M41e Branches of the Resonance Family

The Resonance family continuously interpolates between the two operand orders of exponentiation, a^b and b^a, with the symmetric operation Cpow at the midpoint. M41e studies what happens when these formulas are continued into domains where logarithms and fractional powers become multivalued. In conventional terms, it classifies the branches and monodromy of a particular interpolating family of complex functions and corrects its evaluation below one. It does not claim a general classification of multivalued operations. For positive a and b, define the chiral loads: p = b ln(a), q = a ln(b), and the family: L_r(a,b) = exp(p^(1-r) q^r). It satisfies: L_0(a,b) = a^b, L_(1/2)(a,b) = Cpow(a,b), L_1(a,b) = b^a, and: L_r(a,b) = L_(1-r)(b,a). The parameter r is the chirality dial. On a,b > 1, the whole family is single-valued and real-analytic. Outside that domain, its logarithms and fractional powers introduce nontrivial branch structure. M41e classifies that structure. The principal result is the denominator law. Independent choices of logarithms for p and q multiply the exponent by: exp(2 pi i r t), t in Z. Thus the multiplier group depends only on r: M_r = {exp(2 pi i r t) : t in Z}. If: r = m/n in lowest terms, the family has exactly n branches. If r is irrational, the multipliers are dense on the unit circle and the branch set never closes into a finite collection. Chirality therefore measures not only the position between a^b and b^a, but also the ramification order of the operation. This law singles out the commutative midpoint r = 1/2 in three independent ways: It has the smallest possible interior branch count, namely two. Its two branches are exchanged by the reciprocal involution because the multiplier -1 is available. For operands lying on opposite sides of one, it is the unique member whose value has unit modulus. Cpow is therefore distinguished not only as the geometric midpoint between the two chiral exponentiations. It is also the unique interior member selected by minimal ramification, reciprocity, and unit modulus. M41e also corrects the familiar closed formula: Cpow(a,b) = exp(sqrt(ab ln(a) ln(b))). Read with the principal square root, this formula loses the separate signs of the two chiral loads. On the diagonal below one, it returns: x^(-x) instead of the required: x^x. The two values are reciprocals. The problem disappears when Cpow is evaluated through the Resonance-family form, keeping p and q separate and taking their principal arguments independently. On (0,1)^2, both loads have argument pi, so the midpoint carries the required negative square root automatically. Accordingly, the radical expression is safe only with a signed determination. When h(t) = t ln(t) and h(a), h(b) have the same sign, the required exponent is: K = sgn(h(a)) sqrt(h(a) h(b)). The usual bracketing of Cpow between the two chiral powers is likewise restricted to a,b > 1. It is not a global positive-real statement. Over the complex plane, the determinations of Cpow are indexed by: Z^2 x {+1,-1}. There are two branch loci of different types. The point a = 0 is logarithmic, producing an infinite lattice shift. The point a = 1 is a square-root branch point, and continuation around it acts by inversion: Cpow -> 1/Cpow. The resulting monodromy group is: Z^2 x Z/2, and is abelian. The branch point coincides with the absorbing element 1 because the two sheets exp(+K) and exp(-K) can meet only at a fixed point of inversion. This also clarifies the role of identity elements. At r = 0 and r = 1, the ordinary chiral exponentiations retain a neutral operand in one slot. For every interior chirality: L_r(1,b) = L_r(a,1) = 1. Thus one is absorbing rather than neutral. Moving away from either endpoint simultaneously removes the identity and creates the branch point. The branch structure passes to the rank-three storage mean. The equation: x^x = y does not have a unique positive solution below one. It has two positive solutions for: exp(-1/e) < y < 1, one solution at y = exp(-1/e), and none below that value. Correspondingly, the mean: M_3 = exp(W(K)) has two real determinations, obtained from W_0 and W_(-1), whenever: -1/e < K < 0. The operation and its mean therefore have different branch points: the operation branches at the absorbing unit, while the mean branches at the Lambert critical value K = -1/e. M41e reveals that chirality is simultaneously an interpolation coordinate and a branch-order coordinate. Rational chirality produces finite cyclic coverings whose order is the denominator; irrational chirality produces a dense, non-closing family of determinations. At the midpoint, commutativity is achieved with the least possible branching. The result is intentionally restricted to the rank-three chirality axis. It does not identify this denominator law with the separate denominator law for fractional rank, and it does not prove that the rank-four Caterpillar family obeys the same rule. Corrections to the earlier corpus The principal-root closed form for Cpow gives the reciprocal of the intended value on (0,1)^2. The exponent must retain the common sign of the two chiral loads. The standard bracketing inequality for Cpow applies to a,b > 1, not to the entire positive domain. The degree-two superroot is not uniquely positive below one. It has two positive branches for exp(-1/e) < y < 1.