Local Einstein–scalar Boundary Branches and a Positive Volume Response

Model and boundary closure-----------------------------Let Ω = B₁(0) ⊂ ℝⁿ, S = ∂Ω, m = n − 1, V₀ = Volδ(Ω), κ > 0, μ > 0, and φ = φ₀ + u. The potential and stress tensor are \[V(φ)=μ/2(φ-φ_0)^2, S=-dφ⊗ dφ+(12|∇φ|_g^2-V(φ))g.\]The ghost-sign equations and the comparison standard-sign equations are, respectively, \[G_g=κ S, Δ_gφ+V'(φ)=0; Ric_g=-κ du⊗ du+κμ/n-2u^2g, (Δ_g+μ)u=0,\]\[Ric_g=κ du⊗ du+κμ/n-2u^2g, (Δ_g-μ)u=0(μ>0).\]The gauge and boundary data are \[W(g)^=gⁱʲ(Γ(g)^ᵢⱼ-Γ(δ)^ᵢⱼ), tfγ_0g^T=g^T-{trγ_0g^T}{n-1}γ_0,\]\[W(g)|_S=0, tfγ_0g^T=0, g(ν,ν)=1, u|_S=t.\]The full ambient Bianchi one-form, including its normal component, is prescribed; tangential scale and mixed components remain free. This is not full Dirichlet data. The off-shell conservation identity is \[∇^iSᵢⱼ=-(Δ_gφ+V'(φ))∂_jφ.\]The scalar equation must still be imposed where φ is constant. 2. Full-space isomorphism and kernel calculation------------------------------------------------For a symmetric tensor h, let \[(β_δ h)_j=∂^ihᵢⱼ-12∂_jtrh, Lfullh=(-12Δ h,(β_δ h)|_S,tfγ_0h^T,h(ν,ν)|_S),\]\[ S_μχ=((Δ+μ)χ,χ|_S), L= Lfull⊕ S_μ.\]For n ≥ 3, k ≥ 2 and 0 < α < 1, the metric operator is a full Hölder-space isomorphism with a Schauder estimate.For the ghost scalar operator this is true on 0 < μ < Mₙ, Mₙ = nπ²/4, and \[(Δ+μ)ψgh=0, ψgh|_S=1, L⁻¹(0,0,0,0;0,1)=(0,ψgh).\]The Lopatinski–Shapiro condition holds; the Fredholm index is 0 by Anderson after matching dimensions, signs and reference data. For h in the kernel, with τ = tr h and hᵀ = fγ₀, \[Δ h=0, β_δ h=0, h(ν,ν)=0, divh=12dτ, Δτ=0, τ|_S=mf.\]Set p = h(ν,·)ᵀ, a = (∇νh)(ν,ν), Θ = x·∇ and \[η(x)=12∫_0^1sn/2-1τ(sx)\,ds, Δη=0, (2Θ+n)η=τ.\]The normal Bianchi condition and the two independent formulas for a are \[0=a+div_S p-mf-12∂_ντ, a=mf-Δ_Sf=τ-1mΔ_S(τ|_S), a=τ-(n-2)η.\]Consequently \[Δ_S(τ|_S)=m(n-2)η|_S, τ=2∂_νη+nη S,\]\[∫_S|Dτ|^2+m(n-2)(2∫_Ω|∇η|^2+n∫_Sη^2)=0.\]Thus τ = 0 and f = 0. The final tangential Bianchi relation and energy identity are \[q+np=0, 0=∫_Ω|∇ h|^2+2n∫_S|p|^2,\]so h = 0. This proves injectivity without symmetry. Cube Poincaré gives Mₙ, a nonoptimal bound below the first ball eigenvalue λ₀. 3. Local branches, parity and symmetry--------------------------------------For sufficiently small |t|, contraction yields a unique smooth local branch satisfying \[\|z_t\|_X≤2b_0|t|, g_t≥12δ.\]The radius is symbolic, not numerical or maximal. Gauge recovery follows from Bianchi gives βgE = κ(Δgu + μu)du = 0. The DeTurck one-form solves a homogeneous Dirichlet equation for P_gω = ∇ⁱ∇ᵢω + Ric(ω); invertibility in the small ball gives W(g) = 0 and recovers the original equations. Uniqueness and O(n)-equivariance imply \[σ^ g_t=g_t, u_t∘σ=u_t, g₋ₜ=g_t, u₋ₜ=-u_t,\]\[g_t=A_t(r)dr^2+B_t(r)r^2γ_{ Sⁿ⁻¹}, \|g_t-δ\|k+2,α=O(t^2).\]For general small boundary data φ, the solution map is smooth and equivariant: \[g[-φ]=g[φ], u[-φ]=-u[φ], z[φ∘σ]=σ^ z[φ].\]The standard-sign branch exists locally for each fixed μ > 0, with even metric and odd scalar. 