At the wall.

The example is the tangent bundle of X = ℙ_B(O ⊕ L) ×_B ℙ_B(O ⊕ L), with B = ℙ¹ × ℙ¹ and L = O(1, −1). This is a smooth projective weak-Fano fourfold, not a Fano fourfold. Its tangent sequence has factors R₁, R₂, Q₁, Q₂, all of rank one.

Set x = σ(p, 1 − p) and y = σ(q, 1 − q). The stable inequalities are x₁, x₂, y₁, y₂ > 0, with x₁ + x₂ = y₁ + y₂ = σ. The leading edge weights are k_ij = x_iy_j/σ. The explorer diagonalizes L = D diag(k) Dᵀ on the subspace whose coordinates sum to zero. All quantities displayed are computed from this finite operator.

For the actual Hermitian–Einstein metric in the strict chamber, λ_j = ν_j(L)(1 + O(σ)), j = 1, 2, 3, while the remaining spectrum has a uniform positive gap. In the paper’s slope coordinates and normalized L² norms, the three metric-response singular values are ν_j⁻¹(1 + O(σ)). These are singular values, not an identification of every response direction with a plotted graph eigenvector.

The theorem is uniform through separated positive scales. It supplies no numerical value for the error constant displayed here, so σ is not a certified percentage error. On a nonsplit semistable boundary bundle this stationary Hermitian–Einstein metric is not asserted; boundary graphs are limiting finite models only. This is an endomorphism Dolbeault spectrum, not the full Yang–Mills Hessian.

Set x₁ = t⁶, x₂ = t², y₁ = t⁴, y₂ = t² − t⁴ + t⁶. These positive demands are realized by the paper’s ample chart on the fixed tangent bundle. Its actual eigenvalues satisfy λ₁ = (4/3)t⁶(1 + O(t²)), λ₂ = (3/2)t⁴(1 + O(t²)), λ₃ = 2t²(1 + O(t²)), and λ₄ ≥ γ > 0. The plotted solid curves are graph eigenvalues; the dashed curves are their asymptotic leading terms. The wider end of the plot illustrates the finite model without asserting a numerical geometric error bound.

Both graphs have unit vertex masses and demands b = ρ(2, 1, −1, −2), with ρ = 0.02. The tree weights are ρ(1, 1, 0, 1). For the cycle, k = ρ[[a, 2 − a], [1 − a, a]], where 0 < a < 1 solves a²/((2 − a)(1 − a)) = θ. Thus every row and column sum stays fixed.

The ratio θ = c₁₁c₂₂/(c₁₂c₂₁) comes from harmonic extension data, invariant under factor rescaling. This is an instance of Paper III’s general cyclic mechanism. It is not a free control on the fixed tangent bundle in experiments 1–2: that geometric example has a reference symmetry fixing its leading cycle ratio.

P_δ(τ) = δ / (1 − exp(−2δτ)),   P₀(τ) = 1/(2τ).

These are exact evaluations of the profile in Paper V, not numerical solutions of the Hermitian Yang–Mills PDE. For the paper’s fixed two-factor extension and any smooth positive initial-metric family, each tolerance admits one sufficiently late time and then one parameter neighborhood such that the actual smallest eigenvalue divided by this profile stays within that tolerance for every subsequent time. Both signs of δ and δ = 0 are included.

For δ > 0, P tends to δ. At the wall it equals 1/(2τ). For δ < 0, P ∼ |δ| exp(−2|δ|τ). The experiment does not claim that a stationary Hermitian–Einstein metric continues across the wall, or that this scalar formula describes a general many-factor flow.