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Let:
𝒏ₜ := ⨁𝒉ₖ(𝒖ᵢ) × 𝒅ₖ
# Collective need at time t (aggregated function of individual need by demographic class)
𝒔ₜ := ∑𝒖ᵢ in 𝒮
# Stock of existing goods
𝒒ₜ := Output quantity determined by plan
Planning Function:
𝓟ₜ(𝒖ᵢ) := f(𝒏ₜ, 𝒔ₜ) → 𝒒ₜᵢ
# Planned quantity is function of current need and inventory
Feedback Detection:
Δ𝒏 = 𝒏ₜ₊₁ − 𝒏ₜ
Δ𝒔 = 𝒔ₜ₊₁ − 𝒔ₜ
Control Equation:
𝓕ₜ := δ(Δ𝒏, Δ𝒔) → {𝒒ₜ₊₁ᵢ↑, ↓, =}
# Adjustment function modifies next period's production based on deviations
Convergence Criterion:
limₜ→∞ |𝒒ₜ − 𝒏ₜ| < ε
# Feedback loop aims for asymptotic convergence between need and provision (ε → negligible mismatch)
Failure Set:
𝓔 := {𝒖ᵢ | Δ𝒔 > threshold ∧ Δ𝒏 < 0}
# Items overproduced without sustained need enter error set for downregulation
Operator:
𝒑ₐᵣₜᵧ = δ(𝓔, 𝓕ₜ)
# Invariant operator applies regulation recursively — not subjectively