M43 F09 The Number Sets - A Census from the Integers to the Hermit Units

The Theory of Operations assigns operations to ranks, rooms, half ranks, branch sectors and transported charts. Each operation may carry an arithmetic with its own units, atoms and notion of size.

F09 "The Number Sets - A Census from the Integers to the Hermit Units" calls such a ring or field a number-set and asks what genuinely new number-sets the architecture produces beyond the classical tower Z subset Q subset R subset C. Its answer is deliberately restrictive. The Theory creates no field properly extending the complex numbers. Instead, it distinguishes transported copies of classical arithmetic, special constants selected by operational closure equations, periods computed by half-rank protocols, algebraic points at dyadic addresses and invariants of the ladder’s analytic flow. The census is organised by a closure dichotomy: rational dials close after finitely many steps, while irrational dials generate infinite equidistributed orbits.

At addition and multiplication, where the HyperCore and Symmetric Core coincide, the number-sets are the classical integers, rationals, reals and complex numbers. The classical extension process stops because C is algebraically closed. Finite-dimensional associative real division algebras stop with the quaternions, while positive-definite normed composition algebras stop with the octonions.

The Theory’s branch-history algebras do not evade this boundary. A binary branch algebra of length n is the group algebra of (Z/2)^n over C and is isomorphic to a product of 2^n copies of C. It is commutative and contains zero divisors. Its full algebra automorphism group permutes the primitive idempotents, while the smaller groups preserving the branch basis arise from linear and affine transformations over F_2.

Attaching a dial to a complex number likewise produces an algebra rather than a larger field. The group algebra C[R/Z] has zero divisors whenever the dial has finite order, and its augmentation quotient is C again. A branch script or Configuration Number records a chosen sequence of logarithmic sheets; it is an address for a multivalued orbit, not a new number-set.

Closure and transported arithmetic

For a dial kappa on R/Z, the orbit of repeated addition closes after q steps when kappa=p/q is rational in lowest terms. When kappa is irrational, the orbit is infinite and equidistributed. This classical dichotomy organises the census:

• rational dials produce finite cyclic structures, discrete spectra and finitely closing branch data;
• irrational dials produce nonclosing orbits whose natural description is distributional rather than finite.

A transported number system consists of a set with an operation and a bijection to the positive integers that carries the operation to ordinary multiplication. It inherits unique factorisation, primes, Euler products and prime counting exactly from the integer labels.

This preservation is both its strength and limitation. If a transported zeta function is read through the labels, it is precisely the classical zeta(s). The chart changes representation but moves no arithmetic or spectral data.

Reading instead the real positions of transported primes produces a different Euler product. On the rank-3/2 ascent orbit

x_1=2,

x_(n+1)=AGM(x_n+2,2*x_n),

the ratios satisfy

x_(n+1)/x_n -> AGM(1,2)

= 1.4567910310...,

and

x_n/AGM(1,2)^n -> 3.0910046662....

The positions therefore grow geometrically. An Euler product over the prime-indexed positions converges absolutely and is zero-free on Re(s)>0. Transport preserves the zeros only when it preserves the labels; moving the prime positions rapidly enough to improve convergence removes the zeros from the strip.

Hermit units

The genuinely distinguished numbers of the census are the Hermit units. For an operation circ, a Hermit unit is a nonzero complex number satisfying

u circ u = i*u.

These numbers already lie in C, but each operation selects them through its own closure equation.

For room k, writing w=Log(u), the Hermit condition becomes

H_k(w) = i*pi/2+2*pi*i*m

on logarithmic sheet m, where

H_k(w) = exp^2*ln^[k-1](w)-w.

The first rooms give:

• room one: the unique unit i;
• room two: twelve units in six companion pairs;
• room three: eight principal units;
• room four: infinitely many units on the principal sheet, with no known canonical selection rule.

Room two has the quadratic equation

w^2-w = i*pi/2+2*pi*i*m.

On the principal sheet,

w_tau = (1+sqrt(1+2*pi*i))/2

= 1.4593138910+0.8187082151*i.

Its companion is 1-w_tau, and the corresponding units satisfy

tau*tilde_tau = e.

All twelve room-two units lie on the same logarithmic hyperbola. None lies on the balance circle |u|=sqrt(e).

Other columns and constructions select different units. The HyperCore’s rank-three operation Cpow has equation

w*(exp(w)-1) = i*pi/2

and principal unit

tau_HC = 1.8995894558+1.2639471298*i.

The AscentCore midpoint CatH has

w*exp(w) = i*pi,

while the Symmetric-Core midpoint MultPow has

w^2 = i*pi.

