MDL Selects the Wolfram Rule: Z_7 Dynamics, Algebraic Structure, and Standard Model Encoding of the GTE Polynomial

Stephen Wolfram's Physics Project identifies the universe with a specific hypergraph rewriting rule but provides no principled derivation of which rule nature selects from an astronomically large rule space. We provide the missing selection principle: the Minimum Description Length (MDL) criterion, combined with the Perfect Self-Containment (PSC) orbit constraint, uniquely selects a k=7, r=1 string substitution rule with polynomial form p(L,C,R) = C+R-CR-LCR (mod 7) from 7^{343} ≈ 10^{290} candidates (Theorem T96-02, machine-certified in Lean 4, zero sorry). We present the first systematic study of this polynomial as a standalone dynamical system (Object 0 in the GTE derivation tower), distinct from the previously studied f_MDL lookup table (Object 1 / CMCA); the Schwartz-Zippel theorem proves f_MDL is not a polynomial of degree ≤3 over GF(7) (CatA), establishing that the two objects differ categorically, not merely quantitatively. Five algebraic results are established: Rule 110 arises as the binary restriction of p (CatAL); the invariant subset lattice of p over GF(7) contains exactly three non-empty elements, {0}, {0,1}, and Z_7 --- so Rule 110 is the unique maximal proper invariant sub-CA (CatAL, zero sorry, exhaustive native_decide); p is the unique degree-≤3 polynomial over GF(7) reproducing Rule 110 on binary inputs (CatA); the uniqueness of {0,1} as binary floor is explained by N_fam = 5 being a quadratic non-residue mod 7 --- the Standard Model family count is the algebraic obstruction to any intermediate invariant sublattice (CatAL); and f_MDL is provably non-polynomial (Schwartz-Zippel, CatA). The polynomial's local WolframModel update structure produces a 1023-event fractal binary tree causal graph at 10 generations (CatAL, gte_rulegte_ten_generations); the binary branching arises from the topological degeneracy of GEN_2 and GEN_3 (CatAL, gen2_gen3_topology_degenerate). This causal tree is structurally equivalent to the Lindenmayer system F→F[+F][-F] modelling Daucus carota compound umbels (Apiaceae), with matching Horton ratio r_B = 2 (CatAL) and Strahler number 10. A three-tape extension via the Dimensional Protocol Principle yields a connected 3+1D causal bulk with 40% cross-tape (gravitational) edge fraction for N_TAPES = 3.