higher_berry_lowers_freeEnergy
plain-language theorem explainer
Higher Berry values reduce semantic free energy for non-negative Berry weight. Researchers modeling recognition thermodynamics and load-ratio control would cite this monotonicity. The proof is a one-line wrapper that preserves the inequality under non-negative multiplication then applies linear arithmetic after unfolding the definition.
Claim. Let $F(C,S,A,w,B) = C - A S - w B$. For $w ≥ 0$ and $B_1 ≤ B_2$, $F(C,S,A,w,B_2) ≤ F(C,S,A,w,B_1)$.
background
The Critical Recognition Loading module develops structural lemmas for the load ratio rho = R_dem / R_max. Healthy regimes lie in the sub-saturation band rho_min < rho < 1, with stability assessed on the 360-tick supervisory horizon from lcm(8,45). Semantic free energy is the query-level quantity refCost minus attention times entropy minus berryWeight times berry. Upstream entropy definitions treat entropy as total defect in the initial-condition setting and as k beta averageEnergy plus k log partitionFunction in the Boltzmann and partition-function settings.
proof idea
Apply mul_le_mul_of_nonneg_left to the hypotheses 0 ≤ berryWeight and berry₁ ≤ berry₂ to obtain berryWeight berry₁ ≤ berryWeight berry₂. Unfold the definition of semanticFreeEnergy and finish with linarith.
why it matters
The result supplies a monotonicity lemma inside the critical-recognition-loading module that sketches control for recognition-bandwidth operating regimes. It shows that larger Berry values lower semantic free energy, aligning with the Berry creation threshold in the Recognition Science framework. No downstream theorems are recorded yet; the lemma remains available for future control theorems on the sub-saturation band.
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