M35d The Solvability Line of Tetration - Its Fixed Point Is a Lambert Function; No Cycle of Period Two or More Is

M35d The Solvability Line of Tetration proves that the closed-form solvability boundary for the cycles of tetration lies exactly between period one and period two. For the principal exponential map f_λ(z) = exp(λz) = B^z,λ = Log(B), a fixed point is given by x = −W(−λ)/λ. Thus period one requires precisely the Lambert function and is already non-Liouvillian. The monograph proves that every cycle of exact period n ≥ 2 lies beyond not merely the Lambert W function, but every finite tower assembled from algebraic operations, exponentials, logarithms and exponential-rational inversions. The solvability line of tetration is therefore drawn by period: the fixed point is Lambert-solvable, while the first genuine cycle is not. The work is a phoenix of the Cycle Apparatus from the abandoned M01 and the successor of M35c, The Solvability Ladder, Rebuilt. M35c asked which objects are needed to solve polynomial equations of increasing degree. M35d asks the corresponding question for tetration’s own equations. The analogy is structural: radicals suffice through polynomial degree four and fail generically from degree five, while Lambert-type functions suffice for tetrational period one and fail at every period from two onward. In the polynomial case the obstruction is group-theoretic; for tetration it is a differential-transcendence count based on Ax’s theorem. The first theorem states that no cycle is Liouvillian. For every period n ≥ 1, no cycle point can be expressed using a finite combination of algebraic operations, exponentials and integrals. The proof applies Rosenlicht’s theorem to the cycle equations exp(u_j) = u_(j+1)/λ. A cycle’s points and their exponentials generate the same differential field. If one point were Liouvillian, Rosenlicht’s theorem would force the cycle points to be algebraic over C(λ), while Ax’s theorem excludes the simultaneous algebraicity of a nonconstant function and its exponential. For period one, this recovers the known non-Liouvillianity of Lambert’s W. To distinguish the fixed point from higher cycles, M35d introduces exponential-rational towers. An exponential-rational step adjoins a solution t of an equation exp(t) = R(t), where R is rational over the field already constructed. This vocabulary contains all branches of Lambert’s W, Wright’s omega function, the r-Lambert functions and Mezö’s generalised Lambert functions, as well as their finite algebraic, exponential and logarithmic combinations. The relevant claim is therefore stronger than non-reducibility to ordinary W: higher cycles escape a broad class of Lambert-type special functions. The central technical result is the Tightness Lemma. In every exponential-rational tower, the transcendence degree equals one plus the rational dimension of the tower’s exponents modulo constants. Its corollary, Exponent Capture, says that whenever a tower contains both s and an exponential of s, the quantity s must be a rational linear combination of exponents already introduced by the tower. Each genuinely new step therefore contributes exactly one new exponential direction. This count proves the monograph’s main theorem. Suppose that a point of an n-cycle belonged to such a tower. Exponent Capture would place all cycle points in the rational span of the tower’s exponents. Consider the highest new exponent appearing in those expansions. An ordinary exponential step immediately pushes all cycle points into the preceding level and contradicts its maximality. At an exponential-rational step, comparison of divisors forces the rational function defining the step to have exactly one zero and one pole. The highest step is therefore a Lambert step in disguise. Consistency around the cycle then forces every cycle point to have the same coefficient and the same zero, so all points coincide. Such a step can support a fixed point, but not a cycle of period two or more. Consequently, no genuine tetrational cycle is expressible through any finite stack of algebraic, exponential, logarithmic, Lambert, Wright-omega, r-Lambert or generalised-Lambert operations. The theorem also characterises the exceptional period-one case: any top exponential-rational step capable of carrying a fixed point reduces, after algebraic changes of variables, to a Lambert step. The Lambert function is therefore not merely sufficient for the fixed-point equation but canonical within the tower vocabulary considered. The monograph rebuilds the earlier cycle apparatus before proving this boundary. For a cycle with points x_1,...,x_n, elementary symmetric sum σ_1, product σ_n and multiplier Λ, it proves σ_n = exp(λσ_1) = B^(σ_1),Λ = λ^n exp(λσ_1). Thus the product of the cycle points is the base raised to their sum, and the multiplier depends only on the cycle’s centroid. Attracting and indifferent cycles are consequently separated by a linear condition on the real part of λσ_1. The additional “Lambda