M51d The Derivative Étages - Isolation, the Bottom End, the Chirality Charts and Cycles at Every Level

The Theory of Operations associates a calculus with every operational chart. Ordinary subtraction gives Newton’s derivative, multiplication gives Euler’s operator and multiplicative calculus, while the rank-four Half-Caterpillar chart gives the ascent derivative. M51d "The Derivative Étages" studies what relates these calculi across ranks. It proves that the interior rank-four derivatives are mutually isolated from the classical room derivatives, classifies their geometry near the absorbing unit, compares the rank-three chirality charts with established deformed logarithms, determines the degeneration of the rank-four basal charts into tower levels, and proves the existence and nonsolvability pattern of cycles at every level. Charts and derivative étages If phi is an increasing chart of an interval, its invariant derivative field is X_phi = (1/phi')*d/dy. The chart satisfies X_phi(phi)=1 and transports the complete calculus of the real line to its domain. Addition uses the chart y, multiplication uses ln(y), and associative power uses ln(ln(y)). At rank four, the calculus becomes a one-parameter family. Fix a base B>1, put b=ln(B), and write the exponent coordinate as U=log_B(y). The étage line consists of the steps U -> U^(1-t)*B^(U^t), 0<=t<=1. In the internal coordinate w = t*ln(U), the step becomes F_t(w) = (1-t)*w+t*b*exp(w). The endpoints and centre are: t=0: dilation and the chart log_B(log_B(y)); t=1/2: the Half-Caterpillar and the TetraEtage chart hlog_B; t=1: tetration and the superlogarithmic chart slog_B-1. Each interior member admits a basal chart normalised at the absorbing unit y=1. Below Euler’s wall, it also admits a regular Koenigs chart normalised at the attracting fixed point. These two continuations differ by a periodic comma. Isolation of the derivative étages Inside one chart, every calculus identity follows by transport. The central question is what can be said about derivatives belonging to different charts. M51d proves that every interior rank-four field generates an infinite-dimensional Lie algebra with: Newton’s field d/dy; Euler’s field y*d/dy; the room-two field y*ln(y)*d/dy; the room-three field y*ln(y)*ln(ln(y))*d/dy; the superlogarithmic field at the right endpoint; every other interior member of the étage line. This remains true for every real-analytic continuation under the stated hypotheses. In particular, the ascent derivative at t=1/2 does not belong to any finite-dimensional Lie algebra jointly generated with the classical room fields. The proof uses the classification of finite-dimensional Lie algebras of vector fields on a line. In the chart of one member, any partner belonging to a finite-dimensional algebra would have to be an exponential polynomial. The actual partner fields have incompatible asymptotic behaviour, including double-exponential growth or corrections smaller than every exponential. Each member has an affine partner at either end. At the top, the member and its Koenigs translation field generate aff(1). At the bottom, the member and its Böttcher translation field also generate aff(1). These two affine structures do not close together. One member therefore carries two local linear geometries that are individually simple but mutually incompatible. The bottom end and its moduli Near the absorbing unit, the inverse step expands the internal coordinate by the Böttcher degree d_t = 1/(1-t). Let Lambda = -ln(ln(y)). The basal field has the asymptotic form X_t = -ln(d_t)*(X_3-ln(b)*X_2)*(1+o(1)). Thus every member approaches the same underlying field at the bottom, multiplied by the scalar ln(d_t). The field isomorphism between two members satisfies -ln(phi(y)-1) = C*(-ln(y-1))^gamma*(1+o(1)), where gamma = ln(d_(t'))/ln(d_t). The Böttcher degree is therefore a Hölder modulus of the floor. Members with different parameters are not bi-Hölder equivalent at the absorber. For equal t, the base becomes a second modulus. If the bases are B and B', the conjugacy has the form phi(y)-1 = C*(y-1)^mu*(1+o(1)). The exponent mu changes strictly with the base, so different bases are not C^1-equivalent at the bottom. The absorbing end consequently determines the complete pair (B,t): first the member through its Hölder class, then the base through its differentiable class. The same basal geometry appears in the rank-three operand dial. Under an explicit lift, the operand-fluent step is conjugate to the corresponding member of the rank-four étage line, and their basal Abel coordinates agree up to normalisation. The two dials therefore share one bottom-end geometry. The left end and formal production At t=0, the étage line begins at the elementary room-two chart. The family has a unique formal expansion A_t(a) = a+sum_(n>=1) t^n*delta_n(a), where every delta_n is a polynomial determined by the Abel equation. The first coefficient is