Table A5. Production and scaling (Appendix B)

Symbol	Definition
Φ: X → ℝd	Decoding map: for an admissible specification, the system reliably recovers the intended meaning; Φ belongs to the decoding system and does not depend on i (B.1); below the bar no decoded intent stands (§5.1).
(x̃) ≔ Φ(x) ∈ ℝd	Decoded intent, with ‖(x̃) ‖ > 0 on admissible episodes; the content the system works from, parallel to the criterion x* = Γ(x) (B.1).
Ψ:(0, ∞) × ℝd → ℝd	Execution map: R(S) ≔ Ψ(S,(x̃); no structural form assumed beyond the regularities of B.2 (B.1).
[Smin, Smax]	Operating range, 0 < Smin ≤ Smax; the regularities of B.2 are assumed only on this range (B.2).
λQ > 0, εQ ∈ [0,1)	Per-unit rate and relative error bound of quality-magnitude scaling with its zero branch (B.2).
ηQ (S, x̃)	Relative quality deviation, ∥ Π Q ( R ( S ) ) ∥ λ Q S ∥ Π Q ( x ˜ ) ∥ − 1 , defined for ‖ΠQ (x̃) ‖ > 0; the assumption reads |ηQ| ≤ εQ; audited in B.4 (B.2).
λT > 0, εT ∈ [0,1)	Per-unit rate and relative error bound of nondegenerate directional scaling with its zero branch (B.2).
ηT (S,(x̃)	Relative directional deviation, ∥ Π T ( R ( S ) ) − λ T S Π T ( x ˜ ) ∥ λ T S ∥ Π T ( x ˜ ) ∥ ; the assumption reads ηT ≤ εT; audited in B.4 (B.2).
uS, u0	Unit directions of ΠT (R(S)) and of ΠT (x̃); ‖uS − u0 ‖ ≤2εT on the range (B.3).
S req   ≔   θ Q λ Q   ∥ Π Q ( x ˜ ) ∥	Nominal required scale at which the quality magnitude reaches θQ; bracketed by the sufficient and necessary levels of B.3 (B.3).