The claim
A34 §4 posed the test: under a budget r, is there a d with ‖d‖_𝓜 ≤ r and −⟨g_i, d⟩ ≥ ε_i for all i? A35 §3.1, at the owner’s U35 (“agree that this is our top priority, flesh this out”), defined the residual
Φ(θ; 𝒰, ε) = min_{d∈𝒰} max_i (ε_i + g_iᵀd),
with g_i the gradient of demand i and ε_i > 0 the improvement it requires over the next local step. Φ ≤ 0 when some admissible update meets every requirement to first order; Φ > 0 when every admissible update fails one. A35 §3.2 gave the dual for 𝒰_r = {d : ‖d‖ ≤ r}. Both are checked here.
Proof
Let ‖·‖ be a norm on ℝᵖ and ‖h‖* = max{‖d‖≤1} hᵀd its dual. Write Δ for the simplex in ℝⁿ.
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Primal as a saddle problem. max_i a_i = max_{λ∈Δ} Σλ_ia_i, so Φ_r = min_{‖d‖≤r} max_{λ∈Δ} [Σλ_iε_i + (Σλ_ig_i)ᵀd]. The function is affine in d and linear in λ, and both sets are compact and convex.
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Exchange. By the minimax theorem for such functions (von Neumann’s, or Sion’s), min and max commute: Φ_r = max_{λ∈Δ} min_{‖d‖≤r} [Σλ_iε_i + (Σλ_ig_i)ᵀd].
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Inner minimum. min_{‖d‖≤r} hᵀd = −r‖h‖_*, by the definition of the dual norm. So
Φ_r = max_{λ∈Δ} [ Σλ_iε_i − r‖Σλ_ig_i‖_* ].
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The certificate. The minimum in the primal is attained (continuous function, compact set), so a d with ‖d‖ ≤ r and −g_iᵀd ≥ ε_i for all i exists iff Φ_r ≤ 0, iff Σλ_iε_i ≤ r‖Σλ_ig_i‖* for every λ ∈ Δ, and, since both sides are positively homogeneous in λ, for every λ ≥ 0. That is A34’s form. A λ ≥ 0 with Σλ_iε_i > r‖Σλ_ig_i‖* certifies infeasibility by itself: any feasible d would give Σλ_iε_i ≤ −(Σλ_ig_i)ᵀd ≤ r‖Σλ_ig_i‖_*.
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Governance regret. For an update d^actual the organization actually makes, Γ = max_i(ε_i + g_iᵀd^actual) − Φ_r. If d^actual is admissible, it is one of the d the minimum ranges over, so Γ ≥ 0. Γ = 0 exactly when the actual update is a minimizer.
The identity was also checked numerically when this was filed: on 2,000 random instances with the ∞-norm ball (dual norm ℓ₁), both sides solved as linear programs agreed to within 10⁻¹⁴.
The worked case
K07 (CASE-115), A35 §3.4: L₁ = (θ − 1)², L₂ = (θ + 1)², θ ∈ ℝ, at θ = 0, with ε₁ = ε₂ = ε. Here g₁ = −2, g₂ = 2, and ‖·‖_* = |·|. The dual value at λ = (½, ½) is ε − r·0 = ε, so Φ_r ≥ ε. Every d gives max(ε − 2d, ε + 2d) ≥ ε, with equality at d = 0, so Φ_r = ε for every r ≥ 0. No budget helps: this is structural frustration (CLAIM-357). After K07’s split into θ_A, θ_B, at (0, 0) with the Euclidean norm, Φ_r = ε − √2·r, which is positive only while r < ε/√2.
An application of a known theorem
A39 §2A: if 𝒰 ⊆ ℝᵖ is convex, each A_i = {d ∈ 𝒰 : −g_iᵀd ≥ ε_i} is convex, so by Helly’s theorem, when the demands cannot all be met, “some subset of at most p+1 demands already witnesses the incompatibility”. A39 said what it is: “an application of an established theorem, not a new theorem”. In the terms of CLAIM-347, the minimal non-faces of the complex have at most p + 1 members, and fewer when 𝒰 has lower affine dimension. The bound needs convexity, and A39: “nonlinear, discrete, or nonconvex learning constraints need not obey this same bound.”
What it does not say
- That the multipliers are endorsement weights. A35: “Importantly, these dual multipliers are not endorsement weights.” They certify difficulty; the weights governance actually assigns are another matter (CLAIM-357, QUESTION-126).
- That the magnitude means anything without units. A36 §2D: “Even when all objectives are numerical, their units matter. Rescaling one loss function can change the scalar diagnosis unless we rescale and normalize the corresponding requirement consistently.” The sign of Φ_r, feasible or not, survives a consistent rescaling of g_i and ε_i together; its size does not. Per A37 the residual is “a quantitative refinement of a more basic order-theoretic feasibility problem, not the definition of frustration itself”, which is why it rests on CLAIM-347.
- Anything beyond the convex local model. A34: it “applies to this local linearized, norm-constrained problem; additional nonconvex architectural or discrete update constraints require different analysis.”
- That a change of chart changes it. With the metric pulled back it is invariant (CLAIM-366); with the Euclidean norm taken afresh in each chart it is a different problem.