Prime Decomposition and Operator Responses

Prime factorization is studied on the generated arithmetic family and on the linear spaces induced by it. The point is to retain the relation between algebraic irreducibility, decomposition histories and operator localization.

Three linked descriptions of prime directions

Prime modes are the nonunit elements irreducible under cascade composition. Their residue classes form the canonical basis of the augmentation cotangent space. The corresponding one-word bar vectors span the degree-one kernel of the decomposition Hodge operator. These are identifications between the appropriate sets, quotients and subspaces.

On the mode indexed by a positive integer n greater than one, the degree-one operator has eigenvalue d(n)-2, where d(n) is the number of positive divisors. Its kernel is therefore precisely the prime subspace. The associated heat projection has sharp operator-norm error exp(-tau); the resolvent supplies a second quantitative prime projector.

Occupation, parity and Gaussian scale

Unique factorization gives prime-occupation coordinates. Symmetric occupation traces recover the Euler product for zeta, while graded exterior traces recover its reciprocal in the domain of absolute convergence. Square-free selection and exterior parity realize the Moebius coefficients. These occupation spaces are canonically linked to the natural-mode family.

Creating a prime direction changes both occupation parity and quadratic Gaussian scale. Exact signed dilation identities retain both effects. Finite prime-direction transfers preserve the complete two-parity response before an unsigned capacity bound is taken.

Relative inputs survive addition

Addition merges several ordered input pairs into a single output value. A faithful additive dilation records the relative input in a defect space. The central-parity intertwiner expresses precisely how multiplicative parity is transported through this addition map. Common-factor and primitive-pair decompositions give a second interface between additive observation and multiplicative generation.

These operators and identities make it possible to formulate questions about the complete signed scale response without discarding signs or cross terms. The counting applications recover Bertrand's theorem and the prime-number asymptotic within the constructed system. The Riemann-hypothesis research interface concerns uniform control of the full signed response; the identities and prime projectors specify the objects on which such an estimate must act.

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Keywords: prime numbers, unique factorization, Hodge Laplacian, bar complex, Fock space, Riemann zeta function, Moebius function, Gaussian dilation.

Research subjects and results

Primary subjects

Prime numbers; Prime factorization; Prime number theorem; Hodge Laplacian; Spectral gap; Prime projection; Möbius function; Riemann zeta function; Fock space; Heat semigroups; Dirichlet series; Arithmetic quantum statistical mechanics.

Methods

Homological algebra; Operator semigroups; Resolvent approximation; Spectral projections; Heat trace estimates; Gaussian regularization; Occupation-number representation; Graded traces; Dilation identities; Common-factor decomposition; Charge-defect identities; Fixed-point theorems; Interval enclosures; Mellin transforms; Tauberian theory.

Results and research questions

Prime numbers as degree-one Hodge zero modes; Sharp heat and resolvent approximations to the prime projection; Exact unit spectral gap in a decomposition Laplacian; Euler products and Möbius coefficients from occupation traces; Additive charge defects and multiplicative parity; Common-factor scale identities for signed arithmetic responses; Stable fixed point of the full Gaussian endpoint response; Three-dimensional Weyl counting from square energy and quadratic shell capacity.

Chinese subject terms

素数; 素因数分解; 素数定理; 霍奇拉普拉斯算子; 精确谱隙; 素数投影; 莫比乌斯函数; 黎曼ζ函数; 福克空间; 欧拉乘积; 分解同调; 平方自由数.