Equations with their premises

01 · Static geometric branch

g = −dt² + h; β = n♭; h(n,n) = 1; r ≠ 0
Ric_h = r(h − β ⊗ β); H = div_h n
∇ₙn = 0; n(r) + Hr = 0; div_h(rn) = 0
Ω = |r| iₙvol_h; dΩ = 0

Contracted Bianchi gives geodesicity and along-flow curvature transport. It does not force transverse constancy of r or eliminate twist.

Additional selection hypothesis

δΩ = 0 ⇔ d(|r|β) = 0
⇔ X(r) = 0 for every X ⟂ n, and dβ = 0

This extra condition selects a twist-free branch. Its missing historical B4 derivation remains open.

02 · Shear and twist transport

D = (H/2)I + S + W; W² = −ω²I
H′ = −H²/2 − tr(S²) + 2ω²
S′ = −HS; W′ = −HW; r′ = −Hr
(S/r)′ = 0; (W/r)′ = 0

Primes denote projected covariant differentiation along n in a transported transverse frame. These necessary identities do not replace transverse Ricci equations. In the selected co-closed branch W = 0.

03 · Exact non-umbilic family

h = dρ² + (ρ−c₊)²dx² + (ρ−c₋)²dy²
r = −1 / [(ρ−c₊)(ρ−c₋)]; c₊ ≠ c₋
H = 1/(ρ−c₊) + 1/(ρ−c₋)
σ = [1/(ρ−c₊) − 1/(ρ−c₋)]/2
H² = 4σ² − 4r

Use regular local patches avoiding either vanishing scale factor. The separate umbilic branch has h = dρ² + a(ρ)²q₀, a″ = 0 and r = [K₀ − (a′)²]/a² ≠ 0.

04 · Added magnetic matter sector

I = ½F_μνF⁽μν⁾ = r² > 0; F = Ω; κ > 0; Λ = 0
S_action = ∫√−g [R/(2κ) + L(I)]
T_μν = Lg_μν − 2L_I F_μρ F_ν⁽ρ⁾
G_μν = κT_μν; dF = 0; d(L_I *F) = 0
L_s(I) = −s√I/κ; s = sign(r)

Vary F = dA independently. Einstein matching selects this square-root form on a fixed-sign branch. Its gauge equation enforces dβ = 0, but does not force X(r) = 0. This is a separate added source model.

05 · Chosen DBI scalar action

Signature (−+++); X = −½g⁽μν⁾∂_μφ ∂_νφ
P(X) = −M⁴√Δ; Δ = 1 − 2X/M⁴ > 0
P_X = Δ^(−1/2)
∇_μ(P_X ∇⁽μ⁾φ) = 0

The scalar equation is conditional on the chosen action. It does not itself establish the missing unified action, the full coupled perturbation determinant or biological collapse dynamics.

Counterexamples are part of the proof boundary

Transverse variation: h = dz² + exp(x²)(dx² + dy²), n = ∂z, r = −exp(−x²). The Ricci pattern holds while transverse r varies.

Twist: h = cosh²(u)dx² + dy² + dz², u = y cos(kx) + z sin(kx), n = −sin(kx)∂y + cos(kx)∂z. Here r = −1 but β ∧ dβ ≠ 0 for k ≠ 0. The record credits the Brooks comparison.