Planck-Scale Optimization Framework for Sub-3nm Semiconductor Design: Sign-Change Monotonicity, Constraint Independence, and Power Model Necessity

This paper develops a self-contained framework linking constraint structure to optimization behavior in semiconductor scaling. We establish the Sign-Change Monotonicity-Induced Extremum Lemma: when a continuously differentiable function's derivative exhibits sign reversal due to strictly increasing derivative structure, a unique global extremum necessarily exists. Applied to semiconductor physics, we establish two fundamentally equivalent constraints from first principles—Constraint C1 (quantum stability: lgate ≥ λC = ℏ/ (mₑff c) ) and Constraint C2 (information stability: lgate ≥ ξₜh) —at the same theoretical level. We show that constraints impose necessary structural properties on the objective function, determining the qualitative behavior of efficiency in the feasible region. The power consumption model Pₜotal (l) = A·exp (-B√l) + C/l² + D/l represents the minimal functional structures consistent with the underlying physical mechanisms. Under monotonic derivative conditions, a unique extremum exists, and its location is governed by the interaction of lower-bound constraints. When constraints are of comparable magnitude (λC ≈ ξₜh), the extremum is located in the constraint-intersection regime. At this point, Pₛtatic ≈ Pquantum ≈ Pdynamic ≈ Pₜotal/3. DFT verification confirms S/N = 1. 000000 at lₒpt. Industrial data (R² > 0. 93) validates the framework. Current 3nm nodes operate at 40% theoretical efficiency; 2. 5× improvement is achievable. The framework is logically consistent and self-contained: constraint interaction determines the qualitative structure of the optimization problem.