These equations share a general Hermit form but define distinct constants. They must not be identified merely because they occupy neighbouring architectural positions.

The sideways morph produces star units. In room one, its principal star unit satisfies

ln(i*) = pi*(2-i)/10

and

(i*)^10 = -exp(2*pi).

This unit is transcendental. The room-three cascade produces a separate constant, the cascade discrepancy,

kappa_d = -0.0011728652305-0.0005369995329*i,

which is the unique small fixed point of an explicit contraction. It is neither a branch flux nor a multiplier.

Periods and dyadic points

The next class of entries consists of periods rather than operational units. Gauss’s arithmetic-geometric mean computes elliptic periods:

integral_0^(pi/2) dtheta

/sqrt(a^2*cos(theta)^2+b^2*sin(theta)^2)

= pi/(2*AGM(a,b)).

This establishes the genus-one address at rank 3/2. The corpus labels genus g by the half rank

g+1/2.

The label is a theorem only at genus one. The genus-two interpretation of the Symmetric Hyperelliptic Mean at rank 5/2 and the higher-genus continuation remain conjectural.

The solvability ladder associated with this label concerns period data and modular functions, not new fields beyond C. General quintics and sextics can be reconstructed from genus-two period matrices and theta constants without radicals. Whether a rank-5/2 operational construction can perform this inversion directly from symmetric polynomial data remains open.

Dyadic addresses between the integer ranks supply algebraic steps. The half-lattice uses successive roots of the basic quarter-turn step. A step of length 2^(-j) has algebraic degree 2^j for j>=1. Only the first two steps lie in the Gaussian dyadic ring; every later refinement doubles the degree.

The Hermit units of log-linear paths also form arcs through the dyadic ranks. The Symmetric-Core arc from multiplication to Apow begins at i*pi/2, passes through sqrt(i*pi) at the midpoint and ends at w_tau. At every rational dial position, its logarithmic unit is quadratic over Q(pi*i). The corresponding AscentCore arc is transcendental and passes through the Lambert-W value associated with CatH.

Copies and Lie structure

Combining labelled copies of one element under a commutative operation produces native powers, not a non-abelian algebra. Any bracket defined by

[x,y] = x circ y-y circ x

vanishes identically when circ is commutative. Therefore copies of a Hermit unit do not generate u(1), su(2), su(3) or g_2 through the operation alone.

The non-abelian structure of the Symmetric Core instead belongs to its room fields. The infinitesimal translation fields of adjacent rooms satisfy

[X_k,X_(k+1)] = X_k

and generate the affine Lie algebra aff(1). Their finite projective completion is sl(2,R), while three consecutive room fields generate an infinite-dimensional algebra.

Thus the census separates numbers singled out by operations from Lie structures carried by the actions of their charts.

The clock’s two ends

The final entry is not a number-set but an invariant of the map

phi(t)=t+ln(t),

which carries the rank-two storage generator to the rank-three generator.

The map has two canonical regular coordinates:

• the ladder clock, an Abel coordinate regular at infinity;
• Koenigs’s coordinate, regular at the repelling fixed point t=1, whose multiplier is 2.

Their difference is invariant under phi and therefore periodic in the ladder clock. Its mean is

0.5002756566697444...,

while its total range is only

5.01071048144*10^(-13).

The transition is extremely close to constant but numerically nonconstant. The moduli of its Fourier coefficients measure the difference between the two regular continuations. They are invariants of one dynamical map, not Hermit units, branch fluxes or numbers obtained by cloning an operation.

The resulting census

The corrected census contains:

1. the classical tower Z subset Q subset R subset C;
2. transported copies of integer arithmetic;
3. Hermit units selected by the operations;
4. star units and the cascade discrepancy;
5. periods computed by half-rank protocols;
6. algebraic steps and Hermit arcs at dyadic addresses;
7. room fields carrying the genuine Lie structure;
8. transition invariants measuring the ladder clock’s two regular ends.

The census contains no field extending C. Its new content consists of distinguished numbers and structures within complex arithmetic, together with transported copies, periods and flow invariants.

F09 replaces the first edition’s universal-closure picture with a restrained census organised by the closure dichotomy. The former single quadratic, branch-addressed field, Galois-rank seam and cloning law are withdrawn. What remains is more precise: operationally selected units inside C, transported arithmetics that preserve or erase spectral data according to how they are read, classical periods attached to half-rank protocols, algebraic dyadic points and a measurable transition between the two regular coordinates of the ladder flow.

F papers belong to the M43 chain.