delta_1(a) = (1/2)*(b-1)*a*(1-a). To second order, the transported operation is a plus_t a' = a+a'+k(t)*a*a' + t^2*(b^2/2)*a*a'*(a+a')+O(t^3), with k(t)=t*(b-1)-t^2*b^2/2. The first-order law is the Koenigs-Abel law and, equivalently, Tsallis’s deformed addition: a plus_q a' = a+a'+(1-q)*a*a'. At second order, the étage line leaves this family through an explicit cubic coboundary. The rank-four dial is therefore intrinsically curved in its parameter. This contrasts with the rank-three operand dial L_r(x,y) = exp((y*ln(x))^(1-r)*(x*ln(y))^r). After two logarithms, the rank-three family is exactly affine in r. Rank three supplies a straight chart dial; rank four supplies a genuinely curved family of charts. Continuation dependence Below Euler’s wall, the basal and regular charts of an étage member agree at integer heights but differ between them by a periodic comma. M51d quantifies the order at which this difference becomes visible in height derivatives. For the TetraEtage centre, the continuation-free order decreases as the base approaches the wall. At small bases, derivatives through approximately orders six or seven remain insensitive at the one-percent level. Near the wall, sensitivity may appear already around orders two or three. The following remain independent of continuation: integer-height values; the domain; the top multiplier; the Böttcher degree; the asymptotic growth rates at both ends; the bottom Hölder class; the isolation of the derivative fields. Values between integer heights, sufficiently high derivatives and exact differentiable bottom moduli can depend on the selected continuation. The calculus therefore has a continuation-free low-order core surrounded by continuation-sensitive higher-order data. Chirality charts and their Schwarzian profile Fix x>1. The inverse of the slot map y -> L_r(x,y) is the chirality chart phi_r, interpolating between phi_0(z)=log_x(z) and phi_1(z)=z^(1/x). For 0<r<1, the chart is expressed through Lambert’s W function. Its Schwarzian derivative is governed by one rational profile independent of the fixed operand x. Writing L = ln(phi_r(z)) and N = (1-r)*L+r, the profile satisfies (ln(z))^2*(z^2*S(phi_r)(z)-1/2) = Phi_r(L), where Phi_r(L) = (1/2)*(1-L^2/N^2) - r*(4*N-3*r)/(2*N^4). At its two ends, the chirality chart approaches Hölder-conjugate power classes. Operand exchange interchanges these asymptotic ends. The profile has a single-turn shape below the threshold r_0 = sqrt(6)/(1+sqrt(6)) = 0.710102051443..., and is strictly decreasing above it. The chirality charts are not Möbius-equivalent to the standard deformed logarithms of Tsallis, Kaniadakis, Abe, Schwämmle-Tsallis or Hanel-Thurner. Although some of these functions share leading power behaviour, they lack the logarithmic correction appearing in the chirality Schwarzian. Shannon and Tsallis tangents The two principal dials leave their logarithmic endpoints along two classical deformations of additivity. For the rank-three chirality charts, the first variation of the transported operation is the coboundary of delta(a) = a*ln(a)-a*ln(ln(a))+linear terms. Under scaling, this coboundary approaches Shannon’s entropy cocycle: cob(delta)(nu*a,nu*a') = -nu*(a+a')*H(p) * (1-1/ln(nu)+O(1/ln(nu)^2)), where p=a/(a+a') and H(p)=-p*ln(p)-(1-p)*ln(1-p). The rank-four étage line instead leaves its left endpoint along Tsallis’s addition: a plus_t a' = a+a'+(1-q)*a*a'+O(t^2), with 1-q=t*(ln(B)-1). Thus rank three has an extensive Shannon tangent with a logarithmic correction, while rank four has a nonextensive Tsallis tangent. These are precise first-order deformation laws, not identifications of the complete operational families with entropy calculi. The right end and the tower-level staircase The regular charts reach the superlogarithm as t -> 1. The basal charts have a different limit. Away from the tower points 1, B, B^B, B^(B^B), ..., the basal family converges to floor(slog_B(y))-1. Equivalently, A_t^bas(y) -> TTL(y)-2, where TTL(y) is the number of base-B logarithms needed to bring y down to 1. The regular chart retains both the integer tower level and its fractional part. The basal chart retains the level but discards the fraction. Their periodic comma consequently tends to the sawtooth {slog_B(y)}. The first Fourier amplitude of the comma approaches 1/pi. The renormalised basal field remains continuous between tower points and vanishes exactly where the tower-level staircase jumps. The Böttcher-normalised family therefore degenerates into a discrete counting function, while its rescaled derivative survives as a piecewise dynamical field. The operand-fluent family reaches the same staircase under the one-dial correspondence. Complex members and half étages Below Euler’s wall, the regular continuation extends holomorphically to a disk in the complex parameter t. The top multiplier is q_t = 1-t+t*ln(L). The disk is defined by |q_t|<1. Its centre is a member with a superattracting fixed point of degree