hierarchy” proposed in the old M01 is retired because only the multiplier has an established dynamical meaning. The cycle polynomial’s discriminant is also reassessed. Its vanishing detects satellite bifurcations, where an n-cycle collapses onto a lower-period cycle, but it does not detect primitive births. If a parent cycle has period d and the satellite has period n=dq, the discriminant vanishes to order d(q−1). At a primitive root or co-root, two distinct cycles meet while each cycle retains distinct internal points. Their individual discriminants therefore remain nonzero, and the correct detector is the resultant of the two cycle polynomials. This repairs the old “discriminant sieve.” M35d gives a complete one-parameter uniformisation of every complex 2-cycle. With α(w) = 2w/(1−exp(−2w)),β(w) = 2w/(exp(2w)−1), the scaled cycle points and parameter are u = −β(w),v = −α(w),λ(w) = −α(w) exp(β(w)) = −β(w) exp(α(w)). Every complex 2-cycle arises in this way, uniquely up to w ↦ −w, which exchanges the two points. Its multiplier is Λ = (w/sinh(w))^2. The root occurs at w=0, corresponding to λ=−e and B=e^(−e). The co-roots satisfy sinh(w)=±w, while period-doubling satellites satisfy w/sinh(w)=±i. The monograph also proves that the λ-plane contains exactly one hyperbolic component of period two. The two points of every 2-cycle are two branches of Lambert’s function evaluated at one common argument. Since u exp(u) = v exp(v) = Λ/λ, one may write u = W_j(Λ/λ),v = W_k(Λ/λ),j ≠ k. For the real attracting 2-cycle below B=e^(−e), these are the branches W_0 and W_−1. This does not contradict the solvability theorem: the cycle is Lambert-simple as a function of its common argument, but that argument is not Lambert-type as a function of the base. The transcendental obstruction lies in the inverse map from λ to the cycle parameter. The period-three cycle curve is likewise simple in its own coordinates. After introducing differences r=u_1−u_3 and s=u_2−u_3, the cycle relation becomes s(1−exp(s)) = r(exp(r)−exp(s)). For fixed s, this is an r-Lambert equation in r. Again, the curve is accessible once its internal parameter is known, while the inverse dependence of that parameter on the base lies beyond exponential-rational towers. M35d proves that this period-three parameter is itself not exponential-rationally expressible in λ. Several further non-solvability results follow. For the 2-cycle, the multiplier, centroid, product, common Lambert argument and uniformising parameter w are all non-expressible in exponential-rational towers. The depth-two hyper-Lambert functions that represent genuine periodic points cannot be reduced to finite expressions in ordinary or generalised Lambert functions. For higher periods, the monograph proves that the centroid, product and multiplier stand or fall together, while determining exactly which individual symmetric invariants escape remains open. The result is sharply chiral. The right-caterpillar map z ↦ B^z encounters its solvability wall at period two. By contrast, the left-caterpillar map z ↦ z^B becomes linear in the logarithmic coordinate, and all its periodic points are exponentials of rational functions of B, once the branch integers are fixed. The two bracketings of rank four thus have opposite solvability behaviour: left iteration remains elementary at every period, while right iteration leaves every Lambert-type tower at its first genuine cycle. M35d interprets the theorem as a boundary of naming, not a boundary of computation. The 2-cycle is analytic on its regular domains, can be computed to arbitrary precision and possesses an elementary uniformisation by w. What fails is its expression in the vocabulary that names the fixed point. “Closed form” is therefore treated as a relation between an object and a declared vocabulary. The fixed point is reached by the Lambert vocabulary; all genuine cycles are walled from it. Extending the vocabulary will require a step that couples more than one exponential unknown, such as a genuine period-indexed or hyper-Lambert construction. The monograph leaves a focused research programme. Open problems include towers combining integration with exponential-rational steps; a possible hierarchy of special functions indexed by cycle period; irreducibility of the exact-period curves; transcendence properties of cycle values at algebraic bases; the Riemann surface of the inverse of w/sinh(w); solvability lines for higher-rank step maps; and corresponding boundaries for Cpow, resonance families and the TetraEtage. M35d establishes a new solvability theorem for tetration. The fixed point requires exactly Lambert’s function, but every cycle of period two or more exceeds every finite stack of algebraic, exponential, logarithmic and exponential-rational inversions. The obstruction is not numerical complexity or a failure of parametrisation: it is a theorem about the expressive reach of an entire vocabulary of special functions. For tetration, the first cycle is the first unfulfilled inverse.