Cosmology Sakharov From Ledger

The three conditions

In 1967, the physicist Andrei Sakharov listed three conditions a universe must meet to end up with more matter than antimatter. First, some process must change baryon number, the count of protons and neutrons. Second, that process must treat matter and antimatter differently, a property called CP violation. Third, the universe must be out of thermal equilibrium, so the reactions do not simply run backward and erase the asymmetry. These three conditions, taken together, are the standard recipe for baryogenesis, the creation of the matter-antimatter asymmetry we observe.

Recognition Science (RS) starts from a different place: a ledger, a discrete record of events, where every change is balanced by an equal and opposite entry. The framework's central result is that any cost function for recognition satisfying five plain conditions must equal a specific form, and from that form a chain of results forces the golden ratio, an eight-tick cycle, and three spatial dimensions. The module SakharovFromLedger asks whether this ledger structure, with no imported physics, can reproduce Sakharov's three conditions. Its answer is a machine-checked derivation, not a postulate.

The first condition, baryon number violation, comes from the ledger's geometry. On a three-dimensional lattice, baryon number is a winding charge, a count of how many times a path loops around an axis. A sphaleron-like process, a collective rotation of all three axes at once, changes this winding number by exactly 3. The result sphaleron_changes_B_by_3 proves this, and the number 3 is not assumed; it falls out of the fact that there are three dimensions, which itself is forced by the framework. The second condition, CP violation, is proved to be nonzero through the Jarlskog invariant, a measure of CP violation in the standard model. The result cp_source_positive establishes this, and the asymmetry parameter is defined as this positive quantity.

The third condition, departure from equilibrium, is the least formally developed. The module defines a first-order electroweak phase transition as a proposition and proves it by trivial, meaning the statement is assumed true by construction. The docstring explains the physics: above a critical temperature the Higgs field has zero average value, below it acquires a nonzero value, and the transition proceeds by bubble nucleation, which creates out-of-equilibrium conditions. But the proof of the transition's first-order nature is a definitional choice, not a derived result. The structure SakharovConditions packages all three conditions, and the result baryogenesis_possible assembles them into the claim that baryogenesis is possible in this framework. The module derives the first two conditions from its own structure; the third is a declared property, not yet a forced consequence.

theorem sphaleron_changes_B_by_3 : deltaB_per_sphaleron = 3 := rfl

theorem cp_asymmetry_nonzero : cp_asymmetry_parameter ≠ 0 :=
  ne_of_gt jarlskog_positive

/-- The Higgs VEV in RS-native units is on the φ-ladder.
    v_EW = 246 GeV, and φ^{rung} × (anchor) gives the scale.

    The critical temperature for the EW phase transition is:
    T_EW ∝ v_EW ∝ φ^{rung_EW}

    At T > T_EW: symmetric phase, sphalerons active
    At T < T_EW: broken phase, sphalerons exponentially suppressed -/
def ew_transition_is_first_order : Prop :=
  True

/-- The three Sakharov conditions for baryogenesis. -/
structure SakharovConditions where
  b_violation : deltaB_per_sphaleron = 3
  cp_violation : cp_asymmetry_parameter ≠ 0
  out_of_eq : ew_transition_is_first_order

What this page does not claim

The module does not prove the electroweak phase transition is first-order; it declares it. The module does not derive the baryon-to-photon ratio, only the possibility of baryogenesis. The module does not prove baryon minus lepton number conservation, which its own docstring marks as open.

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/SakharovFromLedger.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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