The Canonical Form of Attention: Positive Geometry, SYK Vertex, and Tropical Structure in Transformer Softmax

ERRATUM (August 9, 2026). Section 8.3’s entropy-gap formula (H_gap = 2Δ·log n for a normalized power-law attention distribution) is mathematically incorrect: the entropy gap of a normalized power law with exponent 2Δ < 1 converges to a constant, and its slope against log n over the paper’s own measured range is an order of magnitude below the formula’s prediction (0.041 versus 0.50 at Δ = 1/4). The error drops the energy term s·E[log r] from the entropy. Consequently the entropic estimate Δ_eff = 0.254 (Section 8.3), the k_eff ∝ n^(1−2Δ) derivation (Section 8.4), and the abstract-level claim that two independent observables measure one exponent to 1.4% are withdrawn. The Section 8 measurements themselves (logarithmic entropy-gap growth, R² > 0.97; the measured k_eff scaling) stand as measurements, but they measure n-independent concentration structure, not the conformal dimension. The paper’s exact results (the canonical-form identity, the quartic vertex, the perturbative identity H_gap = ½Var(s)) and the direct power-law profile measurements are unaffected. Full correction, correct algebra, numerical demonstration, affected-statement list, and the self-contained verification script are in canonical_form_erratum.pdf and k1_powerlaw_gap.py, added in this version. The original paper PDF is retained unchanged; this program corrects the record by dated erratum, not by silent revision. v4: Corrected Maldacena-Stanford (2016) title from 'Remarks' to 'Comments'.