4. Volume response------------------For the ghost branch, ψgh is radial, ψgh > 1 in Ω and ψgh′(r) < 0. The relative volume satisfies \[vgh(t)={Volg_t(Ω)}{V_0}=1+c_2ᵍʰt^2+O(t^4),\]\[c_2ᵍʰ=κ/4V_0[2μ/n-2∫_Ω(1-|x|^2)ψgh^2\,dx+n∫_Ω(ψgh^2-1)\,dx]>0.\]At small mass, \[ψgh(r)=1+μ/2n(1-r^2)+O(μ^2), c_2ᵍʰ=κ n/2(n-2)(n+2)μ+O(μ^2).\]For standard sign, (Δ − μ)ψstd = 0, ψstd|S = 1 and 0 < ψstd < 1, and \[c2, std=κ/4V_0[∫_Ω(1-|x|^2)|∇ψstd|^2\,dx+nμ/n-2∫_Ω(1-|x|^2)ψstd^2\,dx]>0.\]This positivity is proved only for constant standard-sign data. For general small boundary data φ, let (Δ + μ)ψφ = 0 with ψφ|S = φ. Then \[v[φ]=1+Q_μᵍʰ(φ)+O(\|φ\|^4), D^2v(0)(φ,φ)=2Q_μᵍʰ(φ),\]\[Q_μᵍʰ(φ)=κ/4V_0[2μ/n-2∫_Ω(1-|x|^2)ψ_φ^2\,dx+n∫_Ωψ_φ^2\,dx-∫_Sφ^2\,dA].\]The derivation uses \(∫_Sτ\,dA=(n-2)∫_Ωτ\,dx\) and \(∫_Ω(1-|x|^2)Δτ\,dx=-4∫_Ωτ\,dx\). At quadratic order, volume depends on φ only through Q. 5. Modal spectrum and sign thresholds-------------------------------------For spherical harmonic degree ℓ, set νℓ = ℓ + n/2 − 1 and \[f_(r)={r1-n/2Jν_(√μ r)}{Jν_(√μ)}, q_ᵍʰ=κ/4V_0B_(μ),\]\[B_(μ)=∫_0^1[n+2μ/n-2(1-r^2)]f_(r)^2rⁿ⁻¹\,dr-1, Q_μᵍʰ(φ)=∑≥0q_ᵍʰ\|φ_\|L^2(S)^2.\]At μ → 0⁺, \[B_(0)=-2/2+n, q_0ᵍʰ→0, q_ᵍʰ→-κ/2V_0(2+n)\ (≥1).\]Let λℓ = j²ᵥℓ,₁. The spectral-window lemma gives Mₙ < λ₀ ≤ λℓ and fℓ(r,μ) > 0 for 0 ≤ μ ≤ Mₙ. The threshold theorem proves \[B_'(μ)>0\ (0≤μ<λ_), B_0(μ)>0\ (μ>0), 0<μ_1ᵗʰ<μ_2ᵗʰ<⋯,\]where every ℓ ≥ 1 has one simple zero μℓᵗʰ in (0, λℓ). The Wronskian identity is \[ W=rⁿ⁻¹(f_ f₊₁'-f₊₁f_'), W'=(2+n-1)rⁿ⁻³f_ f₊₁>0.\]A threshold is in (0,Mₙ) exactly when Bℓ(Mₙ) > 0. With d = 2ℓ + n, the uniform high-mode bound is \[B_(μ)≤{n+4M_n/(n-2)(d+2)}{d(1-M_n/2d)^2}-1-1(→∞).\]Thus Q is indefinite for every 0 < μ < Mₙ: q₀ > 0 and sufficiently high modes are negative. In dimension three, M₃ = 3π²/4 and \[0<μ_1ᵗʰ<μ_2ᵗʰ<7 0, B_3(15/2)≤-547/6435<0.\]Only degrees one and two change sign once on this interval. Thresholds ≈ 2.207 and ≈ 5.468 are not certified; higher-dimensional decimals are exploratory. Extension to μ < λ₀ = π² requires reclassification; mode three appears to cross near 9.76. 6. Amplitude, curvature and invariant check-------------------------------------------For small 0 ≤ r₁ < r₂, \[{c_2ᵍʰ}2(r_2^2-r_1^2)≤ vgh(r_2)-vgh(r_1)≤{3c_2ᵍʰ}2(r_2^2-r_1^2),\]\[|r_2-r_1|≤√{2{c_2ᵍʰ}}|vgh(r_2)-vgh(r_1)|1/2.\]Volume recovers |t|, never its sign. The inverse is Hölder-1/2 at the vacuum, and that exponent is optimal. Along tφ the same holds if Q(φ) ≠ 0. For any solution, \[Ric_g=κ(-dφ⊗ dφ+2V(φ)/n-2g),\]\[λ_∥=κ(2V/n-2-|∇φ|^2), λ_⊥=2κ V/n-2, R_g=κ(2nV/n-2-|∇φ|^2).\]Hence λ⊥ > 0 if V > 0, λ∥ > 0 iff V > (n−2)|∇φ|²/2, and R > 0 iff V > (n−2)|∇φ|²/(2n). The constant-data branch is non-flat for t ≠ 0 but conformally flat, W(gₜ) = 0. On the invariant subspace, regular harmonic symmetric tensors are spanned by E₀ = δ and E₁ = x♭⊗x♭ − (r²/n)δ, with \[{pmatrix}w_h\(ν,ν){pmatrix}=M{pmatrix}c_0_1{pmatrix}, M={pmatrix}0&(n+2)(n-1)/n\\1&(n-1)/n{pmatrix}, M=-{(n+2)(n-1)}n≠0.\]This is only an invariant-subspace check; unrestricted invertibility relies on the kernel proof.