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 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 lookup table (Object 1 / CMCA); the Schwartz-Zippel theorem proves is not a polynomial of degree 3 over 7 (), 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 7 contains exactly three non-empty elements, \0\, \0,1\, and - 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 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 is provably non-polynomial (Schwartz-Zippel, ). The QNR obstruction is then traced to its source: the master quadratic m(x) = x^2+x-1 (discriminant 5 = N_fam) is the diagonal factor of p over every commutative ring, split over R - where its positive root is the SRRG fixed point seeding the electroweak-scale derivation - and inert at 7, where rootlessness at every 7-adic depth protects the binary floor (CatAL): one Z-quadratic, two completions, with an all-q splitting dichotomy governed by q 5. The same quadratic controls the dynamics: the vacuum is the unique temporally fixed configuration at every ring size (Vacuum Uniqueness Theorem, CatAL), proved by reducing fixed configurations to the golden M"obius map on P^1(7), and the Artin-Mazur zeta functions of the ring dynamics factorize exactly. Eisenstein arithmetic supplies the complementary structure: the internal symmetry group F_21 is the norm-7 residue-field quotient (Z[]/(3+))^+ _3 (CatAL); the zero variety satisfies the motivic identity [V(p)] = _6(L) and carries a torus action whose unique fixed zero is the ether point; the golden and Eisenstein rings are the two quadratic subfields of Q(-3,5), of conductor 15 = N_gen N_fam; and the full symmetry group of V(p) is AGL(1,7) of order 42, whose reflection element swaps Rule 110 with Rule 124 - the chiral pair of the CMCA architecture (CatAL). The period-475 attractor of the 5-cell ring is resolved by an Attractor Factorization Theorem (475 = 5 5 19: ring, drift, and a drift-invariant inert clock carried by 7^3), with the value 19 shown to be enumerative rather than mechanism-forced (CatAD). The polynomial's local WolframModel update structure produces a 1023-event fractal binary tree causal graph at 10 generations (, gte\_rulegte\_ten\_generations); the binary branching arises from the topological degeneracy of GEN_2 and GEN_3 (, 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 () and Strahler number 10. A three-tape extension via the Dimensional Protocol Principle yields a connected 3+1D causal bulk with (-1)/(+2) = 40\% cross-tape (gravitational) edge fraction for = 3.