D2b — From Ω₇ to the Discrete Fano Derivation Algebra 𝒈𝑻𝒓𝒊𝒏: Construction and Exceptional Import M3'(F)

D2b — From the Fano Plane to the Trinotonian Derivation Algebra: Discrete Construction and Structural Identification D2b develops the discrete algebraic construction arising from the Fano-plane structure selected by the Triniton discrete framework. The document begins from the discrete chain leading to the unique Steiner system (S(2,3,7)), equivalently the Fano-plane incidence structure (\Omega_7), obtained from the non-freezing principle MP(III) together with the economy criterion (C_{\mathrm{PE}}). It then constructs the associated seven-dimensional Fano product and the corresponding finite-dimensional Lie algebra through its skew derivations. The principal discrete construction is \operatorname{Der}_{\mathrm{skew}}(V,\times_F).] The document distinguishes strictly between the discrete Fano-product construction and its subsequent identification with the standard octonionic structure. In particular, the Cayley–Dickson identification is treated as the explicit conditional bridge H6 rather than as an implicit consequence of the Fano combinatorics. The resulting algebra is reported to have dimension 14. Its semisimplicity is treated according to the status of the underlying finite computations, while stronger claims such as simplicity are not asserted without the corresponding missing lemma or exact certificate. The relation with the continuous exceptional Lie group (G_2) is therefore presented with its complete dependency chain: discrete Fano construction, H6 identification with the octonionic product, standard octonion/Lie theory, and the classical (G_2)-stabilizer results. The discrete automorphism group of the Fano plane is kept strictly distinct from the continuous Lie group (G_2). D2b is restricted to the discrete regime. Continuous and cosmological arguments are excluded from its derivation unless explicitly introduced as external or conditional bridges. The document is accompanied by D2b-C, an evidentiary companion that records the provenance, logical role, verification mode, conditional dependencies, and limitations of the claims used in D2b.