The claim
A33 §5: “Zero aggregate gradient does not imply absence of frustrated demands.” Say demands i = 1, …, n with losses L_i are jointly stationary at θ when every g_i = ∇L_i(θ) is zero: no demand has a first-order improving direction. Say common first-order descent is infeasible there when some g_i ≠ 0 and no direction lowers every L_i (TERM-060, local form).
Three things are kept apart, and only the first two are about derivatives:
- Stationarity describes derivatives: whether the g_i vanish.
- Compatibility describes the joint existence of acceptable outcomes: whether ⋂ A_i ≠ ∅ for the declared acceptance conditions (CLAIM-347). Common first-order descent is its local, linearized diagnostic for demands of the form “improve”.
- Satisfaction describes whether the selected outcome meets the requirements.
Stationarity is neither necessary nor sufficient for satisfaction. Not sufficient: a point where every g_i vanishes can be a maximum or a saddle of some L_i. Not necessary: a demand stated as a threshold, L(θ) = θ² ≤ 1, is satisfied at θ = 0.5, where L = 0.25 and ∇L = 1 ≠ 0. Vanishing gradients are necessary only for unconstrained interior optima of differentiable losses, which is a narrower proposition. (The first filing called stationarity necessary for reconciliation; the owner’s second review corrected it.) The claim is therefore about what the aggregate gradient can show, not about when demands are satisfied, and the ordinal account (CLAIM-347) decides the latter.
The construction (CASE-115). L₁(θ) = (θ − 1)², L₂(θ) = (θ + 1)², θ ∈ ℝ. At θ = 0, g₁ = −2 and g₂ = 2. With weights ½, ½ the aggregate gradient is ½(−2) + ½(2) = 0. The aggregate L = ½L₁ + ½L₂ = θ² + 1 is not only stationary at θ = 0 but globally minimized there. Yet L₁ = L₂ = 1, each is minimized elsewhere (at +1 and −1), and any d ≠ 0 lowers one to first order and raises the other: g₁d = −2d and g₂d = 2d have opposite signs. The demands are frustrated, not reconciled.
The general step. Let w_i > 0 and Σ w_i g_i = 0. Then either every g_i = 0, or the nonzero g_i satisfy Σ λ_i g_i = 0 with λ_i = w_i / Σ_j w_j over them, so 0 ∈ conv{g_i}. In the second case no d has g_iᵀd < 0 for all i, since Σ λ_i g_iᵀd = 0. So a vanishing positive aggregate is compatible with joint stationarity and with infeasible common descent, and nothing in the aggregate gradient or the aggregate loss separates them. With a zero weight it says less: if w₂ = 0, stationarity of the aggregate constrains g₁ only, and demand 2 can be anything.
The converse case in A33 §5 is the same fact from the other side: if endorsement weights task 1 heavily, the aggregate improves while task 2 is sacrificed. “So aggregate optimization performance may conceal the distribution of frustration among constituent objectives.”
The interpretive reading, not part of the claim
A29, before frustration replaced distress (CLAIM-368): “the unity of the distress signal need not arise from the unity of the underlying causes. Many independently generated problems can be summarized by one regulatory alarm or evaluation channel.” A33 §5 drew the same moral for a monitor: “A scalar training metric could therefore function as an illusory unifier of evaluative success”, where the illusion “would belong to an interpreter or self-model that treats aggregate stability as evidence that all relevant demands have been reconciled”. These say what a reader of the scalar might wrongly infer. The demonstration says only that the inference is invalid. Whether any organization’s monitor makes it is not shown. It is the learning-system case of CLAIM-060 (an encompassing success does not establish its constituents’) and of CLAIM-276 (stability is no evidence of fidelity).
What it does not say
- That the learning objectives have interests. A33: “the analogy must not be taken as a claim that learning objectives have interests or rights.”
- That a zero aggregate gradient indicates frustration. It is compatible with both cases; per-demand gradients are needed to decide.
- Anything beyond first order. At a point where every g_i = 0 the demands may still be frustrated at second order or globally. In the construction the frustration is also global: L₁ + L₂ = 2θ² + 2 ≥ 2, so no θ makes both losses less than 1.