The Action of Calibration: The Reduction of Spherically Symmetric Gravity to Calibration Dynamics, and the Identity of the Volume Term

The metric paper showed that the metric decomposes into direction (the conformal structure, the domain of Ψ) and calibration (the conformal factor, the domain of Λ), and gave the calibration field θ the geometric identity of the conformal factor. Two items remained unconstructed: the identification of the potential of the calibration field with the gravitational action (metric paper, row 11), and the dynamics of the directional side (row 12; consolidated row 27). This paper constructs the first and closes the spherically symmetric sector of the second, at the level of the action. The Einstein equations are neither derived nor modified; the dynamics of Weyl (non-spherical, gravitational waves) remains unconstructed. The content is in four stages. First, three two-dimensional identities: the Einstein tensor vanishes identically (Gₐb ≡ 0 — the dynamics of direction does not exist in two dimensions) ; the Einstein–Hilbert Lagrangian density is a total derivative (√−g R = −2□_ηθ — the 2D EH action is topological, Gauss–Bonnet cited, and yields no equation of motion) ; and the cosmological term is a potential for the calibration field (√−g Λ = Λe^2θ = Λλ² — the identity of the volume term, paired with the metric paper's identity of θ). Second, the spherical-reduction identity: for ds² = e^2θ (t, x) (dt² − dx²) − r (t, x) ²dΩ², the scalar curvature reduces exactly (in the sign conventions of the paper) to R₄ = R₂ − (4/r) □₂r − (2/r²) (∇r) ²₂ − 2/r², machine-verified by direct computation from the four-dimensional Christoffel symbols. Third, the reduction of the action: √−g₄R₄ = sinϑ·√−g₂r²R₂ + 2 (∇r) ² − 2 + total derivative — the coupling of the two calibrations (√−g₂ r²R₂ = −2r²□_ηθ), the kinetic term of the areal radius, and a constant potential supplied by the sphere's directional curvature. All three are terms of the calibration fields (θ, r) ; the directional side supplies no dynamical term. In the spherically symmetric sector, the gravitational action is entirely an action of calibration fields. The Schwarzschild cross-check (R₄ = 0 from the reduction identity, in the tortoise gauge with θₓ = f′/2 and R₂ = f″) confirms consistency. Fourth, the Landau bridge: the volume term e^2θ alone is asymmetric under the reversal ι: θ ↦ −θ, but the quartic Taylor expansion of its ι-symmetrization (λ² + λ⁻²) /2 = cosh (2θ) agrees exactly with the Landau form (μ/2) θ² + (g/4) θ⁴ of the calibration paper, with μ = 4Λ and g = 8Λ/3 (an identification: the symmetrization is a choice; the coefficient agreement under the choice is a theorem). For a single cosh term, μg = 32Λ²/3 ≥ 0 holds identically, so the double well (μ 0) is unrealizable — a single volume term yields only the order-two phase, and the action-level realization of the order-four phase remains unconstructed. The value of Λ is an external scale (consolidated row 29, T2): the dimensionless shape cosh (2θ) does not depend on Λ, and μ/g = 3/2 is a dimensionless number. All theorem-level claims are machine-verified by the accompanying script (verifycalibrationₐction. py, 28 checks, all passing). The demarcation table (12 rows) states what is a theorem, what is a citation (Gauss–Bonnet; the technique of spherical reduction; the whereabouts of Weyl), what is an identification (the Landau bridge), and what is unconstructed (the order-four realization; the dynamics of direction).