The Formal Debt

This paper derives the form of the baryon asymmetry ratio. It does not derive the value. The form is η = E(ε) / (1 + E(ε)). The value requires deriving E(ε) from first principles.

This is a formal debt.

6.1 — What must be computed

The debt is: derive the dimensionless value E(ε) from the self-consistency conditions established in AP24. The target is E(ε) ≈ 6 × 10⁻¹⁰, which would yield η ≈ 6 × 10⁻¹⁰ (since E(ε) ≪ 1).

6.2 — Two paths to the value

Path 1 (Direct computation). Derive the full Standard Model particle spectrum as harmonics of ε. Write the renormalisation group equations with the derived spectrum.

Run the couplings from the Planck scale (where E(ε) → 1, the break total in the limit) to low energies.

The unique low-energy value of E(ε) that produces self-consistent values for α_em ≈ 1/137 and m_e ≈ 0.511 MeV is the answer. This path requires deriving the particle spectrum first (a major undertaking).

Path 2 (Fixed-point method). AP24 identifies a fixed-point structure: ε = f(ε), where f encodes the self-consistency conditions. Formulate f explicitly and solve for the non-trivial fixed point.

If f has a unique stable fixed point in (0, 1), that fixed point is E(ε). This path is more elegant but requires formulating f, which itself depends on the coupling structure.

6.3 — The energy-to-number coefficient

The mapping from the energy ratio E(ε) to the observed number ratio η may carry a thermodynamic coefficient g_eff related to the effective degrees of freedom at the annihilation epoch.

Determining whether g_eff = 1 (the direct mapping) or g_eff ≠ 1 (requiring a correction) is part of the debt.

6.4 — What this debt does not include

This debt is about the numerical value of E(ε).

The following are NOT debts — they are derived results: the form of the baryon asymmetry ratio (Proposition 1), the topological protection of the surplus (Proposition 2, from AP22), and the connection to fundamental constants via AP24’s self-consistency. The numerical value of E(ε) — and the reason the energy face differs from the coupling face α — is the debt (KS-60).

The structural account is complete. The universe has a baryon asymmetry because the axiom is 1:1 + 1×ε @ AS, not 1:1. The form is forced. The value is the remaining calculation.