M12d Operational Arithmetic, Rebuilt - Prime Cycles and Hyperintegers

M12d develops two independent approaches to operational arithmetic. The first studies how prime numbers are distinguished among cycle periods in the complex dynamics of the exponential map z ↦ B^z. The second defines and counts atoms of higher operations, including exponentiation and tetration, directly on their natural integral carriers. The resulting theory includes a prime-satellite theorem for hyperbolic components, a Dirichlet series and asymptotic law for satellite chains, exact counting formulas for operational atoms, uniqueness theorems for exponential and tetrational factorisation, and arithmetic results for higher operational inverses. The first part asks where the primes occur in the parameter plane of f_λ(z) = exp(λz),λ = Log(B). Every positive integer occurs as a cycle period, so primality cannot be identified with the mere existence of a cycle. M12d instead separates satellite components, which bifurcate from components of lower period, from primitive components. It proves the Prime-Satellite Theorem: N_sat(n) = n − 1 if n is prime,N_sat(n) = ℵ_0 if n is composite. Thus prime periods are exactly those carrying finitely many satellite components. Every satellite of prime period p descends directly from the fixed-point component, and there are precisely φ(p)=p−1 of them. If n=dq is composite, every component of period d≥2 carries infinitely many satellites of period n, one for each lift of every rotation number of denominator q. Primality appears dynamically as finiteness. The mechanism is the contrast between multiplier maps. The fixed-point component has a one-to-one multiplier parametrisation, so it contributes exactly φ(q) children of period q. Every component of period at least two has a multiplier map that universally covers the punctured disc. Each rational rotation number therefore has infinitely many lifts, producing infinitely many distinct children. M12d proves that the lifts of a higher-period component are disjoint and uses this to pass rigorously between the periodic covering plane and the λ-plane. The result concerns satellites, not all hyperbolic components. At every prime period p≥3, only p−1 components are satellites, while countably many components exist in total; consequently, infinitely many prime-period components are primitive. Primitive components are expected at composite periods as well, but that expectation is not proved. The theorem therefore replaces the naive claim that all new cycle directions are born at primes with the sharper statement that primes are exactly the periods whose entire satellite population descends from the fixed point. To obtain a finite census of the satellite tree, M12d selects the canonical sheet, retaining at each component the children whose preferred internal angles lie in (0,1). A chain of satellite bifurcations corresponds to an ordered factorisation n = q_1 q_2 ... q_k, and each step contributes φ(q_i) possible rotation numbers. The resulting count is ν(n) = Σ ∏ φ(q_i), summed over all ordered factorisations of n. Equivalently, ν(1)=1,ν(n)=Σ_(d|n, d>1) φ(d)ν(n/d). This is also the main-molecule count studied for the Mandelbrot set. M12d proves that ν(n)=φ(n) exactly when n is prime. The census has Dirichlet series N(s) = Σ_(n≥1) ν(n)n^(−s)= ζ(s)/(2ζ(s)−ζ(s−1)). Its abscissa of convergence is the unique real number ρ>2 satisfying ζ(ρ−1)=2ζ(ρ), numerically ρ = 2.47875078573396026067.... By the Wiener-Ikehara theorem, M12d obtains Σ_(n≤x) ν(n) ~ Kx^ρ, where K = ζ(ρ)/(ρ(2ζ'(ρ)−ζ'(ρ−1)))= 0.15581316765167740408.... The error term admits no power saving because the zeros of 2ζ(s)−ζ(s−1) approach the line Re(s)=ρ from the left. The controlling zeros are therefore not the nontrivial zeros of Riemann’s zeta function. M12d then proves two structural no-go theorems for attempts to recover ζ from cycle counts. A free census of words with non-negative letter weights can never equal ζ(s), because equality would require letter weights −μ(n), which are negative for many squarefree composite integers. A commutative census with unit weights equals ζ(s) if and only if its primitive lengths are exactly {log(p) : p prime}. Integer cycle periods instead produce partition generating functions. For example, a system with one primitive p-cycle for each prime has Artin-Mazur zeta function ∏_p (1−z^p)^(−1), the generating function for partitions into primes, not the Riemann zeta function. A genuine dynamical route to ζ would therefore require positive primitive objects of lengths log(p) assembled commutatively, as in the requirements formulated by Deninger and Connes. The second part rebuilds the theory of hyperintegers. M12d distinguishes three notions that had previously shared the name. Inverse-coordinate lattices merely install ordinary arithmetic through a chart and have either one atom or transported classical primes. Integral chiral carriers use operations such as a^b and right-caterpillar