CASE-115: Two incompatible quadratic demands on one shared parameter, L₁ = (θ − 1)² and L₂ = (θ + 1)², frustrated at θ = 0 for every update budget, and the split of θ into task-routed θ_A and θ_B that makes both satisfiable

The construction

Before. One scalar parameter θ, two demands with losses L₁(θ) = (θ − 1)² and L₂(θ) = (θ + 1)² (A33 §2; A38 §4 writes them L_A, L_B). At θ = 0:

• g₁ = −2, g₂ = 2. Any d ≠ 0 lowers one loss to first order and raises the other.
• With weights ½, ½ the aggregate gradient is 0, and the aggregate ½L₁ + ½L₂ = θ² + 1 is at its global minimum, while L₁ = L₂ = 1 (A33 §5).
• Globally, L₁ + L₂ = 2θ² + 2 ≥ 2, so no θ makes both losses less than 1. For any threshold τ < 1 the acceptance sets {L₁ ≤ τ} = [1 − √τ, 1 + √τ] and {L₂ ≤ τ} = [−1 − √τ, −1 + √τ] are disjoint: the compatibility complex of the two demands is {∅, {1}, {2}}.
• For required local improvements ε₁ = ε₂ = ε, the residual is Φ_r = min_{|d|≤r} max(ε − 2d, ε + 2d) = ε, attained at d = 0, for every r ≥ 0. The dual multiplier λ = (½, ½) gives λ₁g₁ + λ₂g₂ = 0 and the same value (A35 §3.4).

After. A38 §4: “Now allow the organization to differentiate into two context-sensitive processes”, θ_A and θ_B, with the task context available for routing, so L₁ depends on θ_A and L₂ on θ_B. Then (θ_A, θ_B) = (1, −1) meets both demands, and the complex is all of {∅, {1}, {2}, {1, 2}}, a strict enlargement for the same declared demands. Locally at (0, 0), with g₁ = (−2, 0), g₂ = (0, 2) and the Euclidean norm, the residual is Φ_r = ε − √2·r (at λ = (½, ½), attained by d = (r, −r)/√2), so a budget r ≥ ε/√2 meets both: what was structural becomes at worst capacity-limited. Checked numerically.

A38: “The important change isn’t just more parameters. It’s the emergence of a division of functional jurisdiction.”

What it can show

1. A zero aggregate gradient with frustrated demands (CLAIM-351).
2. Structural frustration that no budget relieves, and its dual certificate (CLAIM-365, CLAIM-357).
3. Relief as enlargement for fixed demands (CLAIM-347): the demands before and after are the same two, with the same acceptance sets, so the comparison is CLAIM-361’s honest case, with a correspondence that is the identity.
4. Differentiation as a source of relief. In the terms of CLAIM-360 the split enlarges the realizable function class, since the routed system computes a function of the task context that the shared parameter cannot. It is an illustration of that effect, not evidence about deep networks.

What it cannot show

• That anything individuated. A38: “It does not automatically mean that two new autonomous individuals have formed.” Whether θ_A and θ_B are subsystems of one more differentiated organization or new individuals is the individuation question, and nothing here bears on it.
• That the split is endogenous. The construction stipulates it. Nothing in it shows a learning organization reaching it, maintaining it, or governing it: A38’s caption, “This is not, by itself, a model of endogenous self-maintenance.”
• That more parameters relieve frustration in general. The split works because the context is routed. Without routing, a second parameter both demands must pass through relieves nothing. A38: “Not every form improves performance. Some merely partition the problem, reproduce conflict elsewhere, or restrict information flow.”
• Anything about real networks. Fictional by construction.