QUANTITATIVE INSTABILITY OF ALGEBRAIC REPRESENTATIONS

The paper’s Theorem 1.2 states exactly the inequality the model proves, with v_j highest-weight vectors for k-fundamental representations and nonnegative rational coefficients in the k=Q case. The authors’ proof proceeds by (i) bounding the shrink-rate function by a Busemann function (Theorem 5.1), (ii) expressing Busemann functions as nonnegative linear combinations of fundamental highest-weight lengths (Theorem 3.5, and in the k-rational form Theorem 6.7), and (iii) identifying the fastest-shrinking geodesic with Kempf’s optimal 1-PS to obtain k-rationality and rational coefficients (Theorem 6.12 and Lemma 6.13), completing the proof in Section 7. The model’s solution mirrors these steps, with the same geometric–algebraic synthesis; minor differences are cosmetic (e.g., a brief mis-citation and an implicit reliance on norm equivalence). Overall, the logic and ingredients match the paper’s argument closely and correctly. Key references: statement of Theorem 1.2 and its proof outline ; Busemann lower bound (Theorem 5.1) ; Busemann-as-fundamental-lengths (Theorem 3.5, with the decomposition χ_a = Σ a_j χ_j ensuring nonnegativity) ; Kempf’s 1-PS and fastest geodesic (Theorem 6.12, Lemma 6.13) ; k-rational version (Theorem 6.7) and rational coefficients in Section 7 .