The Form

Proposition 1 (The Surplus Ratio). Let E(ε) be the dimensionless energy of the break. The baryon asymmetry ratio takes the structural form η = E(ε) / (1 + E(ε)).

Proof. The core axiom partitions reality into two components: the symmetric (1:1) and the break (1×ε). The symmetric component produces equal quantities of matter and antimatter. Upon annihilation, this component converts entirely to radiation (photons).

The break component produces a surplus that has no antimatter counterpart (AP22: the break has no σ-image) and therefore survives annihilation as baryonic matter.

The total energy budget is E_total = E_symmetric + E_surplus = 1 + E(ε) (normalising the symmetric component to 1).

After annihilation, the energy in radiation is E_radiation = 1 (from the symmetric annihilation) and the energy in surviving baryons is E_baryons = E(ε) (from the surplus). The baryon-to-photon energy ratio is:

η_E = E_baryons / E_radiation = E(ε) / 1 = E(ε).

The surplus as a fraction of total energy (surplus-to-total ratio) is:

η = E(ε) / (1 + E(ε)).

For E(ε) ≪ 1 (which is established: the break is small), these two expressions are equivalent to leading order: E(ε) / (1 + E(ε)) ≈ E(ε). ∎

3.1 — The energy-to-number mapping

Proposition 1 derives an energy ratio. The observed baryon-to-photon ratio η ≈ 6 × 10⁻¹⁰ is a number ratio (baryons per photon), not directly an energy ratio.

The relationship between the two depends on the thermodynamics of the annihilation epoch.

At the annihilation epoch (~1 GeV), the average photon energy and the baryon mass are of comparable order, so the energy ratio and number ratio approximately coincide.

However, the precise mapping may carry a coefficient g_eff related to the effective number of relativistic degrees of freedom g* at the annihilation temperature. The general form is:

η = E(ε) / (g_eff · (1 + E(ε)))

where g_eff absorbs the thermodynamic factors that convert the energy partition into a particle number ratio. In the simplest case, g_eff = 1 and Proposition 1’s form is exact.

In the general case, g_eff is a calculable number that depends on the particle content at the segregation epoch.

This is not a weakness of the derivation — it is a precision question. The structural form (surplus fraction of total energy) is derived and locked.

The coefficient g_eff determines whether the mapping is direct or carries a thermodynamic correction. Determining g_eff requires knowing the segregation epoch and the particle spectrum at that epoch. This is part of what KS-60 must settle, not a separate debt.