tetration a↑↑b directly on integers. Diagonal orbits iterate the HyperCore diagonal from 2 and form a rank-indexed family with the common beginning 2,4. Only the latter two notions contain genuinely rank-native information. The diagonal orbits separate at their third term. At ranks 1, 3/2, 2, 5/2, 3, 4 the third terms are respectively 8,11.654328248375...,16,62.499895774392...,256,2^(2^69). The rank-3/2 value is AGM(8,16) and is proved transcendental using Gauss’s elliptic-integral representation and Chudnovsky’s theorem on elliptic periods. At rank 5/2, the algebraic nature of the corresponding SHM value remains open. For the rank-four diagonal at powers of two, M12d gives an exact 2-adic history of the iterated exponents. If n=2^m and v=v_2(m), the first non-integral exponent occurs at height m + v_2(m) + 3. The following exponent lies in Q(sqrt(2)), and the next is transcendental by Gelfond-Schneider. This proves that the rank-four diagonal is integral at n=2 and n=4, while for larger powers of two the first non-integral exponent occurs well before the final height. However, Gelfond-Schneider does not decide the final diagonal value. In particular, the rank-four orbit reaches the integer 2^(2^69) at its third term, but the arithmetic nature of its fourth term remains open. The monograph next defines operational atoms directly. On their natural integral carriers: addition has the single atom 1; multiplication has the classical primes; exponentiation has the integers that are not perfect powers; tetration has the integers that are not towers a↑↑k with a,k≥2. Every integer n≥2 has a unique expression n=m^k with m not a perfect power. Hence the number of rank-three atoms up to x is A_3(x) = Σ_(k≥1) μ(k)(floor(x^(1/k))−1), with deviation from its smooth version bounded by log_2(x). This yields the identity ζ(s)−1 = Σ_(m atom) 1/(m^s−1) and recovers the Goldbach-Euler theorem that the sum of 1/(q−1) over distinct perfect powers q equals 1. Rank-three factorisation becomes ordinary ordered factorisation in the exponent. If n=m^j with m a rank-three atom, then the left-bracketed exponentiation factorizations of n correspond exactly to ordered factorizations of j. Their number is Kalmár’s factorisatio-numerorum function. This gives a precise form to the earlier intuition that higher-rank factorisation morphs into the factorisation of a lower-rank parameter. At rank four, M12d proves that integer towers are injective: a↑↑k = b↑↑l ⇒ (a,k)=(b,l) for a,b,k,l≥2. Every integer consequently has a unique nested tower form terminating at a rank-four atom. If s_k(x) denotes the real kth super-root of x, the atom count is A_4(x) = floor(x)−1 − Σ_(k≥2)(floor(s_k(x))−1)^+. Its deviation from a smooth main term is smaller than log*(x)+1. The rank-three error is at most logarithmic, and the rank-four error at most iterated logarithmic, whereas the rank-two atom count is π(x) and carries the oscillations associated with the zeros of ζ. This difference has a structural explanation. At ranks three and four, integers decompose additively over the images of their atoms, and counting is reduced to finite sums and elementary inversion. At multiplication, integers decompose multiplicatively over their atoms. Undoing the product requires log ζ, and the explicit formula introduces the zeros of ζ. In this precise sense, multiplication is the uniquely irregular rank: it is the only rank considered that is simultaneously symmetric, integral and multiplicative. M12d also reconstructs hyperrationals and hyperradicals. The rank-three analogues of fractions are radicals: Rad = {x>1 : x^k is an integer for some k≥1}. Every such number has a unique representation m^q, where m is a rank-three atom and q is a positive rational. Each atom therefore carries its own copy of Q_(>0), with exponentiation acting as multiplication of rational exponents. At rank four, the height-two base inverse is the super-square root ssrt(N) = exp(W(ln N)). M12d proves that the super-square root of an integer is either an integer or transcendental. For the super-cube equation x^(x^x)=n, the real solution is unconditionally either an integer or irrational, and under Schanuel’s conjecture either an integer or transcendental. The fluent inverse between root and logarithm passes through Lambert-function formulas, but the arithmetic classification of its values at integer data remains open. Below addition, the deterministic recurrence x_(k+1) = x_k + ln(x_k),x_1=2 provides the mean-gap skeleton of Cramér’s probabilistic prime model. M12d proves li(x_n) = n − (1/2)ln ln(x_n) + c_* + O(1/(n ln n)) with c_* = −0.1442491 ± 2·10^(−7), and therefore x_n − li^(−1)(n) = −(1/2+o(1)) ln(n) ln ln(n). The deterministic orbit follows the average scale of the primes but trails the inverse logarithmic integral by a growing logarithmic correction much smaller than the primes’ own fluctuations. The same lower-rank region admits a probabilistic bridge from maximum to addition. Ind