Null Models for Arithmetic Coincidences in Lie-Theoretic Model Building

Methodological paper: numerical relations among the invariants of exceptional Lie algebras (Coxeter numbers, ranks, dimensions, torsion primes) are routinely presented as evidence that a particular algebra — usually E8 — is "singled out". This paper argues that such observations carry no evidential weight without a null model, and supplies one. A fixed grammar over fifteen standard invariants generates all 106,030 equations of depth ≤ 2; of the 17,450 with a unique solution, 371 single out E8 robustly across rank cut-offs 30, 50 and 80. Finding one more such relation is therefore not evidence. Applied to four relations proposed as characterising E8, the method yields three corrections: dim = rk·(h+1) is a universal identity (all 199 simple algebras tested, zero information); the apparent uniqueness of h = 30 as the largest solution of φ(n) = d(n) is a rank-cut-off artefact (B15, C15, D16 and A29 also have h = 30); and a list of solutions of φ(n) = d(n) in circulation contains four false entries and two omissions — the correct set is {1, 3, 8, 10, 18, 24, 30} (OEIS A020488). Only h = ∏Bad(g) isolates E8, and it does so without auxiliary hypotheses — but it is one of the 371. Second application: the canonical breaking cascade E8 → E6 → SO(10) → SU(5) is not the unique chain removing torsion primes one at a time in decreasing order; the route through E7 is equally monotone (54 spine chains, exactly two monotone). What excludes E7 is chirality, not arithmetic. The corrected claims circulate in earlier documents of this same research program; this deposit is the program auditing itself. All computations use only the Python standard library and run in seconds; sources available upon request.