Symmetric Informationally Complete Measurements as an Operator Basis for the Complexified G₂ Lie Algebra

Working over the dimension-d=7 symmetric informationally complete (SIC) reference measurement constructed from the exact Appleby–Bengtsson–Grassl–Harrison–McConnell algebraic fiducial under a unique Fano-compatible sign-flip correction, we establish three structural theorems characterizing the relationship between the SIC operator basis and the seven-dimensional defining representation of the exceptional Lie group G₂. Theorem 1 establishes that the complexified Lie algebra g₂ (C) embeds isometrically as a 14-dimensional subspace of gl (7, C), spanned by the 49 SIC projectors, with isometric scale 8/7, explicit closed-form coefficient tensor over Q (√2, i), and dense purely imaginary entries. Theorem 2 establishes that the 147-element SIC symmetry group WH (7) ⋊C₃ contains as an index-7 subgroup the Frobenius group F₂₁ = Z₇⋊Z₃ — the cyclic-axis subgroup of PSL (2, 7), which is itself the discrete Fano-orientation subgroup of G₂. The descent is realized canonically (up to global phase) under a unique Fano-compatible sign-flip correction to the ABGHM fiducial: exactly 2 of 2⁷ = 128 candidate corrections succeed. Theorem 3 establishes that on the seven-point X-subgroup orbit of the corrected ABGHM fiducial, the SIC triple product T₈₉₊ on Fano-line triples (those with all-QR or all-NQR ordered displacement differences in Z₇*) has the closed form T = a + ibφ, with a = (√2 - 1) /16, b = (√2 - 1) √ (2 + 4√2) /32, and φ ∈ +1, -1 tracking the cyclic / anti-cyclic Fano orientation. A separate Lemma establishes the universal-magnitude consequence |T|² = 1/512 = (1/8) ³ for all WH-distinct SIC triples, immediate from the SIC overlap condition. The non-Fano (mixed-residue-class) triple-product ratio b'/b = (1+√2) /2 is verified numerically at 50-digit precision and stated as a conjecture; its closed-form analytic proof is left open. The constants a, b are derived in closed form over Q (√2) from the autocorrelation function of the corrected ABGHM fiducial. All theorem statements are verified at 50-digit precision in independent computational implementations. The d=7 SIC operator frame, constructed from the exact Stark-unit fiducial, encodes the algebraic, group-theoretic, and orientational structure of the seven-dimensional defining representation of the smallest exceptional Lie group. Paper 10 of the PCI/PME Framework series. Full source, computational verification, Model Council adversarial reviews (GPT-5. 5, Opus 4. 7, Gemini 3. 1 Pro), and revision history at github. com/MartinLGraise/PCI-Framework on the paper7-foundation branch.