleron = 3 ∧                   -- B violation
    eta_B_structural > 0 :=                      -- matter exists (sign)
  ⟨rfl, cycle_is_chiral, jarlskog_positive, rfl, eta_B_positive⟩

/-- The structural η_B: proportional to J_CP / g_★.
    SCAFFOLD ONLY — numerically ≈ 3×10⁻⁷, about 500× the observed value,
    because the order-one-suppressed washout constant c is not derived.
    Used solely for the sign theorem (η_B > 0) and the smallness bound. -/
noncomputable def eta_B_structural : ℝ := jarlskog_structural / g_star

/-- The φ-rung exponent for the baryon asymmetry scale.
    φ⁴⁴ ≈ 1.568 × 10⁹, so φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰.
    The observed η_B ≈ 6.1 × 10⁻¹⁰ is within ~4.5% (before the
    (1−φ⁻⁸)² prefactor). HYPOTHESIS-grade rung assignment. -/
def eta_B_rung : ℤ := -44

What this page does not claim

The magnitude of the baryon asymmetry is not derived; the structural scaffold is about 500 times too large. The φ⁻⁴⁴ rung match is a hypothesis with a falsifier, not a theorem. The integer 44 has multiple re-expressions, but none of them is an independent derivation of the rung. The chain does not derive the fine-structure constant alpha; its seed is an identification, not a derived coupling.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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