The Paper and the Pen. From a Constant to a Counted Debt: Dynamical F₄ by Conjugacy with the Four-Volume — Derivation, the Λ(V) Equations, and a Complete Numerical Check

> *Taken as a constant, the positive four-form \(F_4 = f vol_4\) gives \(f = M_P H_0\) with \(H_0\) put in by hand, so that \(ρ_F/M_P^4 = 1/2N\) is an identity, not a prediction. Here \(F_4\) is made dynamical by exhibiting its canonical conjugate: in minisuperspace \((φ, f) ↦ (V_4, ρ_F)\) is a canonical map with \(\{V_4, ρ_F\} = 1\), a local chart on the positive branch and a global pair in the signed (unimodular) sector.*> *A counting hypothesis (the four-volume is a count of cells of volume \(v_0 _P^4\) with variance equal to the mean), together with the identification of the quantum spread of the conjugate variable with the counting noise, gives \(Δ Λ ≈ κₜₕ/√V\), \(κₜₕ = 4π/√v_0\) (the scaling of Sorkin, with the coefficient of Das–Nasiri–Yazdi).* Main claim (the HB settled-once law, HB–A′): *the signed vacuum-energy variable is settled once, as one draw from a centred law of width \(σ(V_*) = κₜₕ M̄_P^2/√V_*\), and is constant afterwards (\(Λ\)CDM). The observed density \(ρ_Λ/M_P^4 = 1.13× 10⁻¹²³\) is a \(0.057σ\) draw at \(V_* = V_0\) with \(v_0 = 1\); the tuning of \(10⁻¹²²\) is traded for one explicit assumption that carries all of it (the state is centred) and one calibrated number, \(W = v_0 V_*/V_0\). Neither is derived.* *A zero-mean law cannot be assigned to the positive four-form energy itself (a theorem), so the law is stated on the signed variable. The deterministic law \(Λ = κ/√V\) along the causal past is excluded in the schemes tested (open problem, route B).* *This is a scale relation with named assumptions, not a complete theory; every statement is classified (Derived, Hypothesis, Reading, Witness, Calibration, Open). \(H_0\) remains an input. Interpretive material the derivation does not use is collected in a separate annex, "Readings". Files: the paper (17 pp), the annex (8 pp).*